Developing economic production quantity models for items that exhibit delay in deterioration with reliability consideration

Thesis Overview

<p> GENERAL INTRODUCTION<br>1.0. INTRODUCTION<br>Inventory consists of materials, commodities, products, etc, which are usually carried in stock in<br>order to be consumed or benefited from when needed. An Economic Order Quantity (EOQ)<br>model sometimes referred to as Economic Order Lot-size model is an inventory control model,<br>which determines the optimal quantity to be ordered so as to meet a deterministic demand over a<br>planned period of time in order to minimize cost. An Economic Production Quantity (EPQ) or<br>Economic Production Lot-size model is an inventory control model which determines the<br>optimal quantity to be produced so as to meet a deterministic demand with the objective of<br>minimizing cost. Thus, EPQ model is an offshoot of the well known EOQ model.<br>In several articles in the literature on inventory models (focusing on EOQ and EPQ), it is<br>assumed that items can be stored for a long period of time for future use without spoilage.<br>However, it is a general knowledge that almost all items on inventory deteriorate over time.<br>Deterioration can be referred to as depression in quality/quantity of items kept on inventory for<br>certain purpose. An item on inventory becomes reliable, if it satisfies the probability that it will<br>adequately perform its specified purpose, for a specified period of time, under specified<br>environmental conditions. Thus reliability is influenced by the decisions made during the design<br>and manufacturing of the product.<br>2<br>In this thesis, we study some EPQ models of deteriorating items which exhibit delay in<br>deterioration with reliability consideration. These are items which do not start deteriorating<br>immediately they are stored, until later. Such items include potatoes, yam, bread, cakes, to name<br>a few. Two modifications to existing models are presented; the first modification assumes a<br>constant demand both before and after deterioration begins, while the second assumes a linearly<br>time dependent demand after deterioration commences.<br>1.1 COMPONENTS OF INVENTORY MODELS<br>The profit of a production is affected by the policies on inventory; as such the choice among<br>inventory policies depends upon their relative profitability. Profitability is determined by the<br>following factors: the costs of ordering or production set-up costs (in case of production),<br>shortage costs, holding costs, salvage costs, revenues and discount rates.<br>• Cost of Ordering: This is the cost of placing an order to an outside supplier or releasing a<br>production order to a manufacturing shop.<br>• Set-up cost: This is the cost incurred in preparing a machine or process for manufacturing<br>an order. It includes the design cost, location of machinery, employee hiring, research<br>and development expenses, and labor cost for cleaning and changing tools or holders.<br>• Shortage Cost: Shortage cost (sometimes called the unsatisfied demand cost) is incurred<br>when the amount of the commodity required (demand) exceeds the available stock.<br>3<br>• Holding Cost: Holding cost (sometimes called the storage cost) represents all the costs of<br>capital tied up, space, insurance, protection, and taxes attributed to storage. The holding<br>cost can be assessed either continuously or on a period-by-period basis.<br>• Salvage Value: Salvage value of an item is the value of a leftover item when no further<br>inventory is desired. The salvage value represents the disposal value of the item to the<br>firm, perhaps through a discounted sale. The negative of the salvage value is called the<br>salvage cost. If there is a cost associated with the disposal of an item, the salvage cost<br>may be positive.<br>• Revenue: Revenue may or may not be included in the model. If both the price and the<br>demand for the product are established by the market and so are outside the control of the<br>company, the revenue from sales (assuming demand is met) is independent of the firm’s<br>inventory policy and may be neglected. However, if revenue is not neglected in the<br>model, the loss in revenue must then be included in the shortage cost whenever the firm<br>cannot meet the demand and the sale is lost.<br>• Discount Rate: discount rate takes into account the time value of money. When a firm<br>ties up capital in inventory, the firm is prevented from using this money for alternative<br>purposes.<br>1.2 A GENERALIZED INVENTORY MODEL<br>The ultimate objective of an inventory model is to answer two questions.<br>1. How much to order/produce?<br>2. When to order/produce?