Dynamic Inventory Lot Size - Assignment Example

DYNAMIC INVENTORY LOT SIZE List of Figures Figure 1: EOQ model graph[Chr09] 6 Figure 2: Classification of inventory model [Ull10]. 12 TABLE OF CONTENTS List of Figures 1 TABLE OF CONTENTS 2 CHAPTER 1: INTRODUCTION 1 1.1. Background Information 1 CHAPTER 2: LITERATURE REVIEW 4 2.1. Introduction 4 2.2. Traditional Inventory models 4 2.2.1. Traditional Economic Order Quantity (EOQ) model 4 2.2.2. Economic Production Quantity (EPQ) model 7 2.3. Dynamic lot sizing problem (DLS) 8 2.3.1. Varying inventory variables 9 2.4. Number of item/product involved 11 2.4.1. Single product 13 2.4.2. Multi-product/item 14 2.5. Quantity discount model 14 2.6. Model Solution Technics 14 2.6.1. Heuristic / Mathematical Algorithms Solutions 14 2.6.2. Non-Linear Optimization 16 2.6.3. New Lagrangian Heuristic Solution 16 CHAPTER 3: CONCLUSION 18 REFERENCES 20 CHAPTER 1: INTRODUCTION 1.1. Background Information Inventory are commodities, materials that are carried in stocks for consumption [Bal09]. Inventory plays a significant role in meeting anticipated customer demand hence are referred to anticipated stocks [Ste02]. Further, inventory aids in smoothing production requirements besides decoupling operations. The primary usefulness of inventory is in the manufacture, wholesale and retail sectors. Consequently, the need for extensive research on inventory is paramount to improve efficiency and performance regarding cost. Nonetheless, the research if done well would save a lot of money for development. Lot size, on the other hand, refers to terms of decisions that pertain to inventory levels, the cost of setup or ordering, capacity requirement and availability [Yil82]. Such decisions are impacted by the cost of setup, the cost of carrying an item inventory. There are many reasons why lot sizing is needed. These reasons are either business or production specific. However, in stock, lot sizing is necessary majorly in the supply chain to control with the sole role to match supply and demand. Inventory theory, on the other hand, handles management of goods with the aim to monitor satisfaction of demand for commodities [HWa]. The element factors in inventory models are quantity and replenishment criteria. The success of the models depends on the complexity of the assumptions on demand, cost and the general characteristics of the system. The real problems on inventory control involve multiple products [HWa]. Notwithstanding, problems involving single products lead to many possible models since the assumptions made on the key variables are enormous. Furthermore, inventory models are distinguished based on the assumptions on various aspect of time and logistic on the model. There are different types of lot sizing techniques: Fixed Order Quantity (FOQ); Economic Order Quantity (EOQ); Lot-for-Lot (LFL); Periods of Supply (POS); Period Order Quantity (POQ); Least Unit Cost (LUC); Least Total Cost (LTC) and Part Period Balancing (PPB) [Had89]. In FOQ, a given number of items may be ordered whenever an order is put on a particular item, and the amount may be arbitrary or EOQ [Had89]. EOQ is where an order volume that governs whether the quantity of an item to be purchased is fixed with the goal to minimize the joint cost of the order and obtaining inventory. EOQ can be represented as where is the annual demand for the item, is the order quantity, is the cost of order preparation or setup cost, is the inventory carrying cost per unit per year. LFL also known as discrete order quantity is where the planned order is generated in quantities equal to the net requirements in each period[Yil82]. In this case, there is no extra available inventory. This is often used mostly in perishable food items where market fluctuations are widely. POS is where lot size is equivalent to net requirements for specific quantity of future time [Yil82]. In POQ, EOQ is used to calculate a fixed number of periods required for each order. LUC is dynamic lot sizing technique. This method sums ordering cost and inventory cost for each lot trial size, then divide by the number of units in the lot size and the lot size with the lowest unit cost number is selected. LTC is also a dynamic lot sizing technique used to order quantity by calculation of carrying cost and ordering cost for various lot size and the costs with an almost equal lot is selected. PPB is where the variation of LTC with back and forward presumptions. The major calculation involves determination of economic part period (EPP), where EPP can be calculated by The requirement periods is added until the period approximate the EPP, where Part period is the quantity of units of inventory held for specific time [Bac13]. Dynamic lot size in literature is the generalization of the economic order quantity inventory theory where the models consider the demand for the product as a function of time [Lee01]. Dynamic lot sizing tends to unravel glitches related to dynamic demand for a particular item over a finite horizon against space: facility, warehouse or retail [Lee01]. In the next Chapter, an illustration of a comprehensive review of literature is done. The Chapter is split in into three main sections: Traditional Inventory model which include the Economic Order Quantity (EOQ) model and Economic production quantity model EPQ; the Dynamic lot sizing problem (DLS); showing how varying model parameters in dynamic nature can affect modeling and solution; and finally, the next chapter also illustrate the different design solution techniques that are used in literature to solve inventory model with different natures to achieve a solution for the lot sizing problem. CHAPTER 2: LITERATURE REVIEW 2.1. Introduction This Chapter discusses the various Inventory models and their solution techniques. The major areas covered are the Traditional Economic Order Quantity (EOQ) Model and Economic Production Quantity (EPQ) Model: -The Dynamic Lot Sizing Model (DLS) while only the demand is varied and when other inventory variables are varied. Finally the different solution techniques for inventory models such as Heuristic/ Mathematical Algorithm, Non-linear Optimization, and the New Lagrangian Heuristic solution. 