USF Libraries
USF Digital Collections

Fuzzy C-means clustering approach to design a warehouse layout

MISSING IMAGE

Material Information

Title:
Fuzzy C-means clustering approach to design a warehouse layout
Physical Description:
Book
Language:
English
Creator:
Naik, Vaibhav C
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
Publication Date:

Subjects

Subjects / Keywords:
warehouse design
dedicated storage policy
data clustering techniques
fuzzy theory
FCM
linguistic variables
Dissertations, Academic -- Industrial Engineering -- Masters -- USF   ( lcsh )
Genre:
government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
ABSTRACT: Allocation of products in a warehouse is done by various storage policies. These are broadly classified into three main categories: dedicated storage, randomized storage, and class-based storage. In dedicated storage policy a product is assigned a designated slot while in random storage policy incoming product is randomly assigned a storage location close to the input/output point. Finally, the class-based storage is a mixed policy where products are randomly assigned within their fixed class. Dedicated storage policy is most commonly used in practice. While designing large warehouse layout, the product information in terms of throughput and storage level is either uncertain or is not available to the warehouse designer. Hence it is not possible to locate products on the basis of the throughput to storage ratio method used in the above mentioned storage location policies. To take care of this uncertainty in product data we propose a fuzzy C-means clustering (FCM) approach. This research is mainly directed to improve the efficiency (distance or time traveled) by designing a fuzzy logic based warehouse with large number of products. The proposed approach looks for similarity in the product data to form clusters. The obtained clusters can be directly utilized to develop the warehouse layout. Further, it is investigated if the FCM approach can take into account other factors such as product size, similarity and/or characteristics to generate layouts which are not only efficient in terms of reducing distance traveled to store/retrieve products but are effective in terms of retrieval time, space utilization and/or better material control.
Thesis:
Thesis (M.S.I.E.)--University of South Florida, 2004.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Vaibhav C. Naik.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 76 pages.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001478793
oclc - 56563774
notis - AJS2483
usfldc doi - E14-SFE0000437
usfldc handle - e14.437
System ID:
SFS0025129:00001


This item is only available as the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 001478793
003 fts
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 040811s2004 flua sbm s000|0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0000437
035
(OCoLC)56563774
9
AJS2483
b SE
SFE0000437
040
FHM
c FHM
090
T56 (ONLINE)
1 100
Naik, Vaibhav C.
0 245
Fuzzy C-means clustering approach to design a warehouse layout
h [electronic resource] /
by Vaibhav C. Naik.
260
[Tampa, Fla.] :
University of South Florida,
2004.
502
Thesis (M.S.I.E.)--University of South Florida, 2004.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 76 pages.
520
ABSTRACT: Allocation of products in a warehouse is done by various storage policies. These are broadly classified into three main categories: dedicated storage, randomized storage, and class-based storage. In dedicated storage policy a product is assigned a designated slot while in random storage policy incoming product is randomly assigned a storage location close to the input/output point. Finally, the class-based storage is a mixed policy where products are randomly assigned within their fixed class. Dedicated storage policy is most commonly used in practice. While designing large warehouse layout, the product information in terms of throughput and storage level is either uncertain or is not available to the warehouse designer. Hence it is not possible to locate products on the basis of the throughput to storage ratio method used in the above mentioned storage location policies. To take care of this uncertainty in product data we propose a fuzzy C-means clustering (FCM) approach. This research is mainly directed to improve the efficiency (distance or time traveled) by designing a fuzzy logic based warehouse with large number of products. The proposed approach looks for similarity in the product data to form clusters. The obtained clusters can be directly utilized to develop the warehouse layout. Further, it is investigated if the FCM approach can take into account other factors such as product size, similarity and/or characteristics to generate layouts which are not only efficient in terms of reducing distance traveled to store/retrieve products but are effective in terms of retrieval time, space utilization and/or better material control.
590
Adviser: Suresh K. Khator.
653
warehouse design.
dedicated storage policy.
data clustering techniques.
fuzzy theory.
FCM.
linguistic variables.
690
Dissertations, Academic
z USF
x Industrial Engineering
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.437



PAGE 1

Fuzzy C-Means Clustering Appro ach to Design a Warehouse Layout by Vaibhav C. Naik A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering Department of Industrial and Ma nagement Systems Engineering College of Engineering University of South Florida Major Professor: Suresh K. Khator, Ph.D. Grisselle Centeno, Ph.D. Qiang Huang, Ph.D. Date of Approval: July 8, 2004 Keywords: warehouse design, dedicated storage policy, data clustering techniques, fuzzy theory, FCM, linguistic variables Copyright 2004, Vaibhav C. Naik

PAGE 2

i TABLE OF CONTENTS LIST OF TABLES.............................................................................................................iv LIST OF FIGURES............................................................................................................v ABSTRACT....................................................................................................................... vi CHAPTER 1 INTRODUCTION......................................................................................1 1.1 Functions of Warehouse.........................................................................................1 1.2 Activities in Warehouse..........................................................................................2 1.3 Types of Warehouse...............................................................................................4 1.4 Material Handling...................................................................................................6 1.4.1 Material Handling Equipment...........................................................................6 1.4.2 Order Picking Equipment.................................................................................6 1.4.3 Receiving/Shipping Equipment........................................................................8 1.5 Receiving /Shipping Dock Design..........................................................................9 1.5.1 Number of Docks..............................................................................................9 1.5.2 Location of Docks.............................................................................................9 1.5.3 Types of Docks...............................................................................................10 1.5.4 Dock Productivity...........................................................................................10 1.6 Summary...............................................................................................................11 CHAPTER 2 LITERATURE REVIEW.........................................................................12 2.1 Warehouse Design: Proble ms and Methodologies...............................................12 2.2 Warehouse Design Resear ch Classification..........................................................13 2.3 Information Systems in Warehouse Design..........................................................13 2.4 Type of Storage Policy..........................................................................................14 2.4.1 Dedicated Storage...........................................................................................14 2.4.2 Randomized Storage.......................................................................................15

PAGE 3

ii2.4.3 Class Based Storage........................................................................................15 2.5 Product Allocation Using Different Policies........................................................16 2.6 Summary...............................................................................................................17 CHATER 3 CLUSTERING ALGORITHMS................................................................18 3.1 Classical Sets........................................................................................................18 3.2 Fuzzy Sets and Membership Function..................................................................19 3.3 Data Clustering Algorithms..................................................................................19 3.3.1 K means Clustering Algorithm.......................................................................20 3.3.2 Hierarchical Clustering Algorithm.................................................................20 3.3.3 Fuzzy C-means Clustering Algorithm............................................................21 3.3.4 Fuzzy Factor....................................................................................................23 3.3.5 Ideal Number of Clusters ‘c’...........................................................................23 3.3.6 Significance of Membership Function in Cluster Analysis............................23 3.4 Fuzzy C-means Clustering Application in Facilities Design................................25 3.5 Summary...............................................................................................................25 CHAPTER 4 PROB LEM DEFINITION........................................................................26 4.1 Design Model for Dedicated Storage Policy by (T/S) Approach.........................26 4.2 Motivation for Research.......................................................................................27 4.3 Proposed Fuzzy c-Means Model...........................................................................27 4.4 Use of Linguistic Variables for Uncertain Data...................................................28 4.5 Step by Step Methodology for the Warehouse Layout.........................................28 4.6 Example 1: A Small Warehouse...........................................................................29 4.6.1 Example 1 Solved by T/S Method..................................................................31 4.6.2 Example 1 Solved by FCM Method Using Crisp Data...................................32 4.6.3 Example 1 Solved by FCM Method...............................................................33 4.7 Summary...............................................................................................................35

PAGE 4

iiiCHAPTER 5 RESULTS AND ANALYSIS...................................................................36 5.1 Example Problem 2: A Medium Warehouse........................................................36 5.1.1 Layout by T/S Method....................................................................................39 5.1.2 Example 2 Solved by FCM Method Using Crisp Data...................................39 5.1.3 Comparison of Layout and Total Expected Distance.....................................43 5.1.4 Example 2 Solved by FCM Method Using Fuzzy Data.................................43 5.1.5 Layout for Fuzzy Data by FCM Method........................................................44 5.2 Principles for Warehouse Design..........................................................................45 5.3 Example 2 with Volume Information...................................................................47 5.3.1 Layout Based on Fuzzy Data by FCM Method..............................................47 5.4 Example Problem 3: A Large Warehouse............................................................49 5.4.1 Layout by T/S Method....................................................................................52 5.4.2 Example 3 Solved by FCM Method Using Crisp Data...................................52 5.4.3 Comparison of Layout and Total Expected Distance.....................................55 5.4.4 Example 3 Solved by FCM Method Using Fuzzy Data.................................57 5.4.5 Layout for Fuzzy Data by FCM Method........................................................58 5.5 Example Problem by FCM for Fuzzy Data: 3 Features........................................60 5.5.1 Layout for Fuzzy Data by FCM Method for 3 Features.................................61 5.6 Sensitivity of Generated Layouts..........................................................................63 5.7 Effect of Number of Clusters on Total Expected Distance...................................63 5.8 Research Contributions.........................................................................................64 5.9 Summary...............................................................................................................64 CHAPTER 6 SUMMARY AND CONCLUSIONS.......................................................65 6.1 Summary and Conclusions..................................................................................65 6.2 Scope for Future Research....................................................................................66 REFERENCES.................................................................................................................67

PAGE 5

iv LIST OF TABLES Table 4.1 Cluster Output for Crisp Data – Cluster 1, 2 and 3...........................................33 Table 4.2 Total Expected Distance by FC M Method for Crisp Data: Example 1............33 Table 4.3 Fuzzy Data: Example 1.....................................................................................34 Table 4.4 Cluster Output for Fuzzy Data – Cluster 1, 2 and 3.........................................34 Table 4.5 Comparison of Results: Example 1..................................................................34 Table 5.1 Product Data for Example 2..............................................................................37 Table 5.2 Cluster Output for Crisp Data: Example 2........................................................41 Table 5.3 Total Expected Distance Traveled Per Day......................................................41 Table 5.4 Comparison of Results: Example 2..................................................................43 Table 5.5 Fuzzy Product Data for Example 2...................................................................44 Table 5.6 Total Expected Distance Traveled....................................................................45 Table 5.7 Fuzzy Throughput, Storage a nd Volume Data for Example 2.........................47 Table 5.8 Total Expected Distance Traveled....................................................................49 Table 5.9 Crisp Product Data for Example 3....................................................................50 Table 5.10 Cluster Output fo r Crisp Data: Example 3......................................................54 Table 5.11 Total Expected Distance Traveled Per Day....................................................55 Table 5.12 Comparison of Results: Example 3................................................................55 Table 5.13 Fuzzy Product Data for Example 3.................................................................57 Table 5.14 Total Expected Distance Traveled Per Day....................................................58 Table 5.15 Fuzzy Product Data for Example 3.................................................................60 Table 5.16 TED by FCM Method for 3 Features: Example 3..........................................61 Table 5.17 Effect of Random Product Data on Expected Distance..................................63 Table 5.18 Analysis of Number of Clus ters on Total Expected Distance........................63

PAGE 6

v LIST OF FIGURES Figure 1.1 Warehouse Roles in th e Distribution Network..................................................2 Figure 1.2 Activities in Warehouse....................................................................................3 Figure 3.1 Membership Function for FCM Algorithm.....................................................24 Figure 4.1 Rectilinear Distance Traveled: Example 1......................................................31 Figure 4.2 Layout by T/S Method.....................................................................................32 Figure 4.3 Layout by FCM Method Fuzzy Data...............................................................35 Figure 5.1 Rectilinear Distance Traveled for Example 2.................................................38 Figure 5.2 Layout by T/S Method.....................................................................................40 Figure 5.3 Layout by FCM Method Crisp Data................................................................42 Figure 5.4 Layout by FCM Method Fuzzy Data...............................................................46 Figure 5.5 Layout by FCM Method Fuzzy Data with 3 Features.....................................48 Figure 5.6 Rectilinear Distance Traveled in Warehouse: Example 3...............................51 Figure 5.7 Layout by T/S Method.....................................................................................53 Figure 5.8 Layout by FCM Method Crisp Data................................................................56 Figure 5.9 Layout by FCM Method Fuzzy Data...............................................................59 Figure 5.10 Layout by FCM Method Fu zzy Data with 3 Features...................................62

PAGE 7

vi FUZZY C-MEANS CLUSTERING APPROACH TO DESIGN A WAREHOUSE LAYOUT Vaibhav C. Naik ABSTRACT Allocation of products in a warehouse is done by various storage policies. These are broadly classified into three main categories: dedicated storage, randomized storage, and class-based storage. In dedi cated storage policy a product is assigned a designated slot while in random storage policy incoming pr oduct is randomly assigned a storage location close to the input/output point Finally, the class-based stor age is a mixed policy where products are randomly assigned within their fixe d class. Dedicated st orage policy is most commonly used in practice. While de signing large warehouse layout, the product information in terms of throughput and storage le vel is either uncertain or is not available to the warehouse designer. Hence it is not pos sible to locate products on the basis of the throughput to storage ratio method used in the above mentioned storag e location policies. To take care of this uncertainty in produc t data we propose a fuzzy C-means clustering (FCM) approach. This research is mainly directed to improve the efficiency (distance or time traveled) by designing a fuzzy logic based warehouse w ith large number of products. The proposed approach looks for similarity in the product da ta to form clusters. The obtained clusters can be directly utilized to de velop the warehouse layout. Further, it is investigated if the FCM approach can take into account other factor s such as product size, similarity and/or characteristics to generate layouts which ar e not only efficient in terms of reducing distance traveled to store/retr ieve products but are effective in terms of retrieval time, space utilization and/or be tter material control.

PAGE 8

1 CHAPTER 1 INTRODUCTION Warehousing is a complex of facilities and activities in support of the manufacturing enterprise with impact throughout the corpora tion. Within the supply chain, warehousing is an important activity in the distribution of materials from raw materials, and work in process, to the finished goods. Hence, wa rehouse system design has acquired lot of importance in the supply chain cycle. The goal of warehouse design is to minimize the existing cost of establishing and operating a warehouse. The key goal of a Warehouse system is to maintain and store stock of parts ready for distribution, so that at all times the demand for items is met. Another important goal of a warehouse is to assemble product batches prior to delivery, to stockpile critical parts, and to facilitate regional distribution network for quick and cost efficient delivery. The following sections are an attempt to explain the key feature and im portance of a warehouse facility. 1.1 Functions of Warehouse A warehouse environment may serve any of the following requirements. Figure 1.1 gives a brief idea of roles of a wa rehouse in distribution network. 1. Bufferit holds inventory that is used to balance and buffer the variation between production schedules and demand. For this application, the warehouse is located close to the manufacturing facility. A warehouse that serves these demands is replenished on monthly to quarterly basis. 2. Consolidation – A warehouse may be used to accumulate and consolidate products from various areas of manufacturing within a single firm or from many firms. It facilitates combined shipment to common customers. This type of warehouse may be located central to the production location or th e customer base. This type of facility responds to regular week ly or monthly orders.

