An Economic Production Quantity Inventory Model With .
Scientia Iranica E (2016) 23(2), 736{746Sharif University of TechnologyScientia IranicaTransactions E: Industrial Engineeringwww.scientiairanica.comAn economic production quantity inventory model withbackorders considering the raw material costsE.A. Pacheco-Vel azqueza and L.E. C ardenas-Barr onb; a. Department of Industrial and Systems Engineering, Tecnol ogico de Monterrey, Campus Ciudad de M exico.b. School of Engineering and Sciences, Tecnol ogico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo Le on,M exico.Received 1 October 2014; received in revised form 20 February 2015; accepted 4 May 2015KEYWORDSAbstract. The classical Economic Production Quantity (EPQ) inventory model does not1. Introductiondetermines the optimal order quantity to be purchased.Conversely, the EPQ inventory model calculates theoptimal production quantity to be manufactured. Sincethat the EOQ/EPQ inventory models appeared, manyresearchers and academicians have been constantlystudying and extending these inventory models in orderto model real life constraints. Two years ago, theEOQ inventory model celebrated its 100th anniversary.According to C ardenas-Barr on et al. [4], Ford WhitmanHarris is the Founding Father of Inventory Theory.There is a vast literature on inventory modelsthat considers raw materials. For example, inventorymodels considering raw materials that satisfy the needsof a production process were proposed by Banerjee etal. [5], Golhar and Sarker [6], Jamal and Sarker [7],Sarker and Golhar [8], Sarker and Parija [9], Sarker etal. [10], Sarker and Parija [11], Sarker and Khan [12],Khan and Sarker [13], just to name a few works. Conversely, there exists also a rich literature on inventorymodels for multiple products on one machine. PerhapsEPQ;Inventory models;Raw materials;Manufacturingsystem.consider ordering and holding costs of raw materials. In this direction, this paper considersthe ordering and holding costs for both raw materials and nished product. Basically,four EPQ inventory models are developed from an easy perspective that has not beenconsidered before. It was found that the ordering and holding costs of raw materials mustbe taken into account, because they signi cantly impact on the optimal production lot sizeof the nished product in both EPQ without shortages and EPQ with shortages inventorymodels. Furthermore, an EPQ inventory model that determines the optimal lot size fora product that requires more than one raw material, and an EPQ inventory model thatobtains the optimal batch size for multiple products, which are manufactured with multipleraw materials, are proposed. Numerical examples are presented in order to illustrate theuse of the proposed inventory models. 2016 Sharif University of Technology. All rights reserved.In recent years, a signi cant progress has been made ininventory management. Management of the inventoriesis a mandatory activity that any company must do inthe best way. Therefore, the inventory has becomea key challenge for every production manager. It iswell known that the two classical inventory modelsof Economic Order Quantity (EOQ) and EconomicProduction Quantity (EPQ) have been proposed byHarris [1] and Taft [2], respectively. Afterwards, theconsultant, Wilson [3], made the EOQ popular, because he applied it in practice in several companies. Itis important to remark that the EOQ inventory model*. Corresponding author. Tel.: 52 81 83284235;Fax: 52 81 83284153E-mail addresses: epacheco@itesm.mx (E.A.Pacheco-Vel azquez); lecarden@itesm.mx (L.E.C ardenas-Barr on)
Pacheco-Vel azquez and C ardenas-Barr on/Scientia Iranica, Transactions E: Industrial Engineering 23 (2016) 736{746Eilon [14] and Rogers [15] were the rst researcherswho studied the multi products-single manufacturingsystem. Later, this type of the problem was treated extensively in the works of Bomberger [16], Madigan [17],Stankard and Gupta [18], Hodgson [19], and Baker [20],just to name a few pioneer works that address multiproducts on a single machine. Later, Davis [21],Fransoo et al. [22], Sarker and Newton [23], Cooke etal. [24], and Hishamuddin et al. [25] continued studyingthis problem. The problem of multi products in onemachine is still being studied by several researchers.For example, Taleizadeh et al. [26] proposed an inventory model that considered multi products singlemachine production system with stochastic scrappedproduction rate, partial backordering, and service levelconstraint. Their inventory model determined, foreach product, the optimal production quantity, theallowable shortage level, and the period length. Inthe same year, Taleizadeh et al. [27] developed anEPQ inventory model with backorders to determine theoptimal lot size and backorders level for multiproductmanufactured in a single machine. Also, Taleizadeh etal. [28] derived an EPQ inventory model with randomdefective items, service level constraints, and