2.2. Traditional Inventory models This section illustrates the formulation and assumptions of the different traditional inventory models available in the literature which includes the Economic Order Quantity model (EOQ) and the Economic Productions Quantity (EPQ) model. The models are illustrated as follows: 2.2.1. Traditional Economic Order Quantity (EOQ) model The economic order quantity (EOQ) model discovered in 1913 by Hari's as the foundation of inventory models. Many pieces of literature inside the inventory management have been keen on the development of lot sizing models that are overboard the limiting assumptions of the EOQ model. For instance, the EOQ assumption of perfect quality is not realistic in extreme industrial applications hence might lead to wrong results during the determination of order sizes and related inventory control [Har13]. EOQ can be defined in simple terms as that optimal quantity of orders where total variable costs are minimized to order and hold inventory [Vuj96]. EOQ model considers the tradeoff between storage and ordering cost to select a quantity to use in replenishing item inventories. Ordering frequency is determined by the amount of the ordering cost, for instance, a larger order quantity shall reduce the ordering frequency which shall also reduce the ordering cost [Sch08]. The major decisions that make managers want to use this model are when they are faced with a decision on how much or how many of their production do they have to make or buy to meet the level of satisfaction of an item [Goh94]. The decision is often made while neglecting the cost functionality of the problem. However, this model uses the following assumptions [Kha11]: There is uniformity, constant or continuity over time of demand; the lead-time is often constant; the order size is unlimited; the cost of placement of an order independent of size of order, and whenever a given amount is ordered, the same quantity ordered is stored. EOQ model has found its use regularly in many organization whenever they are faced with decisions on how much inventory they have to order each month. Therefore, EOQ becomes vital when making decisions on the amount of inventory one should order. EOQ has three types of cost [Tri12]: unit cost: in this case, the cost of units is assumed to be fixed and does not depend on the units ordered, this is referred to no Quantity discount. Inventory holding cost, which is the cost of holding a unit in inventory. The holding cost in its broad term can be said to contain the following: opportunity cost; deterioration costs; the physical space that provides the rent, insurance, taxes or utility cost and cost of handling the items. Furthermore, ordering cost is all the expenses that are associated with creation or processing of an order to a supplier. This cost may include purchase requisition cost, purchase order cost, the cost of inspection of the goods, invoice supply cost, the cost of storing the goods by the supplier and payment value. Moreover, shortage cost is the cost associated with having a shortage arising from the inability to meet demand for the stock. Nonetheless, fixed-order quantity can be used by management to determine an optimal order quantity besides minimizing the sum of the annual costs of holding inventory and ordering inventory. It is often known as an administrative cost. Therefore the total cost curve will have to superimpose the order cost and the holding cost, as presented in the graph below. Figure 1: EOQ model graph[Chr09] The main aim of EOQ model is to identify a fixed order size that will result in the minimum annual cost of holding inventory and ordering inventory [Ste02]. Worth noting is that the unit purchase price is not included in the total cost since the unit price is unaffected by the order size unless quantity discount is a factor. If the holding cost is represented as a percentage of the unit cost, then the unit cost is included indirectly as the total cost of the holding cost. Receipt of order, units is the inception of the inventory cycle, and is assumed to be drawn at a constant rate over time. is submitted to the supplier whenever the quantity at disposal is equivalent to the demand during the lead time. The optimal order quantity indicates the equilibrium of carrying cost and ordering costs. The average inventory on hand is half the order quantity since the amount of unit in hand is decreasing from Thus the average inventory in hand can be numerically represented as if H is the average annual carrying cost per unit, the total carrying cost can then be given by . If the annual demand is given by then the number of orders per year will be given by Therefore, the annual ordering cost can be represented as where is ordering cost per order. Since the number or orders per year decreases as ordering quantity increases and annual ordering cost is inversely related to order size, hence can be represented as where are in the same units, months, year. Using calculus, the optimal order quantity, , can be represented as . 