IAAST ONLINE ISSN 77-1565 PRINT ISSN 0976-488 CODEN: IAASCA International Arcive of Applied Sciences and Tecnology IAAST; Vol 4 [3] September 013: 09-15 01 Society of Education, India [ISO9001: 008 Certified Organization] www.soeagra.com/iaast/iaast.tm ORIGINAL ARTICLE Using Matrix to Solving te Probabilistic Inventory Models (Demand Model) P. R. Sarma and S. Sukla Department of Matematics, University of Rajastan, Jaipur (India) Email Id: sailjpr@yaoo.com ABSTRACT An inventory can be defined as a stock of goods wic is eld for te purpose of future production or sales. An inventory problem exists if te amount of te goods in stock (i.e., inventory) is subject to control and if tere is at least one cost tat decreases as inventory increases. Inventory models are of two types, one is inventory models wit deterministic demand and oter is inventory models wit probabilistic demand. Aim of te paper is to investigate a new approac to solve single period discrete probabilistic inventory models. Tis metod proposes momentous advantages over similar metods. Keywords: Probabilistic Inventory Model, Deterministic Demand Model, Pay-off Matrix. Received 1.08.013 Accepted 08.09.013 Society of Education, India INTRODUCTION Inventories occupy te most strategic position in te structure of working capital of most business enterprises. It constitutes te largest component of current asset in most business enterprises. In te spere of working capital, te efficient control of inventory as passed te most serious problem to te cement mills because about two-tird of te current assets of mills are blocked in inventories. Te turnover of working capital is largely governed by te turnover of inventory. It is terefore quite natural tat inventory wic elps in maximize profit occupies te most significant place among current assets. Meaning and Definition of Inventory:In dictionary meaning of inventory is a detailed list of goods, furniture etc. Many understand te word inventory, as a stock of goods, but te generally accepted meaning of te word goods in te accounting language, is te stock of finised goods only [1,]. In a manufacturing organization, owever, in addition to te stock of finised goods, tere will be stock of partly finised goods, raw materials and stores. Te collective name of tese entire items is Inventory [3,4]. Te term inventory refers to te stockpile of production a firm is offering for sale and te components tat make up te production [5]. Te inventory means aggregate of tose items of tangible personal property wic (i) Are eld for sale in ordinary course of business. (ii) Are in process of production for suc sales. (iii) Tey are to be currently consumed in te production of goods or services to be available for sale. Inventories are expandable pysical articles eld for resale for use in manufacturing a production or for consumption in carrying on business activity suc as mercandise, goods purcased by te business wic are ready for sale. It is te inventory of te trader wo dies not manufacture it. Finised Goods: Goods being manufactured for sale by te business wic are ready for sale. Materials: Articles suc as raw materials, semi-finised goods or finised parts, wic te business plans to incorporate pysically into te finised production. Supplies: Article, wic will be consumed by te business in its operation but will not pysically as tey are a part of te production. Te sort inventory may be defined as te materials, wic are eiter saleable in te market or usable directly or indirectly in te manufacturing process [6,7,8]. It also includes te items wic are ready for making finised goods in some oter process or by comparing tem eiter by te concern itself and/or by outside parties. In oter words, te term inventory means te material aving any one of te following caracteristics. It may be 1. Saleable in te market,. Directly saleable in te manufacturing process of te business, IAAST Vol 4 [3] September 013 9 P a g e Society of Education, India
3. Usable directly in te manufacturing process of te undertaking, and 4. Ready to send to te outside parties for making usable and saleable productions out of it. In te present study raw materials, stores and spare parts, finised goods and work-in-process ave been included inventories. Firm also manufactures inventory to supplies. Supplies included office and plant cleaning materials (soap, brooms etc. oil, fuel, ligt bulbs and te likes). Tese materials do not directly enter into te production process, but are necessary for production process. Inventory constitutes te most significant part of current assets of a large majority of companies in India. For example, on an average inventories are more tan 57 per cent of current assets in public limited companies and about 60.5per cent in government companies in India. Terefore it is absolutely imperative to manage inventories efficiently and effectively in order to avoid unnecessary investment in tem. An undertaking neglecting te management