PAGE 9

23. Rapid ResponseRapid response in a warehous e is an important aspect to shorten transportation distances to permit easy access to customer’s demand. Figure 1.1 Warehouse Roles in the Distribution Network 1.2 Activities in Warehouse As a part of product storage there are many activ ities that occur in th e process of getting material into and out of the warehouse. Disc ussed are some of the important activities involved in a warehouse. Re fer figure 1.2 for a block diagram of activities in a warehouse. 1. Receivingbegins with advance notifi cation of arrival of the goods to the warehouse. Conceptually, it is a collection of activities that involve the orderly receipt of all materials in to the warehouse. This activ ity provides the assurance that the quantity and quality are according to the order, and helps to disburse material to storage or other organiza tional functions needing them. Products arrive in large pallet loads and so labor requirement are not high. Hence receiving accounts for a low operating cost in a warehouse. 2. Prepackagingin a warehouse when products are received in bulk from a supplier then packaging is performed which is p ackaging the products subsequently single MANUFACTURER MANUFACTURER CONSOLIDATION WAREHOUSE LOCAL WAREHOUSE CUSTOMER CUSTOMER CUSTOMER WAREHOUSE MANUFACTURER

PAGE 10

3packages or in combination of other products to form kits or assortments. When packaging greatly increases the storage cube size requirements or when a part is common to several assortments, either th en the entire receipt of merchandise is processed at once, or a portion is kept in bulk form to be processed later. 3. Put Awaythe process or act of placing merchandise in storage that includes transportation and placement is comm only known as put-away. Before product can be put away an appropriate storag e location must be determined. The importance of this task is that it determin es how quickly and at what cost the item is later retrieved for the customer. Wh en a product is put away, the storage location should also be scanned to record where the product has been stored. Putaway generally requires a larg e amount of labor as the pr oducts are required to be moved a large distance to their respective locations. Figure 1.2 Activities in Warehouse RECEIVING SHIPPING CROSSDOCKING ACCUMULATION, SORTATION & PACKING RESERVE STORAGE AND PALLET PICKING CASE PICKING BROKEN CASE PICKING PUTAWAY TO RESERVE REPLENISHMENT REPLENISHMENT

PAGE 11

44. Storage-While merchandise is waiting fo r demand; the physical containment of that merchandise is called as Storage. Various forms of storage depend on the size and quantity of the items in invent ory and the handling characteristics. 5. Order PickingOrder Picking is the se rvice that the warehouse provides for the customers. It is the process or act of removing merchandise items from storage to meet a specific demand. It is one of th e most important activ ities as it is the function around which most of the warehouses are designed. 6. Sortation-When an order has more than one item and the accumulation is not done as the picks are made; then the Sortation of batch picks into individual orders and accumulation of distributed picks into orders must be done. 7. Packing and ShippingIt is a combina tion of various activities following order picking and package. Some key aspects are as mentioned below. A. Packaging of items in appropriate shipping containers. B. Preparation of shipping documents C. Checking orders for completeness an d weighing to calculate charges. D. Accumulating orders by outbound carriers. E. Loading trucks may or may not be a pa rt of it as in many cases this is carrier’s responsibility. 8. Cross Dockingcross-docks are high-sp eed warehouses. If an arriving item has been requested by a customer there is no n eed to store it as anticipated inventory, instead the items can move directly from receiving to shipping, without intermediate storage and retrieval. Thus the item can move much more quickly through the facility and the costly pa rt of warehouse labor can be avoided. 9. Replenishment primary locations fr om the reserve storage location. 1.3 Types of Warehouse 1. Factory Warehousea factory warehouse in terfaces production with wholesalers. Such warehouses have following important features. A. A comparatively small number of or ders are picked up on daily basis.

PAGE 12

5B. For a factory warehouse advance information about the order composition is required. C. Focus on cost and order accuracy is also high. D. The responsiveness depends h eavily on production schedules. 2. Retail Distribution warehouseit serv es a number of captive retail units. Following are the main features of a retail distribution warehouse. A. Advance info about order composition is needed. B. Carton and item picking is done from a forward area. C. More orders per shift than consolidation/shipping lanes D. It focuses on cost, accuracy and fill rate of the packages. E. Responsiveness depends heavily on truck routing schedules F. The only critical point is that if th e retail units are not captive, then responsiveness become s a crucial issue. Catalog Retailer warehousethis type of warehouse deals with filling orders from catalog sales. Main features are as follows. A. A large number of small; frequently single-line orders are picked up. B. Item and, sometimes, carton picking C. Daily compositions of orders are usually unknown. D. Only statistical information available. E. Like factory warehouse and retail ware house, the emphasis is on cost and response time. 4. Support of manufacturing operations ware housethis type of warehouse serves the purpose of a stock room providing raw material and work-in-process items to manufacturing operations. The main feat ures of this type of warehouse are as mentioned. A. Contains many small orders. B. Only statistical information available about order composition. C. Stringent time requirements for response time. D. Focus on response time but also accuracy and cost.

PAGE 13

61.4 Material Handling Material Handling is defined as “... providing the right amoun t of the right material, in the right condition, at the right place, at the ri ght time, in the right position, in the right sequence, and for the right cost, by using th e right methods.” (Tompkins, 1996) In a warehouse, it is the material handling syst em that makes possible the materials flow specified in the layout of th e facility. Material handli ng system tasks facilitate distribution of material to th e plant cell; implementation of planned flow paths in the layout and controls the flow of parts within and between the departments. The three major activities in material handling that in clude rest other sub activities are receiving, order picking, and shipping. Material handli ng has several key functions; the important amongst them is the setting up of directed flow paths among carriers and buffering between the staging area and st orage area. In addition, mate rial handling facilitates the continuous flow without excessive congestion or idleness in the warehouse. Further, it helps in maintaining safety and good housekeeping in the warehouse. 1.4.1 Material Handling Equipment Material Handling Equipments are broadly cl assified into two categories namely order picking equipment and receiving/shipping equipmen t. There is a variety of equipment to reduce labor cost and to incr ease space utilization. This equipment’s are discussed in details below. 1.4.2 Order Picking Equipment As with the picking methods, the picking equipm ent used will also depend on a variety of factors. Below mentioned is a consolidated list of material handling equipment’s and its application in various picking environment. 1. Static shelving the most common equipment for st orage in piece pick operations, static shelving is designed with depths from 12” to 24”. Product is placed either directly on the shelving or in corrugated or plastic parts bins. Static shelving is economical and is the best method where there are few picks per SKU or where parts are very small.

PAGE 14

72. Carton Flow RackCarton flow rack is similar to static shelving with the exception that rather than shelves there are small sections of gravity conveyor mounted at a slight angle. Product is stoc ked from the rear of the flow rack and picking is done from the face. Product can be stocked in cartons or small totes or bins, as a carton or tote is emptied, it is removed from the rack, and another one will roll into place. Carton flow rack is most useful where there are a very high number of picks per SKU 3. Carousels Horizontal Carousels are a version of the same equipment used by dry cleaners to store and retrieve clothing; they have racks hanging from them that can be configured to accommodate vari ous size storage bins Generally, an operator will run 2 to 4 ca rousels at a time avoiding th e need for the operator to wait while one unit is turning. Picking is usually performed in batches with orders downloaded from the host system to the carousel software. Horizontal carousels are most common in picking operations with very high number of orders, low to moderate picks per or der, and low to moderate picks per SKU Horizontal carousels provide very high pick rates as well as high storage density. Pick-to-light systems are often integrated into carousels. Vertical Carousels are frequently used in laboratories and sp ecialty manufacturing operations and are rarely used in regular order picking operations. 4. Automatic Storage and Retrieval Systems (AS/RS) An AS/RS is a system of rows of rack, each row having a dedicated retrieval unit that moves vertically and horizontally along the rack picking and pu tting away loads. ASRS systems are available in Mini-load types that store and transfer product on some type of tray or in bins and Unit-load type s that transfer and store pa llet loads. In addition to the automation features, AS/RS units can provide extremely high storage density with capabilities to work in racking up to 100 feet hi gh. The high costs of AS/RS equipment and the length of the retrieval times make it difficult to incorporate into a piece picking operation. 5. Automatic Picking MachinesFully auto mated picking machines (such as Aframes) are rare and are used only wher e very high volumes of similar products

PAGE 15

8are picked, or where high volume in combination with high accuracy requirements exists. 6. Pick to lightPick to light systems cons ists of lights and LED displays for each pick location. The system uses software to light the next pick and display the quantity to pick. Pick-to-light systems have the advantage of not only increasing accuracy, but also increasi ng productivity. Since hard ware is required for each pick location, pick-to-light systems are co stly and are suitable where very high picks per SKU occur. Carton flow rack and horizontal carousels are good applications for pick to light. In batch picking, pick-to-light is also incorporated into the cart or rack that hol ds the cartons or totes that you are picking in to. The light will designate which order you s hould be placing the picked items in. 7. Voice Directed PickingVo ice technology has come of age in recent years and is now a very viable solution for piece picks, case picks, or pallet picks operations. 8. Automated Conveyor and Sorting System sAutomated Conveyor systems and sorting systems are an integral part of any large-scale piece picks operation. The variety of equipment and system designs is enormous. 1.4.3 Receiving/Shipping Equipment Also known as the material transport equipmen t they differ from the other category of material handling equipment by their primary function namely material transport. Some of the widely used materials hand ling equipment’s are listed below. 1. Conveyorsconveyors are used when material is to be moved more often between locations. They are mostly used for a fixed path traverse. Hence there must be a sufficient volume of product movement to implement conveyor type handling. Conveyors are mainly classified based on th e product type and the location of the conveyor. Some common types are chute, be lt, roller, wheel, a nd chain and trolley type conveyors. 2. Industrial Vehiclesis the simplest mode of materi al transport in a warehouse. The main advantage that they provid e in a warehouse is maneuvering and transportation. Industrial vehi cles are broadly classified as hand trucks, pallet jacks, and powered industrial trucks.

PAGE 16

93. Automated Storage and Retrieval Machines -this type of storage system uses a fixed path storage and retrieval machines running on rails between storage racks. These systems handle loads in excess of 1000 pounds. 4. Automated Guided Vehiclesas the name suggests these are driverless industrial trucks and they follow a predefined path in an aisle. The path followed is a simple loop or a complex network with many designated load/unload stations. 1.5 Receiving /Shipping Dock Design The most valuable area in a warehouse is th e receiving/shipping dock area. Every item in the warehouse comes some time or the other through the dock. Everything that leaves the warehouse goes out across the dock. Hence, no activity in a warehouse is complete without the dock area. Unfortunately, load ing docks generally receive less thought and foresight in the layout and de sign efforts. The key factor s while designing dock area is the selection of right number of docks, location of the docks, and the type of dock to be used. Other important factors considered are productivity of the docks, safety features, and dimensions of the receiving/shipping areas. 1.5.1 Number of Docks The number of docks required is determined by a combination of factors namely; number of receipts and shipments, type of loading and unloading, types and sizes of vehicles, number and timing of carriers, and different areas in which materials will be utilized, stored or prepared for shipment. Ba sed upon the various characteristics one dock position should be allowed for each seven hours of planned activity per shift. The greater the number of operating shifts for shipping/r eceiving lower the total number of doors that are required. 1.5.2 Location of Docks Traditionally docks were located in the rear of a facility and out of sight. Generally receiving and shipping docks were all located in the same area, in order to reduce the need for duplicate supervision. In some larger facilities, shipping w ould be at one end of the building and receiving at th e other, in order to create a straight through material

PAGE 17

10movement. However, today, given the move to reduced inventories and the tendency for shipments to be in close proximity to th e manufacturing location, more facilities are being constructed with multiple shipping and receiving docks. These multiple docks drastically reduce the flow of materials within a facility. 1.5.3 Types of Docks Various types of docks are used as per the application for which they are designed. Saw tooth docks are useful when a site does not have sufficient exterior area to maneuver vehicles in and out of the docks. They op timize the amount of distance from the edge of the building to the end of the property line or the end of the paved area. This reduces the number of docks and interior space that can be used around the docks because of the sawtoothed pattern. Straight docks, on the other hand, optimize interior space. Open docks are impractical in most environments, and even where they can be used, they need to be evaluated as to their benefit versus potentia l theft and malicious damage. Interior docks provide protection from the exte rnal conditions and protect pr oducts from potential loss. Interior docks, however, come at a considerab le cost of lost space and increased energy consumption. 1.5.4 Dock Productivity Warehouse productivity, which includes dock pr oductivity, is a very complicated issue because of the differences of activities, t ypes of receiving and shipping units, types of material handled, sizes of individual receiv ed/shipped items, and types of loading and unloading equipment’s. Even in compar ing two warehouses in the same company, productivity will differ due to the differences in volumes and types of activity in each facility. The only accurate productivity benc hmarks are those developed specifically for that operation based upon the applicable activ ities, type, volume, and equipment used. Thus, dock space forms an important aspect of warehouse design and must be carefully designed for getting the maximum cost benefit and throughputs.

PAGE 18

111.6 Summary This chapter discussed the basics of warehous e management principals, types of material handling equipment used for vari ous activities in the warehous e, factors related to dock design. The rest of the thesis is organized as follows. In chapter 2 we discuss the research in the field of warehouse design and research in product allocation policies. We also discuss the storage location models for warehouse design. Chapter 3 deals with the concept of fuzzy data sets and data cl ustering algorithms. Fuzzy C-means (FCM) clustering approach forms the basis of solvi ng the warehouse layout problem dealt in this thesis. Chapter 4 explains an existing dedica ted storage model in detail with an example problem. We also explain the steps to fo llow for solving the warehouse design problem by FCM approach. Chapter 5 reviews the resu lts and analysis of the warehouse layout problem solved by FCM. In chapter 6 we conc lude and summarize the results obtained to achieve the goal of FCM approach to design a Fuzzy logic based warehouse.

PAGE 19

12 CHAPTER 2 LITERATURE REVIEW In this literature review, we discuss the various approaches to warehouse design and the framework for classification of warehouse design problems. The problems encountered during the design of warehouse and a system atic approach for warehouse design is stressed upon. We explain the storage location policies in brief. Later the review focuses upon the research done in the field of assi gnment of products with three assignment policies namely; dedicated, randomized and class based storage policy. The papers reviewed helped in unders tanding the current appro aches to warehouse design. 2.1 Warehouse Design: Problems and Methodologies This section discusses factor s that must be kept in mi nd while designing a warehouse. Rounwenhorst, et al (2000) discuss a reference fr amework and classification of warehouse design and control problems. Th e authors emphasize a need for designoriented studies as opposed to strong analysis oriented research on isolated sub problems that are dominant in current papers. Barthol di and Hackman (1998) analyze problems that are encountered during the design of a warehous e and its subsystem. A design-oriented approach primarily aims at a synthesis of a large number of both technical systems and planning and control procedures. The au thors develop a methodology for systematic warehouse design. The paper broadly discusses the concept of three different axes along which warehouses may be viewed upon namely process, resources, and organization. Further, they discuss performance criteria and process of warehouse design on a strategic, tactical, and operational level. The problem in each area and suggestion for improvement in the problem areas is made.