2.2.2. Economic Production Quantity (EPQ) model EPQ model is where the quantity of a company or retailer is determined to minimize the total inventory costs by balancing the inventory holding cost and setup cost (items will be produced internally) [Cár02]. Economic production quantity model states in its assumptions that item produced is superior, and the production rate is constant with the production quality depending on the state of the process [Tsa12]. Therefore, imperfect quality items may lead to deterioration of the modeling process as a result of shifting of the production process. The shifting of the process can be reduced by reworking on the factors thereby reducing production-inventory cost. The major assumption of EPQ model is that the output of production is of perfect quality and that the production rate is determined in advance and is fixed [Tsa12]. This is not always the case since in reality perfection is far from being achieved due to the occurrence of errors from human and other mistakes. Therefore, generation of imperfect quality is inevitable [Tsa12]. Production firms allow for reworking and repairing of imperfect items hence reducing the production-inventory costs. In summary, the assumption for EPQ is similar with the difference being that the production or delivery is not instantaneous. The production capacity is also finite with limited production rate If the production rate is given by and the production cycle time is , withdrawal cycle, , total inventory cycle, maximum inventory, , average inventory and the number of orders per unit time, . Then the cost of production associated with EPQ is given by following mathematical formulation [Ste02]: If the total holding cost , total ordering/ setup cost , total production or purchase cost , total cost and Unit cost . Then the EPQ will be given by differentiation of the total cost function with respect to quantity, Q, hence the differentiation can be represented as , whose solution yields which can be written as , therefore, if holding cost 2.3. Dynamic lot sizing problem (DLS) DLS is an extension of the EOQ model that considers product demand continuously over time. Furthermore, DLS is a form of planning model known for its finite time horizon and discrete time scale [Par13]. Initially, at its inception, dynamic lot sizing was meant for one item. The most important issue in lot sizing is supplier selection due to its probable difference between purchasing costs. In DLS model, when some quantity discount is considered and that the order made by the buyer is huge, the supplier will reduce the purchase price per unit by the schedule [Par13]. Consequently, suppliers often proposals some price breaks which ultimately reduces the unit purchase price as the order quantity increases. Therefore, dynamic lot sizing is affected by the inclusion of quantity discount and selection of the supplier. Dynamic lot sizing models can be grouped according to variation in inventory variables. The subsection that follows discusses how change in various inventories can affect DLS. 2.3.1. Varying inventory variables Time-varying demand is where only demand is changing in the DLS model. However, the major assumption in dynamic lot sizing is that the level of demand satisfaction within a given period is known. The demand cannot be delivered within specified time if backlogging is allowed [Lee011]. DLS model with variation in demand has been used in reuse and remanufacture models [Bak14]. In Baki, et al., (2014) paper, the major focus was the fluctuation in demand for an item. In that article, though some factors are said to vary, demand variation was stated as the central aspect of the research. In their formulation of the models, the major difficulty was the selection of the items to manufactured and remanufactured. This ought not to be the problem since the remanufactured items should be selected from the items with defects. The writers noted that the problem posed is hard to solve due to lack of no known algorithm that can find the optimal solution in polynomial time. Heuristic solution technic was hence employed to solve the problem Generally, DLS has been considered as the representation of the model with the only variation in demand. As stated earlier in the introduction, DLS was initially considered as a one item model. The model major solution method was regarded as a heuristic. When quantity discount was considered in the model, the solution was proposed to be presentable in Fordyce–Webster Algorithm (FWA). Contrary to Wagner and Whitin (1958) optimal solution, the solution presented by Fordyce–Webster Algorithm (FWA) was non-optimal. The solution presentation by matrices in the algorithm is easiest to explain than any other heuristic [Lee13]. The major elements in DLS are the supplier selection that arises from the difference in purchasing cost resulting from different suppliers if they were selected [Lee13]. Further, the supplier unit price is dependent on the buyer’s demand and quantity discount, since as the consumers demand increases, the supplier reduces the purchase price by specified schedule. Moreover, price break may be offered by the supplier as per unit purchase price as the order quantity increases [Lee13]. According to Parsopoulos, et al., (2015) inventory management and control is a fundamental aspect of the product returns. In this study (study by Parsopoulos, et al., (2015)), Dynamic or Economic Lot Sizing (ELS) is the most researched topic since many countries have strict laws on environmental management thereby need for remanufacturing. Many variations of ELS has been studied under the remanufacturing problem. For instance, the classical Wagner-Whitin model has been extended to introduce a remanufacturing process. This study showed that the existence of the optimal solution is a zero-inventory policy [Ric00]. According to Gicquel, et al., (2014) when production are run using large lot sizes to reduce changeover costs, the changeover cost does not depend on the number of products processed. This has been noted to be less effective since the total cost increases as inventory holding cost is generated [Gic141]. Though their experiment on the same proved through a computation that the varying other variables other than the demand lead to significance changes in the model, further work is required. However, in this work, the others choice of computation (model solution technique) though efficient, would have been solved quickly by another method. In general, varying other inventory variables has been discussed in depth by many writers indirectly. For instance EOQ and Economic Lot Scheduling Problem (ELSP) are considered as an infinite time horizon continuous time scale and with demand regarded as constant [Maz15]. 