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An inventory model with two truckload transportation and quantity discounts

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An inventory model with two truckload transportation and quantity discounts
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Santhanam, Ramesh T
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Logistics cost
Optimal ordering quantity
Freight transportation
TL
Quantity discounts
Dissertations, Academic -- Industrial Engineering -- Masters -- USF   ( lcsh )
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
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ABSTRACT: Transportation plays a vital role in the movement of raw materials and finished goods from one place to another. Trucks play a vital role in the movement of materials and are indispensable part of almost every shipment, both domestic and international. On the average, thirty-nine percent of the total logistics cost is spent on transportation. Therefore reducing the transportation cost may significantly reduce the total logistics cost. The total annual logistics cost considered in this research includes ordering cost, material cost, transportation cost and inventory holding cost. The main objective of this research is to develop algorithms for finding the optimal ordering quantity that minimizes total annual logistics cost, when the suppliers offer -No quantity discounts -All-unit quantity discounts -Incremental quantity discounts This research considers truckload transportation where two truck sizes are available.The algorithm developed in this research will identify the optimum ordering quantity and the optimum number of trucks required to ship the ordering quantity. MATLAB programming of the algorithm will analyze the factors that affect that the total annual logistics cost.
Thesis:
Thesis (M.S.I.E.)--University of South Florida, 2005.
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Includes bibliographical references.
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by Ramesh T. Santhanam.
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Title from PDF of title page.
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An Inventory Model With Two Truckloa d Transportation and Quantity Discounts by Ramesh T. Santhanam 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: Michael X.Weng, Ph.D. Grisselle Centeno, Ph.D. Qiang Huang, Ph.D. Date of Approval: October 28, 2005 Keywords: Logistics Cost, Optimal Orderi ng Quantity, Freight Transportation, TL, Quantity Discounts. Copyright 2005, Ramesh T. Santhanam

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Dedication I dedicate this thesis to my Mom, Dad, Grand Mother, Jayashree, Karthik and Vaishnavi

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Acknowledgements I would like to thank Dr. Michael X. Weng, without his thoughtful insights, patience and guidance this thesis would have ju st been a dream. I would like to take this opportunity to thank my committee members fo r their valuable suggestions and help towards completion of this thesis. I would like to thank my friends for all th eir help, it was them who made me feel as if I was in my home with my family, dur ing my stay in Tampa. I would like to name every one in person but that would take two mo re pages, so I am naming a few. Thiag, Mathesh, Baskar, Deepak, Soji, Anand Babu, Anand Manohar, Shiva Sr., Ravikumar, Naveen, Senthil, Sunil, Shiv Ram, Ranga, Krishna. Special thanks to Radhakrishnan Narasi mman for all his he lp and support. I would also like to thank Nancy Coryell fr om USF Golf course, Toufic and Larry from USF physical plant for all their help and support during my stay in Tampa. I would like to thank god for showering his divine blessings and grace on me, without which none of these would have been possible.

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i Table of Contents List of Tables................................................................................................................. ....iv List of Figures................................................................................................................ .....v Abstract....................................................................................................................... .......vi Chapter 1 Introduction........................................................................................................1 1.1 What is a Supply Chain.......................................................................................1 1.2 Objective of a Supply Chain...............................................................................2 1.3 Supply Chain Drivers..........................................................................................3 1.3.1 Inventory.....................................................................................................4 1.3.2 Transportation.............................................................................................4 1.3.3 Facilities......................................................................................................4 1.3.4 Information.................................................................................................5 1.4 Different Modes of Trans portation in Supply Chain..........................................5 1.5 Road Transportation............................................................................................7 1.6 Truckload Transportation and Less-than-Truckload Transportation..................7 1.7 Research Objective.............................................................................................8 1.8 Thesis Layout......................................................................................................9 Chapter 2 Literature Review.............................................................................................11 2.1 Inventory Systems.............................................................................................11 2.2 Transportation Systems and Price Discounts....................................................14

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ii Chapter 3 No Unit Quantity Discounts.............................................................................17 3.1 Introduction.......................................................................................................17 3.2 Transportation Cost...........................................................................................18 3.3 Total Annual Logistics Cost.............................................................................19 3.4 Development of an Optimal Algorithm............................................................20 3.5 Optimal Ordering Quantity Algorithm.............................................................22 3.6 Optimal Ordering Quantity Algorithm when Q* > R ........................................23 Chapter 4 Quantity Discounts...........................................................................................24 4.1 Introduction.......................................................................................................24 4.2 All-unit Quantity Discounts..............................................................................25 4.2.1 Optimal Ordering Quantity Algorithm fo r All-unit Quantity Discounts...................................................................................................26 4.2.2 Optimal Ordering Quantity Algorithm fo r All-unit Quantity Discounts when Q* > R .............................................................................30 4.3 Incremental Quantity Discounts.......................................................................34 4.3.1 Optimal Ordering Quantity Algorithm for Incremental Quantity Discounts...................................................................................................35 4.3.2 Optimal Ordering Quantity Algorithm for Incremental Quantity Discounts when Q* > R .............................................................................39 Chapter 5 Numerical Study...............................................................................................43 5.1 Introduction.......................................................................................................43 5.2 Optimal Solution by MATLAB........................................................................44 5.2.1 Impact of Discount Percentage and Annual Demand on the Optimal Ordering Quantity.....................................................................................44 5.2.2 Impact of K on the Ordering Quantity......................................................50

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iii 5.2.3 Impact of WL on the Optimal Ordering Quantity and Number of Trucks.......................................................................................................56 5.2.4 Impact of C on the Optimal Ordering Quantity........................................59 Chapter 6 Conclusions and Future Directions..................................................................64 6.1 Conclusions.......................................................................................................64 6.2 Summary of Contributions................................................................................65 6.3 Future Directions..............................................................................................65 References..................................................................................................................... ....67

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iv List of Tables Table 5.1 Quantity Discount Structure.............................................................................44 Table 5.2 Impact of R and Discount % on Ordering Quantity..........................................45 Table 5.3 Impact of K on the Ordering Quantity..............................................................51 Table 5.4 Ratio of Capacity of Small Truck to Capacity of Large Truck........................57 Table 5.5 Impact of WL on the Total Annual Logistics Cost............................................57 Table 5.6 Impact of C on the Ordering Quantity..............................................................60

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v List of Figures Figure 1.1 Example of a Supply Chain...............................................................................2 Figure 1.2 Domestic Primary Freight Market by Mode.....................................................6 Figure 1.3 General Freight Sh ipments by Carrier Type.....................................................8 Figure 3.1 Transportation Choice Selections....................................................................19 Figure 4.1 Unit Price with Incremental Quantity Discounts.............................................34 Figure 5.1 Impact of R on Q* for All-unit Quantity Discounts........................................47 Figure 5.2 TC ( Q* )/ R vs. % Discount for All-unit Quantity Discounts...........................48 Figure 5.3 Impact of R on Q* for Incremental Quantity Discounts..................................49 Figure 5.4 TC ( Q* )/ R vs. % Discount for Incremental Quantity Discounts.....................50 Figure 5.5 Impact of K on Q for All-unit Quantity Discounts........................................53 Figure 5.6 TC ( Q* )/ R vs. % Discount for All-unit Quantity Discounts...........................54 Figure 5.7 Impact of K on Q for Incremental Quantity Discounts.................................55 Figure 5.8 TC ( Q* )/ R vs. % Discount for Incremental Quantity Discounts.....................56 Figure 5.9 WS / WL vs. Q *................................................................................................59 Figure 5.10 Impact of C on Q* for All-unit Quantity Discounts......................................62 Figure 5.11 Impact of C on Q* for Incremental Quantity Discounts...............................63

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vi An Inventory Model with Two Truckload Transportation and Quantity Discounts Ramesh T. Santhanam ABSTRACT Transportation plays a vital role in the movement of raw materials and finished goods from one place to another. Trucks play a vital role in the movement of materials and are indispensable part of almost every sh ipment, both domestic an d international. On the average, thirty-nine percent of the total logistics cost is spent on transportation. Therefore reducing the transporta tion cost may significantly reduce the total logistics cost. The total annual logistics cost considered in this research includes ordering cost, material cost, transportation cost and invent ory holding cost. The main objective of this research is to develop algorithms for finding the optimal ordering quantity that minimizes total annual logistics cost, when the suppliers offer No quantity discounts All-unit quantity discounts Incremental quantity discounts This research considers truckload tr ansportation where tw o truck sizes are available. The algorithm developed in this research will identify the optimum ordering quantity and the optimum number of trucks required to ship the ordering quantity.

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vii MATLAB programming of the algor ithm will analyze the factors that affect that the total annual logistics cost.

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1 Chapter 1 Introduction This chapter gives a brief idea of what a supply chain is and discusses some important links that forms the chain. This chapter also discusses the importance of transportation in a supply chain. The current problem under inve stigation is explained in detail and the initial ideas about the proposed solution are specified in this chapter. This chapter explains the goals of the thesis and the last section of this chapter gives an overview of the layout in which the thesis is organized. 1.1 What is a Supply Chain All the processes involved from procuremen t of a raw material, transportation of raw materials to the faciliti es, transformation of raw mate rials into finished goods, transportation of finish ed good to the retailers, and finally to the consumers form the basis of a supply chain. For example let us consider an apparel manufacturing firm, the primary source for cloth is cotton, th e cotton is obtained from co tton fields, raw cotton thus obtained is then transported to the cotton gin to remove burs and leaf trash, the processed cotton fibers are then sent to the thread ma king facility, where these cottons are made in to bundles of thread. These threads are tran sported to dying industry where threads are dyed in to different colors. Colored threads ar e then transported to knitting facility, where the threads of different colors are knitted to form a cloth. The cloth thus obtained is shipped to distribution center, which in turn ships the cl oth to various retailers, based on the demand. Cotton fields, threading facility, dy ing facility, knitting facility, distribution

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2 centers (DCs), and retailers, are referred to as the stages of the supply chain. Each and every stage has multiple supplie rs and consumers, which can be clearly viewed from the above example. If you look at th e dying industry, their suppliers include dye manufacturing, and these dye manufacturing industry have chemi cal industry as their suppliers. Therefore each and every stages of the supply chain have multiple suppliers and multiple consumers. It is easy to identify a s upply chain in a manufacturing ente rprise, but the complexity of the chain may vary from industry to industr y or even company to company. A simple schematic representation of a suppl y chain is shown in Figure 1.1 [30]. Figure 1.1 Example of a Supply Chain 1.2 Objective of a Supply Chain The objective of every supply chain is to maximize the overall value generated. The difference between, the cost incurred for the final product by the customer, and the cost incurred by all stages of the supply chain in fulfilling the custome rs request is called the value of the supply chain. For example, a customer purchasing a watch from a showroom pays $2,500, to purch ase it. This $2,500 the customer pays, represents the

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3 revenue the supply chain receives. The manufactu rer of the watch and the other stages of the supply chain incur cost to produce components, convey information, store them, transport them and transfer funds and so on. The differe nce between the $2,500 that the customer paid and the sum of all costs incurred by the supply chain to produce and distribute the watch represents the supply chain profitability. Supply chain profitability is the total profit to be shared across all stag es of the chain. The higher the supply chain profitability, the more succe ssful the supply chain. Supply chain success should be measured in terms of supply chain profitabil ity and not in terms of the profits at an individual stage [9]. Co-ordination within the stages of the chain by, optimization of the resources needed to fulfill the customers request increases the supply chain profitability. Optimization promises to improve a company s supply chain performance in a variety of areas: Reduced supply costs Improved product margins Increase production Better return assets 1.3 Supply Chain Drivers Drivers are the compelling forces that facilitate the movement of material, information and resources in the supply chai n. The four major drivers of supply chain are inventory, transportation, facilities and inform ation. These drivers not only determine the supply chains performance in terms of respons iveness and efficiency, they also determine

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4 whether strategic fit is achieved across the su pply chain [24]. Let us define each driver and its impact in the performance of the supply chain. 1.3.1 Inventory Inventory is all raw materials, work in process, and finished goods within a supply chain. Inventory is an essential aspect of materials management, knowledge about in-hand inventory and demand rate of the material is important for successful handling of the inventory. Improper inventory management w ill reduce the responsiveness and efficiency of the supply chain. 1.3.2 Transportation In todays fast moving world, products ar e produced in one region and consumed in other, due to cheap produc tion cost, availabili ty of cheap labor, etc. The product produced in one region have to reach its cons umers who are located in different regions, because of this reason transportation plays a major role in supply chain for transporting raw materials and finished goods from suppliers to retailer. Transportation choices have a large impact on supply chain re sponsiveness and efficiency. 1.3.3 Facilities A facility is a general term for a fixed location where the l ogistics activities are carried out. The two major types of facilities are manufacturing locations and warehouses. Decisions regarding location, capacity and flexib ility of facilities have a significant impact on the performance of the supply chain.