<br>4<br>The answer to the first question (how much to order/produce) is expressed in terms of what we<br>call the order/production quantity and the second question (when-to-order/produce) is the<br>inventory level at which a new order/product should be placed/produced usually expressed in<br>terms of re-order point.<br>According to Hadley and Whitin (1963), one can summarize the total cost of a general inventory<br>model as a function of its principal components in the following manner:<br>Total inventory cost = purchasing cost + setup cost (or ordering cost) + holding cost<br>+ shortage cost (if shortages are allowed)<br>1.3 TYPES OF INVENTORY MODELS<br>Basically, all inventory models (EOQ/EPQ) are classified into two categories:<br>ï‚· Deterministic model and<br>ï‚· Stochastic model<br>1.3.1 DETERMINISTIC DEMAND<br>This is a situation where by the demand rate is known with certainty. It can be further classified<br>into uniform (constant) demand and time-dependent demand. The time-dependent demand is also<br>classified into discrete time dependent demand and continuous time dependent demand. The<br>time-dependent demand may be:<br>o linearly increasing given by (t)  a  bt ; a  0, b  0 ,<br>o linearly decreasing given by (t)  a  bt ; a  0, b  0 ,<br>5<br>o it may be exponentially increasing given by (t)  aebt ; a  0, b  0 , or<br>o exponentially decreasing with ( ) ; 0, 0  t  aebt a  b  and so on.<br>1.3.2 STOCHASTIC DEMAND<br>This is a situation where by the demand rate follows a statistical distribution which may be a<br>known probability distribution or an arbitrary probability distribution.<br>Inventory models can also be classified based on their current mode of supervision: periodic<br>review and continuous review. In periodic review, the level of the inventory is to be checked at<br>discrete intervals, e.g., at an interval of one month, and decisions on ordering are to be made only<br>at these times (an interval of one month) even if the inventory level is below the reorder point<br>between the current and preceding review times. In continuous review, placement of an order is<br>done as soon as the stock level falls down to the prescribed reorder point. (Hillier and<br>Lieberman, 2001)<br>1.3.3 DETERMINISTIC CONTINUOUS-REVIEW MODEL<br>Manufacturers/producers, retailers as well as wholesalers face a common inventory scenario; that<br>items in stock are exhausted/drained over time and they are refilled/replaced by the arrival of<br>new manufactured items. An inventory control model describing such situation in a production<br>environment is the economic production quantity (EPQ) model, which is sometimes referred to<br>as the production lot-size model.<br>In the EPQ model, the demand rate is constant. The inventory is replenished when required by<br>producing a batch of fixed size (Q units), where all Q units are produced at the desired time. For<br>6<br>this case of basic economic production quantity (EPQ) model, the following costs are<br>considered:<br>K = set-up cost per production run<br>c= unit cost of the item<br>h= inventory carrying cost in a production cycle.<br>The main objective is either to minimize the total inventory cost per unit time or to maximize the<br>profit. (Paknejad et al., 1995)<br>1.3.3.1 THE BASIC EPQ MODEL<br>The basic EPQ model is an offshoot of the well known Economic Order Quantity (EOQ) model<br>and the two (EPQ and EOQ) have similar assumptions. The EPQ as an inventory control model<br>is usually based on the following assumptions:<br>i. Constant deterministic demand rate per unit time.<br>ii. If inventory level drops to 0, then the production quantity (Q) to replenish inventory is<br>produced all at once.<br>iii. Planned shortages are not permitted.<br>The total cost of production per unit time T is computed from the following components.<br>Production cost per cycle =K+cQ.<br>The average inventory level during a cycle is Q/2 units, and the corresponding cost is hQ/2 per<br>unit time. Since the cycle length is Q/a,<br>7<br>Holding cost per cycle<br>2<br>2<br>hQ<br>a<br><br>Then,<br>Total cost per cycle<br>2<br>2<br>K cQ hQ<br>a<br>  <br>And so, the total cost per unit time T is<br>2 2 <br>2<br>K cQ hQ a aK hQ T ac<br>Q a Q<br> <br>   <br>The value of Q, say Q*, that minimizes T is found by setting the first derivative of T with respect<br>to Q to zero (provided that the second derivative is positive).<br>i.e. 2 0<br>2<br>dT aK h<br>dQ Q<br>   <br>giving Q* 2aK<br>h<br> (1.1)<br>which is the well-known EPQ formula. The corresponding cycle time, say t*, is<br>t*<br>Q* 2K<br>a ah<br>  (1.2)<br>(Nahmias, 2009)<br>1.3.4 DETERMINISTIC PERIODIC-REVIEW MODEL<br>The assumptions in the basic EPQ model are not always realistic. This is why several authors<br>modified the model over time to reflect several realistic scenarios. When the assumption of<br>constant demand is relaxed for instance i.e. when the amounts that need to be withdrawn from<br>8<br>inventory are allowed to vary from period to period, the EPQ formula no longer ensures a<br>minimum-cost solution, for all cycles.