2.4. Number of item/product involved There are many ways of classifying inventory models. However, in terms of production planning, models can be divided into nine categories as seen in Figure 2. The sections that follow will cover these two areas to (or “intending to”) distinction and understanding the models. Figure 2: Classification of inventory model [Ull10]. The subsections that follow aims at discussing single and multiple products as a classification of the inventory models. 2.4.1. Single product In a single product lot size, a single product is produced to satisfy the demand. In the single product lot size, the excess products are stored for the satisfaction of the demand for the next demand periods. Consequently, the Production costs and holding costs are piecewise linear functions [Lut01]. If negative inventory levels are allowed with positive holding cost, the backlogging is permitted. If the production device runs in the present period as opposed to previous, then additional cost is expected to be introduced. Therefore, to prevent start-up costs, the production equipment has to be kept running in periods with no production [Lut01]. However, in single product lot size, the additional fix production costs is often missing during production. In a study by Sbihi & Eglese, (2007) where a single product was considered for finite time horizon, the dynamic lot-sizing problem was discussed for a single facility. In the study, the manufacturer was assumed for an unlimited quantity of the product with fixed ordering cost. Shortages at the retailer were allowed by shortages. The decision on quantification of the order at the facility that will satisfy all the demand at minimum cost. The study by Ganas & Papachristos, (2005) considers a single-product lot-sizing problem over a finite planning horizon. In this study, the demand for each period was constant. However, the excess was completely backlogged. Notwithstanding, holding and backlogging costs were proportionate to the quantity of inventory backlogged. At the same time, ordering cost was fixed and independent of the quantity ordered. In this model, the primary objective was to minimize the total relevant costs over the planning horizon. The study resulted in an algorithm that is easier to implement and hence was considered suitable for practical use. 2.4.2. Multi-product/item According to Sultana Parveen & Ahsan, (2009) the use of multi-item in DLS modeling has increased complexity in the formulation of inventory problem models. Furthermore, the solution of the problems often becomes too complicated to solve whenever the assumption of multi-item is removed. However, its definition is simple as outlined by Gicquel, et al., (2008) in the multi-level lot sizing problem, the production planning is not only considered for the end products, but also for the components and sub-assemblies that are needed to make the end products. 2.5. Quantity discount model By definition, a quantity discount model is a form of EOQ model that considers quantity discounts. In their basic forms, they are price reductions designed to induce large orders [Ada16]. Quantity discount model argues that whenever discounts are accessible, the buyer must weigh the potential gains of the low purchase price and fewer orders alongside the rise in carrying costs caused by greater average inventories. Hence, the customer’s objective, in this case, is to choose the order quantity that will lessen total costs [Glo14]. 2.6. Model Solution Technics 2.6.1. Heuristic / Mathematical Algorithms Solutions Heuristic solutions are based on Fordyce-Webster Algorithm (FWA). The model can be used to solve a single-supplier case as used by Mazdeh, et al., (2015). In this case, the heuristic method was used to solve the multi-supplier problem with the aid of a third dimension being added to the matrices to facilitate the FWA [For84]. In their methodology on the use of the solution method, Fordyce, and Webster (1984) noted the two primary advantages of this approach as a clear demonstration of computational algorithm steps and the dynamic programming is done easily without in-depth calculations. Furthermore, the solution exhibited optimality if the discount is not considered. Moreover, the method adopted in this case was most suitable for the use of the illustrative matrix is easier to explain and understand. In this model, the method entails the use of several matrices with each demonstrating specific cost. This might be confusing for non-experts or starters. Mazdeh, et al., (2015) proposed a model known as 3DFWA with the third dimension added to the known FWA heuristic solution method to take care of the supplier selection problem into consideration. This process was regarded as a fast algorithm since its run time was far less than any other algorithm runtime. Furthermore, Mazdeh, et al., (2015) concluded that such solutions, those obtained within low computational times, are of use in upper bounds for complex problems. This method, though better with regards to time, might be very complicated for simple problems. The methodology for doing this was not also clearly outline in the text. Paper presented by Baki, et al., (2014) stated that the problem proposed by the manufacture and remanufacture difficult to solve due to lack on known algorithm. Consequently, a heuristic method to solve the problem was paramount. A heuristic that exploits and mimics the structure was proposed. To solve the proposed problem, efficient heuristic making use of dynamic programming and Wagner-Whithin algorithm was used. The result yielded high-quality solutions which could be embedded with other programs to speed up the optimization process [Bak14]. 