of inventories will be jeopardizing its long run profitability and may fail ultimately. It is possible for a company to reduce its level of inventories to a considerable degree e.g. 10 to 0 per cent witout any adverse effect on production and sales. OBJECTIVES OF INVENTORY MANAGEMENT Te primary objectives of inventory management are: (i) To minimize te possibility of disruption in te production scedule of a firm for want of raw material, stock and spares. (ii) To keep down capital investment in inventories. So it is essential to ave necessary inventories. Excessive inventory is an idle resource of a concern. Te concern sould always avoid tis situation. Te investment in inventories sould be just sufficient in te optimum level. Te major dangers of excessive inventories are: (i) Te unnecessary tie up of te firm s funds and loss of profit. (ii) Excessive carrying cost, and (iii) Te risk of liquidity. Te excessive level of inventories consumes te funds of business, wic cannot be used for any oter purpose and tus involves an opportunity cost. Te carrying cost, suc as te cost of sortage, andling insurance, recording and inspection, are also increased in proportion to te volume of inventories. Tis cost will impair te concern profitability furter. Problems faced by management: (i) To maintain a large size inventories for efficient and smoot production and sales operation. (ii) To maintain only a minimum possible inventory because of inventory olding cost and opportunity cost of funds invested in inventory. (iii) Control investment in inventories and keep it at te optimum level. Inventory management, terefore, sould strike a balance between too muc inventory and too little inventory. Te efficient management and effective control of inventories elp in acieving better operational results and reducing investment in working capital [9-11]. It as a significant influence on te profitability of a concern. MATHEMATICAL FORMULATION AND ANALYSIS Probabilistic Inventory Models: Tese are te models were te demand is probabilistic or random variable. Following eiter a discrete or continuous probability distribution. Tere would be single period model and multi period models. Single period model is discussed in tis paper. Single period discrete probabilistic demand model: Tese models deal wit te inventory situation of te item requiring one time purcase only, suc as news papers, perisable goods, spare parts and seasonal goods. Tese items sould be stocked at te beginning of a given time period. Te demand is unknown, but te probability distribution of demand is given. In tese cases purcases are made only once, i.e. no reordering is possible during te period, if more units are required tan stocked. Tus, te lead time factor and ordering costs are least important in tese models. In single period models, te problem can be studied using marginally (or incremental) or payoff matrix metod. Te decision procedure consists of a sequence of steps. In suc cases, tere are two types of costs involved, namely (a) over-stocking cost and (b) under-stocking cost representing opportunity losses incurred wen te number of units stocked is not exactly equal to te number of units actually demanded. IAAST Vol 4 [3] September 013 10 P a g e Society of Education, India
Te objective is to determine te initial inventory level wic will optimize te expected return, taking into account factors suc as unit cost, carrying or olding cost, selling price, sortage cost and salvages value. Te models are discussed using te following notations: D = number of unite of te item demanded (a random variable) = te number of units of te item stocked C = unit cost price C = te unit carrying cost for te entire period C s = P = S = C = 1 C = te sortage cost te unit selling price te salvage value over-stocking cost or over-ordering cost, (i.e. an opportunity loss associated wit eac unit left unsold) C C V under-stocking cost or under-ordering cost, (i.e. an opportunity loss due to not meeting te demand), P C C / C S Single period probabilistic demand model can be discrete or continuous. Te discrete problem can be solved troug incremental analysis metod and payoff matrix metod, as discussed below Incremental Analysis Metod for Single Period Discrete Probabilistic Inventory Model: Te cost equation may be developed as following: (a) If only D units are consumed, for any quantity in te stock, for te cost associated wit units in stock for te specified period of time, is eiter: ( D) C1, wen D or ( D) C, wen D. (1) (b) Since, te demand D is random variable; its probability distribution is known. If p(d) denotes te probability of demand (D units), suc tat total probability is 1, i.e. p(0) p(1)... p(d)... p(d) 1.. () D0 (c) Te sum of expected under stocking cost and overstocking cost would be te total expected cost, say F(), wic is given by F( ) C ( D) p(d) C ( D ) p(d). 1 D0 D1 (3) (d) If quantity stocked is optimal, tan, te total expected cost F( ) will be minimum. Tus, if one unit more or less tan te optimal quantity is stocked, te total expected cost will be iger tan optimal. For minimum of F ( ), te condition F( 1) 0 F( ) must be satisfied. Te equation (3) can be differenced under te summation sign, for D C [( 1) D]p(D) C [ D ( 1) p(d)]. (4) 1 Tis is obviously satisfied ere. Now taking first difference of equation (3), following is obtained 1 1 D0 D 1 F( ) C [( 1) (D) ( D)]p(D) C [D ( 1) (D )]p(d) F( ) C1 p(d) C p(d) p(d) D0 D 1 D0 IAAST Vol 4 [3] September 013 11 P a g e Society of Education, India