repairfailure. Basically, their inventory model obtained theoptimal cycle length, optimal lot size, and optimalbackordered level. Chiu et al. [29] obtained the optimalreplenishment lot size and shipment policy for an EPQinventory model with multiple deliveries and rework ofdefective products. Later, Taleizadeh et al. [30] solvedthe multiproduct single machine problem with andwithout rework considering backorders. Taleizadeh etal. [31] addressed the multi-product, multi-constraint,single period problem considering uncertain demandsand an incremental discount situation. On the otherhand, Sepehri [32] addressed a multi-period and multiproduct problem in a multi-stage with multi-membersupply chain. Subsequently, Taleizadeh et al. [33] developed an EPQ inventory model with rework processfor multi products in one machine and determined theoptimal cycle length as well as the optimal productionquantity for each product. In the same year, Ramezanian and Saidi-Mehrabad [34] presented a Mixed Integer Nonlinear Programming (MINLP) model to solvea multi-product unrelated parallel machines scheduling problem considering that the production systemcould manufacture imperfect products. Afterwards,Taleizadeh et al. [35] proposed an EPQ inventory modelwith random defective items and failure in repair formultiproduct in one machine environment. Later,Taleizadeh et al. [36] optimized a joint total cost foran imperfect, multi-product production system withrework subject to budget and service level constraints.In a subsequent paper, Taleizadeh et al. [37] developed and analyzed an EPQ inventory model withinterruption in process, scrap, and rework. Their737inventory model considered multiple products and allproducts were processed in one machine. Recently,Holmbom et al. [38] developed a solution procedurethat solved the well-known Economic Lot SchedulingProblem (ELSP) when the machine had high utilization. Later, Holmbom and Segerstedt [39] gave ahistorical summary from Harris's [1] EOQ formulae tothe ELSP. Basically, they presented the complexitiesand di culties in scheduling several products on onemachine subject to capacity constraint. Pal et al. [40]developed a stochastic inventory model that consideredtwo di erent markets to sell the products: 1) for goodquality products, and 2) for average quality products.This inventory model also considers that the usedproducts' recovery rates from consumers are randomvariables and recovery products are put in storage intwo warehouses. After that, Roy et al. [41] proposed aneconomic production lot size model for a manufacturingsystem that produced defective products. The defective products were accumulated and then reworked.This inventory model also permits shortages and thepartial and full backordering situations are analyzedand compared. Other relevant and related studies arethe research works of Hsu [42], Hsu and Hsu [43],Sana [44], Kumar et al. [45], Tripathi [46], Sana [47],Sana et al. [48], and Farughi et al. [49].The rest of the paper is organized as follows. Section 2 introduces the problem and establishes the notation that is used through the whole paper. Section 3presents the mathematical formulation of the EPQ inventory model with raw material costs for multiple rawmaterials and one nished product. Also, a comparisonwith the classical EPQ inventory model is made.Section 4 develops two EPQ inventory models. Therst one is for multi-products without shortages withthe constraint that all products share the same setupcost for the production run. The second one is for thesituation when the products are comprised of multipleraw materials. Finally, Section 5 provides a conclusion.2. Problem de nition and notationRegularly, the EPQ inventory model (see Figure 1)considers that at the beginning of the production run,there is not inventory cost due to the fact that there arenot nished products. However, in the real world, thereare some costs that must be considered, because rawmaterials are procured with anticipation. Thus, theordering and holding costs of raw material are incurredbefore starting a production run. In this direction, thisresearch work has the main goal to include these costsin the development of the inventory models.In any manufacturing process, at the beginningof every production run, it is necessary to prepare allthe raw materials required to complete lot size of thenished products. This also means that an ordering