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5 1.3.4 Information Though in most cases these drivers are not clearly visible, they are potentially the biggest driver that drives th e supply chain. Synchronization of all the other drivers is made possible with the help of information exch ange. Efficiency of the supply chain will decrease without proper communi cation between the stages of the chain. The concept of supply chain management would have been impossible without information exchange between the stages of the chain. Responsiv eness and efficiency of any supply chain depends on how good the information is tran sferred, any small miscommunication may lead to heavy loss. Measures have to be ma de for synchronized transfer of information from one stage to the other, while designing the supply chain. 1.4 Different Modes of Transp ortation in Supply Chain There are various modes of transportation av ailable in th e supply chain to ship raw materials and finished goods. Air Truck Rail Water Pipeline Each mode incorporates specific advantages and disadvantages that determine its usefulness within any given industry. There is no best mode for a given firm, the shipper can select one, all or any combination of the above mentioned modes based on their preferred choice.

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6 The preferred choice is a func tion of many factors such as The type of industry The location of the firm Location and distribution of suppliers Marketing Area Availability of various transportation modes The following figures gives information a bout volume of freight shipped and the revenue earned from each of the above mentioned modes. Freight Volume % (Total = 11 bln tons) Truck ( 59.5 % ) Air ( 0.10 % ) Rail ( 17.4 % ) Water ( 9.9 % ) Pipeline ( 13.1 % ) Freight Revenue % (Total = $457 bln) Truck ( 81.3 % ) Air ( 3.5 % ) Rail ( 8.9 %) Water ( 1.7 % ) Pipeline ( 4.6 %) Figure 1.2 Domestic Primary Freight Market by Mode From the figure above it is cl early visible that trucks ar e the most preferred mode of transportation [30]. Since the majority of freights are shipped through trucks, the trucks yield more revenue than any other modes of transportation. Trucks however, possess significant advantages over other modes. The capital cost of vehicl es is relatively small High relative speed of vehicles

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7 Flexibility in route choice and flex ibility in loading and unloading of shipments Door-to-Door delivery is made po ssible with the help of trucks 1.5 Road Transportation Road transportation has gained its popularity due to the following reasons. It gives very high reliability in delivering the goods on-time to the doors of the customers with less damage to the shipment, by giving more options and flexibility in shipping, at a moderate cost. Road transportation mainly uses trucks as carriers. The role of trucking varies depending on the region. In large, sparsely populated areas where railroads are well developed, trucks would be used for local delivery and defer to the rail for long distance trips. In areas where the railroad is not so well developed or the ma rket area is heavily populated, trucks become more useful. 1.6 Truckload Transportation and Less -than-Truckload Transportation There are two different truck shipments, namely truckload (TL) and less-thantruckload (LTL) shipments. Shipment that is charged by its maxi mum capacity, either by weight or cube is called a TL shipment. Trucking company wh ich dedicates trailers to a single shippers cargo is called a TL carrier. TL carriers charge for the full truck irrespective of the quantity shipped by the shipper. Carriers give a rate reduction for shipping a TL size shipment and the rates vary with distan ce. The general rule, which influence the transportation co st for any mode of transpor tation is higher the quantity shipped, lower will be the transportation cost [24]. The quantity of freight required in

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8 filling a trailer to truck lo ad capacity is usually more than 10,000 pounds. In less-thantruckload the cost of the frei ght will usually depe nd upon the weight of the freight shipped and the cost of shipment varies with dist ance. Trucking company th at consolidates LTL cargo for multiple destinations on one vehicle is called LTL carriers [17]. Unit shipping cost is less for TL if the truck is filled to its maximum capacity. Th e unit shipping cost of LTL is bit high when compared to TL. TL is more profitable for long distance shipment. The volume of shipment and the revenue obt ained from the TL and LTL is shown in the below figure [30]. Many large companies use th eir own trucks for transportation and it is indicated as Private in the below figure. Volume Private ( 52 % ) TL ( 45 % ) LTL ( 3 % ) Revenue Private ( 47 % ) TL ( 37 % ) LTL ( 16 % ) Figure 1.3 General Freight Shipments by Carrier Type 1.7 Research Objective The objective of this research is to formulate an algorithm for finding the optimal ordering quantity, that minimizes the total an nual logistics cost. Th e total annual logistics cost includes ordering cost, material cost, tr ansportation cost and inventory holding cost. The ordering cost is fixed for each order, irrespective of the quantity ordered. The material cost depends upon the ordering quantity. The transportation cost varies depending upon

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9 the transportation choices made for shippi ng the freight. The ordering quantity will determine the average inventory for which an inventory carrying co st will be charged. This research considers two types of transportation choices, they are Large truckload Small truckload Both the transportation choices offer fixed transportation cost. The carrying capacity of the large truckload is greater than the carrying capacity of the small truckload. The unit transportation cost for using a large truc kload is less than that of small truckload. This research also analyzes the effect of price discounts on th e ordering quantity. The price discounts considered in this research are All-units quantity discount s: A one time price reduction for all the units ordered, based on the number of units ordered or the size of the order are called all-units quantity discounts Incremental quantity discounts: The first se t of ordered quantities will be given at particular price and the remaining qua ntities are given at a reduced price 1.8 Thesis Layout In Chapter 2, a brief review on inventory management and impact of transportation cost and quantity discounts on inventory d ecision is provided. Chapter 3, presents a methodology to find the optimal ordering quantity that minimizes the total annual logistics cost, with transportation cost consideration but no quantity discounts. In Chapter 4, an algorithm for finding the order quantity that minimizes the total annual logistics cost, when (a) all-unit quantity discounts, and (b) in cremental quantity discounts are offered by

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10 the supplier. Chapter 5 pres ents the results based on th e methodology used. Chapter 6 contains the concluding remarks and outlin es the potential research extensions.

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11 Chapter 2 Literature Review A brief review of literature associated with this research is presented in this chapter. An overview of different authors t hose who have researched on inventory models is discussed in section 2.1 and section 2.2 discu sses the literature invo lved in the field of transportation systems and quantity discounts. 2.1 Inventory Systems Many authors have researched inventory models, considering various assumptions and solved the inventory problem by differe nt methods. In 1913 Harri s [11] addresses a practical industrial problem of, finding the lot size of each order such that the overall costs associated with manufacturing a unit of the product is minimized. The formula developed by Harris forms the basis of all economic orde r quantity models. Vassian [34] finds an optimal inventory policy for periodic inventor y models to satisfy the requirement of a particular management. Morse [25] extende d the work done by Vassian [34] by assuming the system to be stochastic and analyzed th e effect changes in the inventory policy. Morse [25] discusses periodic review inventory model and uses Markovs process for solving following situations, When the size of the replenishment orde r is equal to the number of demands arriving at the last period When the order is quantized in multiples of some lot-size q

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12 When the replenishment order is delivered within the next period When the replenishment order is delayed for one or more periods The author analyzes the inventory policy when the replenishment order is delayed for one or more periods. Hadley and Whitin [12] extended the work done by Morse [25] by assuming a stock-out cost for any replenishment order which is delayed for one or more periods. The stock-out cost considered by Hadley and Whitin [12] is a sum of fixed cost per unit for the quantity that is out of stock and a variable cost which is proportional to the time period for which that particular st ock-out quantity. They also find the total cost expression, by considering, Poisson demand with fixed and gamma lead times. Veinott Jr. [35] considers the same model as Hadley and Whitin [12], but makes a assumption that the demand in each period ar e independent and identically distributed random variables and the lead time is constant Veinott Jr. [35] considered a policy in which the inventory is reviewed at the begi nning of each period. If the stock on-hand and on-order is less than the fixed inventory level k, and a quantity Q is ordered which will bring the combination of the on-hand inventory and on-order inventory to a level greater than or equal to inventory level k. No or der will be placed if the stock on-hand and onorder is greater than the fixed inventory level k during the revi ew made at the start of the each period. The author named this policy as (k, Q) policy and proved that the (k, Q) policy is optimal for the finite and infinite models. Lippman [18] finds a optimal inventor y policy for a discrete review, single product, dynamic inventory model by assuming th e ordering cost as a multiple set-up cost. Lippman [19] assumes that the holding cost in each period i is a non-decreasing, left continuous function of the invent ory level at the end of period of i and the ordering cost

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13 function of the inventory level at the period i is neither con cave nor convex. Lippman [19] establishes the existence of an optimal production schedule for each period and also studies the stationary, infinite horizon ve rsion of the multiple set-up cost problem. Lippman [20] extends Lippm an [19] to a case in which the holding cost is proportional to the inventory level while th e ordering cost constitutes of a cost proportional to the amount orde red and a set-up cost independent of the ordering quantity. Iwaniec [16] modifies the ordering cost assu mptions made by Lippma n [18] in finding the optimal ordering quantity. The author also de rives a solution algorithm for the case of periodic review inventory policy, and assumes that the ordering cost consists of linear purchase cost and a fixed cost (truckload co st) for each vehicle used in shipping the quantity. The author examines a policy, if the stock level at the beginning of period n, does not exceed a critical amount Tn then order a the smallest number of full vehicle loads which will raise the inventory level just above the critical amount Tn. No order is placed if the stock level is above the critical level Tn. Since the solution algorithm developed by Iwaniec [16] is difficult to understand and use, Aucamp [1] derives an easy to use algorithm for the multiple set-up cost by using continuous review inventory po licy. The application of th e algorithm extends beyond the situation of fixed carload charges as given by [16]. Extension of the classical economic order quantity model to ec onomies of scale is performed by Buffa and Miller [6], McClain and Thomas [26], Silv er and Peterson [33]. While Lee [21] considers freight discount in stead of quantity discount, he considers the practical situation where the freight cost is discounted whenever a large shipment is placed. The assumptions made by Lee [21] ar e same as the classical economic order

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14 quantity model developed by Harris [11], except for the set-up cost structure. The author considers the set-up cost a sum of fixed co st and freight cost with discounts, which implies that the set-up cost depends upon the quantity ordered. The higher the quantity the lower will be the set-up cost. The authors of [21] and [13] discuss about the individua l impact of price discount and freight discount on i nventory policy. Hwang, Moon, and Shinn [14] discuss the combined effect of price and freight discount on an inventory polic y. Whenever an order is placed price discount is offe red, so it is profitable to buy large quantities of the product, at the same time the freight cost also decrea ses due to large shipment size. Providing price and freight discount will incr ease the ordering quantity, ther e by having a heavy impact on inventory policy. 2.2 Transportation Systems and Price Discounts The literature in this area discusses mainly about the truckload inve ntory models, less-than-truckload inventory models, economies of scale in quantity price, and discounted freight cost. Lancaster [22] and Quandt [31], discuss about various transportation choices, their advantages, disadvantages and common practic es in the shipping industry. Baumol and Vinod [7] explains the importance of trans portation choices made by shippers, whereby order quantity and trans portation alternative can be jointly determined. The optimal choice of mode is shown to involve a trade-off among freight rates, speed, dependability, and en-route losses. They also prove that faster and more dependable service simply reduces the safety stock and the in-transit inventory fo r a shipper or receiver The authors develop a model which will help in statistical comparison of the different modes of transportation

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15 by using the four attributes mentioned be low, thereby the ordering quantity and the transportation alternative is jointly determ ined. The four attributes used in the development of the model are: Shipping cost per unit (including freight rate, insurance, etc.) Mean shipping time Variance in shipping time Carrying cost per unit of time while in transit (interest on capital, pilferage and deterioration) Wehrman [36] finds the minimum total co st including, freight cost, ordering cost, inventory carrying cost and mate rial cost. Freight cost has si gnificant impact on total cost incurred in procuring a material The author constructs the m odel for freight cost, which is substituted in the total cost function to find the minimum total cost for various quantities. Larson [23] makes changes to the wo rk done by Baumol and Vinod [7] by considering the safety stock equal to the in-transit inventory, in finding the economic transportation quantity. Das [10] presents a mo del for finding the economic order quantity when the supplier offers quantity discounts wh ile formulating the i nventory holding cost. The author considers only the cost incurred due to the in-transit inventory and cycle inventory and does not consider the cost incurr ed due to safety stock. This model is also studied from the suppliers point of view by Monahan [28], who designs the procedures for determining the optimum discount schedule for the supplier. Model developed by both Das and Monahan assumes that the demand ra te for the product is known and constant. Abad [2] assumes the demand to be stochastic and develops a model for determining the

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16 optimal selling price and lot-size, while all-unit quantity discount s are offered by the supplier. The author also analyzes the problem for incremental quantity discounts and discusses in Abad [3]. Abdelwahab and Sargious [4] consider both the mode of transportation and the shipment size in determining the optimal shipment size. The author also extends the selection of transportation mode [4], to the selection of transportation carrier within the mode and discusses in [5]. Abdelwahab and Sargious [5] also examine the nature of dependency between the unit freigh t charge and shipment size. Benton and Park [8] give a classification l iterature on all the research done in the field of quantity discounts. Munson and Rose nblatt [29] analyze thirty nine firms that receives/offers quantity discounts. The result of the study indica tes that eighty three percent of the buyers receive quantity discoun ts for most of the items they purchase, which illustrates the prevalence and impor tance of quantity discounts in practice. Rieksts and Ventura [32] determines the optimal inventory policy with two modes of freight transportation. The author consid ers two transportation choices, i.e. truckload (TL) and less-than-truckload (LTL) transpor tation. In truckload transportation, there is fixed cost per load up to given capacity irresp ective of the quantity shipped. For quantities that are less than a full truckload are shi pped using LTL transportation with the cost of shipment depends upon the quantity shipped. Me ndoza and Ventura [27] extends the work by Rieksts and Ventura [32] to all-unit quan tity discounts and finds the optimum ordering quantity that minimizes the total annual cost.