<br>Suppose planning is to be done for the next n periods regarding how much (if any) to produce to<br>replenish inventory at the beginning of each of the periods. The demands for the respective<br>periods are known (but not the same in every period) and are denoted by<br>ô€Žô€¯œ = demand in period ô€…, for ô€… = 1,2,3,…, ô€Š<br>The EPQ in this case is given by<br>* 2 i<br>i<br>Q rK<br>h<br> for ô€… = 1,2,3,…, ô€Š (1.3)<br>and<br>*<br>* 2 i<br>i<br>i i<br>t Q K<br>r rh<br>  for ô€… = 1,2,3,…, ô€Š (1.4)<br>(Hadley and Whitin, 1963; Hillier and Lieberman, 2001)<br>1.3.5 A STOCHASTIC CONTINUOUS-REVIEW MODEL<br>In a stochastic continuous-review inventory system for a particular item, there are two factors to<br>be considered, namely:<br>R = reorder point,<br>Q = order quantity.<br>9<br>For a retailer or wholesaler (or a manufacturer replenishing its raw materials inventory from a<br>supplier), the purchase order for Q units of the product is the order quantity. On the other hand,<br>for a manufacturer managing its finished products on inventory, the production run of size Q is<br>the order quantity.<br>Inventory policy based on these factors (R and Q) is as follows: an order for Q more units is to be<br>placed to replenish the inventory, if the inventory level of the product drops to R units. Such a<br>policy is sometimes called reorder point, order quantity policy or (R, Q) policy. [Consequently,<br>the overall model might be referred to as the (R, Q) model. Other modifications such as (Q, R)<br>model, (Q, R) policy, and so on, are also used.] (Paknejad et al., 1995)<br>1.3.5.1 CHOOSING THE ORDER QUANTITY Q<br>The approach used in formulating Q for stochastic continuous-review model is as follows:<br>Total cost=setup cost + purchase cost + holding cost +shortage cost<br>2  2<br>2 2<br>hR p Q R T K cQ<br>a a<br><br>    (1.5)<br>where p is the shortage.<br>Total cost per unit time T(Q,R) is given as<br>   2 2<br>,<br>2 2<br>aK hR p Q R T Q R ac<br>Q Q Q<br><br>    (1.6)<br>taking the partial derivative of (1.6) with respect to Q and R and set the result to zero, we have<br>10<br>R* 2aK p and Q* 2aK p h<br>h p h h p<br><br> <br><br>(1.7)<br>(Ra’afat, 1991)<br>1.3.6 STOCHASTIC PERIODIC-REVIEW MODELS<br>This is a situation when we assume that the demand is uncertain. However, in contrast to the<br>continuous-review inventory system, we now assume that the system is only being monitored<br>periodically. At the end of each period, when the current inventory level is determined, a<br>decision is made on how much to order (if any) to replenish inventory for the next period. Each<br>of these decisions takes into account the planning for multiple periods into the future.<br>1.4 ORDER POINT AND SAFETY STOCK<br>The economic production quantity model indicates how many units to produce. Practitioners are<br>also concerned with the order point. This quantity reflects the level of inventory that triggers the<br>start of set up for additional units.<br>Determination of the order point is based on three factors:<br>ï‚· usage (quantity of inventory used or sold each day),<br>ï‚· lead time (is the time it takes from the start of set up to when the goods are produced),<br>and<br>ï‚· safety stock (The quantity of inventory kept on hand by a company in the event of<br>fluctuating usage or unusual delays in lead time).<br>11<br>Order point=(usage per unit time lead ï‚´ time)+safety stock<br>If usage per unit time is entirely constant and lead time is known with certainty, the order point is<br>equal to usage per unit time multiplied by lead time:<br>Order point=usage per unit timeï‚´lead time<br>i.e. there is no need for safety stock. Note that in the EPQ case, lead time is zero and safety stock<br>is also zero, so the addition of these two in the basic EPQ model is one of the ways in which the<br>model is modified to reflect some realistic situations. (Ra’afat, 1991)<br>1.5 THE FINITE PRODUCTION RATE MODELS WITH DETERIORATION<br>Misra (1975) developed the first production lot size model in which both a constant and variable<br>rate of deterioration were considered and obtained approximate expressions for the production<br>lot size with no backlogging. For the case of Weibull distribution deterioration, no closed<br>expression for the lot size and the average total cost was possible. However, for the case of<br>exponential distribution (i.e. constant deterioration rate), through a series of approximations,<br>Misra (1975) calculated the optimal production lot size to be<br> <br> <br>0.5<br>3<br>1<br>C<br>Q = 1 + Q ,<br>Cp E<br>d<br>   <br>  <br>  <br>, with<br> <br> <br>0.5<br>2<br>1<br>2C<br>Q<br>C p d E<br> dp    <br>   <br>,<br>where<br>E Q is the production lot size for items without decay,<br>12<br>p is the constant production rate,<br> is the constant rate of decay,<br>d is the constant demand rate and<br>1 2 3 C , C , and C are inventory carrying cost, ordering cost and deteriorating cost respectively.