738Pacheco-Vel azquez and C ardenas-Barr on/Scientia Iranica, Transactions E: Industrial Engineering 23 (2016) 736{746Figure 2. Inventory behavior of (a) raw material withEOQ, and (b) nished product with EPQ withoutshortages.Figure 1. The classical inventory behavior in the EPQinventory model without shortages.cost of the raw material is incurred when one placesan order on the supplier. Additionally, the holdingcost of raw material must be considered too. Thesecosts typically are ignored in the traditional EPQinventory model. Therefore, the cost of ordering theraw material and its holding cost must be consideredin the development of the inventory model.Obviously, with the inclusion of the raw materialcosts, the optimal lot size of the nished productwill change considerably. Consequently, in this paper,we propose an Economic Production Quantity (EPQ)without shortages and with shortages considering fullbackordering when the inventory of raw materials isinvolved.The notations that will be used in this paper arede ned and given below:DDemand rate (units/time unit);PProduction rate (units/time unit);APSetup cost of the production run( /setup);hPHolding cost of the nished product( /unit/time unit);CPProduction cost per nished product( /unit);AMOrdering cost of raw material( /order);uUnits required to produce one unit ofnished product (units);hMHolding cost of raw material( /unit/time unit);CMRaw material cost ( /unit); Fixed backordering cost ( /unit); tLinear backordering cost ( /unit/timeunit); QProduction lot size (units);b Backorders level (units);KTotal cost.3. Development of the inventory models3.1. An EPQ inventory model with rawmaterial costs and without shortagesFirst of all, it is necessary to develop an inventorymodel that considers the replenishment of raw materialand the manufacturing process, jointly. Basically, thismeans that the ordering and holding costs of rawmaterial must be included in the modelling of theinventory model (see Figure 2). It is assumed thatthe amount of raw materials ordered before must meetthe requirements for the production lot size within anyproduction cycle. Mathematically speaking, this canbe expressed as follows.K (Q) AP DQ hP1Q2DD AMPQQD:(1)2PAlgebraically, the above expression is reduced to: uhM D QDDK (Q) (AP AM ) hP 1 uhM:Q 2PP (2)In obtaining the global minimum optimal solution, thetotal cost function of the EPQ inventory model isoptimized via di erential calculus. It is easy to seeand show that K (Q) is a convex function in Q. Then,by applying the optimization technique via di erentialcalculus, one gets the optimal production lot size, givenby Eq. (3):Q s2(AP AM )D D:hP 1 DP uhM P(3)It is obvious that if the ordering (AM ) and holding(hM ) costs of raw material are not considered, thenthe production lot size decreases immediately to theclassical EPQ without shortages.3.2. An EPQ inventory model with rawmaterial costs and shortagesNow, this subsection presents an EPQ inventory modelwith raw material costs and shortages. Here, we
Pacheco-Vel azquez and C ardenas-Barr on/Scientia Iranica, Transactions E: Industrial Engineering 23 (2016) 736{746consider that the shortages occur only for the nishedproduct. It is assumed that all shortages are backordered. The total cost of the inventory model is givenbelow: K (Q; b) AP t b 22Q 1Q 1D hPQ2QD P D b 2P 1 DP bDDQD AM uhM: (4)QQ2PThus: Q 1D hPQ2QK (Q; b) (AP AM ) t b22Q 1D P D b 2P 1 DP bDQD uhM:Q2P(5)It is required to minimize the function K (Q; b) todetermine Q and b. It is easy to show that K (Q; b) is aconvex function in Q and b. Therefore, it is necessaryto di erentiate K (Q; b), partially, with regard to Q andb. Hence, optimizing Eq. (5), one obtains the optimalproduction lot size and the optimal backorders level,which are given by Eqs. (6) and (7), respectively:Q s2(AP AM )D(hP t ) 2 D2 1 uDhP t 1 DP hM P (hP t )D P ;(6) (h Q D) 1 DPb P:(7)hP tIt is easy to see that the above expression for theproduction lot size immediately transforms into thewell-known equation of the EPQ with shortages whenboth ordering (AM ) and holding (hM ) costs of rawmaterial are zero. In other words, it is when thesecosts are not considered.3.3. Comparison with EPQ with and withoutshortagesIn this section, a numerical comparison is made between the results of the proposed inventory model andthose of the classical EPQ inventory model with andwithout shortages.Numerical examples 1 and 2.The data for Examples 1 and 2 are given in the rstcolumns of Tables 1 and 2, respectively. The resultsof the comparison with the EPQ without shortages arepresented in Table 1. In Table 2, they are given forthe comparison with the EPQ with shortages. In bothtables, we use the expression of w uCM CP . Thismeans that the raw material cost is a fraction of thecost of the nished product. Typically, the holdingcosts are calculated as a percent of the cost (i.e., hP iCP yhM iCM ); then, the term w can be estimatedas w uhM hP .Here, note that according to the results of Tables 1and 2, the optimal production lot size is sensible tothe fraction of cost of raw material in both inventorymodels. If the fraction increases, the production lotsize decreases.3.4. An EPQ inventory model with multipleraw materials and one nished productIt is now appropriate to discuss that in many situationsof the real life, a product is comprised of several rawmaterials. Although there is a unique nished product,it is made of several raw materials; therefore, onecan apply the inventory models developed in the Sections 3.1 and 3.2. There are always welcome practicalTable 1. Comparison of the proposed inventory model with EPQ without shortages.Q Q % ofAP hP AM w DPEPQProposed inventory di 2.92191.9412.823.773.399.2514.1Table 2. Comparison of the proposed inventory model with EPQ with shortages.Q b Q b QbAM w t DPEPQProposed inventory% of di 2.255.7312.0917.3226.744.8111.9725.3536.34