PAGE 20

132.2 Warehouse Design Research Classification Warehouse design problems can be posed in a nu mber of ways. In this literature review, two major categories of design problems ar e studied. The first category addresses the overall design problem and concentrates on the formulation of top-down; iterative, optimization-based approaches (Ash ayeri and Gelders, 1985 and Gray, et al ., 1992). The overall design problem is a complex problem and has many aspects to it. These models provide a basic conceptual framework for th e design problem. Even in the case when a proposed design procedure is appl ied to a case study as in Gray et al (1992), it is not always clear how results can be vali dated beyond the case study structure. The second category addresses specific desi gn problems like design of a storage system or an order-picking system. The papers th at discuss this issu e are Bozer and White (1996); Goetschalckx (1992); Jarvis and McDowell (1991); Rosenblatt, et al (1993) and Yoon and Sharp (1996). The models discussed in these papers are useful but it is difficult to integrate models for different problems into an overall design procedure due to different assumptions or data representations. In reality ex pert practitioners rarely use the results of the extensive resear ch done in the warehouse design area. Rather, they rely on their experience and expertise. 2.3 Information Systems in Warehouse Design Information is the key in designing warehouses with large number of product data. In the warehouse design field, this information, expe rience and knowledge may be applied by the use of a well-esta blished design procedure such as Systematic Layout Planning, Muther (1973). In any case, over time an expert develops methods for decision-making, as well as specific information requirements that are integral to design decisions. Information is the key to the design deci sion-making process (Hazelrigg, 1996). In the warehouse design domain, today’s computeri zed information systems provide the designer with large historical da tasets that can be used in the design process. The authors seek to formalize the decision-making pro cess and sequence, the information used, the criteria applied, and the evaluation methods utilized. Green (1992) outlines a number of

PAGE 21

14relevant attributes possessed by experts na mely supplying context, ordering decisions, abstracting parameters, a nd classifying heuristics. 2.4 Type of Storage Policy Type of Storage policy decides how to allocate the various storage lo cations of a uniform storage medium to a number of SKU These are broadly classified in to three main categories mentioned as below. 2.4.1 Dedicated Storage Also referred to as fixed slot storage, involves the assignment of specific storage locations for each product stored. As storag e location is assigned or dedicated to a specific product, the term “dedicated storage” is used. Every SKU is assigned a particular number of storage locations, exclusively alloca ted to it. The number of storage locations allocated reflects its maximum storage n eeds and is determined through inventory activity profiling. Two variati ons in dedicated storage polic y are commonly used. Part number sequence storage is frequently used due to its simplicity. In this type, the storage location of a product is based entirely on the part number assigned to it. Low part numbers are assigned to the best location in the warehouse. Hence, a part with a large part number gets a poor storage location. This type of storage policy does not take in to account the activity level of the parts. Throughput based dedicated storage is an alternative to part number se quence storage. Such a stor age policy gives a thought to activity levels and storage requirements among products to be stored. Throughput-based storage is preferred to the part number sequencing storag e when there are significant differences in either the activity level or th e inventory level for pr oducts being stored. With dedicated storage, the number of storag e locations assigned to product must be capable of satisfying the maximum storage requirement for the product. With multiproduct storage, the storage space require d is the sum of the maximum storage requirements for each of the product.

PAGE 22

152.4.2 Randomized Storage Also referred to as floating slot storage, allows the storage location for a particular product to change over a period. When a produc t arrives for storage it is placed in the closest location available and retrieval occurs on a first in first out basis. For warehouses with more than one I/O points, the storage lo cation selected is the one nearest to the I/O point. With randomized storage, products can be stored in any availa ble storage location. Hence, the storage space requirement will be equal to the maximum of the aggregate storage requirements for the products. Due to the dynamic conditions in the replenishment of products, it is very difficu lt to forecast the exact storage requirements for this storage policy. Hence, storage capac ity levels are decided by treating inventory levels of the products as random variables. The randomized storage model is explained in brief as below. For given n storage spaces required we have to determine the storage space layout that minimizes the total expected travel distance between each storage space and m I/O points. The sum of the distances of storage space j from each I/O point is given by the equation m k kjd1. Arrange the spaces in ascending order of the sum of these distances, and pick the n closest storage spaces. Here n depends on the inventory le vels of all the items, so the total number of spaces n is less than that required under the dedicated policy. The basic assumptions for the randomized storage m odel are that every empty slot is equally likely to be selected for storage and each unit of a particular product is equally likely to be retrieved when multiple storage locations exist, Heragu (1997) and Francis (1992). 2.4.3 Class Based Storage This storage location policy is midway between dedicated storage policy and the randomized storage location policy. The class based storage policy is based on Pareto’s law with respect to storage and retrieval (S/R ) activity level generate d by different items. “In a warehouse 80% of the S/R activity is dir ected at 20% of the items, 15% at 30% of the items and the remaining 5% of the S/R ac tivity at 50% of the items,” Heragu (1997). Incoming items are thus classified into three cl asses as A, B, and C, based on the level of

PAGE 23

16S/R activity (from high to low) they generate Thus to minimize the time/distance spent in storage and retrieval, Class A items must be stored closest to the input/output point, Class B next closest a nd Class C the farthest. 2.5 Product Allocation Using Different Policies In this section we will discu ss the research done in the fiel d of dedicated and class based storage allocation policies, that forms the basi s of the existing approach in this thesis. In dedicated storage policy pr oducts are assigned to storage /retrieval locations in an attempt to minimize the time/distance required to perform the storage and retrieval operations. For dedicated storage to be feas ible there should be sufficient number of storage slots to be dedicated to the products The basic aim is to minimize the distance traveled to store and retrieve the assigne d products. Tompkins (1996) suggest the T/S approach where T is the throughput of products to be placed and S is the storage level. However for large warehouses, the input data available is fuzzy and so T/S approach cannot be used. Also this approach does not consider the product ch aracteristics that are important to a warehouse designer. Malmborg and Bhaskaran (1990) evaluate the Cube per Order Index storage policy for different kinds of warehouses, based on an alytical expressions for the maximum throughput. Park and Webster (1989) derive analytical expressions for the maximum throughput of multiple three-dimensional storag e systems with the cubic-in-time storage policy. Kaylan and Medeiros ( 1988) evaluate storage policies for a miniload system with multiple I/O points and suggest storage algorithms for the Deep Lane Storage System that minimizes the number of relocations. Further, Hausman et al. (1976) analyze class-based storage in an AS/RS, assuming single comma nds. They develop analytical methods to determine the optimal dimensions of the zones, considering storage capacity and maximum throughput. Goetschalckx and Ratlif (1990) evaluate st orage policies for block storage through an analytical study. Jarvis and McDowell (1991) propose a heuristic for the storage policy in a conventional warehouse. Roll and Rose nblatt (1983) analyze th e storage capacity of

PAGE 24

17conventional warehouses with alternative stor age policies, using si mulation techniques. Wilhelm and Shaw (1996) present an empirical study concerning the storage policy of an AS/RS. This research was helpful in unders tanding the selection of appropriate storage policy and the various appro aches to implement them. 2.6 Summary In this chapter we reviewed the varied re search done in the field of warehouse layout design. We also learnt the di fferent storage location models and how they help design a warehouse layout. The dedicated storage appro ach uses crisp information of the input data namely throughput and storage level, that in most cases is not available to the warehouse designer for large number of parts that are stored even in a small sized warehouse. We would like to in corporate the fuzzy logic appr oach to design a warehouse layout where the input data is large and fuzzy in nature. To our knowledge, no research effort has been reported in the literature for designing warehouse layouts based on fuzzy logic approach. In chapter 3 we would discuss the fuzzy Cmeans clustering approach (FCM) and how it helps to solve the fuzzy nature of the input variables. An attempt will be made in this thesis to use a fuzzy logic based method which will hopefully give good results in comparison with existing T/S method.

PAGE 25

18 CHATER 3 CLUSTERING ALGORITHMS This chapter introduces the basic definitions a nd concepts of fuzzy data sets and Fuzzy cmeans clustering technique that will be need ed in the further chapters. Gradually the chapter shifts to Fuzzy c-means clustering, which forms the basis for solving the warehouse layout problem. 3.1 Classical Sets A classical set is a set that has a crisp boundary. For example, a classical set X of real numbers greater than 6 is expressed as A= {xx > 6} In this set of real numbers there is a clear unambiguous boundary 6 such that if x is greater than this number. In this case x either belongs to this set ‘A’ or it does not belong to this set. These types of sets are called Clas sical Sets and the elements in this set are a part of the set or they are not a part of the set. Classical sets are an important tool in mathematics and computer science but they do not reflect the nature of human concepts and thought. In contrast to a classical se t, a fuzzy set is a set without crisp boundaries. That is, the process of an element “belongs to a set” to “ does not belong to a set” is gradual. This transition is decided by the membership functi on of a fuzzy dataset. Real life problems have data which most of the time has a degree of “trueness” or “falsene ss” that is the data cannot be expressed in terms of classical set. A good example of this is; the same set A is a set of tall basketball pl ayers. According to the classical set logic a player 6.01 ft tall is considered to be tall whereas a player 5.99 ft tall is considered to be short.

PAGE 26

193.2 Fuzzy Sets and Membership Function Membership functions give fuzzy sets th e flexibility in modeling commonly used linguistic terms such as “the water is hot” or “the temperature is high.” Zadeh (1965) points out that, this imprecise data set info rmation plays an important role in human approach to problem solving. It is important to note that fuzziness in a dataset comes does not come from the randomness of the elemen ts of the set, but from the uncertain and imprecise nature of the abst ract thoughts and concepts. If X is a collection of objects denoted by x, then a fuzzy set ‘A’ in ‘X’ is defined as a set of ordered pairs A= {( x, A ( x )) x X }, Where A (x) is called the membership function (MF) for the fuzzy set A. The membership function maps each element of X to a membership grade between 0 and 1. If the value of the membership function is restricted to either 0 or 1, then A is reduced to a classical set and A (x) is the characteristic function of A. Usually X is referred to as the universe of discourse and may consist of discrete objects or continuous space. 3.3 Data Clustering Algorithms Clustering of numerical data forms the ba sis of various classi fication and system modeling algorithms. The purpose of clustering is to identify natura l groupings of data from a large data set to produce a concis e representation of a system's behavior. Clustering algorithms are not only used to orga nize and categorize data, but are helpful in data compression and model construction. Cluste ring partitions the da ta set into several groups such that the similar ity within a group is larger than among the groups. To achieve such partitions it is essential to ha ve a similarity metrics that takes two input vectors and returns a value reflecting their similarity. As most of the similarity metrics are sensitive to the range of elements in the input vectors, each of the input variables must be normalized or scaled down. Clusteri ng techniques are broadly classified as hard clustering and fuzzy clustering.

PAGE 27

203.3.1 K means Clustering Algorithm The K-means clustering, also known as C-mean s clustering, has been applied to variety of areas, including image and speech data compression. This technique is based on randomly choosing k initial cluster centers, or means. These initial cluster centers are updated in such a way that after a number of cycles they repres ent the clusters in the data as much as possible. A drawback of the k-means algorithm is that the number of clusters is fixed; once k is chosen it always remains k cluster centers. The K-means algorithm circumvents the problem by removing the redund ant clusters. Whenev er a cluster centre is not assigned enough samples, it may be remove d. In this way one is left with a more or less optimal number of clusters. The problem of choosing the initial number of clusters still remains unsolved, but by taking k large enough this will us ually not be a problem. 1. The algorithm starts out with initializing Ci this is achieved by randomly selecting C points from among all the data points. 2. Determine the membership matrix U, where the element uij is 1 if the jth data point xj belongs to the group I and 0 otherwise. 3. Compute the cost function by the equa tion given below. Stop if the value of cost function is below a certain threshold value. 2 1, 1( Ci X Ji Jc iGi Xk k k c i ) 4. Update the clusters center centers Ci and determine the new U matrix. The K-means algorithm is mainly iterative, and hence hard to predict its convergence to optimum solution. The performance of th e K-means algorithm depends on the initial position of the cluster centers. Hence, initia l clusters centers are predicted by a front-end tool, which generates clus ter centers iteratively. 3.3.2 Hierarchical Clustering Algorithm In hierarchical clustering the data is not partitioned into a particular cluster in a single step. Instead, a series of partitions takes place that run from a single cluster containing all objects to N clusters each containing a single object. Hierarchi cal clustering is further classified as agglomerative method, whic h proceed by series of fusions of the N objects into groups, and divisive method, which separate N objects successively into finer

PAGE 28

21groupings. Hierarchical clus tering may be represented by a two-dimensional diagram known as dendrogram, which illustrates the fusi on or divisions made at each successive stage of analysis. Given a set of N items to be clustered, and an NxN distance matrix, the basic process of Johnson's (1967) hierarchical clustering is briefly explained below. 1. The algorithm starts by assigning each ite m to its own cluster, such that for N items, we have N clusters, each containing just one item. Let the distances between the clusters equal the distances between the items they contain. 2. Find the closest (most similar) pair of clusters and merge them into a single cluster, so that we have one less cluster. 3. Compute distances between the new clus ter and each of the old clusters. 4. Repeat steps 2 and 3 until all items ar e clustered into a single cluster of size N. Step 3 can be done in different ways, whic h is what distinguis hes single-link from complete-link and average-li nk clustering. In single-li nk, clustering (also called the connectedness or minimum method); we consid er the distance between one cluster and another cluster to be equal to the shortest di stance from any member of one cluster to any member of the other cluster. If the data consist of similarities, we consider the similarity between one cluster and another cluster to be equal to the largest similarity from any member of one cluster to any member of the other cluster. In complete-link, clustering (also called the diameter or maximum method); we consider the distance between one cluster and another cluster to be equal to th e longest distance from any member of one cluster to any member of the other cluster. In average-link clustering, we consider the distance between one cluster and another cluster to be equal to the average distance from any member of one cluster to any member of the other cluster. 3.3.3 Fuzzy C-means Clustering Algorithm Fuzzy C-means clustering (FCM) algorithm, also known as fuzzy Isodata, is a data clustering algorithm in which each data point be longs to a cluster to a degree specified by a membership grade. Bezdek proposed this algorithm in 1973 as an improvement to K-

PAGE 29

22means algorithm also known as the hard Cmeans algorithm. Hard k-means algorithm executes a sharp classification, in which each obj ect is either assigned to a class or not. The application of fuzzy clustering to the data set function allows the class membership to have several classes at the same time but w ith different degrees of membership function ranging from 0 to 1. Fuzzy c-means (FCM) is a method of clustering which allows one piece of data to belong to two or more clusters. It is based on minimization of the following objective function 2 11 j i N i C j m ij mc x u J Where m the fuzzy factor is any real number greater than 1, j is the number of cluster decided by the user, uij is the degree of membership of xi in the cluster j, xi is the ith of ddimensional measured data namely throughput, storage level and volume, cj is the ddimension center of the cluster, and 2 j ic x is any norm expressing the similarity between the measured data (throughput, storag e level and volume) and the center. Fuzzy partitioning is carried out th rough an iterative optimizati on of the objective function shown above, with the update of membership matrix uij and the cluster centers cj by, C k m k i j i ijc x c x u1 1 / 2 2 21 and, N i m ij N i i m ij ju x u c1 1. This iteration will stop when ) ( ) 1 (, maxk ij k ij iju u u where is a termination criterion between 0 and 1 and usually se t to 0.02 (Zimmermann, (1990) whereas k is the iteration steps. This procedure converges to a local minimum or a saddle point of Jm. The algorithm is composed of the fo llowing steps mentioned below. 1. Initialize U=[uij] matrix, U(0) 2. At k-step, calculate the centers vectors C(k)=[cj] with U(k)

PAGE 30

23 N i m ij N i i m ij ju x u c1 1. 3. Update U (k), U (k+1) C k m k i j i ijc x c x u1 1 / 2 2 21 4. If || U (k+1) U (k) ||< then STOP, otherwise return to step 2. As indicated earlier, the value of lies between 0 and 1 and 0.02 is the commonly used value. The number of iterations for the algorithm to reach a local minimum will be decided by the value of 3.3.4 Fuzzy Factor The fuzzy factor ‘m’ was introduced by Bezdek ( 1974) and is also known as ‘fuzzifier’ As the value of m approaches 1 the clusters fo rmed tend to be hard and as the value of m tends to infinity the obtaine d clusters tend to go in a the fuzziest state. There is no theoretical justification on the value of ‘m ’ but is usually set to 2 and in a more generalized form tends to be between 1.5 and 3 (Zimmermann, 1990). 3.3.5 Ideal Number of Clusters ‘c’ From the research on decision of ideal number of clusters for the FCM algorithm, we find out that there is nothing called as an ideal number of clusters (Zimmermann, 1990). The number of clusters for a certain type of data will vary based on the data partition desired. The number of clusters can vary between 2 to infinity. In Chapter 5 we will discuss the effect of varying the number of clusters on the total expected distance traveled. 3.3.6 Significance of Membership F unction in Cluster Analysis As discussed in the earlier section, data are bound to each cluster by means of a membership function, which represents the fuzzy behavior of this algorithm. To do that, we build an appropriate matrix named U w hose factors are numbers between 0 and 1, and

PAGE 31

24represent the degree of membership between data and centers of clusters. In the FCM approach, instead, the same given datum doe s not belong exclusively to a well-defined cluster, but it can be placed in a middl e way. In the case of FCM, the membership function follows a smoother line to indicate that every datum may belong to several clusters with different values of the membership coefficient. Figure 3.1 Membership Function for FCM Algorithm In figure 3.1 (George and Yuan, 1995), the da tum shown as a red marked spot belongs more to the cluster B rather than the clus ter A. The value 0.2 of membership function indicates the degree of membership to A fo r such datum. Now, instead of using a graphical representation, we introduce a matrix NxCU whose factors are the ones taken from the membership functions. The number of rows and columns depends on how many data and clusters we are consid ering. Here C (columns) is th e total number of clusters and N (rows) is the total data points.