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17 Chapter 3 No Unit Quantity Discounts 3.1 Introduction The problem addressed in this chapter can be described as follows. There is an annual demand R of a single product. The various cost associated with each ordering quantity Q are fixed ordering cost K and a transportation co st that varies with Q The Q will also determine the average inventory for wh ich an inventory carrying cost is charged. What is to be determined is the ordering quantity Q that yields mini mum total logistics cost. The total annual logistics cost includes material cost ordering cost, transportation cost and inventory holding cost. In partic ular, this chapter considers the following transportation scenario. There are two truck sizes: large and small. A large truck has a capacity of WL and charges a fixed price of CL, regardless of actual quantity loaded. Similarly, a small truck has a capacity of WS and charges a fixed price of CS, regardless of actual load (not exceedi ng its capacity). Depending upon the ordering quantity Q it is necessary to use a combination of JL large trucks and JS small trucks, for some JL 0 and JS 0. It is assumed that S S L LW C W C (i.e., if both large and small trucks are fully loaded, the unit shipping cost for a large truck is smaller than that for a small truck). The main objective of this chapter is to present an algorithm that identifies an ordering quantity Q that minimizes the total annual logistics cost.

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18 3.2 Transportation Cost This section describes th e transportation cost. Let Y indicate the least integer that is greater than or equal to Y and Y the largest integer that is lesser than or equal to Y. We now derive the optimal LJ and* SJ, for a given order size Q. Let ) (*Q JLand ) (*Q JS be the optimal number of large and small trucks used to transport quantity Q. Then the corresponding transportation cost ) ( ), ( ,* *Q J Q J Q TS L=S S L LC Q J C Q J ) ( ) (* *. It is clear that) (*Q JL LW Q. Since S S L LW C W C it must be true that) (*Q JS S LW W, and) (*Q JL LW Q. Define A LW Q, (1) B S LW AW Q, and (2) n = S LC C. (3) Then it can be shown that the following must be true. 1 , ) (*n B if A n B if A Q JL (4) 0 , ) (*n B if n B if B Q JS (5)

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19 And the optimal transportation cost is ) 1 ( , ) ( ), ( ,* *n B if C A n B if BC AC Q J Q J Q TL S L S L (6) That is, it is optimal to fill A large trucks, and the remaining part LW A Q will be shipped either by B small trucks if no more than n small trucks are n eeded, or by another large truck, otherwise. This optimal transpor tation cost is depicted in Figure 3.1. Figure 3.1 Transportation Choice Selections 3.3 Total Annual Logistics Cost The total annual logistics cost is the sum of ordering cost, holding cost, material cost and transportation co st. Given order quantity Q, let TC (Q) denote the corresponding total annual logistics cost. Then ) ( 2 ) (Q T Q R RC Q h C K Q R Q TC (7)

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20 Substituting Eq. (6) in to Eq. (7), yields Eq. (8) ) 1 ( 2 ), ( 2 ) , (* *n B if C A Q R RC Q hC K Q R n B if BC AC Q R RC Q hC K Q R J J Q TCL S L S L (8) 3.4 Development of an Optimal Algorithm The total cost given by (8) is a function of ordering quantity Q. Let the derivative TC (Q) = 0. Then we can get the following ordering quantity Q. ) ( 2 , ) ( 2 n B if hC C AC K R n B if hC BC AC K R QL L S L (9) In Eq. (9), however, A and B are also functions of Q, as defined by (1) and (2). Therefore, we cannot use Eq. (9) to find the true optimal Q* that minimizes the total cost given by (8).To find the true optimal Q*, we will consider all combinations of JL and JS. That is, for any given JL and JS, we consider all S S L L S S L LW J W J W J W J Q ) 1 ( to find the optimal Q* (JL, JS) in the range. It can be shown that TC (Q) = 0 leads to the following. ) ( 2 , ) ( 2 ) ( n J if hC C C J K R n J if hC C J C J K R J J QS L L L S S S L L S L (10) The corresponding optimal Q* can be determined by Eq. (11) and Eq. (12) presented below.

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21 If JS n, then the optimal Q* is given by ; ) 1 ( ; ) 1 ( 1 ) 1 ( ) (* S S L L S S L L S S L L S S L L S S L L S S L L S LW J W J Q if W J W J W J W J W J W J Q if Q W J W J Q if W J W J J J Q (11) If JS n, then the optimal Q* is given by ) 1 ( ) 1 ( ; ) 1 ( , ; 1 ) (* L L L L L L S L L S L L S L L S LW J Q if W J W J nW W J Q if Q nW W J Q if nW W J J J Q (12) Note that if JS > n, then it is optimal to use JL +1 large trucks by Eq. (6). Consequently, the corresponding optimal cost can be determined by ) 1 ( ) ( ) ( 2 ) ( ), ( ) ( ) ( 2 ) ( ) (* * * *n J if C J J J Q R RC J J Q hC K J J Q R n J if C J C J J J Q R RC J J Q hC K J J Q R J J Q TCS L L S L S L S L S S S L L S L S L S L S L (13) The optimal order size Q* = Q*) (* S LJ J and the corresponding optimal total cost is T C ( Q*), where ) (* S LJ J = argmin 1 ,.., 2 1 ; ,.., 1 0 ) (*n J W R J J J Q TCS L L S L (14) If Q* > R the optimal number of trucks required can be determined by ) (* S LJ J = argmin 1 ,.., 2 1 ; ) 1 ( ,.., 1 ) (*n J W R t W R t W R t J J J Q TCS L L L L S L (15) The discussion in section 3.2 indicates that if SJ > n then it is optimal to use 1*LJ large trucks. Therefore the optimal order size Q* is given by

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22 ) ( 2 ) * ( 2 ) * ( n S J if hC L C L C L J K R n S J if hC S C S J L C L J K R S J L J Q (16) And the optimal cost is given by ) 1 ( ) ( ) ( 2 ) ( ), ( ) ( ) ( 2 ) ( ) (* * * * * * * * * * * * * *n J if C J J J Q R RC J J Q hC K J J Q R n J if C J C J J J Q R RC J J Q hC K J J Q R J J Q TCS L L S L S L S L S S S L L S L S L S L S L (17) 3.5 Optimal Ordering Quantity Algorithm The algorithm given below, gives a step-b y-step approach for finding the optimum ordering quantity that minimizes the total annual logistics cost. Algorithm A START For ..., 2 1 0 L LW R J For 1 ..., 2 1 n n JS Compute S LJ J Q by Eq. (10); and ) (* S LJ J Q by Eq. (11), or (12) Compute ) (* S LJ J Q TC by Eq. (13); End End Determine the optimal number of large trucks* LJ and small trucks* SJ by Eq. (14), Q* by (16) and ) (*Q TC by (17). STOP

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23 3.6 Optimal Ordering Quantity Algorithm when Q* > R Algorithm A gives an optimal ordering quantity that is no more than the annual demand. In some cases, this may not be true This section provides an optimal ordering quantity that may be more than R. Algorithm B Step 1: START Step 2: Initialize t =0 Step 3: For L L L LW R t W R t W R t J ) 1 ( ......., 1 For 1 ..., 2 1 n n JS Compute S LJ J Q by Eq. (10); and ) (* S LJ J Q by Eq. (11), or (12) Compute ) (* S LJ J Q TC by Eq. (13); End End Determine the optimal number of large trucks* LJ and small trucks* SJ by Eq. (15), Q* by (16) and ) (*Q TC by (17). Step 4: If L LW R t J ) 1 (* and1* n JS, go to Step 5, Else go to Step 6 Step 5: Increment t by 1 and go to Step 3 Step 6: STOP

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24 Chapter 4 Quantity Discounts 4.1 Introduction When specifying the material cost, the Algorithm A presented in Chapter 3 assumes that the unit cost is constant regard less of the quantity ordered. However, there are many instances in which the pricing schedule yields economies of scale, with prices decreasing as lot size is increased. This form of pricing is very common in business-tobusiness transactions. Quantity discounts are generally provided for: Increasing the sales of the product Reducing the in-hand inventor y, by increasing the sales Better production planning Lower order processing cost Reducing the transportation cost, by maki ng use of the discounts offered by the trucking industry A discount is lot size-based if the pricing schedule o ffers discounts based on the quantity ordered in a single lot. A discount is volume-based if the discount is based on the total quantity purchased over a gi ven period (e.g. a year), regardless of the number of lots purchased over that period. Chapter 3 disc usses the case of volum e-based discount (i.e.

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25 the annual demand R determines the unit price C ). This chapter deals with the case of lot size-based discount. Two commonly used lot-size based discount schemes are the following. All-unit quantity discounts Incremental quantity discounts This chapter analyses th e effect of lot size-base d quantity discounts on the inventory policy. The main objective of this chap ter is to present algorithms that identifies an ordering quantity Q that minimizes the total annua l logistics cost, when all-unit quantity discounts and incremental quantity discounts are offered. 4.2 All-unit Quantity Discounts In all-unit quantity discounts, the entire orde r is charged with the same unit price, which is, however, a function of the actual ordering quantity Q In particular, the pricing schedule contains specified break points M0, M1, M2, ...., Mk -1, Mk with 0 = M0 < M1< M2< M3 < Mk, such that Q k M if k C k M Q k M if k C M Q M if C M Q M if C M Q M if C Q C price Unit 1 1 . . . . 3 2 2 2 1 1 1 0 0 ) ( (18) where C0 > C1 > > Ck. In this case the material cost for ordering Mj+1 units may be smaller than that for ordering Mj units. In general, materi al cost of purchasing Mj units is

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26 greater than that of purchasing Mj+1 units, if 1 ,1 j C C C Mj j j j. If the ordering quantity falls in the interval ( Mj, Mj +1], then the unit cost is Cj. 4.2.1 Optimal Ordering Quantity Algorithm for All-unit Quantity Discounts The transportation cost and the total annual logistics co st remains the same as discussed in Chapter 3. For convenience th e transportation cost is restated here. ) 1 ( , ) ( n J if C J n J if C J C J Q TS L L S S S L L (19) The total annual logistics cost is computed by ) ( ) ( 2 ) ( ) ( Q T Q R Q RC Q Q hC K Q R Q TC (20) To find the optimal Q*, we will consider all combinations of JL and JS. If, for the given JL and JS, there exists a single j (0 j k ) such that Mj ( JLWL + ( JS 1) WS, JLWL + JSWS], then the unit material cost for a ny ordering quantity in the range ( JLWL + ( JS 1) WS, Mj] is set as Cj-1, and the unit material cost for any ordering quantity in the range ( Mj, JLWL + JSWS] is set as Cj. There is no Mj such that Mj ( JLWL + ( JS 1) WS, JLWL + JSWS], for the given JL and JS. Then the unit material cost Cj for all Q ( JLWL + ( JS 1) WS, JLWL + JSWS], is determined by finding the largest value of j that satisfies the condition Mj JLWL + ( JS 1) WS. The third case is that there are more than one Mj that belongs to ( JLWL + ( JS 1) WS, JLWL + JSWS]. Without loss of generality, assume Mj, Mj +1, ., Mj+g belong to ( JLWL + ( JS 1) WS, JLWL + JSWS], where g 1. Then each interval ( JLWL + ( JS 1) WS, Mj], ( Mj, Mj +1],

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27 ( Mj +1, Mj +2], .,( Mj+g, JLWL + JSWS] is considered separately and the corre sponding unit material cost for each interval is Cj -1, Cj, Cj +1, ., Cj+g, respectively. The procedure for obtaining the optimal ordering quantity remains the same as Algorithm A, except for the unit cost structur e. The following algorithm identifies the unit material cost Cj for any given JL and JS, and then finds the optimal ordering quantity. Algorithm C START For ..., 2 1 0 L LW R J For JS = 1, 2, ......, n n +1. If JS n If S S L L S S L L jW J W J W J W J M ) 1 (, for all j = 1, 2, ., k Find the largest j such that S S L L jW J W J M ) 1 ( and set unit price as Cj and computeS L jJ J Q ,, S LJ J Q ,* and ) (* S LJ J Q TC, by j S S L L S L jhC C J C J K R J J Q ) ( 2 ; , 1 , ; 1 , 1 1 ,* S S L L S L j S S L L S S L L S S L L S L j S L j S S L L S L j S S L L S LW J W J J J Q if W J W J W J W J W J W J J J Q if J J Q W J W J J J Q if W J W J J J Q ). ( , 2 ) (* * S S L L S L j S L j S L S LC J C J J J Q R C R J J Q C h K J J Q R J J Q TC Else there exists some j1 and g0 such that Mj, Mj +1, ., Mj+g ( JL WL +( JS 1) WS, JL WL + JS WS,], S S L L jW J W J M ) 1 (1 andS S L L g jW J W J M 1. Then consider each interval ( JLWL + ( JS-1) WS, Mj], ( Mj, Mj +1], ...., ( Mj+g, JL WL + JS WS] separately as follows. For all Q ( JLWL + ( JS-1) WS, Mj], set unit price as Cj-1 and computeS L jJ J Q,1, S L jJ J Q ,* 1 and) (* 1Q TCj by 1 1) ( 2 j S S L L S L jhC C J C J K R J J Q.