F( ) (C1 C ) p(d) C D0 From equation (4), 1 D0 (Because F( ) 0 (C C ) p(d) C 0 D0 1) (5) C or p(d). (6) (C C ) D0 1 Tus, te optimum value of stock level can be obtained by te relationsip 1 C p(d) p(d). (7) D0 (C1 C) D0 For practical application of equation (7), te tree step procedure is as follows: Step-I. A table sowing p(d), te probability and te cumulative probability p( D ) for eac reasonable value of D is prepared. C Step-II. Te ratio known as service level is computed. (C1 C ) Step-III. Te value of satisfying te equation (7) is obtained. Example-1. A trader stocks woollen sweaters at te beginning of winter and cannot re-order. Te item costs im Rs. 50 eac and e sells at Rs. 100. For eac sweater tat cannot be met on demand, te trader loses goodwill of Rs. 30. Eac unsold piece will ave salvage value of Rs. 0. Holding cost during te period is 10 persent of te item cost. Te probability of demand is as follows: Unit stocked (in undreds): 4 6 8 10 1 Demand probability p( D ) : 0.30 0.0 0.30 0.15 0.05 Compute te optimal number of item to be stocked. If a stock level is to be maintained 10 units, find te values of under stocking cost ( C ). Solution: Metod-I: Te data regarding demand distribution given in problem is tabulated in Table 1: Unit stocked (in undreds) 4 6 8 10 1 Demand probability p(d=) 0.30 0.0 0.30 0.15 0.05 0.30 0.50 0.80 0.95 1.00 Cumulative probability p( D ) Table-1: Probability Distribution of demand of woollen sweaters In te problem, P=100, C=50, C =0.10 50=5.0, C =30, S=0 s Terefore, C 1 =C+ C -S=50+5.0-0=35 C C =P-C- + C 5.0 s =100-50- +30=77.5 C 77.5 Tus, = =0.69. (C1 C ) (35 77.5) Observing te table 1, tis ratio lies between cumulative probability of 0.50 and 0.80 wic in turn, reflect te values of as 6 and 8. Tat is, p( D 6) =0.50<0.69<0.80= p( D 8). Terefore, te optimum number of units to stock is 4 units. Cost of under stocking can be calculated as follows: From te problem, following inequality can be found p( D 8) C (35 C ) p( D 10) or 0.80 Te minimum value of C is determined by letting C 0.95. (35 C ) IAAST Vol 4 [3] September 013 1 P a g e Society of Education, India
C =0.80 or C = (0.80)(35) = Rs. 140. (35 C ) (1 0.80) Te maximum value of C is determined by letting C =0.95 or C = (0.95)(35) = Rs. 665 (35 C ) (1 0.95) Terefore, 140 C 665. Metod-II: Matrix Metod Tis metod can be explained by considering again te above example. Te trader as five options. i.e. five reasonable strategies. He can stock te items from 4 to 1 units. Tere is no reason to stock more tan 1 units as e can never sell more tan 1 and tere is no possible reason for ordering less tan. Since tere are five alternative courses of action for stocking and five levels of demand, it follows tat tere are 5 combinations of one strategy and one level of demand. For tese 5 combinations. Te trader s payoff can be determined in te form of payoff matrix. As per te cost information given, te payoffs are determined for te two situations, i.e. (a) for te demand not more tan te stock level and (b) for te demand more tan te stock quantity, i.e.. Payoffs for D <D Cost of item -50-50 Sale of item 100D 100 Goodwill cost - -30(D-) Salvage value 0(-D) - Carrying cost -5(-D)-5D/ -5/ Total payoff -35+8.5D 77.5-30D Te payoff matrix will be five by five. Eac element of te matrix can be determined by above total payoffs for demand less tan equal to or greater tan te order size. Wen demand is less tan or equal to te order size, following contribution would be obtained to te payoff. (a) Te trader purcases te items for Rs. 50. (b) He sells D of tem for Rs. 100D. (c) He earns salvage of Rs. 0(-D) for items not sold. (d) He acquires te carrying cost of Rs. (0.10)(50)(-D) on unsold items, and (e) He acquires an average carrying cost of (0.10) (50) (D/) on te items sold during te period. Te total payoff, tus comes out to be as (a) -35+8.5D for demand less or equal to order size. (b) 77.5-30D for te demand more tan te order size. Te payoff matrix is given as in table : Unit Demanded (D) Units stocked or order size () 3 4 5 6 4 Rs. 190 130 70 10-50 6 10 85 5 165 105 8 50 15 380 30 60 10-0 145 330 475 415 1-90 75 40 405 570 Probability of demand 0.30 0.0 0.30 0.15 0.05 Table- Te expected payoff can now be determined for te order size. Te procedure for calculating te expected values is as following: For any given strategy, eac possible payoff for tat strategy is multiplied by te corresponding probability of te given level of demand and all tese products are added up. Tus, for first strategy of order size 4 units, te expected value of payoff is: (190)(0.30)+(130)(0.0)+(70)(0.30)+(10)(0.15)+(-50)(0.05)=Rs. 144 Proceeding similarly, all te expected values can be calculated as following: Order Size 4 6 8 10 1 Expected value (Rs.) 144 190.5 33 14 149.15 Table-3 Te objective is to select te strategy tat provides te igest payoff. Hence te trader sould order for 8 units for te igest expected payoff of Rs. 33. IAAST Vol 4 [3] September 013 13 P a g e Society of Education, India