PAGE 32

253.4 Fuzzy C-means Clustering App lication in Facilities Design Cell formation, one of the most import ant problems faced in designing cellular manufacturing systems, is to group parts with similar geometry, function, material and process into part families and the correspondi ng machines into machine cells. There has been an extensive amount of work in this area and, consequently, numerous analytical approaches have been developed. On e common weakness of these conventional approaches is that they assume that disjoint part families exist in the data; therefore, a part can only belong to one part family. In reality, it is clear that some parts belong to more than one part family. Chu (1991) and Unde (2003) propose a fuzzy c-means clustering algorithm to formulate the problem. This approach offers a special advantage over conventional clustering. It not only shows the specific part family that a part belongs to, but also provides the degree of memb ership of a part asso ciated with each part family. This information allows users flexibility in determining to which part family a part should be assigned so that the worklo ad balance among machine cells can be taken into consideration. The author devel ops computer program to simplify the implementation and to study the impact of the model's parameters on the clustering results. 3.5 Summary The concept of hard and fuzzy clustering algor ithms was introduced in this chapter. The chapter also dealt with basic information on fuzzy data sets. Further the fuzzy c-means algorithm was explained in detail. This FCM algorithm will form the basis for solving the warehouse layout problem with fuzzy product data of throughput and storage requirement.

PAGE 33

26 CHAPTER 4 PROBLEM DEFINITION In this chapter, we discuss the existing T/S approach to dedicated storage location problem in a warehouse and the assumptions made to solve this problem. We also discuss the fuzzy c-means approach to so lve the warehouse stor age location problem. Further we introduce the concept of fuzzy data also known as linguistic variables. A step by step approach to designing a fuzzy based warehouse layout is also given. We solve a small warehouse problem with T/S method a nd FCM method and comp are the results for the total expected distance traveled in the warehouse. 4.1 Design Model for Dedicated Storage Policy by (T/S) Approach Francis (1992) suggests a generalized mode l for dedicated storage policy. A warehouse has m I/O points through which n items enter and leave the warehouse. The items are stored in one of s storage spaces or locations. Each location requires the same storage space, and it is known that item j requires Sj storage spaces.jT is the throughput requirement level for product j in number of storage/ retrieval per unit time and pi,j is the percent of storage/retrieval trips for product j from I/O point i. The distance traveled between I/O point I and storage/retrieval locations k is given by dj,k. Hence we can express f(x) as the expected distance traveled between storage location k and the I/O point i required to fulfill the throughput requi rement for the warehouse facility. m i n j k j k i j i s k j jx d p S T x f11 , 1) ( min Subject to n j k jx1 ,1, for k = 1, 2… s and, j s i ijS x 1 for j = 1, 2… n

PAGE 34

27xj,k = (0, 1) for all j and k m i k i j i s k k j n j j jd p x S T x f1 , 1 1) ( min Based on this formulation, Tompkins (1996) gives a T/S method to minimize the total expected distance traveled approach mentioned as below. 1. Rank the products in the descending order of their j jS T 2. Compute the distance traveled ) ( x f for all the slots in the warehouse. 3. Assign the products with the highest T/ S ratio to the slot with the least ) ( x f and so on. 4.2 Motivation for Research The main motivation for this research was to assist in the development of efficient warehouse layout in the absence of precise in formation about the throughput and storage levels for large number of products found in a modern warehouse. Furthermore, it was of value to investigate if in addition to throughput and storage, other pr oduct attributes such as product similarity, characteristics or volume could be taken into account while developing the layout which will give good result for the ex pected distan ce criterion and at the same time generate a layout which will reduce storage/retrieval time, improve space utilization or yield better material control. 4.3 Proposed Fuzzy c-Means Model The proposed approach involves the applicati on of the FCM algorithm to solve the layout design problem for dedicated storage locat ion problem and the class based storage location problem. As discussed in Chapter 2 class based storage operates as a dedicated storage for the formation of the classes and ra ndomized storage within the formed class. The approach as formulated earlier in this chapter works fine when the input information is crisp. But when the throughput and storag e requirement informati on is fuzzy, i.e. in the form of “High” and “low”, the T/S met hod does not work. Hence, for a warehouse with fuzzy input information of product data fuzzy c-means algorithm generates clusters of similar data. These clusters can be used as groups or classes for a storage policy. The

PAGE 35

28fuzzy cmeans algorithm was discussed in details in Chapter 3. Thus, the obtained cluster information helps in designing the warehouse layout. The generated layout should result in values comparable to T/S method for the total distance/time traveled. The validity of this method lies in the fact that, clustering tries to iden tify the relationships among patterns in a data set by organizing the pa tterns into a number of clusters, where the patterns in each cluster show a cert ain degree of closeness or similarity. 4.4 Use of Linguistic Variables for Uncertain Data The set of data that are defined on the set of ‘R’ real numbers are known as fuzzy sets. Membership functions of these numbers have a quantitative meaning and are viewed as fuzzy numbers or fuzzy intervals. These fuzzy numbers are numbers that are close to a real number. The concept of fuzzy numbers helps in characterizing many applications like states of fuzzy control, decision-maki ng, approximate reasoning, optimization, and statistics with imprecise probabilities, (K lir and Yuan, 1995). When the fuzzy numbers represent linguistic concepts li ke very small, small, medium, and high and so on. These variables are set as per the user’s discre tion and are known as ‘Linguistic Variables’. Each linguistic variable is defined in terms of a base variable the values of which are real numbers within a specific range. A base variable is a variable in the classical sense, like in our case throughput, storage requirement and Volume. This concept of linguistic variables will be used to solv e the warehouse layout problem. 4.5 Step by Step Methodology for the Warehouse Layout The steps to be followed for running the FCM algorithm for calculation of the total expected distance traveled in th e warehouse are as given below. 1. Input. The user enters the number of clus ters, throughput levels in terms of number of input/output trip s per unit of time, storage and volume requirement. The input data for storage, throughput and volume is fuzzy and in levels such as very low, low, medium low, medium, me dium high, high and very high. The user, based on the product data size, decides the levels of the fuzzy variables. Each level of fuzzy data for st orage and throughput has a fixe d range and divided in to equal intervals. The user also decides th e number of clusters based on the size of

PAGE 36

29product data. The number of iterations is achieved by running the FCM algorithm till it achieves the condition || U (k+1) U (k) ||< (refer section 3.3.3). 2. Distance Calculations for Storage Slots Based on the dimensions of the warehouse and the probability of throughput for each port the rectilinear distance traveled in the warehouse for each slot is calculated (refer section 4.5). 3. Conversion of Linguistic Cate gories to Numeric Values. The program for linguistic to numerical data converter converts the lingu istic data to numerical data. The data is randomly generated w ith a fixed range for each linguistic variable of throughput (T) and storage (S). This numerical data is the input to the FCM algorithm. Several repli cations for generating random data is done to see the effect of change in data on the total expected distance 4. Normalization of Storage Requirements. The values of randomly generated data of storage levels are normalized to equa l the total number of available storage bays. 5. Cluster Generation. Clusters are generated by the FCM algorithm. After obtaining the clusters they are ranked in the des cending order based on the ratio of cluster center distance. 6. Cluster Ranking. The cluster with highest rank gets the closest slots to the I/O port and within the cluster the product with highest T/S ratio is placed first and so on in. (For generating layout with 3 features, e.g. throughput, storage and volume the cluster with highest weight is identified based on the cluster center information for each cluster. Within the cluster the products are ranked in the descending order of T/S ratio.) 7. Total Distance Calculation. Based on the obtained layout, the total expected distance traveled for storage/retrieval in the warehouse is calculated (Refer section 4.5). 4.6 Example 1: A Small Warehouse This problem has been taken from Francis (1992). The problem deals with a small warehouse with only four diffe rent products. The warehouse has separate I/O ports for receiving and shipping items with variable amount of activity from these ports. There is

PAGE 37

30small variation in terms of activity levels of different products. The storage requirements of these products, however, vary greatly. Problem Data 1. Warehouse dimensions are 20ft x 20ft. 2. Total number of slots is 50. 3. Receiving ports are port numbers 4 and 5. 4. Shipping ports are 1, 2, and 3 with th e middle port more likely to be used. 5. Probability of activity level from each port p1= 0.15, p2= 0.20, p3= 0.15, p4= 0.25 and p5= 0.25 6. Number of products is 4 namely A, B, C and D. 7. Throughput information for the 4 products is 60, 70, 80 and 90 trips per day. 8. Storage requirement for the 4 products is 20, 10, 15 and 5 bays. Assumptions 1. Assume rectilinear travel at constant sp eed within the warehouse and is assumed to originate at the centroid of the bay. 2. Full units are assumed to be Received/Sh ipped and the number of loads received equals the number of loads shipped. Sample Distance Calculations for Storage Slots to I/O Points Rectilinear distance tr aveled in the warehouse for slot number 50 is calculated below ) ( ) ( ) ( ) ( ) ( ) 50 (5 5 4 4 3 3 2 2 1 1f p f p f p f p f p f Where, ip is the probability of pro ducts entering through port i. if is the distance of a slot from port i. The total distance of slot 50 is given as, ) 20 3 ( 25 0 ) 20 2 ( 25 0 ) 20 3 ( 15 0 ) 20 5 ( 20 0 ) 20 7 ( 15 0 ) 50 ( x x x x x f ) 50 ( f= 75 ft.

PAGE 38

31Figure 4.1 shows the distance calculations for all the storage slots in a warehouse with 5 I/O ports. From the figure we observe that the distance for slots closer to the I/O ports is less than that for slots away from the I/O por ts for example the distance for slot 40 is 75 ft and that for slot 1 is 205 ft The distan ce calculations are done from each I/O port to the centroid of the storage slot. 4.6.1 Example 1 Solved by T/S Method T/S Ratio for the 4 products A, B, C and D is 3, 7, 5.3 and 18. Arranging the products in descending order of T/S the new sequence is D, B, C and A. The product with highest T/S gets the closest slot ava ilable. In this manner all th e products are placed in the warehouse. Figure 4.2 shows a layout for a de dicated storage warehouse. From the figure we see that product D, which ha s the highest T/S ratio is allo cated the closest slot in the warehouse. Similar allocation is followed for all the products. The products can be shifted to the next best slot in order to obtain a rectangul ar or ‘L’ shaped pattern for similar products. However, this will affect the total expected distance (TED) traveled in the warehouse. 1 205 2 185 3 165 4 148 5 134 6 124 7 118 8 115 9 115 10 115 11 185 12 165 13 145 14 128 15 114 16 104 17 98 18 95 19 95 20 95 21 170 22 150 23 130 24 113 25 99 26 89 27 83 28 80 29 80 30 80 31 165 32 145 33 125 34 108 35 94 36 84 37 78 38 75 39 75 40 75 41 165 42 145 43 125 44 108 45 94 46 84 47 78 48 75 49 75 50 75 Figure 4.1 Rectilinear Dist ance Traveled: Example 1 IO 1, p1=0.15 IO 2, p2=0.2 IO 3, p3=0.15 IO 5, p5=0.25 IO 4, p4=0.25

PAGE 39

32 3 A 3 A 3 A 3 A 3 A 3 A 3 A 2 C 2 C 2 C 3 A 3 A 3 A 3 A 2 C 2 C 2 C 2 C 2 C 2 C 3 A 3 A 3 A 2 C 2 C 2 B 2 B 2 B 2 B 2 B 3 A 3 A 3 A 2 C 2 C 2 B 2 B 2 B 1 D 1 D 3 A 3 A 3 A 2 C 2 C 2 B 2 B 1 D 1 D 1 D Note: The numbers in top right corner denote the cluster number and the numbers in the left bottom corner denote the product assigned to that slot. Figure 4.2 Layout by T/S Method Total Expected distance travel ed in the Warehouse per day day ft x f / 824 29 20 60 205 185 185 170 165 165 165 165 150 148 145 145 145 134 130 128 125 125 124 118 15 80 115 115 115 114 113 108 108 104 99 98 95 95 95 94 94 10 70 89 84 84 83 80 80 80 78 78 75 5 90 75 5 ) ( 4.6.2 Example 1 Solved by FCM Method Using Crisp Data In order to check the results for the FCM method that has been develope d in this thesis, the example problem would be run with the crisp values given earlier. The distance comparison would then be indicative of th e performance of the FCM method. The cluster output for crisp product data for exampl e 1 is given in table 4.1. This cluster information is used to design the warehouse layout. Total number of clusters for this IO 1, p1=0.15 IO 2, p2=0.2 IO 3, p3=0.15 IO 4, p4=0.25 IO 5, p5=0.25

PAGE 40

33 problem is set to 3. The steps given in section 4.4 ar e followed except for steps 4 and 5 which deal with converting linguistic values to numerical data. Fo r this small problem both methods result in same layout (as s hown in figure 4.2) an d hence the distance traveled will be the same. Table 4.2 shows th e expected distance traveled for each of the 3 clusters. Table 4.1 Cluster Output for Crisp Data – Cluster 1, 2 and 3 Cluster 1 Product Throughput Storage D 90 5 Cluster 2 Product Throughput Storage B 70 10 C 80 15 Cluster 3 Product Throughput Storage A 60 20 Table 4.2 Total Expected Distance by FCM Method for Crisp Data: Example 1 Cluster Number Expected Distance Traveled in ft/day 1 6,750 2 14,008 3 9,066 Total 29,824 4.6.3 Example 1 Solved by FCM Method We use this approach to de sign a fuzzy based warehouse, where the input information is in the form of fuzzy data. The data for the throughput and storage were converted as high, medium and low as given in table 4. 3. To generate numerical values for the converted data, a range for each linguistic vari able is set. Within this range a randomly generated numerical value is our input to the FCM algorithm.