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28 ; , 1 , ; 1 , 1 1 ,1 1 1 1 1 j S L j j j S S L L S L j S L j S S L L S L j S S L L S L jM J J Q if M M W J W J J J Q if J J Q W J W J J J Q if W J W J J J Q ). ( , 2 ) (* 1 1 1 1 1 1S S L L S L j j S L j j S L j jC J C J J J Q R C R J J Q C h K J J Q R Q TC For all Q( Mi, Mi +1], i = j j +1, ., j + g -1, set unit price as Ci. Note that this case disappears if g = 0. Compute S L iJ J Q ,, S L iJ J Q ,* and) (*Q TCi, by i S S L L S L ihC C J C J K R J J Q ) ( 2 ; , , , ; , 1 ,1 1 1 i S L i i i i S L i S L i i S L i i S L iM J J Q if M M M J J Q if J J Q M J J Q if M J J Q ). ( , 2 ) (* * *S S L L S L i i S L i i S L i iC J C J J J Q R C R J J Q C h K J J Q R Q TC For all Q (S S L L g jW J W J M,], set unit price as Cj+g and computeS L g jJ J Q ,, S L g jJ J Q ,* and) (*Q TCg j, by g j S S L L S L g jhC C J C J K R J J Q ) ( 2 ,. ; , , , ; , 1 ,* S S L L S L g j S S L L S S L L g j S L g j S L g j g j S L g j g j S L g jW J W J J J Q if W J W J W J W J M J J Q if J J Q M J J Q if M J J Q ). ( , 2 ) (* * S S L L S L g j g j S L g j g j S L g j g jC J C J J J Q R C R J J Q C h K J J Q R Q TC Find u = argmin g j j j j i Q TCi ...., 1 , 1 ) (*, and set ) ( ) (* S L u S LJ J Q J J Q and ) ( ) (* *Q TC J J Q TCu S L. Else If JS > n If L L S L L jW J nW W J M ) 1 ( , for all j = 1, 2, ...., k Find the largest j such that S L L jnW W J M and set unit price as Cj and computeS L jJ J Q ,, S LJ J Q ,* and ) (* S LJ J Q TC, by j L L L S L jhC C C J K R J J Q ) ( 2

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29 ) 1 ( , ) 1 ( ; ) 1 ( , , ; , 1 ,* L L S L j L L L L S L L S L j S L j S L L S L j S L L S LW J J J Q if W J W J nW W J J J Q if J J Q nW W J J J Q if nW W J J J Q ) 1 ( , 2 ) (* * L L S L j S L j S L S LC J J J Q R C R J J Q C h K J J Q R J J Q TC Else there exists some j1 and g0 such that Mj, Mj +1, ., Mj+g ( JL WL + n WS, ( JL +1) WL], S L L jnW W J M 1,andL L g jW J M ) 1 (1 Then consider each interval ( JLWL + nWS, Mj], ( Mj, Mj +1], ...., ( Mj+g, ( JL+1) WL] separately as follows. For all Q ( JLWL + nWS, Mj], set unit price as Cj-1 and computeS L jJ J Q ,1, S L jJ J Q ,* 1 and) (* 1Q TCj, by 1 1) ( 2 j L L L S L jhC C C J K R J J Q. ; , , , ; , 1 ,1 1 1 1 1 j S L j j j S L L S L j S L j S L L S L j S L L S L jM J J Q if M M nW W J J J Q if J J Q nW W J J J Q if nW W J J J Q ) 1 ( , 2 ) (* 1 1 1 1 1 1 L L S L j j S L j j S L j jC J J J Q R C R J J Q C h K J J Q R Q TC For all Q( Mi, Mi +1], i = j j +1, ., j + g -1, set unit price as Ci. Note that this case disappears if g = 0. Compute S L iJ J Q ,, S L iJ J Q ,* and) (*Q TCi, by i L L L S L ihC C C J K R J J Q ) ( 2 ; , , , ; , 1 ,1 1 1 i S L i i i i S L i S L i i S L i i S L iM J J Q if M M M J J Q if J J Q M J J Q if M J J Q ) 1 ( , 2 ) (* * L L S L i i S L i i S L i iC J J J Q R C R J J Q C h K J J Q R Q TC For all Q (L L g jW J M ) 1 ( ], set unit price as Cj+g and computeS L g jJ J Q ,, S L g jJ J Q ,* and ) (*Q TCg j by g j L L L S L g jhC C C J K R J J Q ) ( 2 ,.

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30 ; ) 1 ( , ) 1 ( ) 1 ( , , ; , 1 ,* L L S L g j L L L L g j S L g j S L g j g j S L g j g j S L g jW J J J Q if W J W J M J J Q if J J Q M J J Q if M J J Q ) 1 ( , 2 ) (* * L L S L g j g j S L g j g j S L g j g jC J J J Q R C R J J Q C h K J J Q R Q TC Find u = argmin g j j j j i Q TCi ...., 1 , 1 ) (*, and set ) (* S LJ J Q = ) (* S L uJ J Qand ) (* S LJ J Q TC = ) (*Q TCu. End End Optimal *,S LJ J = argmin 1 ...., 2 1 ...., 1 0 ) (*n n J W R J J J Q TCS L L S L, * *,S LJ J Q Q and ) ( ) (* * S LJ J Q TC Q TC. STOP 4.2.2 Optimal Ordering Quantity Algorithm fo r All-unit Quantity Discounts when Q* > R Algorithm C may not be true if optimal ordering quantity is more than the annual demand R This section provides an optimal orde ring quantity that may be more than R when all-unit quantity discounts is offered. Algorithm D Step 1 START Step 2 Initialize t = 0. Step 3 For JL L L LW R t W R t W R t ) 1 ( ......., 1 ,. For JS = 1, 2, ......, n n +1. If JS n If S S L L S S L L jW J W J W J W J M ) 1 (, for all j = 1, 2, ., k

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31 Find the largest j such that S S L L jW J W J M ) 1 ( and set unit price as Cj and compute S L jJ J Q ,, S LJ J Q ,* and ) (* S LJ J Q TC, by j S S L L S L jhC C J C J K R J J Q ) ( 2 ; , 1 , ; 1 , 1 1 ,* S S L L S L j S S L L S S L L S S L L S L j S L j S S L L S L j S S L L S LW J W J J J Q if W J W J W J W J W J W J J J Q if J J Q W J W J J J Q if W J W J J J Q ). ( , 2 ) (* * S S L L S L j S L j S L S LC J C J J J Q R C R J J Q C h K J J Q R J J Q TC Else there exists some j1 and g0 such that Mj, Mj +1, ., Mj+g ( JL WL + ( JS 1) WS, JL WL + JS WS], S S L L jW J W J M ) 1 (1 andS S L L g jW J W J M 1. Then consider each interval ( JLWL + ( JS-1) WS, Mj], ( Mj, Mj +1], ...., ( Mj+g, JL WL + JS WS] separately as follows. For allQ ( JLWL + ( JS-1) WS, Mj], set unit price as Cj-1 and computeS L jJ J Q ,1, S L jJ J Q ,* 1 and) (* 1Q TCj, by 1 1) ( 2 j S S L L S L jhC C J C J K R J J Q. ; , 1 , ; 1 , 1 1 ,1 1 1 1 1 j S L j j j S S L L S L j S L j S S L L S L j S S L L S L jM J J Q if M M W J W J J J Q if J J Q W J W J J J Q if W J W J J J Q ). ( , 2 ) (* 1 1 1 1 1 1 S S L L S L j j S L j j S L j jC J C J J J Q R C R J J Q C h K J J Q R Q TC For all Q( Mi, Mi +1], i = j j +1, ., j + g -1, set unit price as Ci. Note that this case disappears if g = 0. Compute S L iJ J Q ,, S L iJ J Q ,* and) (*Q TCi, by i S S L L S L ihC C J C J K R J J Q ) ( 2 ; , , , ; , 1 ,1 1 1 i S L i i i i S L i S L i i S L i i S L iM J J Q if M M M J J Q if J J Q M J J Q if M J J Q ). ( , 2 ) (* * S S L L S L i i S L i i S L i iC J C J J J Q R C R J J Q C h K J J Q R Q TC

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32 For all Q (S S L L g jW J W J M,], set unit price as Cj+g and computeS L g jJ J Q ,, S L g jJ J Q ,* and) (*Q TCg j by g j S S L L S L g jhC C J C J K R J J Q ) ( 2 ,. ; , , , ; , 1 ,* S S L L S L g j S S L L S S L L g j S L g j S L g j g j S L g j g j S L g jW J W J J J Q if W J W J W J W J M J J Q if J J Q M J J Q if M J J Q ). ( , 2 ) (* * S S L L S L g j g j S L g j g j S L g j g jC J C J J J Q R C R J J Q C h K J J Q R Q TC Find u = argmin g j j j j i Q TCi ...., 1 , 1 ) (*, and set ) ( ) (* S L u S LJ J Q J J Qand ) ( ) (* *Q TC J J Q TCu S L. Else If JS >n If L L S L L jW J nW W J M ) 1 ( , for all j = 1, 2, ...., k Find the largest j such that S L L jnW W J M and set unit price as Cj and computeS L jJ J Q ,, S LJ J Q ,* and ) (* S LJ J Q TC, by j L L L S L jhC C C J K R J J Q ) ( 2 . ) 1 ( , ) 1 ( ; ) 1 ( , , ; , 1 ,* L L S L j L L L L S L L S L j S L j S L L S L j S L L S LW J J J Q if W J W J nW W J J J Q if J J Q nW W J J J Q if nW W J J J Q ) 1 ( , 2 ) (* * L L S L j S L j S L S LC J J J Q R C R J J Q C h K J J Q R J J Q TC Else there exists some j1 and g0 such that Mj, Mj +1, ., Mj+g ( JL WL + n WS, ( JL +1) WL], S L L jnW W J M 1,andL L g jW J M ) 1 (1 Then consider each interval ( JLWL + nWS, Mj], ( Mj, Mj +1], ...., ( Mj+g, ( JL+1) WL] separately as follows. For allQ ( JLWL + nWS, Mj], set unit price as Cj-1 and computeS L jJ J Q ,1, S L jJ J Q ,* 1 and) (* 1Q TCj by 1 1) ( 2 j L L L S L jhC C C J K R J J Q.