Out of te above two metods, incremental analysis provides only te optimum level of purcase quantity and does not indicate about te level of expected profit, wereas payoff matrix metod Provides bot te answer, i.e., optimum purcase quantity as well as te optimum expected return. Furter, it is easy to directly convert te payoff matrix to te opportunity cost matrix as following: a. Wen te payoffs are profits, any column of te payoff matrix corresponding to a specific level of demand is taken and te largest payoff to get te corresponding opportunity costs. b. Wen payoffs are costs te smallest payoff is taken and smallest payoff is subtracted from eac payoff in te same column to get te opportunity costs In te above example, te opportunity cost matrix can be formed as in te given table 4: Unit Demanded (D) Units stocked or order size () Sarma and Sukla 4 6 8 10 1 4 Rs. 0 155 310 465 60 6 70 0 155 310 465 8 140 70 0 155 310 10 10 140 70 0 155 1 80 10 140 70 0 Probability of demand 0.30 0.0 0.30 0.15 0.05 Table-4: Payoff matrix for opportunity costs c. Te expected opportunity costs are determined for eac alternative wit te objective to select te strategy giving minimum expected opportunity cost. Te expected opportunity cost for te first alternative is (0)(0.30)+(155)(0.0)+(310)(0.30)+(465)(0.15)+(60)(0.50)=Rs. 194.75 Similarly, all te expected values can be calculated as given below Order Size 4 6 8 10 1 Expected value (Rs.) 194.75 137.5 94.75 119.75 178.5 Table-5 Te decision would be to select te minimum expected costs, i.e. te trader sould store 8 units for te lowest costs of Rs. 94.75. A relationsip between te expected opportunity costs and expected payoffs can be determined as follows: Expected opportunity costs (EOC) = K (constant)-expected payoff (EP) K=sum of te expected value of te largest entries in eac column of te payoff matrix =1900.30+850.0+3800.30+4750.15+5700.50=37.75. (8) Tus EOC= K(37.75)-Expected payoff for eac strategy From te equation (8), it can be observed tat te minimum value of EP will simultaneously produce te minimum of EOC. Tus te two analyses, i.e., te payoff matrix metod and te opportunity cost matrix metod give te same result. If te original payoff matrix is in terms of costs, te relationsip (8) will be of te form: EOC=EP-K, were EP is in te terms of costs. CONCLUSION In tis paper, a new tecnique is introduced to solve inventory problem. Tis metod is applicable to solve probabilistic inventory problems. In tis metod we do not require so muc calculation, so tis tecnique is systematic, easy to apply and consume less time in comparison to anoter tecniques. REFERENCES 1. KAPLAN, R., (1970). A Dynamic Inventory Model wit Stocastic Lead Times. Management Science, 16, 491-507.. Davis, M. H. A. (1993). A Markov Models and Optimization. Capman & Hall, London. 3. Cen, X., Zou, S. X. and Cen, Y. (011). Integration of Inventory and Pricing Decisions wit Costly Price Adjustments. Operations Researc, 59(5), 1144-1158. 4. Cen, F. and Song, J. (001). Optimal Policies for Multi-Ecelon Inventory Problems wit Markov-Modulated Demand. Operations Researc, 49, 6 34, 5. Hennet, J.C. (003). A Bimodal Sceme for Multi-Stage Production and Inventory Control. Automatica, 39, 793-805. 6. Porteous, E. L. (00).Foundations of Stocastic Inventory Teory. Stanford Business Books, Stanford. 7. Hu, W., Janakiraman, G. (008). (s, S) Optimality in Joint Inventory-Pricing Control: An Alternative Approac, Operations Researc, 56(3), 783-790. 8. Pang, Z., F. Y. Cen, Y. Feng (01). A Note on Te Structure of Joint Inventory-Pricing Control Wit Leadtimes, Operations Researc, 60(3), 581-587. IAAST Vol 4 [3] September 013 14 P a g e Society of Education, India
9. Yin, R. (007). Joint Pricing and Inventory Control wit Markovian Demand Model. Euro. J. Operations Researc, 18(1),113-16. 10. Zipkin, P. H. (008). On te Structure of Lost-Sales Inventory Models. Operations Researc, 56(4), 937-944. 11. Bensoussan, A., Cakanyildirim, M., Minjarez-Sosa, J., Royal, A. and Seti, S. (008). Inventory Problems wit Partially Observed Demands and Lost Sales. Journal of Optimization Teory and Applications, 136(1). Citation of Article: P. R. Sarma and S. Sukla. Using Matrix to Solving te Probabilistic Inventory Models (Demand Model). Int. Arc. App. Sci. Tecnol., Vol 4 [3] September 013: 09-15 IAAST Vol 4 [3] September 013 15 P a g e Society of Education, India