<br>Shah and Jaiswal (1976) derived results similar to those of Misra (1975) for a constant<br>deterioration rate and extended the model to include backlogging. By assuming the average<br>carrying inventory to be approximately one-half the maximum level of inventory and using the<br>same notation with Misra, they obtained the following expression for the production lot size as a<br>function of inventory cycle time: Q p ln l d exp T l ,<br>p<br><br><br>                     <br>where T is the<br>inventory cycle time.<br>1.6 THE INVENTORY MODELS WITH DELAY IN DETERIORATION<br>In the EOQ model with constant rate of deterioration, many authors assume that deterioration of<br>the items start from the instance of their arrival in stock. As a matter of fact, many items (for<br>example, firsthand vegetables, fruits, and some items produced in industry like bread, cakes, etc)<br>have a span of maintaining fresh quality or original condition. During that period, there is no<br>deterioration occurring, but after sometime, deterioration begins. Thus it is important to consider<br>inventory problems for non-instantaneous deteriorating items. Ouyang et al. (2006) developed an<br>EOQ model for non-instantaneous deteriorating items with permissible delay in payments and<br>where the demand before deterioration starts is the same as that after deterioration begins. Musa<br>13<br>and Sani (2012) developed an EOQ inventory model for delayed deteriorating items under<br>permissible delay in payments but where the demand before deterioration starts is different from<br>that after deterioration starts. Thus, this paper is a generalization of Ouyang et al. (2006).<br>Similarly, in the context of EPQ model, many authors assume that deterioration start<br>immediately after production, but this is not the usual situation. Some items (for example; bread,<br>cakes, etc) have a span of maintaining their original condition. Hence, it is important to consider<br>inventory problems of delayed deteriorating items. Sugapriya and Jeyaraman (2008a) developed<br>a model to determine a common production cycle time for an economic production quantity<br>model of non-instantaneous deteriorating items allowing price discount and permissible delay in<br>payments. Sugapriya and Jeyaraman (2008b) also developed an EPQ model for noninstantaneous<br>deteriorating items in which production and demand rate are constant, holding cost<br>varies with time, completely deteriorated units are discarded, partially deteriorated items are sold<br>with some discount and no shortage is allowed. Baraya and Sani (2011) developed an EPQ<br>model for delayed deteriorating items with stock-dependent demand rate and linear time<br>dependent holding cost.<br>1.7 JUSTIFICATION FOR THE RESEARCH<br>The economic production quantity (EPQ) model has been widely used in practice because of its<br>simplicity. However, there are some drawbacks in the assumptions of the original EPQ model<br>and many authors have tried to improve it with different assumptions. The assumption of the<br>unconstrained production period length is one of these shortcomings. The classical EPQ model<br>assumes that production period length is unconstrained. However, in real production<br>14<br>environment, this assumption is not always tenable because, it can often be observed that the<br>production period length is constrained due to some technical services reasons. Hence, the<br>inventory policy determined by the conventional model would be inappropriate.<br>Reliability of an item is the probability that it will adequately perform its specified purpose for a<br>specified period of time under specified environmental conditions. Product reliability is<br>influenced by the decisions made during the design and manufacturing of the product. This<br>implies that reliability can be viewed as a link to integrate the different stages of manufacturing –<br>design, engineering, production, marketing, and post sale service – in an effective manner. As<br>such reliability is very important in the context of new products. Recently research articles,<br>emphasize the growing importance of this subject to both consumer and producer (Cheng, 1991).<br>Objective determination of reliability costs will help manufacturers plan operations more<br>effectively since an accurate knowledge of reliability costs allows more accurate profit<br>expectations which may, in turn, lead to some marketing advantages (Cheng, 1991).