PAGE 41

34Table 4.3 Fuzzy Data: Example 1 Product Throughput Storage A M H B H M C H H D H L The cluster output for fuzzy product data for exam ple 1 is given in table 4.4. This cluster information is used to design a warehouse layout with fuzzy data. Total number of clusters for this problem is 3. The steps 1 through 7 to be followed for layout generation are given in section 4.4. Tota l Expected Distance traveled per day is 27,308 ft/day shown in table 4.5. Layout for this problem is show n in figure 4.3. Compar ing this layout with the one developed by T/S method (figure 4.2), we see that due to changes in the number of storage slots calculated by FCM method for linguistic categories, the resulting layout is somewhat different. However, the relativ e location of the products in both the layouts is same. Table 4.4 Cluster Output for Fuzzy Data – Cluster 1, 2 and 3 Cluster 1 Product Throughput Storage D 87 6 Cluster 2 Product Throughput Storage B 77 13 C 63 11 Cluster 3 Product Throughput Storage A 49 20 Table 4.5 Comparison of Results: Example 1 Cluster Number Expected Distance Traveled in ft/day 1 6,525 2 13,379 3 7,404 Total 27,308

PAGE 42

35 3 A 3 A 3 A 3 A 3 A 3 A 3 A 2 C 2 C 2 C 3 A 3 A 3 A 3 A 2 C 2 C 2 C 2 B 2 B 2 C 3 A 3 A 3 A 2 C 2 C 2 B 2 B 2 B 2 B 2 B 3 A 3 A 3 A 2 C 2 B 2 B 2 B 1 D 1 D 1 D 3 A 3 A 3 A 2 C 2 B 2 B 2 B 1 D 1 D 1 D Figure 4.3 Layout by FCM Method Fuzzy Data 4.7 Summary In this chapter we discussed the concept of T/S method to design a warehouse layout for a small problem. We explained the FCM appr oach to the warehouse layout problem. Also the concept of linguistic variables, wh ich forms the basis for converting the crisp data to fuzzy data was explained. We solved a small warehouse problem by T/S and FCM method and compared the results for total expected distance traveled The results obtained were same by both the methods for this small problem. In our attempt to designing an efficient pr actical method for warehouse layout, in Chapter 5 we will follow the FCM method for two la rger problems one with 20 products and 250 storage locations and other with 50 products and 700 locations by both the methods. We will try to explore if FCM method can includ e a third feature e.g. product size (volume) to increase space utilization and/or have better material control. The addition of the third feature can not be incor porated in T/S method. IO 4, p4=0.25 IO 5, p5=0.25 IO 1, p1=0.15 IO 2, p2=0.2 IO 3, p3=0.15

PAGE 43

36 CHAPTER 5 RESULTS AND ANALYSIS In this chapter, we will discuss the re sults obtained by running the FCM algorithm namely cluster output, expected distance trav eled in the warehouse and effect of number of clusters on the expected distance traveled with the help of two additional problems. We will follow the same presentation format for these two problems, namely, first we will solve the problem using T/S method, followed by using FCM method with the same numerical (crisp) data. This will be done to compare the efficiency of the FCM method in terms of the expected distance traveled Finally, we will use the FCM method on a linguistic data used for the throughput and st orage levels and comment on the quality of the generated layout. As mentioned earlier we will explore if FC M method can include information such as product size, similarity or characteristics to be able to increase space utilization, lower storage/retrieval time and/or have better material c ontrol. This fact can not be incorporated in T/S method. We will fu rther do a sensitivity anal ysis for the effect of number of clusters on the total expected distance traveled. 5.1 Example Problem 2: A Medium Warehouse This problem deals with a me dium warehouse with 20 differe nt products consisting of 250 storage slots. The warehouse has 4 sepa rate I/O ports for r eceiving and shipping items with variable amount of activity from these ports. Ther e is large variation in terms of activity levels, storage requirements in this problem. Problem Data 1. Warehouse dimensions are 10ft x 10ft. 2. Total number of slots is 250. 3. Receiving ports are port numbers 1 and 2, both equally likely to be used 4. Shipping port are 3 and 4, both equally likely to be used.

PAGE 44

375. Probability of throughput from each port p1 = 0.25, p2 = 0.25, p3 = 0.20 and p4 = 0.30 respectively. 6. Number of products is 20 namely 1 through 20 7. Throughput and storage information for the 20 products is as shown in table 5.1 below. These values are given as cris p values. However, in real life the numerical data for large number of pr oducts may not be available. We will convert this data into fuzzy linguisti c data and solve it using FCM method. Table 5.1 Product Data for Example 2 Product Throughput (T) Storage (S) T/S Rank Product Throughput (T) Storage (S) T/S Rank 1 2 3 0.67 17 11 60 6 10 2 2 7 2 3.5 9 12 70 15 4.67 6 3 10 30 0.33 19 13 90 25 3.6 8 4 15 7 2.14 13 14 5 21 0.24 20 5 4 9 0.44 18 15 50 8 6.25 5 6 8 12 0.67 16 16 55 1 55 1 7 20 14 1.43 15 17 80 11 7.27 4 8 28 17 1.65 14 18 75 10 7.5 3 9 35 8 4.38 7 19 68 23 2.96 10 10 44 19 2.32 12 20 25 9 2.78 11 Rectilinear Distance Traveled The rectilinear distance travel ed in the warehouse is shown in the figure 5.1 below. The sample calculations for distance calculations are done in section 4.5 Note, the numbers on the upper right corner denote warehouse sl ot number and the numbers on the lower left corner denote the rectilinear distance traveled for that slot.

PAGE 45

38 1 167 2 157 3 147 4 137 5 127 6 117 7 107 8 97 9 87 10 82 11 83 12 84 13 85 14 86 15 87 16 88 17 96 18 106 19 112 20 118 21 128 22 138 23 148 24 158 25 168 26 167 27 157 28 147 29 137 30 127 31 117 31 107 33 97 34 87 35 82 36 83 37 84 38 85 39 86 40 87 41 88 42 96 43 106 44 112 45 118 46 128 47 138 48 148 49 158 50 168 51 167 52 157 53 147 54 137 55 127 56 117 57 107 58 97 59 87 60 82 61 83 62 84 63 85 64 86 65 87 66 88 67 96 68 106 69 112 70 118 71 128 72 138 73 148 74 158 75 168 76 167 77 157 78 147 79 137 80 127 81 117 82 107 83 97 84 87 85 82 86 83 87 84 88 85 89 86 90 87 91 88 92 96 93 106 94 112 95 118 96 128 97 138 98 148 99 158 100 168 101 167 102 157 103 147 104 137 105 127 106 117 107 107 108 97 109 87 110 82 111 83 112 84 113 85 114 86 115 87 116 88 117 96 118 106 119 112 120 118 121 128 122 138 123 148 124 158 125 168 126 167 127 157 128 147 129 137 130 127 131 117 132 107 133 97 134 87 135 82 136 83 137 84 138 85 139 86 140 87 141 88 142 96 143 106 144 112 145 118 146 128 147 138 148 148 149 158 150 168 151 167 152 157 153 147 154 137 155 127 156 117 157 107 158 97 159 87 160 82 161 83 162 84 163 85 164 86 165 87 166 88 167 96 168 106 169 112 170 118 171 128 172 138 173 148 174 158 175 168 176 167 177 157 178 147 179 137 180 127 181 117 182 107 183 97 184 87 185 82 186 83 187 84 188 85 189 86 190 87 191 88 192 96 193 106 194 112 195 118 196 128 197 138 198 148 199 158 200 168 201 167 202 157 203 147 204 137 205 127 206 117 207 107 208 97 209 87 210 82 211 83 212 84 213 85 214 86 215 87 216 88 217 96 218 106 219 112 220 118 221 128 222 138 223 148 224 158 225 168 226 167 227 157 228 147 229 137 230 127 231 117 232 107 233 97 234 87 235 82 236 83 237 84 238 85 239 86 240 87 241 88 242 96 243 106 244 112 245 118 246 128 247 138 248 148 249 158 250 168 Figure 5.1 Rectilinear Distan ce Traveled for Example 2 IO 1, p1=0.25 IO 2, p2=0.25 IO 3, p3=0.20 IO 4, p4=0.30

PAGE 46

39 5.1.1 Layout by T/S Method The layout obtained by T/S method for exampl e 2 with crisp product data is shown in figure 5.2. The numbers in the warehouse slot indicate the product number and the arrows denote the position of the I/O ports with proba bilities of activity level for each port. The products are allocated in the descending order of their T/S ratio. The products have to be rearranged to obtain a modular layout. Rear ranging the products will affect the total expected distance traveled in the warehouse. In normal practice the similar products are arranged so as to form a rectangular or ‘L ’ shaped layout, which is normally desired. The total expected distance traveled in the warehouse by T/S method is 70,818 ft/day. The distance calculations are done by arranging the products in the descending order of the T/S ratio. As the data of products is large th e distance calculati ons are performed by implementing a ‘c’ code. 5.1.2 Example 2 Solved by FCM Method Using Crisp Data From the cluster output for the product data as given in table 5.2, it can be seen that product having similarity in throughput and st orage are grouped together. This cluster information is used to design a warehouse layout with crisp data. Total number of clusters chosen for this problem is 5. The allocation of products is done by followi ng steps followed for example 1. The steps are given in detail in section 4.4. Layout obtained by FCM method for the product data is shown in the figure 5.3. Note the numbers in the slot indicate th e product number. From the layout we can see that th e product allocation similar to T/S method is di sjointed. This is due to the fact that the products in similar cluster try to occupy the least available distance in the warehouse. This problem can be solved by making classes of products and allocating product of similar clus ters to the respective class. This will however affect the total expected distance tr aveled in the warehouse.

PAGE 47

40 14 3 5 7 8 10 20 19 12 16 18 17 15 12 13 13 13 19 10 4 8 6 3 3 14 14 3 5 7 8 10 20 19 9 11 18 17 15 12 13 13 13 19 10 4 7 6 3 3 14 14 3 5 7 8 10 20 19 9 11 18 17 15 12 13 13 13 19 10 4 7 6 3 3 14 14 3 5 7 8 10 20 19 9 11 18 17 15 12 13 13 13 19 10 4 7 6 3 3 14 14 3 5 7 8 10 20 19 9 11 18 17 15 12 13 13 2 19 10 8 7 6 3 3 14 14 3 5 1 8 10 20 19 9 11 18 17 15 12 13 13 2 19 10 8 7 6 3 3 14 14 3 5 1 8 10 20 19 9 11 18 17 12 12 13 13 19 19 10 8 7 6 3 3 14 14 3 5 1 8 4 20 19 9 18 17 17 12 12 13 13 19 19 10 8 7 6 3 3 14 14 3 5 6 8 4 10 19 9 18 17 15 12 12 13 13 19 19 10 8 7 6 3 3 14 14 3 5 6 8 4 10 19 13 18 17 15 12 12 13 13 19 20 10 8 7 6 3 14 14 Figure 5.2 Layout by T/S MethodI/O 3, p3=0.20I/O 4, p4=0.30 I/O 1 p 1=0.25 I/O 2, p2=0.25

PAGE 48

41 Table 5.2 Cluster Output for Crisp Data: Example 2 Cluster 1 Product Throughput (T) Storage (S) 1 2 3 2 7 2 4 15 7 5 4 9 Cluster 2 3 10 30 14 5 21 Cluster 3 13 90 25 19 68 23 Cluster 4 6 8 12 7 20 14 8 28 17 9 35 8 10 44 19 20 25 9 Cluster 5 11 60 6 12 70 15 15 50 8 16 55 1 17 80 11 18 75 10 The total expected distance trav eled in feet per day in the warehouse is calculated. The expected distance is calculated for each clus ter and the sum of these distances for the 5 clusters is the total expected distan ce traveled as shown in table 5.3. Table 5.3 Total Expected Distance Traveled Per Day Cluster Number Expected Dist ance Traveled in ft/day 1 32,572 2 14,267 3 18,142 4 3,879 5 2,367 Total 71,228

PAGE 49

42 14 3 5 6 8 10 20 19 12 16 18 17 15 12 13 13 19 9 10 8 7 4 3 3 14 14 3 5 6 8 10 20 19 13 11 18 17 15 12 13 13 19 9 10 8 7 4 3 3 14 14 3 5 6 7 10 20 19 13 11 18 17 15 12 13 13 19 9 10 8 7 4 3 3 14 14 3 5 6 7 10 20 19 13 11 18 17 15 12 13 13 19 9 10 8 7 4 3 3 14 14 3 5 6 7 10 20 19 13 11 18 17 15 12 13 13 19 9 10 8 7 4 3 3 14 14 3 5 6 7 8 20 19 13 11 18 17 15 12 13 13 19 9 10 8 7 4 3 3 14 14 3 5 6 7 8 10 19 13 11 18 17 12 12 13 19 19 9 10 8 6 4 3 3 14 14 3 5 6 7 8 10 19 13 18 17 17 12 12 13 19 19 20 10 8 6 1 3 3 14 14 3 5 2 7 8 10 19 13 18 17 15 12 12 13 19 19 20 10 8 6 1 3 3 14 14 3 3 2 7 8 10 9 13 18 17 15 12 12 13 19 19 20 10 8 6 1 3 14 14 Figure 5.3 Layout by FCM Method Crisp Data I/O 3, p3=0.20 I/O 4, p4=0.30 I/O 1 p 1=0.25 I/O 2, p2=0.25

PAGE 50

43 5.1.3 Comparison of Layout and Total Expected Distance By observing the layout by T/S and FCM we can see that there are some minor changes in the layout while the relative locations of different products is more or less unchanged. This is due to the difference in allocation t echniques used by the two methods. In the T/S method, the product with the highest T/S rati o gets the closest slot in the warehouse. Where as in the FCM method the clusters with relatively similar product data are clustered together and then th ey are ranked within the cluster. The results for the total expected distance traveled for both the above mentioned cases is give n in table 5.4. The percentage increase in total expected distance by FCM method is 0.58 %. Here we can see that the percentage increase in distan ce traveled by FCM met hod is negligible. Table 5.4 Comparison of Results: Example 2 Total Exp. Distance Traveled by T/S method 70,818 ft/day Total Exp. Distance Traveled by FCM Method 71,228 ft/day 5.1.4 Example 2 Solved by FCM Method Using Fuzzy Data The crisp product information given earlier was coded into five fuzzy levels namely, very low, low, medium, high and very high. The category ranges were found by dividing the highest value of throughput and storage into number of levels. For this example for throughput it will result in 90/5 = 18 and for st orage it will be 30/5 = 6. Therefore the values for throughput in the increasing order fo r very low to very high will be 1-18, 1937, etc. Similarly, values for very low to very high for storage will be 1-6, 7-13, etc. Table 5.5 gives the fuzzy values for the thr oughput and storage levels shown earlier in table 5.4. Total number of clusters for this problem is 5. The output of FCM algorithm generates clusters that are used to design a warehouse layout with fuzzy data. From FCM cluster output below we see that products with similar pattern of data are clustered together, for example products 1 and 2 with very low values of throughput and storage. Similarly products 17 and 18 with very hi gh throughput and low storage are clustered together and so on.