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33 ; , , , ; , 1 ,1 1 1 1 1 j S L j j j S L L S L j S L j S L L S L j S L L S L jM J J Q if M M nW W J J J Q if J J Q nW W J J J Q if nW W J J J Q ) 1 ( , 2 ) (* 1 1 1 1 1 1 L L S L j j S L j j S L j jC J J J Q R C R J J Q C h K J J Q R Q TC For all Q( Mi, Mi +1], i = j j +1, ., j + g -1, set unit price as Ci. Note that this case disappears if g = 0. Compute S L iJ J Q ,, S L iJ J Q ,* and) (*Q TCi, by i L L L S L ihC C C J K R J J Q ) ( 2 ; , , , ; , 1 ,1 1 1 i S L i i i i S L i S L i i S L i i S L iM J J Q if M M M J J Q if J J Q M J J Q if M J J Q ) 1 ( , 2 ) (* * L L S L i i S L i i S L i iC J J J Q R C R J J Q C h K J J Q R Q TC For all Q (L L g jW J M ) 1 ( ,], set unit price as Cj+g and computeS L g jJ J Q ,, S L g jJ J Q ,* and ) (*Q TCg j, by g j L L L S L g jhC C C J K R J J Q ) ( 2 ,. ; ) 1 ( , ) 1 ( ) 1 ( , , ; , 1 ,* L L S L g j L L L L g j S L g j S L g j g j S L g j g j S L g jW J J J Q if W J W J M J J Q if J J Q M J J Q if M J J Q ) 1 ( , 2 ) (* * L L S L g j g j S L g j g j S L g j g jC J J J Q R C R J J Q C h K J J Q R Q TC Find u = argmin g j j j j i Q TCi ...., 1 , 1 ) (*, and set ) (* S LJ J Q = ) (* S L uJ J Qand ) (* S LJ J Q TC = ) (*Q TCu. End End Optimal *,S LJ J = argmin 1 ...., 2 1 ...., 1 0 ) (*n n J W R J J J Q TCS L L S L, * *,S LJ J Q Q and ) ( ) (* * S LJ J Q TC Q TC. Step 4 If L LW R t J ) 1 (* and1* n JS, go to Step 5, Else go to Step 6.

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34 Step 5 Increment t by 1 and go to Step 3. Step 6 STOP 4.3 Incremental Quantity Discounts In the incremental quantity discounts, the unit material cost is incremental and varies with the break point quantities. As for the all-unit discount case, let Mj ( j = 0, 1, k ) represent the j -th break point in the pr icing schedule, with 0 = M0 < M1< M2 .< Mk. Then, the incremental quantity discounts can be depicted as Fi gure 4.1. If the ordering quantity Q M1, the entire order is charged with unit price C0; if M1 Q M2, then the unit price is C0 for the first M1 units and C1 for the rest of the orde r; and so on in general, C0 > C1 > > Ck. Figure 4.1 Unit Price with Incremental Quantity Discounts For an order size of Q ( Mj, Mj +1], the material cost is Vj + ( Q Mj) Cj where Vj = Vj -1 + ( MjMj -1) Cj -1, j = 1,2, ., k ., with V0 = 0. Therefore the annual material cost, annual holding cost and annual logistic s cost are given by, respectively, Annual material cost = j j jC M Q V Q R (21)

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35 Annual holding cost = 2 h C M Q Vj j j (22) Annual logistics cost, ) ( 2 ) ( Q T Q R C M Q V Q R h C M Q V K Q R Q TCj j j j j j (23) substituting Eq. (19) in Eq. (23) yields ) 1 ( 2 ), ( 2 ) ( n J If C J Q R C M Q V Q R h C M Q V K Q R n J If C J C J Q R C M Q V Q R h C M Q V K Q R Q TCS L L j j j j j j S S S L L j j j j j j (24) 4.3.1 Optimal Ordering Quantity Algorithm for Incremental Quantity Discounts The algorithm given below, gives a step-b y-step approach for finding the optimal ordering quantity, that mini mizes the total annual logistics cost, when incremental quantity discounts is offered by the supplier. Algorithm E START For ..., 2 1 0 L LW R J For JS = 1, 2, ......, n n +1. If JS n If S S L L S S L L jW J W J W J W J M ) 1 (, for all j = 1, 2, ., k Find the largest j such that S S L L jW J W J M ) 1 ( and computeS L jJ J Q ,, S LJ J Q ,* and ) (* S LJ J Q TC, by j S S L L j j j S L jhC C J C J C M V K R J J Q ) ( 2

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36 ; , 1 , ; 1 , 1 1 ,* S S L L S L j S S L L S S L L S S L L S L j S L j S S L L S L j S S L L S LW J W J J J Q if W J W J W J W J W J W J J J Q if J J Q W J W J J J Q if W J W J J J Q ). ( ) ( 2 ) ( ) (* * * S S L L S L j j S L j S L j j S L j S L S LC J C J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R J J Q TC Else there exists some j1 and g0 such that Mj, Mj +1, ., Mj+g ( JL WL + ( JS 1) WS, JL WL + JS WS,], S S L L jW J W J M ) 1 (1 andS S L L g jW J W J M 1. Then consider each interval ( JLWL + ( JS-1) WS, Mj], ( Mj, Mj +1], ...., ( Mj+g, JL WL + JS WS] separately as follows. For all Q ( JLWL + ( JS-1) WS, Mj], compute S L jJ J Q ,1,S L jJ J Q ,* 1 and ) (* 1Q TCj, by 1 1 1 1 1) ( 2 j S S L L j j j S L jhC C J C J C M V K R J J Q. ; , 1 , ; 1 , 1 1 ,1 1 1 1 1 j S L j j j S S L L S L j S L j S S L L S L j S S L L S L jM J J Q if M M W J W J J J Q if J J Q W J W J J J Q if W J W J J J Q ). ( ) ( 2 ) ( ) (* 1 1 1 1 1 1 1 1 1 1 1 1 S S L L S L j j j S L j j S L j j j S L j j S L j jC J C J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC For all Q( Mi, Mi +1], i = j j +1, ., j + g -1, set unit price as Ci. Note that this case disappears if g = 0. Compute S L iJ J Q ,, S L iJ J Q ,* and *Q TCi, by i S S L L i i i S L ihC C J C J C M V K R J J Q ) ( 2 ; , , , ; , 1 ,1 1 1 i S L i i i i S L i S L i i S L i i S L iM J J Q if M M M J J Q if J J Q M J J Q if M J J Q ). ( ) ( 2 ) ( ) (* * * S S L L S L i i i S L i i S L i i i S L i i S L i iC J C J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC For all Q(S S L L g jW J W J M ,], compute S L g jJ J Q ,, S L g jJ J Q ,* and *Q TCg j, by

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37 g j S S L L g j g j g j S L g jhC C J C J C M V K R J J Q ) ( 2 ,. ; , , , ; , 1 ,* S S L L S L g j S S L L S S L L g j S L g j S L g j g j S L g j g j S L g jW J W J J J Q if W J W J W J W J M J J Q if J J Q M J J Q if M J J Q ). ( ) ( 2 ) ( ) (* * * S S L L S L g j g j g j S L g j g j S L g j g j g j S L g j g j S L g j g jC J C J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC Find u = argmin g j j j j i Q TCi ...., 1 , 1 ) (*, and set ) (* S LJ J Q = ) (* S L uJ J Qand ) (* S LJ J Q TC = ) (*Q TCu. Else If JS >n If L L S L L jW J nW W J M ) 1 ( , for all j = 1, 2, .., k Find the largest j such that S L L jnW W J M and computeS L jJ J Q ,, S LJ J Q ,* and ) (* S LJ J Q TC, by j L L L j j j S L jhC C C J C M V K R J J Q ) ( 2 . ) 1 ( , ) 1 ( ; ) 1 ( , , ; , 1 ,* L L S L j L L L L S L L S L j S L j S L L S L j S L L S LW J J J Q if W J W J nW W J J J Q if J J Q nW W J J J Q if nW W J J J Q ) 1 ( ) ( 2 ) ( ) (* * * L L S L j j S L j j j S L j S L S LC J J J Q R C M J J Q V Q R h C M J J Q V K J J Q R J J Q TC Else there exists some j1 and g0 such that Mj, Mj +1, ., Mj+g( JL WL + n WS, ( JL + 1) WL,], S L L jnW W J M 1,andL L g jW J M ) 1 (1 Then consider each interval ( JLWL + nWS, Mj], ( Mj, Mj +1], ., ( Mj+g, ( JL+1) WL] separately as follows. For all Q ( JLWL + nWS, Mj], compute S L jJ J Q ,1, S L jJ J Q ,* 1 and) (* 1Q TCj, by 1 1 1 1 1) ( 2 j L L L j j j S L jhC C C J C M V K R J J Q.

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38 ; , , , ; , 1 ,1 1 1 1 1 j S L j j j S L L S L j S L j S L L S L j S L L S L jM J J Q if M M nW W J J J Q if J J Q nW W J J J Q if nW W J J J Q ) 1 ( ) ( 2 ) ( ) (* 1 1 1 1 1 1 1 1 1 1 1 1 L L S L j j j S L j j S L j j j S L j j S L j jC J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC For all Q( Mi, Mi +1], i = j j +1, j + g -1, set unit price as Ci. Note that this case disappears if g = 0. Compute S L iJ J Q ,, S L iJ J Q,* and *Q TCi, by i L L L i i i S L ihC C C J C M V K R J J Q ) ( 2 ; , , , ; , 1 ,1 1 1 i S L i i i i S L i S L i i S L i i S L iM J J Q if M M M J J Q if J J Q M J J Q if M J J Q ) 1 ( ) ( 2 ) ( ,* * * L L S L i i i S L i i S L i i i S L i i S L i iC J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC For all Q(L L g jW J M) 1 ( ], compute S L g jJ J Q,, S L g jJ J Q ,* and *Q TCg j, by g j L L L g j g j g j S L g jhC C C J C M V K R J J Q ) ( 2 ,. ; , ) 1 ( ) 1 ( , , ; , 1 ,* S S L L S L g j L L L L g j S L g j S L g j g j S L g j g j S L g jW J W J J J Q if W J W J M J J Q if J J Q M J J Q if M J J Q ) 1 ( ) ( 2 ) ( ,* * * L L S L g j g j g j S L g j g j S L g j g j g j S L g j g j S L g j g jC J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC Find u = argmin g j j j j i Q TCi ...., 1 , 1 ) (*, and set ) (* S LJ J Q = ) (* S L uJ J Qand ) (* S LJ J Q TC = ) (*Q TCu. End End

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39 Optimal *,S LJ J = argmin 1 ...., 2 1 ...., 1 0 ) (*n n J W R J J J Q TCS L L S L, * *,S LJ J Q Q and ) ( ) (* * S LJ J Q TC Q TC STOP 4.3.2 Optimal Ordering Quantity Algorithm for Incremental Quantity Discounts when Q* > R Algorithm E may not be true for some cas es where optimal ordering quantity is more than the demand. This section provides an optimal ordering qu antity that may be more than R when incremental quantity discounts is offered. Algorithm F Step 1 START Step 2 Initialize t =0. Step 3 For ..., 2 1 0 L LW R J For JS = 1, 2, ., n n +1. If JS n If S S L L S S L L jW J W J W J W J M ) 1 (, for all j = 1, 2, ., k Find the largest j such that S S L L jW J W J M ) 1 ( and computeS L jJ J Q ,, S LJ J Q ,* and ) (* S LJ J Q TC, by j S S L L j j j S L jhC C J C J C M V K R J J Q ) ( 2 ; , 1 , ; 1 , 1 1 ,* S S L L S L j S S L L S S L L S S L L S L j S L j S S L L S L j S S L L S LW J W J J J Q if W J W J W J W J W J W J J J Q if J J Q W J W J J J Q if W J W J J J Q ). ( ) ( 2 ) ( ) (* * * S S L L S L j j S L j S L j j S L j S L S LC J C J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R J J Q TC

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40 Else there exists some j1 and g0 such that Mj, Mj +1, ., Mj+g ( JL WL + ( JS 1) WS, JL WL + JSWS], S S L L jW J W J M ) 1 (1 andS S L L g jW J W J M 1. Then consider each interval ( JLWL + ( JS-1) WS, Mj], ( Mj, Mj +1], ...., ( Mj+g, JL WL + JS WS] separately as follows. For all Q ( JLWL + ( JS-1) WS, Mj], compute S L jJ J Q ,1,S L jJ J Q ,* 1 and ) (* 1Q TCj, by 1 1 1 1 1) ( 2 j S S L L j j j S L jhC C J C J C M V K R J J Q. ; , 1 , ; 1 , 1 1 ,1 1 1 1 1 j S L j j j S S L L S L j S L j S S L L S L j S S L L S L jM J J Q if M M W J W J J J Q if J J Q W J W J J J Q if W J W J J J Q ). ( ) ( 2 ) ( ) (* 1 1 1 1 1 1 1 1 1 1 1 1 S S L L S L j j j S L j j S L j j j S L j j S L j jC J C J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC For all Q( Mi, Mi +1], i = j j +1, ., j + g -1, set unit price as Ci. Note that this case disappears if g = 0. Compute S L iJ J Q ,, S L iJ J Q,* and *Q TCi, by i S S L L i i i S L ihC C J C J C M V K R J J Q ) ( 2 ; , , , ; , 1 ,1 1 1 i S L i i i i S L i S L i i S L i i S L iM J J Q if M M M J J Q if J J Q M J J Q if M J J Q ). ( ) ( 2 ) ( ) (* * * S S L L S L i i i S L i i S L i i i S L i i S L i iC J C J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC For all Q (S S L L g jW J W J M ,], computeS L g jJ J Q,, S L g jJ J Q ,* and *Q TCg j by g j S S L L g j g j g j S L g jhC C J C J C M V K R J J Q ) ( 2 ,. ; , , , ; , 1 ,* S S L L S L g j S S L L S S L L g j S L g j S L g j g j S L g j g j S L g jW J W J J J Q if W J W J W J W J M J J Q if J J Q M J J Q if M J J Q