<br>In case of demand of an item, it is natural that the higher the price of an item the lower the<br>demand and the lower the price of an item the higher the demand of such item, i.e. the unit cost<br>of an item is inversely related to the demand of the product. In general the unit cost of production<br>is directly proportional to the reliability of the product and inversely related to the demand of the<br>product.<br>1.8 THE PROBLEM STUDIED IN THIS THESIS<br>EPQ model has been widely used for more than four decades as an important tool to control<br>inventory. However, as already indicated, EPQ model did not represent the real world problem in<br>some situations. The analysis for finding an EPQ therefore has several weaknesses. This is why,<br>15<br>many authors had to make extensions or modifications in several aspects of the original EPQ<br>model. The quality assurance (reliability) is one good aspect that could be added to the EPQ<br>model since quality assurance plays an important role in the demand of an item in the market.<br>Quality assurance was incorporated in the models of many authors such as Cheng (1991) and<br>Tripathy et al. (2003).<br>Musa and Sani (2009) developed an EOQ model for items that exhibit delay in deterioration. The<br>non-instantaneous deterioration (delay in deterioration) is a situation where items do not start<br>deteriorating immediately they are stocked. During this period, before deterioration sets in,<br>depletion of inventory is dependent on demand only. As deterioration sets-in depletion is then<br>dependent on both demand and deterioration. The items that exhibit delay in deterioration<br>include farm produce such as fruits, potatoes etc. or even fashion goods such as cars, fabrics etc.<br>In this thesis, we intend to make an extension of Musa and Sani (2009) but in the context of EPQ<br>by assuming the unit cost of production of an item to be directly related to reliability (quality<br>assurances in producing the item) and inversely related to demand rates. This is a reasonable<br>assumption because the higher the reliability of an item the higher the price is in many cases, and<br>the lower the reliability of the item the lower the price is in many cases. Also, the higher the<br>price of an item the lower the demand of such item, and the lower the price of the item the higher<br>the demand of such item in many cases.<br>1.9 LIMITATION<br>The applicability of the study is limited to items with delay in deterioration, where the unit cost<br>of production is directly related to reliability and it is inversely related to quantity demanded.<br>16<br>1.10 RESEARCH METHODOLOGY<br>The approach we use in this study will start with the review of existing literature in both<br>Economic Order Quantity (EOQ) model and Economic Production Quantity (EPQ) model. It will<br>also review literature on constant demand, varying demand, constant deterioration, varying<br>deterioration and reliability consideration in EPQ models. Mathematical modeling will then be<br>used to derive the mathematical relationship for the required models under the stated<br>assumptions after which numerical examples will be used to show the application of the models.<br>Sensitivity analyses will also be conducted to see the effect of changes in some of the<br>parameters.<br>1.11 RESEARCH AIMS AND OBJECTIVES<br>The aim of this research is to develop economic production quantity models for items which<br>exhibit delayed deterioration with quality assurance consideration.<br>The objectives of this research are:<br>ï‚· To investigate the effect of quality assurance (reliability) in the EPQ inventory model of<br>items that exhibit delayed deterioration;<br>ï‚· To develop an EPQ model of items which exhibit delayed deterioration with quality<br>assurance consideration and constant demand;<br>ï‚· To develop an EPQ model of items which exhibit delayed deterioration with quality<br>assurance consideration and linear demand (after deterioration begins).<br>17<br>1.12 OUTLINE OF THE THESIS<br>Chapter one deals with the general introduction of the thesis. It starts with the introduction of<br>inventory control theory, components of inventory, generalized inventory model and basic<br>classification of inventories. The chapter goes ahead to discuss the Economic Production<br>Quantity (EPQ) model and some other models that depend on it. The justification of reliability<br>consideration, problem studied, methodology, objectives of the study and limitations of the study<br>are also stated in the chapter.