PAGE 51

44Cluster 11, 2, 4, 5, 6, 7, 8, 9 and 20 Cluster 23 and 14 Cluster 317 and18 Cluster 410, 12, 13 and 19 Cluster 511, 15 and 16 Table 5.5 Fuzzy Product Data for Example 2 Product Throughput (T) Storage (S) Product Throughput (T) Storage (S) 1 VL VL 11 H VL 2 VL VL 12 H M 3 VL VH 13 VH VH 4 VL L 14 VL H 5 VL L 15 M L 6 VL L 16 H VL 7 L M 17 VH L 8 L M 18 VH L 9 L L 19 H H 10 M H 20 L L 5.1.5 Layout for Fuzzy Data by FCM Method Layout obtained by FCM method for the product da ta is shown in figure 5.4. Note the numbers in the slot indicate the product number. The allo cation of products is done by following steps 1 through 7 in section 4.4. Fr om the layout we can see that the product allocation is disjoint. This is due to the fact that the pr oducts in one cluster occupy the least available distance in the warehouse. Comparing the layout with the T/S layout we can see that the pattern of product allocation does not vary much. The small change in expected distance justifies this claim. However, due to the difference to the random generation of storage data the number of products allocated are different. The total expect ed distance traveled in feet per day in the

PAGE 52

45warehouse is calculated. The expected distance is calculated for each cluster and the sum of these distances for the 5 clusters is the total expected distance traveled (Refer table 5.6). Table 5.6 Total Expected Distance Traveled Cluster Number Expected Distance Traveled in ft/day 1 13,873 2 12,940 3 21,825 4 18,705 5 4,328 Total 71,671 5.2 Principles for Warehouse Design The FCM algorithm generates clusters for data point in a n-dimensional space. This aspect of FCM can be used to add more f eatures to design a warehouse layout. This can be done by using critical principles that re ally have an impact on the design of a warehouse. A warehouse designer gets the flex ibility to use these principles to the warehouse that the T/S approach does not have. We brief the main principles that play an important role in warehouse design (Tompkins, 1996) 1. PopularityThe popularity of products is derived from ‘Pareto’s law’ that suggests to place15% of the products clos er to the I/O ports to minimize the distance traveled.. 2. Similaritythe items received/ shipped t ogether should be stored together. By doing this travel times for order receipt and order picking can be minimized. 3. Sizelocating products based on the size or bulk and the space it utilizes. It is a common practice to locate all the bulky items close to the I/O ports to minimize the traveling and maneuvering time. 4. Characteristicsof products like perishab le materials, crushable items, hazardous material, security items and compatibility of products. 5. Space UtilizationSome impor tant factors to be consid ered are conservation of space, limitations of space a nd accessibility of products.

PAGE 53

46 3 14 6 5 7 8 10 13 19 11 15 17 12 12 19 13 13 10 20 7 9 4 14 14 3 3 14 6 5 7 8 10 13 19 16 15 17 12 12 19 13 13 10 20 7 9 4 14 3 3 3 14 6 5 7 8 10 13 19 16 18 17 12 12 19 13 13 10 20 7 9 4 14 3 3 3 14 6 5 7 8 10 13 19 16 18 17 12 12 19 13 13 10 2 7 9 4 14 3 3 3 14 6 5 7 8 1 10 19 16 18 17 12 12 19 13 13 10 2 7 9 4 14 3 3 3 14 6 5 9 8 1 10 19 15 18 17 12 19 13 13 13 10 2 7 9 4 14 3 3 3 14 6 4 9 8 20 10 19 15 18 17 12 19 13 13 13 10 2 7 9 4 14 3 3 3 14 14 4 9 8 20 10 19 15 18 17 12 19 13 13 13 10 8 7 7 4 14 3 3 3 14 14 4 9 8 20 10 19 15 18 12 12 19 13 13 13 10 8 7 7 6 14 3 3 3 14 14 4 9 8 20 10 19 15 18 12 12 19 13 13 13 10 8 7 7 6 14 3 3 Figure 5.4 Layout by FCM Method Fuzzy Data I/O 2, p2=0.25 I/O 1, p1=0.25 I/O 3, p3=0.20 I/O 4, p4=0.30

PAGE 54

47 5.3 Example 2 with Volume Information The product information for throughput, storag e and volume is fuzzy and is defined in five fuzzy levels namely, very low, low, medium, high and very high (refer table 5.7). The ranges for throughput and storage are same as that for the 2 feature problem. In this case the fuzzy data for volume is generate d by considering 1000 as the maximum volume and dividing it in to 5 levels namely 1000/5= 200. So the ranges for the data starting from very low to high will be 1-200, 201-400, etc. To tal number of clusters for this problem is 5. The output of FCM algorithm generates cl usters that are used to design a warehouse layout with fuzzy data. Table 5.7 Fuzzy Throughput, Storage and Volume Data for Example 2 Product Throughput (T) Storage (S) Volume Product Throughput (T) Storage (S) Volume 1 VL VL VL 11 H VL VL 2 VL VL L 12 H M H 3 VL VH M 13 VH VH M 4 VL L VH 14 VL H L 5 VL L VH 15 M L M 6 VL L VH 16 H VL M 7 L M VL 17 VH L L 8 L M H 18 VH L H 9 L L VL 19 H H VH 10 M H H 20 L L L 5.3.1 Layout Based on Fuzzy Data by FCM Method Layout obtained by FCM method for the product data is shown in the figure 5.5. Note the numbers in the slot indicate the product num ber. The allocation of products is done by following steps 1 through 7 in section 4.4. Du e to the addition of the third feature the layout changes. This change is due to the change in cluster formation and the impact ‘volume’ has on the cluster formation. From the layouts for fuzzy 2 feature and 3 feature we observe that due to the addition of the thir d feature (volume) there is change in the

PAGE 55

48 7 14 14 3 17 18 13 10 19 6 4 4 8 8 19 12 10 13 13 15 3 3 14 2 9 7 20 14 3 17 18 13 10 19 6 4 5 8 8 19 12 10 13 13 15 3 3 14 2 9 7 20 14 3 17 18 13 10 19 6 4 5 8 8 19 12 10 13 13 15 3 3 14 7 9 7 20 14 3 17 18 13 10 19 6 4 5 8 19 12 12 10 13 13 15 3 3 14 7 9 7 20 14 3 17 18 13 10 19 6 4 5 8 19 12 12 10 13 13 16 3 3 14 7 9 7 20 14 3 17 18 13 10 19 6 4 5 8 19 12 12 10 13 13 16 3 3 14 7 9 7 20 14 3 11 18 13 10 19 6 4 5 8 19 12 12 10 13 13 16 3 3 14 7 9 9 20 14 3 3 15 13 10 19 6 4 5 8 19 12 12 10 13 13 16 3 14 14 7 9 9 2 14 3 3 15 13 10 19 6 4 5 8 19 12 12 10 13 13 17 3 14 14 7 9 9 2 14 3 3 15 13 10 19 4 4 5 8 19 12 12 10 13 18 17 3 14 14 7 1 Figure 5.5 Layout by FCM Method Fuzzy Data with 3 Features I/O 1, p1=0.25 I/O 2, p2=0.25 I/O 3, p3=0.20 I/O 4, p4=0.30

PAGE 56

49 product allocation. For example product 7, 13, 17 and 18 move away from the I/O ports. This is due to the impact on cluster formation by the comparatively low values of third feature. On the contrary, products 4, 5, 6 a nd 8 move closer to the I/O ports due to the high values of the third feature. The expected distance is calculated for each cluster and the sum of these distances for the 5 clusters is the total expected distance trav eled (Refer table 5.8). The expected distance is larger (by 19.8 %) in this case compared to that for 2 f eature data. This is the tradeoff made by including the third feature in cluster formation. Table 5.8 Total Expected Distance Traveled Cluster Number Expected Distance Traveled in ft/day 1 4,307 2 22,869 3 38,936 4 3,932 5 15,902 Total 85,945 5.4 Example Problem 3: A Large Warehouse This problem has been taken from Francis (1992). The problem deals with a large warehouse with 50 different products. The warehouse has 3 separate I/O ports for receiving and shipping items with variable amount of activity from these ports. There is large variation in terms of activity levels storage and volume requirement for this problem. Problem Data 1. Warehouse dimensions are 10ft x 10ft. 2. Total number of slots is 700. 3. Receiving ports are port numbers 1 and 2, both equally likely to be used 4. Shipping port is 3.

PAGE 57

505. Probability of throughput from each port p1= 0.25, p2= 0.25 and p3= 0.50 respectively. 6. Number of products is 50 namely 1 through 50. 7. Throughput and storage requireme nt for the 50 products is mentioned in table 5.9. Rectilinear Distance Traveled The rectilinear distance traveled in the warehouse is shown in the figure 5.6. The sample calculations for distance calcu lations are done in section 4. 5 Note, the numbers on the upper right corner denote wa rehouse slot number and the numbers on the lower left corner denote the rectilinear dist ance traveled for that slot. Table 5.9 Crisp Product Data for Example 3 Product Throughput (T) Storage (S) T/S RankProduct Throughput (T) Storage (S) T/S Rank 1 4 8 0.5 30 26 3 2 1.5 12 2 5 12 0.4235 27 10 16 0.6326 3 9 4 2.257 28 3 6 0.5 31 4 7 8 0.8622 29 8 4 2 9 5 3 8 0.3636 30 15 13 1.1517 6 9 5 1.8 10 31 10 9 1.1119 7 3 10 0.3 38 32 7 5 1.4 14 8 30 24 1.2516 33 5 6 0.8323 9 2 28 0.0744 34 15 13 1.1518 10 34 12 2.834 35 30 8 3.752 11 12 12 1 20 36 3 4 0.7524 12 13 10 1.3 15 37 10 4 2.5 6 13 1 25 0.0450 38 6 4 1.5 13 14 9 10 0.9 21 39 4 9 0.4432 15 4 2 2 8 40 10 6 1.6711 16 11 20 0.5529 41 3 7 0.4333 17 3 5 0.6 28 42 5 15 0.3337 18 13 19 0.6825 43 50 16 3.133 19 2 40 0.0548 44 10 45 0.2239 20 17 4 4.251 45 4 18 0.2240 21 1 18 0.0646 46 56 20 2.8 5 22 8 19 0.4234 47 3 15 0.2 41 23 1 15 0.0745 48 4 25 0.1642 24 3 50 0.0647 49 1 20 0.0549 25 1 10 0.1 43 50 20 32 0.6327

PAGE 58

51 Figure 5.6 Rectilinear Distance Tr aveled in Warehouse: Example 31 2 3 4 5 6 7 8910111213141516171819 20 2122232425262728293031323334 35 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 36 37 38 39 40 41 42 434445464748495051525354 55 5657585960616263646566676869 70 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 71 72 73 74 75 76 77 787980818283848586878889 90 919293949596979899100101102103104 105 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 106 107 108 109 110 111 112 113114115116117118119120121122123124 125 126127128129130131132133134135136137138139 140 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 141 142 143 144 145 146 147 148149150151152153154155156157158159 160 161162163164165166167168169170171172173174 175 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 176 177 178 179 180 181 182 183184185186187188189190191192193194 195 196197198199200201202203204205206207208209 210 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 211 212 213 214 215 216 217 218219220221222223224225226227228229 230 231232233234235236237238239240241242243244 245 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 246 247 248 249 250 251 252 253254255256257258259260261262263264 265 266267268269270271272273274275276277278279 280 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 281 282 283 284 285 286 287 288289290291292293294295296297298299 300 301302303304305306307308309310311312313314 315 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 316 317 318 319 320 321 322 323324325326327328329330331332333334 335 336337338339340341342343344345346347348349 350 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 351 352 353 354 355 356 357 358359360361362363364365366367368369 370 371372373374375376377378379380381382383384 385 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 386 387 388 389 390 391 392 393394395396397398399400401402403404 405 406407408409410411412413414415416417418419 420 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 421 422 423 424 425 426 427 428429430431432433434435436437438439 440 441442443444445446447448449450451452453454 455 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 456 457 458 459 460 461 462 463464465466467468469470471472473474 475 476477478479480481482483484485486487488489 490 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 491 492 493 494 495 496 497 498499500501502503504505506507508509 510 511512513514515516517518519520521522523524 525 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 526 527 528 529 530 531 532 533534535536537538539540541542543544 545 546547548549550551552553554555556557558559 560 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 561 562 563 564 565 566 567 568569570571572573574575576577578579 580 581582583584585586587588589590591592593594 595 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 596 597 598 599 600 601 602 603604605606607608609610611612613614 615 616617618619620621622623624625626627628629 630 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 631 632 633 634 635 636 637 638639640641642643644645646647648649 650 651652653654655656657658659660661662663664 665 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 666 667 668 669 670 671 672 673674675676677678679680681682683684 685 686687688689690691692693694695696697698699 700 263 253 243 233 223 213 203 193 183 175 170 165 160 155 150 145 145 150 155 160 165 170 175 180 185 190 198 208 218 228 238 248 258 268 278 I/O 3, p3=0.50 I/O 1, p1=0.25 I/O 2, p2=0.25

PAGE 59

525.4.1 Layout by T/S Method The layout obtained for the numerical data of throughput and storage by T/S is shown in figure 5.7. The numbers in the lower right corner of the warehouse slot indicate the product number and the arrows de note the position of the I/O ports with pr obabilities of throughput for each port. The products are alloca ted in the descending order of their T/S ratio. As indicated in the previous problem, the products have to be rearranged to obtain a modular layout. Rearranging the products will affect the tota l expected distance traveled in the warehouse. The total expected distance traveled in the warehouse by T/S method is 80,936 ft/day. The distance calculations are done by arranging the products in the descending order of the T/S ratio. As the data of products is large th e distance calculati ons are performed by implementing a ‘c’ code. 5.4.2 Example 3 Solved by FCM Method Using Crisp Data The cluster output for the problem data based on 10 clusters is given in table 5.10. This cluster information is used to design a wa rehouse layout with cris p data. Layout obtained by FCM method is shown in the figure 5.8. The appropriate steps for allocation of products is followed as given in section 4.4. The total expected distance traveled in th e warehouse is calculated. The expected distance is calculated for each cluster and the sum of these distances for the 10 clusters is the total expected distance tr aveled (Refer table 5.11).

PAGE 60

53 Figure 5.7 Layout by T/S Method 19 21 24 9 48 45 44 5 1 50 33 31 8 40 46 20 20 46 40 8 31 36 50 16 41 22 42 44 47 25 23 24 19 49 13 19 21 24 9 48 45 44 5 1 50 36 31 8 40 46 20 20 46 40 8 31 36 50 16 41 22 7 44 47 25 23 24 19 49 13 19 21 24 9 48 45 44 5 1 50 36 31 8 40 46 35 35 46 26 8 11 18 50 16 41 22 7 44 47 25 23 24 19 49 13 19 21 24 9 48 45 44 5 1 50 18 11 8 26 46 35 35 46 38 8 11 18 50 16 41 22 7 44 47 25 23 24 19 49 13 19 21 24 9 48 45 44 5 1 50 18 11 8 38 46 35 35 46 38 8 11 18 50 16 41 22 7 44 47 9 23 24 19 49 13 19 21 24 9 48 45 44 5 28 50 18 11 30 38 46 35 35 46 32 30 11 18 50 16 41 22 7 44 47 9 23 24 19 49 13 19 21 24 9 48 45 44 42 28 50 18 11 30 32 46 43 43 46 32 30 11 18 50 16 41 2 7 44 47 9 23 24 19 49 13 19 21 24 9 48 45 44 42 28 50 18 11 30 32 46 43 43 46 32 30 11 18 50 16 22 2 7 44 47 9 24 24 19 49 13 19 21 24 9 48 45 44 42 28 27 18 11 30 12 46 43 43 46 12 30 14 18 27 16 22 2 7 44 47 9 24 24 19 49 13 19 21 24 9 48 45 44 42 28 27 18 14 30 12 46 43 43 46 12 30 14 18 27 16 22 2 7 44 48 9 24 24 19 49 13 19 21 24 9 48 45 44 42 28 27 18 14 30 12 37 43 43 37 12 30 14 18 27 16 22 2 7 44 48 9 24 24 19 49 13 19 21 24 9 48 45 44 42 39 27 18 14 30 12 37 43 43 37 12 34 14 18 27 16 22 2 44 44 48 9 24 24 19 49 13 19 21 24 23 48 45 44 42 39 27 50 14 34 12 3 43 43 3 12 34 14 50 27 16 22 2 44 44 48 9 24 24 19 49 13 19 21 24 23 48 45 44 42 39 27 50 14 34 8 3 43 43 3 8 34 4 50 27 16 22 2 44 44 48 9 24 24 19 49 13 19 21 24 23 25 47 44 42 39 27 50 4 34 8 15 10 10 15 8 34 4 50 27 16 22 2 44 44 48 9 24 24 19 49 13 49 19 24 23 25 47 44 42 39 27 50 4 34 8 29 10 10 29 8 34 4 50 27 16 22 2 44 44 48 9 24 24 19 13 13 49 19 24 23 25 47 44 42 39 17 50 4 34 8 29 10 10 29 8 34 4 50 17 16 22 2 44 45 48 9 24 24 19 13 13 49 19 24 23 25 47 44 42 39 17 50 4 34 8 6 10 10 6 8 34 33 50 17 1 22 2 44 45 48 9 24 21 19 13 13 49 19 24 23 25 47 44 42 39 17 50 33 31 8 6 10 10 6 8 31 33 50 16 1 22 5 44 45 48 9 24 21 19 13 13 49 19 24 23 25 47 44 42 39 16 50 33 31 8 6 10 10 40 8 31 33 50 16 1 22 5 44 45 48 9 24 21 19 13 13 I/O 3, p3=0.50 I/O 1, p1=0.25 I/O 2, p2=0.25