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41 ). ( ) ( 2 ) ( ) (* * * S S L L S L g j g j g j S L g j g j S L g j g j g j S L g j g j S L g j g jC J C J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC Find u = argmin g j j j j i Q TCi ...., 1 , 1 ) (*, and set ) (* S LJ J Q = ) (* S L uJ J Qand ) (* S LJ J Q TC = ) (*Q TCu. Else If JS > n If L L S L L jW J nW W J M) 1 ( , for all j = 1, 2, .., k Find the largest j such that S L L jnW W J M and computeS L jJ J Q,, S LJ J Q,* and ) (* S LJ J Q TC, by j L L L j j j S L jhC C C J C M V K R J J Q ) ( 2 . ) 1 ( , ) 1 ( ; ) 1 ( , , ; , 1 ,* L L S L j L L L L S L L S L j S L j S L L S L j S L L S LW J J J Q if W J W J nW W J J J Q if J J Q nW W J J J Q if nW W J J J Q ) 1 ( ) ( 2 ) ( ) (* * * L L S L j j S L j j j S L j S L S LC J J J Q R C M J J Q V Q R h C M J J Q V K J J Q R J J Q TC Else there exists some j1 and g0 such that Mj, Mj +1, ., Mj+g( JL WL + n WS, ( JL + 1) WL,], S L L jnW W J M 1,andL L g jW J M) 1 (1 Then consider each interval ( JLWL + nWS, Mj], ( Mj, Mj +1], ., ( Mj+g, ( JL+1) WL] separately as follows. For all Q ( JLWL + nWS, Mj], computeS L jJ J Q ,1, S L jJ J Q ,* 1 and) (* 1Q TCj, by 1 1 1 1 1) ( 2 j L L L j j j S L jhC C C J C M V K R J J Q. ; , , , ; , 1 ,1 1 1 1 1 j S L j j j S L L S L j S L j S L L S L j S L L S L jM J J Q if M M nW W J J J Q if J J Q nW W J J J Q if nW W J J J Q ) 1 ( ) ( 2 ) ( ) (* 1 1 1 1 1 1 1 1 1 1 1 1 L L S L j j j S L j j S L j j j S L j j S L j jC J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC

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42 For all Q( Mi, Mi +1], i = j j +1, j + g -1, set unit price as Ci. Note that this case disappears if g = 0. Compute S L iJ J Q ,, S L iJ J Q ,* and *Q TCi, by i L L L i i i S L ihC C C J C M V K R J J Q ) ( 2 ; , , , ; , 1 ,1 1 1 i S L i i i i S L i S L i i S L i i S L iM J J Q if M M M J J Q if J J Q M J J Q if M J J Q ) 1 ( ) ( 2 ) ( ,* * * L L S L i i i S L i i S L i i i S L i i S L i iC J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC For all Q (L L g jW J M ) 1 ( ], computeS L g jJ J Q ,, S L g jJ J Q ,* and *Q TCg j, by g j L L L g j g j g j S L g jhC C C J C M V K R J J Q ) ( 2 ,. ; , ) 1 ( ) 1 ( , , ; , 1 ,* S S L L S L g j L L L L g j S L g j S L g j g j S L g j g j S L g jW J W J J J Q if W J W J M J J Q if J J Q M J J Q if M J J Q ) 1 ( ) ( 2 ) ( ,* * * L L S L g j g j g j S L g j g j S L g j g j g j S L g j g j S L g j g jC J J J Q R C M J J Q V J J Q R h C M J J Q V K J J Q R Q TC Find u = argmin g j j j j i Q TCi ...., 1 , 1 ) (*, and set ) (* S LJ J Q = ) (* S L uJ J Qand ) (* S LJ J Q TC = ) (*Q TCu. End End Optimal *,S LJ J = argmin 1 ...., 2 1 ...., 1 0 ) (*n n J W R J J J Q TCS L L S L, * *,S LJ J Q Q and ) ( ) (* * S LJ J Q TC Q TC Step 4 If L LW R t J ) 1 (* and1* n JS, go to Step 5, Else go to Step 6. Step 5 Increment t by 1 and go to Step 3. Step 6 STOP

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43 Chapter 5 Numerical Study 5.1 Introduction This chapter deals with the numerical an alysis of the algorithms developed in Chapters 3 and 4. MATLAB programming of th e algorithms is perf ormed to show the effect of demand, unit material cost, orde ring cost, and truck capacity on the ordering quantity and total logistics cost This chapter analyses seve ral random instances in which the values of K R WL, and C varies for each instances and Q*, TC ( Q*), JL and JS are computed for all the instances, c onsidering the following three cases when no quantity discounts are offered when all-unit quantity discounts are offered when incremental quantity discounts are offered If the quantity discounts are not offered, the unit material cost C is arbitrarily assumed. Whenever quantity discounts are offe red, the unit price of the material depends upon the break point quantities. In all the in stances, we consider four break points M1, M2, M3, and M4 following a continuous discount quantity schedule. In the continuous discount quantity schedule the break points are assumed to be continuous just to make it simple i.e. if the first break point starts at M0 and ends at M1 then the next break point starts at M1 + 1, and ends at M2 and so on. The corresponding unit ma terial cost for any ordering quantity in the range [ M0, M1], ( M1, M2], ( M2, M3], ( M3, M4], and ( M4, ] is C0, C1, C2, C3, and C4, respectively. Values of C0, C1, C2, C3, and C4 are chosen as percentages of C as

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44 shown in Table 5.1. We consid er four scenarios of disco unt percentage. In all these scenarios the difference in the discount percen tage at each break point quantity is 1%, 2%, 3% or 4%. Table 5.1 Quantity Discount Structure Break-point Quantities % Discount 0 Q 400 0%0%0% 0% 400 < Q 800 1%2%3% 4% 800 < Q 1200 2%4%6% 8% 1200 < Q 1600 3%6%9% 12% 1600 < Q 4%8%12%16% 5.2 Optimal Solution by MATLAB MATLAB programming of Algorithm A, B, C, D, E and F are done. All the instances considering three different cases are executed in the program and the results obtained are discussed in the following sections. 5.2.1 Impact of Discount Percentage and Annual Demand on the Optimal Ordering Quantity In this section, we will analyze the impact of di scount percentage and annual demand on the optimal ordering quantity. We consider three demands: 4000, 8000, and 12000 units/year. In addition, we consider K = 500, h = 25%, WL = 800, WS = 600, CL = 820, CS = 700, and C = 20. Table 5.2 shows the op timal solutions obtained from MATLAB, along the actual to tal unit cost, computed by TC ( Q* )/ R for all theses cases: no quantity discounts, all-unit quantity discounts and incr emental quantity discounts.

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45 I-QD 2400 82907 3 0 20.72 6.4 % 4000 158800 5 0 19.85 9.1 % 4800 233630 6 0 19.47 10.1 % 4% A-QD 2200 76984 2 1 19.25 13.0 % 2400 149310 3 0 18.66 14.5 % 2400 221440 3 0 18.45 14.8 % I-QD 2400 84913 3 0 21.23 4.1 % 3200 163590 4 0 20.45 6.3 % 4000 241300 5 0 20.11 7.2 % 3% A-QD 2200 80404 2 1 20.10 9.2 % 2400 155950 3 0 19.49 10.7 % 2400 231280 3 0 19.27 11.0 % I-QD 2400 86920 3 0 21.73 1.8 % 2400 168120 3 0 21.01 3.8 % 3200 248535 4 0 20.71 4.4 % 2% A-QD 2200 83824 2 1 20.96 5.4% 2400 162590 3 0 20.32 6.9 % 2400 241120 3 0 20.09 7.3 % I-QD 800 88190 1 0 22.05 0.5 % 2400 171990 3 0 21.50 1.5 % 2400 255060 3 0 21.25 2.0% 1% A-QD 1400 86766 1 1 21.69 2.1 % 2200 169210 2 1 21.15 3.2 % 2400 250960 3 0 20.91 3.5 % Discount % 0% N-QD 800 88600 1 0 22.15 0 % 1600 174700 2 0 21.84 0 % 1600 260050 2 0 21.67 0 % Q* TC ( Q *) JL JS TC ( Q *) / R % decrease Q* TC ( Q *) JL JS TC ( Q *) / R % decrease Q* TC ( Q *) JL JS TC ( Q *) / R % decrease Table 5.2 Impact of R and Discount % on Ordering Quantity R 4000 8000 12000 N-QD = No Quantity Discounts; A-QD = All-unit Quantity Discounts; I-QD = Incremental Quantity Discounts; % decrease = % decrease in TC ( Q *) with respect to TC ( Q *) at 0% discount rate.

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46 (1) No quantity discounts. As annual demand increases from 4,000 to 8,000, optimal ordering quantity also doubles. However, Q remains the same as R increases from 8,000 to 12,000. From Ta ble 5.2, one can see that TC ( Q *)/ R decreases as R increases, this is because the unit ordering cost a nd the unit holding cost decreases and the unit material cost and unit trans portation cost remains the same as R increases. (2) All-unit quantity discounts. We can see fr om Table 5.2 that at the discount rate of 1%, the optimal ordering quantity increases by 800 units and 200 units, when R increases from 4000 to 8000 and from 8000 to 12000, respectively. For the discount rate of 2%, 3% and 4%, the optimal orderi ng quantity increa ses by 200 units as R increases from 4000 to 8000 and remains the same as R increases from 8000 to 12000. From Table 5.2, one can see that for R = 4000, 8000 and 12000 the tota l annual logistics cost decreases by more than 2%, 3% and 3.5%, re spectively, for every 1% increase in discount rate. Figure 5.1 depicts Q for all discount rates and R considered. One can see that for R =12000, Q* increases as discount rate changes fr om 0 to 1%, but does not change as discount rate furthe r increases. For both R = 8000 and R = 4000, Q increases as the discount rate increases up to 2%, and remain s the same as the discount rate further increases.

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47 Figure 5.1 Impact of R on Q* for All-unit Quantity Discounts Figure 5.2 depicts TC ( Q *)/ R for all discount rates and R considered. The unit total cost decreases almost linearly as discount rate increases, for all R studied. This indicates that, the impact of all-unit quantity discount on the total actual cost per unit decreases as demand increases.

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48 Figure 5.2 TC ( Q* )/ R vs. % Discount for All-unit Quantity Discounts (3) Incremental quantity discounts. We can see from Table 5.2 th at at the discount rate of 1%, the optimal ordering quantity increases by 1600 units as R increases from 4000 to 8000. However, the optimal orderi ng quantity does not increase as R increases from 8000 to 12000. For the discount rate of 2%, th e optimal ordering quantity remains the same as R increases from 4000 to 8000, but the optimal ordering quantity increases by 800 units as R increases from 8000 to 12000. When a di scount rate of 3% is offered, the optimal ordering quantity increases by 800 units as R increases from 4000 to 8000 and from 8000 to 12000. For a discount rate of 4% the optimal ordering quantity increases by 1600 units and 800 units, when R increases from 4000 to 8 000 and from 8000 to 12000, respectively. From Table 5. 2, one can see that for R = 4000, the total annual logistics cost

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49 decreases by more than 0.5% for ev ery 1% increase in discount rate. For R = 8000, the total annual logistics cost decreases by more than 1.5% for every 1% increase in the discount rate. For R = 12000, the total annual logistics cost decreases by more than 2% for every 1% increase in the discount rate. Figure 5.3 depicts Q for all discount rates and R considered. One can see that when R = 12000, Q increases linearly as discount rates increases. We can also see that Q increases linearly as the discount percenta ge increases beyond 2% for R = 8000. For R = 4000, Q increases linearly until discount rate of 2% and does not change as the discount rate increases further. Figure 5.3 Impact of R on Q* for Incrementa l Quantity Discounts

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50 Figure 5.4 depicts TC ( Q *)/ R for all discount rates and R considered. The unit total cost decreases almost linearly as discount rate increases, for all R studied. This indicates that, the impact of incremental quantity disc ount on the total actual cost per unit decreases as demand increases. Figure 5.4 TC ( Q* )/ R vs. % Discount for Incremental Quantity Discounts 5.2.2 Impact of K on the Ordering Quantity In this section we will analyze the impact of K on the optimal ordering quantity. Let the values of h WL, WS, CL, CS, and C be the same as given in Section 5.2.1, consider R = 8,000 and K = 300, 500, and 700. Table 5.3 pres ents the computational results.