<br>Chapter two surveys the existing literature on Inventory Management and Control and<br>particularly the deteriorating inventory. It covers various inventory related problems especially<br>the EOQ/EPQ where EOQ/EPQ models are discussed under different mathematical assumptions.<br>These include inventory models for constant deteriorating items and inventory models for noninstantaneous<br>(delayed) deteriorating items, inventory models for deteriorating items with<br>process reliability, inventory models for items with constant demand, inventory models for items<br>with varying demand, inventory models for deteriorating items with varying demand and<br>inventory models with imperfect quality.<br>Chapter three contains development of the mathematical model (EPQ) on items that exhibit<br>delay in deterioration with reliability consideration and constant demand (both before and after<br>deterioration sets-in). It is assumed in the model that the unit cost of production is directly<br>related to reliability of the product and inversely related to the rates of demand. The chapter then<br>gives some numerical examples to show how the model is applied. The chapter also gives a<br>sensitivity analysis of some important parameters.<br>18<br>Chapter four contains development of the EPQ model of items that exhibit delay in deterioration<br>with reliability consideration but where the demand rate after deterioration sets-in is linearly time<br>dependent. Numerical examples are also given to illustrate the application of the model. The<br>chapter also gives a sensitivity analysis of some important parameters.<br>Chapter five gives the summary of the thesis, contribution of the thesis to research on inventory<br>of deteriorating items and it gives a conclusion of the work. Possible areas of interest where the<br>research could be extended are also given and this is in addition to further recommendations<br>given.<br>1.13 DEFINITIONS OF SOME BASIC TERMS: The following are the definitions of some<br>of the technical terms we will frequently use in this thesis.<br> Backlogging: This is the process of holding customer orders to be filled later when they<br>cannot be settled immediately because of stockouts.<br> Backorder: A customer order that cannot be filled when presented, and for which the customer<br>is prepared to wait for some time.<br> Backorder cost: The cost of handling the backorder (special handling, follow-ups etc.) plus<br>whatever loss of goodwill occurs as a result of having to backorder an item.<br> Demand rate: This is also called the usage rate. It is the number of units demanded by<br>customers of production departments per unit of time. The demand may be constant (static)<br>or variable (dynamic).<br>19<br> Instantaneous inventory receipt: This is the inventory that is received or obtained at one point<br>in time and not over a period of time.<br> Inventory Turnover (or stock turn): This is a ratio showing how many times a company’s<br>inventory is sold and replaced over a period.<br> Lead time: This is the time between ordering a replenishment of an item and actually<br>receiving the item into inventory. The lead-time can be either deterministic (constant or<br>variable) or probabilistic.<br> Instantaneous delivery: If the lead time of an item is zero, then we have a special case of<br>instantaneous delivery where there is no need for placing an order in advance. This occurs in<br>many cases in production industries when the production run is so planned that new items<br>produced become available just as old items finish. This is clearly seen in a bakery for<br>instance. In our own study, we consider the lead time to be zero; therefore we have a case of<br>Instantaneous delivery.<br> Inventory cycle: This is made up of the activities of sensing a need for ordering materials,<br>placing an order, lead time for getting the material delivered, receiving the material and using<br>it.<br> Inventory level: This refers to the amount of materials on hand in inventory that is ready for<br>use, i.e. the current amount of a product that a business has in stock.<br> Inventory carrying (holding) cost: This is the cost a business incurs over a certain period of<br>time, to hold and store its inventory.<br> Order quantity: This is the quantity of material produced each time inventory is replenished.<br>20<br> Planned shortages: This is a situation where stock outs are planned.<br> Set-up cost: This is the cost incurred in preparing a machine or processing for manufacturing<br>an order. It includes the design cost, moving of machinery, employee hiring, research and<br>development expenses, and labor cost for cleaning and changing tools or holders.<br> Time horizon: The period over which the inventory level will be controlled is called the time<br>horizon. It can be finite or infinite depending on the nature of demand.<br>21 <br></p>