PAGE 61

54Table 5.10 Cluster Output for Crisp Data: Example 3 Product Throughput (T) Storage (S) Cluster 1 10 34 12 35 30 8 Cluster 2 15 4 2 17 3 5 26 3 2 28 3 6 33 5 6 36 3 4 38 6 4 Cluster 3 8 30 24 50 20 32 Cluster 4 9 2 28 13 1 25 48 4 25 Cluster 5 16 11 20 21 1 18 22 8 19 23 1 15 42 5 15 45 4 18 47 3 15 49 1 20 Cluster 6 19 2 40 24 3 50 44 10 45 Cluster 7 43 50 16 46 56 20 Cluster 8 1 4 8 2 5 12 4 7 8 5 3 8 7 3 10 25 1 10 39 4 9 41 3 7 Cluster 9 11 12 12 12 13 10 14 9 10 18 13 19 27 10 16 30 15 13 30 15 13 Cluster 10 3 9 4 6 9 5 20 17 4 29 8 4 31 10 9 32 7 5 37 10 4 40 10 6

PAGE 62

55Table 5.11 Total Expected Distance Traveled Per Day Cluster Number Expected Distance Traveled in ft/day 1 9,280 2 15,594 3 12,110 4 13,878 5 4,517 6 8,619 7 5,436 8 6,717 9 3,492 10 1,841 Total 81,484 5.4.3 Comparison of Layout and Total Expected Distance Comparing the layouts for both the cases we obs erve that there is small change in the product placement. This is due to the fact that in T/S approach we locate products in descending order of the T/S ratio, where as with FCM method we rank the clusters and then allocate the products in that cluster. The results fo r the total exp ected distance traveled for both the above mentioned cases is given in table 5.12. The percentage increase in total expected distance by FCM me thod is 0.68 %. Here we can see that the percentage increase in distance trav eled by FCM method is negligible. Table 5.12 Comparison of Results: Example 3 Total Exp. Distance Traveled by T/S method 80,936 ft/day Total Exp. Distance Traveled by FCM Method 81484 ft/day

PAGE 63

56 Figure 5.8 Layout by FCM Method Crisp Data 48 19 24 44 49 23 42 16 41 8 33 18 30 40 46 35 35 46 40 30 18 33 8 4 5 25 22 47 21 44 24 24 19 9 13 48 19 24 44 49 23 45 16 41 8 33 18 30 40 46 35 35 46 32 30 18 36 50 4 5 25 22 47 21 44 24 24 19 9 13 48 19 24 44 49 23 45 16 41 50 36 18 30 32 46 35 35 46 32 30 18 36 50 4 5 25 22 47 21 44 24 19 48 9 13 48 19 24 44 49 23 45 16 41 50 36 18 30 32 46 35 35 46 32 30 18 17 50 1 5 25 22 47 21 44 24 19 48 9 13 48 19 24 44 49 23 45 16 41 50 17 18 30 31 46 10 10 46 31 30 18 17 50 1 5 25 22 47 21 44 24 19 48 9 13 48 19 24 44 49 23 45 16 41 50 17 18 30 31 46 10 10 46 31 30 18 28 50 1 5 25 22 47 21 44 24 19 48 9 13 48 19 24 44 49 23 45 16 41 50 17 18 30 31 46 10 10 46 31 11 18 28 50 1 5 25 42 47 21 44 24 19 48 9 13 9 19 24 44 44 23 45 22 2 50 28 27 11 31 46 10 10 46 31 11 27 28 50 1 7 16 42 47 49 44 24 19 48 9 13 9 19 24 44 44 23 45 22 2 50 28 27 11 31 20 10 10 20 12 11 27 8 50 1 7 16 42 47 49 44 24 19 48 9 13 9 19 24 44 44 21 45 22 2 50 28 27 11 12 20 10 10 20 12 11 27 8 50 1 7 16 42 47 49 44 24 19 48 9 13 9 19 24 44 44 21 45 22 2 50 8 27 11 12 37 43 43 37 12 11 27 8 50 1 7 16 42 47 49 44 24 19 48 9 13 9 19 24 44 44 21 45 22 2 50 8 27 11 12 37 43 43 37 12 11 27 8 50 39 7 16 42 47 49 44 24 19 48 9 13 9 19 24 24 44 21 45 22 2 50 8 27 11 12 3 43 43 3 12 14 27 8 50 39 7 16 42 47 49 44 24 19 48 9 13 9 19 24 24 44 21 45 22 2 50 8 27 11 12 3 43 43 3 30 14 27 8 50 39 7 16 42 47 49 44 24 19 48 9 13 9 19 24 24 44 21 45 22 2 50 8 27 14 30 29 43 43 29 30 14 27 8 50 39 7 16 42 23 49 44 24 19 48 9 13 9 19 24 24 44 21 45 22 2 50 8 15 14 30 29 43 43 29 30 14 15 8 50 39 7 16 42 23 49 44 24 19 48 13 13 9 19 24 24 44 21 45 22 2 50 8 26 14 30 6 43 43 6 30 14 26 8 50 39 7 16 42 23 49 44 24 19 48 13 13 9 19 24 24 44 21 45 22 2 50 8 38 14 30 6 43 43 6 30 18 38 8 4 39 25 16 42 23 49 44 24 19 48 13 13 9 19 24 24 44 21 45 22 2 4 8 38 18 30 6 46 46 40 30 18 38 8 4 39 25 16 42 23 49 44 24 19 48 13 13 9 19 24 24 44 21 47 22 5 4 8 33 18 30 40 46 46 40 30 18 33 8 4 39 25 16 42 23 49 44 24 19 48 13 13 I/O 3, p3=0.50 I/O 1, p1=0.25 I/O 2, p2=0.25

PAGE 64

575.4.4 Example 3 Solved by FCM Method Using Fuzzy Data The product information given earlier (table 5. 9) was converted into seven fuzzy levels namely, very low, low, medium, medium low, medium high, high and very high following the procedure explained in exampl e 2. The highest value for the throughput was 50 and the largest storage requirement was 50 as well. Table 5.13 shows the fuzzy product data for the problem. Total number of clusters assumed for this problem is 10. The output of FCM algorithm generates clus ters that are used to design a warehouse layout with fuzzy data. Table 5.13 Fuzzy Product Data for Example 3 Product Throughput Storage Level Product Throughput Storage Level 1 VL L 26 VL VL 2 VL L 27 L ML 3 L VL 28 VL VL 4 VL L 29 L VL 5 VL L 30 L L 6 L VL 31 L L 7 VL L 32 VL VL 8 M M 33 VL VL 9 VL M 34 L L 10 MH L 35 M L 11 L L 36 VL VL 12 L L 37 L VL 13 VL M 38 VL VL 14 L L 39 VL L 15 VL VL 40 L VL 16 L ML 41 VL VL 17 VL VL 42 VL L 18 L ML 43 VH ML 19 VL H 44 L H 20 ML VL 45 VL ML 21 VL ML 46 VH ML 22 L ML 47 VL L 23 VL L 48 VL M 24 VL VH 49 VL ML 25 VL L 50 ML MH From the cluster output results shown belo w we can see that products 15, 17, 26 and 28 with very low product data ar e in one cluster. Cluster 3 has only one product 8 with medium values, this is due to the large number of clusters there is fine data partition.

PAGE 65

58Cluster 1 10 and 35; Cluster 2 15, 17, 26, 28, 32, 33, 36, 38, 41; Cluster 3 8 Cluster 4 50; Cluster 5 9, 13, 21, 48 and 49; Cluster 6 19, 24 and 44 Cluster 7 43 and 46; Cluster 8 1, 2, 4, 5, 7, 23, 25, 39, 42, 45 and 47 Cluster 9 14, 16, 18, 16, 27, 30, 31 and 34; Cluster 10 3, 6, 11, 12, 20, 29, 37 and 40 5.4.5 Layout for Fuzzy Data by FCM Method Comparing the layout in figure 5.9 with T/S layout we can see that the pa ttern of product allocation in both the layout is similar. The small change in expected distance traveled justifies this claim. The slots occupied however are different due to the random generation of fuzzy storage data for this pr oblem. The products have to be rearranged to obtain a rectangular layout of similar product type. This however will affect the total expected distance traveled in the warehouse. The total expected dist ance traveled per day in the warehouse is calculated (Refer sa mple calculation section 4.5). The expected distance for each of the 10 clusters is calculate d and the total expected distance is the sum of the expected distances of the 10 cluste rs. Refer table 5.14 for the total expected distance traveled by FCM method. Table 5.14 Total Expected Distance Traveled Per Day Cluster Number Expected Distance Traveled in ft/day 1 8,845 2 15,459 3 14,106 4 4,694 5 14,834 6 3,500 7 4,307 8 6,850 9 3,324 10 2,812 Total 78,730

PAGE 66

59 Figure 5.9 Layout by FCM Method Fuzzy Data 13 49 19 24 44 44 5 7 33 27 31 30 8 40 43 10 10 43 40 8 30 31 27 38 42 23 25 45 44 24 19 49 9 21 48 13 9 19 24 44 44 5 7 33 27 31 30 8 40 43 10 10 43 40 8 30 31 27 38 42 23 25 45 44 24 19 49 9 21 48 13 9 19 24 44 44 5 7 33 27 31 16 8 40 43 10 10 43 11 8 16 31 50 32 42 23 25 45 44 24 19 49 9 21 48 13 9 19 24 44 44 5 7 33 50 31 16 8 11 43 10 10 43 11 34 16 31 50 32 42 23 25 45 44 24 19 49 9 21 48 13 9 19 24 24 44 5 7 41 50 31 16 34 11 43 10 10 43 11 34 16 31 50 32 42 23 2 45 44 24 19 49 9 21 48 13 9 19 24 24 44 5 4 41 50 18 16 34 11 43 35 35 43 11 34 16 18 50 32 42 39 2 45 44 24 19 49 9 21 48 13 9 19 24 24 44 5 4 41 50 18 16 34 11 43 35 35 43 12 34 16 18 50 32 42 39 2 45 44 24 19 49 13 21 48 13 9 19 24 24 44 5 4 15 50 18 16 34 12 43 35 35 43 12 34 16 18 50 17 42 39 2 45 44 24 19 49 13 21 48 13 9 19 24 24 44 1 4 15 50 18 16 34 12 43 35 35 43 12 34 16 18 50 17 42 39 2 45 44 24 19 49 13 21 48 13 9 19 24 24 44 1 4 15 50 18 16 34 12 20 46 46 20 12 14 16 18 50 17 42 39 2 45 44 24 19 49 13 21 48 13 9 19 24 24 44 1 4 15 50 18 16 14 12 37 46 46 37 12 14 16 18 50 17 42 39 2 45 44 24 19 49 13 21 48 13 9 19 24 24 44 1 4 15 50 18 16 14 12 37 46 46 29 8 14 16 18 50 17 42 39 2 45 44 24 19 49 13 21 48 13 9 19 24 24 44 1 4 28 50 18 16 14 8 29 46 46 29 8 14 16 18 50 17 42 39 2 45 44 24 19 49 13 21 48 13 9 19 24 24 44 1 4 28 50 18 16 14 8 29 46 46 6 8 14 16 18 50 17 42 7 2 45 44 24 19 49 13 21 48 13 9 19 19 24 44 1 4 28 50 27 16 14 8 6 46 46 6 8 14 16 27 50 17 23 7 2 45 44 24 19 49 13 21 48 21 9 19 19 24 44 1 25 28 50 27 16 14 8 6 46 46 6 8 14 16 27 50 17 23 7 2 45 44 24 19 49 13 48 48 21 9 19 19 24 44 1 25 28 50 27 16 14 8 3 46 46 3 8 30 16 27 50 17 23 7 2 45 44 24 19 49 13 48 48 21 9 19 19 24 44 1 25 28 50 27 16 30 8 3 46 46 3 8 30 16 27 50 33 23 7 5 44 44 24 19 49 13 48 48 21 9 49 19 24 44 1 25 42 50 27 31 30 8 3 46 46 3 8 30 31 27 36 33 23 7 5 44 44 24 19 49 13 48 48 21 9 49 19 24 44 1 25 42 36 27 31 30 8 40 43 43 40 8 30 31 27 36 33 23 7 5 44 44 24 19 49 13 48 48 I/O 3, p3=0.50 I/O 2, p2=0.25 I/O 1, p1=0.25

PAGE 67

605.5 Example Problem by FCM for Fuzzy Data: 3 Features The product information for throughput, storag e and volume is fuzzy and is defined in seven fuzzy levels namely, very low, low, medium, medium low, medium high, high and very high (refer table 5.15). The fuzzy data for volume in this example is generated in the same way as example 2. Total number of clus ters for this problem is 10. The output of FCM algorithm generates cluste rs that are used to design a warehouse layout with fuzzy data. Table 5.15 Fuzzy Product Data for Example 3 Product Throughput Storage Level Volume ProductThroughput Storage Level Volume 1 VL L VL 26 VL VL H 2 VL L L 27 L ML VH 3 L VL VL 28 VL VL M 4 VL L ML 29 L VL MH 5 VL L VL 30 L L VH 6 L VL L 31 L L H 7 VL L ML 32 VL VL ML 8 M M M 33 VL VL L 9 VL M L 34 L L VH 10 MH L L 35 M L ML 11 L L VL 36 VL VL M 12 L L ML 37 L VL L 13 VL M ML 38 VL VL L 14 L L VL 39 VL L VL 15 VL VL L 40 L VL ML 16 L ML M 41 VL VL M 17 VL VL L 42 VL L H 18 L ML ML 43 VH ML VH 19 VL H M 44 L H VL 20 ML VL L 45 VL ML VL 21 VL ML L 46 VH ML VL 22 L ML L 47 VL L VL 23 VL L M 48 VL M L 24 VL H MH 49 VL ML ML 25 VL L ML 50 ML MH M

PAGE 68

615.5.1 Layout for Fuzzy Data by FCM Method for 3 Features The layout obtained by FCM method for 3 features is shown in the figure 5.10. From the layout we can see that the products with high values of volume data are grouped together and are placed in locations close to the I/ O ports (for example product 26, 27, 30 and 31). Similarly the products with low values of vol ume data (product 5, 35, 47 and 49) are placed away from the I/O ports. This justifie s the effect of third feature on the layout obtained. The total expected distance trav eled per day in the warehouse is calculated. The expected distance for each of the 10 clusters is given in table 5.16. The total expected distance (TED) traveled in the warehouse is the sum of expected distance for each cluster. Due to the third feature the cluster output changes a nd this causes the increase in the distance traveled. The distance traveled for this problem is more than that for fuzzy data with 2 features and the increase is 26.8%. The sacrif ice in distance is the gain in better space utilization with 3 features. Table 5.16 TED by FCM Method for 3 Features: Example 3 Cluster Number Expected Distance Traveled in ft/day 1 6,903 2 4,525 3 15,241 4 2,612 5 14,758 6 2,744 7 4,839 8 21,067 9 21,185 10 5,923 Total 99,797