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51 I-QD 4000 158400 5 0 19.80 8.4 % 4000 158800 5 0 19.85 9.0 % 4000 159200 5 0 19.90 9.3 % 4% A-QD 2200 148620 2 1 18.57 14.2 % 2400 149310 3 0 18.66 14.5 % 2400 149970 3 0 18.74 14.7 % I-QD 3200 163090 4 0 20.38 5.7 % 3200 163590 4 0 20.45 6.3 % 3200 164090 4 0 20.51 6.5 % 3% A-QD 2200 155240 2 1 19.40 10.4 % 2400 155950 3 0 19.49 10.7 % 2400 156610 3 0 19.57 10.9 % I-QD 2400 167450 3 0 20.93 3.2 % 2400 168120 3 0 21.01 3.7 % 3200 168710 4 0 21.08 3.9 % 2% A-QD 2200 161860 2 1 20.23 6.5 % 2400 162590 3 0 20.32 6.9 % 2400 163250 3 0 20.40 7.1 % I-QD 2400 171330 3 0 21.42 1.1 % 2400 171990 3 0 21.49 1.5 % 2400 172660 3 0 21.58 1.7 % 1% A-QD 2200 168480 2 1 21.06 2.7 % 2200 169210 2 1 21.15 3.1 % 2400 169890 3 0 21.23 3.3 % Discount % 0% N-QD 800 173200 1 0 21.65 0 % 1600 174700 2 0 21.83 0 % 1600 175700 2 0 21.96 0 % Q* TC ( Q *) JL JS TC ( Q *) / R % decrease Q* TC ( Q *) JL JS TC ( Q *) / R % decrease Q* TC ( Q *) JL JS TC ( Q *) / R % decrease Table 5.3 Impact of K on the Ordering Quantity K 300 500 700 N-QD = No Quantity Discounts; A-QD = All-unit Quantity Discounts; I-QD = Incremental Quantity Discounts; % decrease = % decrease in TC ( Q *) with respect to TC ( Q *) at 0% discount rate.

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52 (1) No quantity discounts. As the ordering cost increases from 300 to 500, the optimal ordering quantity almost doubles. However Q remains the same as K increases from 500 to 700. TC ( Q *) increases by 0.8% and 0.6% as K increases from 300 to 500 and 500 to 700, respectively. This increase in TC ( Q *) is due to the increase in the ordering cost. The total annual logistics cost and the to tal unit cost increases as the ordering cost increases. (2) All-unit quantity discounts. We can see from Table 5.3 that at the discount rate of 1%, the optimal ordering quantity remains the same as K increases from 300 to 500. However, the optimal ordering quantity increases by 200 units as K increases from 500 to 700. For the discount rate of 2%, 3% and 4% the optimal ordering quantity increases by 200 units as K increases from 300 to 500 and remains the same as K increases from 500 to 700. From Table 5.3, one can see that for K = 300, the total annual lo gistics cost decreases by 2.7% as the discount rate increases from 0% to 1%. As the di scount rate increases beyond 1% the total annual logi stics decreases by almost 4% for every 1% increase in discount rate. For K = 500 and 700, as the discount rate increases from 0% to 1% the total annual logistics decreases by 3.1% and 3.3%, respectively. For both K = 500 and 700 as the discount rate increases be yond 1% the total annual logistic s cost decreases steadily by 3.8% for every 1% increase in the discount rate. Figure 5.5 depicts Q for all discount rates and K considered. One can see that for K =500, Q* increases almost linearly as discount rate increases up to 2%, but does not change as discount rate fu rther increases. For both K = 300 and K = 700, Q increases as the discount rate increases up to 1%, and rema ins the same as the discount rate further increases.

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53 Figure 5.5 Impact of K on Q for All-unit Qu antity Discounts Figure 5.6 depicts TC ( Q *)/ R for all discount rates and K considered. The unit total cost decreases almost linearly as discount rate increases, for all K studied. This indicates that, the impact of all-unit qua ntity discount on the total actual cost per unit increases as ordering cost increases.

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54 Figure 5.6 TC ( Q* )/ R vs. % Discount for All-unit Quantity Discounts (3) Incremental quantity discounts. We can see from Table 5.3 th at at the discount rate of 1%, 3% and 4%, the optimal orde ring quantity remains the same for all K studied. For the discount rate of 2%, the optimal ordering quantity remains the same as K increases from 4000 to 8000, but the optimal ordering quantity increases by 800 units as K increases from 8000 to 12000. From Table 5.3, one can see that for K = 300, the total annual logistics cost decreases by 1.1% as the disc ount rate increases from 0% to 1%. As the discount rate increases beyond 1% the total an nual logistics decreases by more than 2% for every 1% increase in discount rate. For K = 500, as the discount rate increases from 0% to 1% the total annual logistics decrea ses by 1.5%. As the discount rate increases beyond 1% the total annual logi stics cost decreases by more than 2.0% for every 1% increase in the discount rate. For K = 700, as the discount rate increases from 0% to 1%

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55 the total annual logistics decr eases by 1.7%. As the discoun t rate increases beyond 1% the total annual logistics cost decreases by more than 2.0% for every 1% increase in the discount rate. Figures 5.7 depicts Q* for all discount rates and K considered. One can see that for K = 300 and 500, Q increases as the discount rate increases from 0% to 1%. The optimal ordering quantity remains the same as the di scount rate increases from 1% to 2% and increases linearly as the disc ount rate increases beyond 2%. For K = 700, the optimal ordering quantity increases linearly as the discount rate increases up to 2%. Q remains the same as the discount rate increases from 2% to 3% and the optim al ordering quantity increases for further increase in discount rate. Figure 5.7 Impact of K on Q for Incremental Qu antity Discounts

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56 Figure 5.8 depicts TC ( Q *)/ R for all discount rates and K considered. The unit total cost decreases almost linearly as discount rate increases, for all K studied. This indicates that, the impact of all-unit qua ntity discount on the total actual cost per unit increases as ordering cost increases Figure 5.8 TC ( Q* )/ R vs. % Discount for Incremental Quantity Discounts 5.2.3 Impact of WL on the Optimal Ordering Quantity and Number of Trucks In this section we will analyze the impact of WS / WL on the optimal ordering quantity, for a given WS. Let the values of h WS, CL, CS, and C be the same as given in Section 5.2.1. Consider R =8000, discount rate of 1%, and WS / WL as given in Table 5.4. Note that the ratio of WS to WL cannot be greater than 1 because of the assumption WS
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57 Table 5.4 Ratio of Capacity of Sma ll Truck to Capacity of Large Truck WS / WL 0.65 0.70 0.75 0.80 0.85 WL 923 857 800 750 706 Table 5.5 shows the computational resu lts for all three cases: no quantity discounts, all-unit quantity discounts and incremental quantity discounts. Table 5.5 Impact of WL on the Total Annual Logistics Cost WS / WL 0.65 0.70 0.75 0.80 0.85 Q* 923 1714 1600 1500 1306 TC ( Q *) 173750 174270 174700 175160 175640 JL* 1 2 2 2 1 JS* 0 0 0 0 1 No Quantity Discounts TC ( Q *) / R 21.72 21.78 21.84 21.89 21.95 Q* 1846 1714 2200 2100 2012 TC ( Q *) 167300 167700 169210 169460 169720 JL* 2 2 2 2 2 JS* 0 0 1 1 1 All-Unit Quantity Discounts TC ( Q *) / R 20.91 20.96 21.151 21.18 21.21 Q* 1846 1714 2400 2250 2118 TC ( Q *) 170870 171540 171990 172740 172990 JL* 2 2 3 3 3 JS* 0 0 0 0 0 Incremental Quantity Discounts TC ( Q *) / R 21.36 21.44 21.50 21.59 21.62 (1) No quantity discounts. As WS / WL increases from 0.65 to 0.70, the optimal ordering quantity increases by 791 units. The optimal ordering quantit y decreases by 114 units as WS / WL increases from 0.70 to 0.75. As WS / WL further increases the optimal ordering quantity decreases. Th is decrease in the optimal ordering quantity is due to the decrease in the capacity of the large truck as WS / WL increases. From Table 5.5, one can

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58 also see that the to tal annual logistics cost increases as WS / WL increases. JL increases from 1 to 2 as WS / WL increases from 0.65 to 0.70. This is increase in JL is due to the increase in the optimal ordering quantity. For WS / WL = 0.70, 0.75, and 0.80 the optimum number of large trucks required to sh ip the quantity remains the same. As WS / WL increases from 0.80 to 0.85, JL decreases by 1. This is due to the decrease in the optimal ordering quantity. JS remains the same as for all values of WS / WL, considered, except for WS / WL = 0.85. JS increases from 0 to 1 as WS / WL increases from 0.80 to 0.85. This increase in JS is due to reduction in capacity of the large truck. (2) All-unit quantity discounts. We can see that from Table 5.5 that the optimal ordering quantity decr eases by 132 units as WS / WL increases from 0.65 to 0.70. When WS / WL increases from 0.70 to 0.75 the optimal ordering quantity increases by 486. As the WS / WL increases from 0.75 to 0.80 and 0.80 to 0.85, the optimal ordering quantity decreases by 100 units and 88 units, respectively From Table 5.5, one can also see that the total annual logistics cost increases as WS / WL increases. This increase in total annual logistics cost is due to the decrease in the capacity of the large truck. JS remains the same for WS / WL = 0.65 and 0.70, however, JS increases by 1 as WS / WL increases from 0.70 to 0.75. In this case, reduction in capacity of the large trucks forces the increase in JS *. The unit total cost increases by less than 1% for every 0.05 increase in the ratio of WS to WL. (3) Incremental quantity discounts. From Ta ble 5.5, one can see that the optimal ordering quantity decr eases by 132 units as WS / WL increases from 0.65 to 0.70. As WS / WL increases from 0.70 to 0. 75 the optimal ordering quan tity increases by 686 units. As WS / WL increases from 0.75 to 0.80 and 0.80 to 0.85, the optimal ordering quantity increases by 750 units and 250 units, respectivel y. The number of large trucks required

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59 increases from 2 to 3 as WS / WL increases from 0.70 to 0.75. The number of large trucks required increases due to the decrease in capacity of the large truck as WS / WL increases. Figure 5.9 depicts the Q* for all WS / WL considered. From Figure 5.9, one can see that as WS / WL increases beyond 0.75, Q decreases almost linearly as WS / WL increases. For WS / WL = 0.70, the optimum ordering quantity re mains the same for all the three cases considered. Figure 5.9 WS / WL vs. Q 5.2.4 Impact of C on the Optimal Ordering Quantity In this section we will analyze the impact of C on the optimal ordering quantity. Let the values of h WL, WS, CL, CS, and K be the same as given in Section 5.2.1, consider R = 8000 and C = 15, 20, and 25. Table 5.6 pres ents the computational results.

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60 I-QD 4000 121400 5 0 15.17 9.2 % 4000 158800 5 0 19.85 9.0 % 4000 196200 5 0 24.52 9.0 % 4% A-QD 2400 114450 3 0 14.31 14.4 % 2400 149310 3 0 18.66 14.5 % 2200 184100 2 1 23.01 14.6 % I-QD 3200 125055 4 0 15.63 6.4 % 3200 163590 4 0 20.45 6.3 % 3200 202125 4 0 25.26 6.3 % 3% A-QD 2400 119430 3 0 14.93 10.7 % 2400 155950 3 0 19.49 10.7 % 2200 192380 2 1 24.05 10.8 % I-QD 3200 128520 4 0 16.06 3.8 % 2400 168120 3 0 21.01 3.7 % 2400 207680 3 0 25.96 3.6 % 2% A-QD 2400 124410 3 0 15.55 7.0 % 2400 162590 3 0 20.32 6.9 % 2200 200650 2 1 25.08 7.0 % I-QD 2400 131460 3 0 16.43 1.6 % 2400 171990 3 0 21.50 1.5 % 2400 212525 3 0 26.55 1.4 % 1% A-QD 2400 129390 3 0 16.17 3.2 % 2200 169210 2 1 21.15 3.1 % 2200 208930 2 1 26.12 3.1 % Discount % 0% N-QD 1600 133700 2 0 16.71 0 % 1600 174700 2 0 21.83 0 % 800 215700 1 0 26.96 0 % Q* TC ( Q *) JL JS TC ( Q *) / R % decrease Q* TC ( Q *) JL JS TC ( Q *) / R % decrease Q* TC ( Q *) JL JS TC ( Q *) / R % decrease Table 5.6 Impact of C on the Ordering Quantity C = 15 C = 20 C = 25 N-QD = No Quantity Discounts; A-QD = All-unit Quantity Discounts; I-QD = Incremental Quantity Discounts; % decrease = % decrease in TC ( Q *) with respect to TC ( Q *) at 0% discount rate.