2.6.2. Non-Linear Optimization In the study done by Gicquel, et al., (2014), where the optimization problem was formulated quadratically constrained binary problem. The method of non-linear optimization was employed to solve the problem. The results showed the solution quality was within the acceptable standards when compared to other solution techniques. This prompted the use of this method as accurate. Non-linear optimization solution might be used to solve simple problems which might entail multi-product multi-period inventory. This is because the problem possesses a greater challenge to solve to optimality [Ane15]. The performance of this solution method was noted to provide superior results as compared to other methods. Furthermore, the experimental results in the research gave similar answers as those generated by process. In another study by Almeder, et al., (2015), other methods such as Multi-level Capacitated Lotsizing Problem (MLCLSP) are infeasible or excessive work that has to be accomplished to attain the desired results. Almeder, et al., (2015) suggested further research on the solution implying that the solution method is limited. 2.6.3. New Lagrangian Heuristic Solution New Lagrangian Heuristic (NLR) solution depends on the decomposition of the algorithm of the primal problem. The primal problem is subdivided into a lot-sizing and a set of single- machine [Xia15]. The solution attains to be improved using annealing heuristic implying the longer solution method which is not suitable for non-experts. Further, Xiao, et al., (2015) noted that the utilization of this approach is greedy heuristic. This is visible in the many numerous iteration steps in the solution. This is a longer action which is very tiresome hence leads to poor results if the programmer is not careful. A study by Eisenhut, (1975) where mixed-integer linear programs for solving economic lot sizes for manufactured products showed that when both a Lagrangian relaxation based heuristic and a commercial mixed-integer linear programming solver are used, the learning capabilities for Lagrangian relaxation was update step size in the subgradient algorithm. The results of the computation show that the Lagrangian heuristic outperforms the solver on different formulations, in particular for large problems with long time horizons. This study revealed that Lagrangian heuristic solution is superior to most model solution techniques. CHAPTER 3: CONCLUSION In this work, inventory aid in smoothing and decoupling of production and operations respectively. Inventory levels decisions are guided by lot sizes. Inventory theory deals with the management of goods to (or “intending to”) monitor satisfaction of demand. It is also noted that various inventory model success depends on the complexity of the assumptions of the models used. Different types of lot sizing techniques were discussed. Of the lot size methods discussed, EOQ was discussed ineptly in this work and is noted to be critical in lot sizing. Dynamic lot size was seen as a generalization of EOQ in inventory theory. EOQ was noted to have been discovered in 1913 and is the foundation of inventory models. EOQ models are a tradeoff between storage and ordering cost since managers regard EOQ in their decision making. Thus, EOQ has found its use more often in organizations. EOQ model minimizes fixed order sizes that result to minimum annual cost for the decision-making process. EPQ on the other hand, minimizes the total inventory by balancing the inventory cost and setup cost. Inventory variables can be varied by some activities: varying demand where only demand is changing. Inventory variables with varying demand only are majorly used in remanufacturing of defective items. In terms of classification, inventory can be classified in terms of some items involved. For instance, in a single product, the items are produced for the satisfaction of demand. The demand is met by either direct production or storage of excess to meet the demand for subsequent periods. Multiple products only increase complexity in DLS modeling formulation of problem models. Other models can be classified in terms of quantity discount where the discount is considered for some quantities and hence are regarded as price reducers. The major solution techniques for models are heuristic which involves mathematical algorithms and are based on FWA. This solution technique is common in solving single-supplier cases. Besides, the techniques are used where the problem is difficult to solve like in cases where remanufacture is considered. Non-Linear Optimization technique though not commonly used, is occasionally used in solving simple but multiple product multiple period inventories since such problems pose a greater challenge to solve to optimality. The performance of the solution technique is said to be superior to other techniques. NLR technique is algorithm dependent, and its utilization is greedy heuristic. When compared to other techniques, it is superior. 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