PAGE 69

62 Figure 5.10 Layout by FCM Method Fuzzy Data with 3 Features 45 33 44 21 48 19 2 7 40 16 50 24 24 23 26 30 30 31 23 24 24 50 16 49 12 22 25 19 9 48 44 46 14 47 39 45 33 44 21 48 19 2 7 40 16 50 24 24 23 31 30 30 31 23 24 24 50 16 49 12 22 25 19 9 48 44 46 14 47 39 45 15 44 21 48 19 2 7 18 16 50 24 24 23 31 30 30 31 23 24 24 50 16 49 12 22 25 19 9 48 44 46 14 1 39 45 17 44 21 48 19 2 7 18 16 50 8 24 23 31 30 30 31 23 24 8 50 16 49 12 22 25 19 9 21 44 46 14 1 39 45 6 44 21 48 19 19 7 18 16 50 8 24 23 31 30 27 31 23 24 8 50 16 49 12 22 25 19 9 21 44 46 14 1 39 45 6 44 44 48 19 19 7 18 16 50 8 24 23 31 27 27 31 23 24 8 50 16 49 12 22 25 19 9 21 44 46 14 1 39 45 6 44 44 48 9 19 4 18 13 50 8 24 43 31 27 27 42 43 24 8 50 13 49 12 22 25 19 9 21 44 46 3 1 39 45 37 44 44 48 9 19 4 18 13 50 8 24 43 42 27 27 42 43 24 8 50 13 49 12 22 25 19 9 21 44 46 3 1 39 45 37 44 44 48 9 19 4 18 13 50 8 24 43 42 27 27 42 43 24 8 50 13 49 12 22 25 19 9 21 44 46 45 1 5 47 11 10 44 48 9 19 4 18 13 50 8 24 43 42 27 27 42 43 24 8 50 13 49 35 22 25 19 9 21 44 46 45 1 5 47 11 10 44 48 9 19 4 18 13 50 8 24 43 42 27 27 42 43 24 8 50 13 49 35 22 25 19 9 21 44 46 45 1 5 47 11 10 44 48 9 19 4 18 13 50 8 24 43 42 27 27 42 43 24 8 50 13 49 35 22 38 19 9 21 44 46 45 1 5 47 11 10 44 48 9 19 4 18 13 50 8 24 43 29 27 34 29 43 24 8 50 13 49 35 22 38 19 9 21 44 46 45 1 5 47 11 10 44 48 9 19 4 18 13 50 8 24 43 29 34 34 29 43 24 8 50 13 49 35 22 38 19 9 21 44 46 45 1 5 47 11 10 44 48 9 19 4 18 13 50 8 24 43 36 34 34 36 43 24 8 50 13 49 35 22 38 19 9 21 44 46 45 39 5 47 11 10 44 48 9 19 4 18 13 16 8 24 43 28 34 34 28 43 24 8 16 13 49 35 22 2 19 48 21 44 20 45 39 5 47 11 10 44 48 9 19 32 18 13 16 8 24 43 28 34 34 28 24 24 8 16 13 40 35 7 2 19 48 21 44 33 45 39 5 47 14 10 44 48 9 19 32 18 13 16 8 24 24 28 34 34 41 24 24 50 16 13 40 22 7 2 19 48 21 44 33 45 39 5 47 14 46 44 48 9 19 32 12 49 16 50 24 24 41 34 26 41 24 24 50 16 49 40 22 7 2 19 48 21 44 33 45 39 5 47 14 46 44 48 9 19 32 12 49 16 50 24 24 41 26 26 41 24 24 50 16 49 40 22 7 2 19 48 21 44 33 45 39 5 I/O 3 p 3=0.50 I/O 2, p2=0.25 I/O 1, p1=0.25

PAGE 70

635.6 Sensitivity of Ge nerated Layouts One of the steps of the FCM method involves assuming range for the linguistic variables used and then randomly generating values for throughput and storage. The random data was generated for five replications for medi um warehouse Example 2. The result of the replications indicated a sma ll variation in the expected distance traveled between the layouts (largest difference of 5.5%, see table 5.17) and hen ce very small changes in the layout On the basis of this problem we can ascertain that the FCM method performs well and is not very sensitive to th e random generation of product data. Table 5.17 Effect of Random Product Data on Expected Distance Replications 1 2 3 4 5 Exp. Distance Traveled ft/day 74,75774,48373,31676,669 72,658 5.7 Effect of Number of Clusters on Total Expected Distance There is no significant research done to study the effect of number of clusters on the cluster formation. The analysis we have done to see the effect of number of cluster on total expected distance (TED) in shown in ta ble 5.18. We can observe that there is no significant change in the total expected dist ance with the change in number of clusters. From table we see that for 10 clusters the TED is the least. To decide the ideal number of clusters for running the FCM algorithm start wi th 3 clusters for products less than 10. For product data ranging from 50 and above use of 10 clusters should give good results. Table 5.18 Analysis of Number of Clus ters on Total Expected Distance Cluster No Total Exp. Distance 20 products Total Exp. Distance 50 Products 2 Features 3 Features 2 Features 3 Features 3 74,698 83,385 79,116 102,602 4 71,648 87,009 81,174 105,936 5 71,671 85,945 79,583 104,167 6 71,949 84,985 78,981 107,058 7 71,179 84,985 78,918 104,696 8 71,179 84,895 79,054 102,700 9 71,156 81,891 78,898 101,352 10 71,503 81,999 78,730 99,797

PAGE 71

645.8 Research Contributions The current state of the warehouse layout te chniques use exact information about the product data which may not be available for large number of products found in today’s warehouse. Furthermore, the existing appr oaches can only take into account the throughput and storage informa tion to yield a layout that will minimize the expected distance traveled. This research effort in our opinion has resulted into the following two contributions in the field of warehouse design. 1. A fuzzy logic based warehouse with uncer tain information of product data was developed that gave excellent results for the layout generated as measured by the total expected distance traveled. 2. It was shown (with the help of two wa rehouse examples) that it is possible to incorporate, in addition to throughput and st orage, another product feature such as volume to generate a layout that will have added flexibility. 5.9 Summary In this chapter we developed layouts for a medium and a large warehouse. We have analyzed the results obtained for total expect ed distance traveled for both the cases with crisp data and fuzzy data. Also the results for total expected distance for the fuzzy data of 3 feature problem is solved and analyzed. Fu rther we analyze the e ffect of number of clusters on the total expected distance trav eled in the warehouse for the two problems with fuzzy data. From the results obtained we can say that the FCM method gives good results for total expected distance traveled. The introduction of the third factor increases the total expected distance but generates a layout that clusters products with more features and that is user friendly. We will summarize the work done and draw appropriate conclusions in the next chapter. Some of th e logical extensions to the problem will also be presented.

PAGE 72

65 CHAPTER 6 SUMMARY AND CONCLUSIONS 6.1 Summary and Conclusions In a warehouse environment, layout design is on e of the important aspe cts. This is due to the fact that the cost involved in storage/retrieva l (S/R) of products is high. A variety of research has been done to minimize the distance traveled for S/R activities. In most of the large warehouses the product information of throughput (T) and storage (S) is not exact and typically available in the form of categori cal data such as low medium and high. This is an ideal environment for exploring the us e of fuzzy logic based method, which looks out for a pattern in the data to get a cluste r of similar data. The existing T/S method for layout design needs exact information of throughput and storage to rank the product on the basis of T/S ratio. A fuzzy c-means (FCM) clustering method was implemented in this thesis to design a ware house layout. Both the existi ng T/S method as well as FCM method was explained with the help of a small warehouse problem. Comparison of expected distance traveled – the perfor mance measure used to judge how good a generated layout was – by both the methods show ed that FCM algorithm resulted in very good layouts. Two more problems, a medium and a large warehouse, were solved by both the methods and the results obtained showed an insignificant (less than 1%) increase in the total expected distance traveled in the warehous e. The problem was also solved by using linguistic variables of product data to design a fuzzy based warehouse using this additional information. Further, the problem was solved by using fuzzy data with 3 features (product volume information was adde d) to obtain a layout. The resulting layout did honor the third feature in layout generation, however, with a tradeoff in the expected distance traveled.

PAGE 73

66The conversion of linguistic variables to numeric values was done by drawing random values within the range assigned for the vari able. The results of several replications indicated very insignif icant difference in the effectiven ess of the generated layout as measured by expected distance traveled. Finall y, the sensitivity of the FCM method to the number of clusters was inves tigated. It was observed that th e method was not sensitive to the number of clusters used. However, it is recommended that with larger number of products more clusters should be used wh ich will reduce the n eed for deciding the allocation of products within a cluster. Thus the overall goal of designing a fuzzy logic based warehouse layout was achieved. 6.2 Scope for Future Research In this research an attempt was made to apply a fuzzy logic based cluster formation technique to develop an efficient warehouse la yout in the absence of precise information regarding throughput and storage levels of large number of products stored in a modern warehouse. The method was validated using se veral examples taken from literature. There are several research extensions (men tioned below) to the approach developed which will improve the applicability of the method even further. 1. In this thesis a third feature, product vo lume, was used to see if a layout can be developed using information in addition to throughput and distance which could be helpful in increased space utilization. Ho wever, it will be interesting to see if the method could use other attributes such as product similarity and see its effect on the generated layout. This will widen the applicability of the FCM method for designing warehouse layouts in such areas as retail and pharm aceutical industry. 2. The sensitivity of the FCM approach to th e number of clusters was investigated in this thesis. Sensitivity of the FCM met hod to the number of classes of linguistic variables perhaps can also impact the la yout which will be helpful in providing guidance while collect ing throughput and storage information. 3. The developed method was tested using se veral problems. The largest warehouse problem was fairly large having 50 pr oducts and 700 locations. However, in comparison to real life warehouse it was still small. The real proof of the applicability of the FCM method will be using it for a real life warehouse layout.

PAGE 74

67 REFERENCES 1. A. Kaylan, D.J. Medeiros, (1988), Analysis of storage polices for miniload AS/AR, Engineering cost and Production Economics, Vol. 13, pp. 311-318. 2. B. Rouwenhorst, J.P. van den Berg, G.J. van Houtum, W.H.M. Zijm, (1996), Performance analysis of a carousel system in: Proceedings of the 1996 International Material Handling Research Colloquium, The Material Handling Industry of America, Charlotte, NC, pp. 495-511. 3. Chu, Chao-Hsien and Hayya, Jack C., (1991), A Fuzzy Clustering Approach to Manufacturing Cell Formation, International Journal of Producti on Research, Vol. 29 (7), pp. 1475-1487. 4. C.J. Malmborg, (1995), Optimization of c ube-per-order index warehouse layouts with zoning constraints, Interna tional Journal of Production Research, Vol. 33 (2), pp. 465482. 5. C.J. Malmborg, K. Bharkaran, (1990), A pplied Mathematical Modeling, A revised proof of optimality for the cube-per-order index rule for stored item location, Vol. 14 (2), pp. 87-95. 6. C.J. Malmborg, K. Bharkaran, (1990), A pplied Mathematical Modeling, A revised proof of optimality for the cube-per-order index rule for stored item location, Vol. 14 (2), pp. 87-95. 7. D.L. van Oudheusden, (1992), W. Zhu, Storage layout of AS/RS racks based on recurrent orders, European Journal of Oper ational Research, Vol. 58 (1), pp. 48-56. 8. G.J. Klir and Bo Yuan, (1995), Fuzzy Sets and Fuzzy Logic: Theory and Application, Prentice Hall, Inc. 9. H.J. Zimmermann, (1990), Fuzzy Set Theory and its Application, Kluwer Academic Publishers, Boston. 10. J.A. Tompkins, J.A. White, et al ., (1996), Facilities Pla nning, John Wiley & sons, New York. 11. J. Ashayeri, L.F. Gelders, (1985), Warehous e design optimization, European Journal of Operational Research Vol. 21, pp. 285-294.

PAGE 75

68 12. J. Ashayeri, L. Gelders, L. van Wa ssenhove, (1985), A microcomputer-based optimization model for the design of automa ted warehouses, International Journal of Production Research, Vol. 23 (4), pp.825-839. 13. J. Ashayeri, R. Heutz, H.C. Veraart, (1996), A new appro ach for the determination of expected traveling time in an AS/RS unde r any assignment policy, in: Progress in Material Handling Research, The Material Handling Industry of Am erica, Charlotte, NC, pp. 51-69. 14. J.C. Bezdek, (1974), Numerical taxonomy w ith fuzzy sets, Journal of Mathematical Biology, Vol. 1, pp. 5771. 15. J.C. Bezdek, Plenum Press, (1981), Pa ttern Recognition with Fuzzy Objective Function Algorithms, New York. 16. J.J. Bartholdi, L.K. Platzman, (1985), Desi gn of efficient bin numbering schemes for automated warehouse carousel storage sy stems, Technical Report MHRC-TR-85-09, Georgia Institute of Technology, Atlanta, GA. 17. J.M. Jarvis, E.D. McDowell, (1991), optim al product layout in an order picking warehouse, IIE Transactions Vol. 23 (1), pp. 93-102. 18. J.P. van den Berg, G.P. Sharp, (1996), Fo rward-reserve alloca tion in a unit-load warehouse operation with picking periods in: Progress in Material Handling Research, The Material Handling Industry of America, pp. 625-638. 19. M.B. Rosenwein, (1994), an application of cl uster analysis to th e problem of locating items within a warehouse, IIE Tran sactions Vol. 26 (1) pp. 101-103. 20. M. Goetschalckx, H.D. Ratlif, (1990), Shar ed storage policies based on the duration stay of unit loads, Management Science Vol. 36 (9), pp. 1120-1132. 21. M. Guenov, R. Raeside, (1992),Zone shap es in class based storage and multicommand order picking when st orage/retrieval machines ar e used, European Journal of Operational Research, Vol. 58 (1), pp. 37-47. 22. M.M. Unde, (2003), A fuzzy logic based desi gn for cellular manufacturing systems, Master’s Thesis, University of South Florida. 23. M.R. Wilhelm, J.L. Shaw, (1996), an empiri cal study of the closes t open location rule for AS/RS storage assignments, in: Progr ess in Material Handling Research, The Material Handling Industry of Amer ica, Charlotte, NC, pp. 639-650. 24. R. Jaikumar, M.M. Solomon, (1990), Dynami c operational policies in an automated warehouse, IIE Transactions, Vol. 23 (4), pp370-376.

PAGE 76

69 25. R.L. Francis, L.F. McGinnis, and J.A. Wh ite, (1992), Facility layout and location: An analytical approach, Prentice Hall, New Jersey. 26. Rouwenhorst et al., (2000), Warehouse desi gn and control: Framework and literature review, European Journal of Operational Research Vol. 122, pp. 515-533. 27. S. Heraghu, (1997), Facilities Design, PWS Publishing Company, New York. 28. W.H. Hausman, L.B. Schwarz, S.C. Graves (1976), Optimal storage assignment in automatic warehousing systems, Management Science, Vol. 22 (6), pp. 629-638. 29. Y.H. Park, D.B. Webster, (1989), Design of class-based storage racks for minimizing travel time in a three-dimensional storag e system, International Journal of Production Research, Vol. 27 (9), pp. 1589-1601. 30. Y. Roll, M.J. Rosenblatt, (1983), Random versus grouped storage policies and their effect on warehouse capacity, Ma terial Flow 1, pp. 199-205.