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61 (1) No quantity discounts. As unit material cost increases from 20 to 25, optimal ordering quantity decreases by 50%. However, Q remains the same as C increases from 15 to 20. The total annual logistic s cost increases by 23% and 19% as C increases from 15 to 20 and 20 to 25, respectively. This increase in total annual logistics cost is due to the increase in material cost and inventory holding cost. (2) All-unit quantity discounts. We can see fr om Table 5.6 that at the discount rate of 1%, the optimal ordering quan tity increases by 200 units when C increase from 15 to 20. However, Q remains the same as C increases from 20 to 25. For the discount rate of 2%, 3% and 4%, the optimal ordering quantity decreases by 200 units as C increases from 20 to 25 and remains the same as C increases from 15 to 20. From Table 5.6, one can see that for C = 15, 20 and 25 the total annual logistics cost decreases by more than 3.2%, 3.1% and 3.2%, respectively, for every 1% increase in discount rate. Figure 5.10 depicts Q for all discount rates and C considered. One can see that for C = 15 and 25, Q* remains the same as discount rate increases beyond 1%. For C =20 the value of Q* increases linearly until the discount percen tage of 2% and remains the same as the discount percentage further increases.

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62 Figure 5.10 Impact of C on Q* for All-unit Quantity Discounts (3) Incremental quantity discounts. We can see from Table 5.6 th at at the discount rate of 1%, 3% and 4%, the optimal orde ring quantity remains the same for all C studied. For the discount rate of 2%, the optimal ordering quantity remains the same as C increases from 20 to 25, but the optimal orde ring quantity remains the same as C increases from 15 to 20. From Table 5.6, one can see that for C = 15, the total annual logistics cost decreases by 1.6% as the discount rate increases from 0% to 1%. As the di scount rate increases beyond 1%, the total annual logistics decreases by more than 2% for every 1% increase in discount rate. For C = 20, as the discount rate increases from 0% to 1% the total annual logistics decreases by 1.5%. As the discount rate increas es beyond 1% th e total annual logistics cost decreases by more than by 2.2% for every 1% increase in the discount rate. For C = 25, as the discount rate increases fr om 0% to 1% the total annual logistics

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63 decreases by 1.4%. As the discount rate in creases beyond 1% the total annual logistics cost decreases by more than 2.0% for every 1% increase in the discount rate. Figures 5.11 depicts Q* for all discount rates and C considered. One can see that for C = 15, Q increases linearly as the discount rate increases from 0% to 2%. The optimal ordering quantity remains the same as the discount rate increas es from 2% to 3%. For C = 20 and 25, the optimal orde ring quantity increases as th e discount rate increases from 0% to 1%. Q remains the same as the discount ra te increases from 1% to 2%. We can also see that, for C = 20 and 25, Q increases linearly as the discount rate increases beyond 2%. Figure 5.11 Impact of C on Q* for Incremental Quantity Discounts

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64 Chapter 6 Conclusions and Future Directions 6.1 Conclusions The main objective of this thesis was to develop algorithms for finding the optimal ordering quantity that minimizes total annual logistics cost, when the suppliers offer No quantity discounts All-unit quantity discounts Incremental quantity discounts The total annual logistics cost considered in this research includes ordering cost, material cost, inventor y holding cost and transp ortation cost. We have considered a fixed ordering cost, the unit price of an item will depend upon the ordering quantity and quantity discounts, the inventor y holding cost is charged ba sed on the average inventory of the system and the transportation co st depends upon the orde ring quantity of each order. This research considers the following tr ansportation scenario. There are two truck sizes: large and small. A larg e truck has a capacity of WL and charges a fixed price of CL, regardless of actual quantity loaded. Similarly, a small truck has a capacity of WS and charges a fixed price of CS, regardless of actual load (not exceeding its capacity). Depending upon the ordering quantity Q it is necessary to use a combination of JL large trucks and JS small trucks, for some JL0 and JS0. It is assumed that S S L LW C W C (i.e., if

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65 both large and small trucks are fully loaded, the unit shipping cost for a large truck is smaller than that for a small truck). MATLAB programming of the algorithm is done. Numerical analysis of various factors that affect the ordering quantity and the total cost ar e analyzed in Chapter 5. The factors that are considered in the numerical analysis are the annual demand, ordering cost, unit price and capacity of the truck. Discount rates of 1%, 2%, 3% and 4% are also considered in determining the impact of quantity on discounts on the ordering quantity. 6.2 Summary of Contributions Developed an optimal ordering quant ity algorithm that considers only truckload transportation for shipments. Extended the optimal ordering quantit y algorithm for all-unit quantity discounts and incremental quantity discounts. 6.3 Future Directions This thesis has presented an inventor y system assuming the demand to be a constant. It would be interesting to formul ate an algorithm assuming the annual demand to be stochastic. The algorithm presented in this research considers only two trucks sizes for transportation. It would also be interesting to formulate an algorithm when there are 3 trucks sizes namely, large, medium and small are available. Even though quantity disc ounts play a vital role in todays buyer-shipper relationship, there are other f actors like speed of delivery, service, and quality should be considered before purchasing an item. By demanding a larger discount, for example, a retailer may have to agree to accept a longer lead time from th e supplier. Future research

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66 could examine these interactions more closely and explore the role and power of quantity discounts as a bargaining chip in the overall buyer-supplier negotiation process.

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67 References [1] Aucamp, D. C., Nonlinear freigh t costs in the EOQ problem, European Journal of Operational Research Vol. 9, No. 1, 1982, pp. 61-63. [2] Abad, P. L., Determining optimal selli ng price and lot size when the supplier offers all-unit quantity discounts, Decision Sciences Vol. 19, No. 3, 1988, pp. 622-634. [3] Abad, P. L., Joint price and lot-size determination when supplier offers incremental quantity discounts, Journal of Operational Research Society Vol. 39, No. 6, 1988, pp. 603-607. [4] Abdelwahab, W. M. and M. A. Sarg ious, A simultaneous decision making approach to model the demand for freight transportation, Canadian Journal of Civil Engineering Vol. 18, No. 3, 1990, pp. 515-520. [5] Abdelwahab, W. M. and M. A. Sargious Freight rate structure and optimal shipment size in freight transportation, Logistics and Transportation Review Vol. 26, No. 3, 1992, pp. 271-292. [6] Buffa, E. S. and J. G. Miller, Production-Inventory Systems: Planning, and Control, 3rd Edition, Richard D. Irwin, Inc., Homewood, Ill., 1979. [7] Baumol, W. J. and H. D. Vinod, An inventory theoretic model of freight transportation demand, Management Science Vol. 16, No. 7, 1970, pp. 413-421. [8] Benton, W. C. and S. Park, A classificati on of literature on determining the lot size under quantity discounts, European Journal of Operational Research Vol. 92, No. 2, 1996, pp. 219-238. [9] Chopra, S. and P. Meindl, Supply Chain Management: Strategy, Planning, and Operation, Prentice-Hall, Upper Sa ddle River, N.J., 2001. [10] Das, C., A unified appro ach to the price-break econom ic order quantity problem, Decision Sciences Vol. 15, 1984, pp. 350-358. [11] Harris, F. W., How Many Parts to Make at Once, Operations Research Vol. 38, No. 6, 1990, pp. 947-950. (Reprint from the factory, The Magazine of Management Vol.10, No.2, 1913, pp.135-136).

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68 [12] Hadley, G. and T. M. Whitin, A family of inventory models, Management Science Vol. 7, No. 4, 1961, pp. 351-371. [13] Hadley, G., and T. M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. [14] Hwang, H., Moon, D. H. and S. W. Sh inn, An EOQ Model with Quantity Discounts for Both Purchasing Price and Freight Cost, Computers & Operations Research Vol. 17, No. 1, 1990, pp. 73-78. [15] Hall, R. W., Dependence between sh ipment size and mode in freight transportation, Transportation Science Vol. 19, No. 4, 1985, pp.435-444. [16] Iwaniec, K., An inventory m odel with full load ordering, Management Science Vol. 25, No. 4, 1979, pp. 374-384. [17] Jones, N. and Natural, E., LTL Trucking Terms & Process Guide, Prepared for Pork Niche Market Working Group, 2004. [18] Lippman, S. A., Optimal policy with subadditive ordering costs and stochastic demand, SIAM Journal on Applied Mathematics Vol. 17, No. 3, 1969, pp.543559. [19] Lippman, S. A., Optimal inventory policy with multiple set-up cost, Management Science Vol. 16, No. 1, 1969, pp. 118-138. [20] Lippman, S. A., Economic order qua ntities and multiple set-up cost, Management Science Vol. 18, No. 1, 1971, pp. 39-47. [21] Lee, C-Y., The Economic Order Quan tity for Freight Discount Costs, IIE Transactions Vol. 18, No. 3, 1986, pp. 318-320. [22] Lancaster, K. J., A new appr oach to consumer theory, Journal of Political Economy 1966, pp.74. [23] Larson, P. D., The economic transportation quantity, Transportation Journal Vol. 28, No. 2, 1988, pp. 43-48. [24] Long, D., International Logistics, Global Supply Chain Management, Kulwer Academic Publishers, 2003. [25] Morse, P.M., Solutions of a class of discrete-time inventory problems, Operations Research Vol. 7, No.1, 1958, pp.67-68. [26] McClain, J. O. and L. J. Thomas, Operations Management, 2nd Edition, PrenticeHall, Inc., Englewood Cliffs, N.J., 1985.

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69 [27] Mendoza, A. and J. A. Ventura, An I nventory Model with Two Transportation Modes and Quantity Discounts, IIE Annual Conference and Exhibition, 2004. [28] Monahan, J. P., A quantity discount pricing model to increase vendor profit, Management Science Vol. 30, No. 6, 1984, pp.720-726. [29] Munson, C. L. and M. J. Rosenblatt, The ories and Realities of Quantity Discounts: an Exploratory Study, Production and Operations Management Vol. 7, No. 4, 1998, pp. 352-369. [30] Nagarajan, A., Mitchell, W., and E. Canessa, E-Commerce and the Changing Terms of Competition: A view from within the sectors, BRIE UC Berkeley, 2003. [31] Quandt, R. E. and W. J. Baumol, The demand for abstract transport modes: Theory and Measurement, Journal of Regional Science Vol. 6, 1966, [32] Rieksts, B. Q. and J. A. Ventura, Op timal inventory policies with two modes of freight transportation, Working paper Department of Indust rial and Manufacturing Engineering, The Pennsylvania State Univ ersity, University Park, P. A., 2003. [33] Silver, E. A. and R. Peterson, Decision Systems for Inventory Management and Production Planning, 2nd Edition, John Wiley & sons, N.Y., 1985. [34] Vassian, H. J., Application of discrete variable theory to inventory control, Operations Research Vol. 3, 1955, pp. 272-282. [35] Veinott Jr., A. F., The optimal i nventory policy for batch ordering, Operations Research Vol. 13, No. 3, 1961, pp. 424-432. [36] Wehrman, J. C., Evaluating tota l cost of a purchase decision, Production and Inventory Management Vol. 25, No. 4, 1984, pp. 86-91.


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ABSTRACT: Transportation plays a vital role in the movement of raw materials and finished goods from one place to another. Trucks play a vital role in the movement of materials and are indispensable part of almost every shipment, both domestic and international. On the average, thirty-nine percent of the total logistics cost is spent on transportation. Therefore reducing the transportation cost may significantly reduce the total logistics cost. The total annual logistics cost considered in this research includes ordering cost, material cost, transportation cost and inventory holding cost. The main objective of this research is to develop algorithms for finding the optimal ordering quantity that minimizes total annual logistics cost, when the suppliers offer -No quantity discounts -All-unit quantity discounts -Incremental quantity discounts This research considers truckload transportation where two truck sizes are available.The algorithm developed in this research will identify the optimum ordering quantity and the optimum number of trucks required to ship the ordering quantity. MATLAB programming of the algorithm will analyze the factors that affect that the total annual logistics cost.
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