740Pacheco-Vel azquez and C ardenas-Barr on/Scientia Iranica, Transactions E: Industrial Engineering 23 (2016) 736{746DjQjperspectives in any company for applying mathematical models. Therefore, in this direction, one needs justto establish the following oversimpli cation, which issimple, easy to apply, and computationally e cient.Set AM 1 ; AM 2 ; ; AMn as the ordering costs ofeach raw material; u1 ; u2 ; ; un as the number ofunits of each raw material required to manufacture oneunit of the nished product; and hM 1 ; hM 2 ; ; hMn asthe holding cost of each raw material. Then, AM anduhM are de ned as AM AM 1 AM 2 AMn anduhM u1 hM 1 u2 hM 2 un hMn , respectively. Asa result, one can use the previous proposed inventorymodels.subject to:4. An EPQ inventory model without shortageswith multi-products and raw material costs4.1. An EPQ inventory model withoutshortages with multi-products and oneraw materialDQDQD QQ2 2 1 ; Q3 3 1 ; ; Qn n 1 ;D1D1D1and:Here, the situation is considered in which there isa unique raw material to process several nishedproducts and these products are manufactured in onemachine or process. Consequently, it is assumed thatall products share one setup cost of the production run.In this type of problem, a schedule is required for thefabrication of products. A common cycle time is usedfor all the products. Figure 3 illustrates the commoncycle time.It is easy to understand that in order to minimizethe holding cost of raw materials, it is necessary thatthe product with a higher consume rate of raw materialmust be scheduled rst.As an illustrative example, consider the situationof two products A and B. Suppose that one unit ofproduct A consumes 6 units of raw material and oneunit of product B requires 4 units of raw material.Moreover, the production rate of product A is 2000units per year and the production rate of product Bis 4000 units per year. Therefore, the consume ratesof raw material for products A and B are 12000 unitsper year and 16000 units per year, respectively. Asmentioned before, product B must be the rst in theproduction schedule (i.e., the sequence is B-A).Then the optimization problem can be expressedas:minK (Q1 ; Q2 ; ; Qn ) AP nX DkPkhkQ 12 kk 1 AMDj (8) hM I;QjD1 D2D n;(9)Q1 Q2Qnwhere I represents the required units of raw materialand is given by Eq. (11). The details of the derivationof Eq. (11) are given in the Appendix.Since:82(10)1390nn Q XDk2 uk nX14 Dk @ XI 1 2uj DjA5 :;2D1 :k 1 PkPk j k 1(11)k 1Eq. (8) can be written as:minK (Q1 ) (AP AM )Q 12D1(nX1nXk 12 hk Dk0 D1Q1D1 kPk nXDk2 uk hMPkk 113)nXD 2 4 k @uj hM Dj A5 :Pkk 1j k 1(12)Optimizing Eq. (12), one can obtain the optimal lotsize for the product 1, which is given by Eq. (13) givenin Box I.The optimal lot sizes for the rest of the productscan be easily calculated by the following equation:DQ i i Q 1 :D1(14)Numerical Example 3Consider a set of products that share the same rawFigure 3. Inventory behavior of (a) raw material, (b) nished product 1, and (3) nished product 2 considering an EPQwithout shortages.
Pacheco-Vel azquez and C ardenas-Barr on/Scientia Iranica, Transactions E: Industrial Engineering 23 (2016) 736{746Q 1 vuuuuPn hthk 1k Dk 1DkPk i2(AP AM )D12 n D2 u hPk k Mk 1Pk 2nP1k 1"DkPk(13)!# :nPj k 1741uj hM DjBox ITable 3. Additional information for Numerical Example 3.Units of raw Total cost of u hUnit ofProduction Demandi Mmaterialraw materialproduction costraterateProduct AProduct BProduct CProduct D12648 120.0060.0040.0080.00 24.0012.008.0016.00material and all of them must be manufactured in aunique machine. The joint setup cost of the productionrate is 1000 per run and the ordering cost of theraw material is 400 /order. The cost of each unitof raw material is 10 per unit. Assume that theannual holding cost is calculated as 20% of the costof raw material and nished product (i.e., hk 0:2Ckand hM 0:2CM ). Additional information for thenumerical example is given in Table 3.An observation is that in order to solve the inventory problem, it is necessary to obtain the consumerate of raw material for each product. Hence, theconsume rates are 240000, 360000, 120000 and 400000for the products A, B, C, and D,
inventory management. Management of the inventories is a mandatory activity that any company must do in the best way. Therefore, the inventory has become a key challenge for every production manager. It is well known that the two classical inventory models of Economic Order Quantity (EOQ) and Economic Production Quantity (EPQ) have been proposed by
13-13 Inventory Management Economic order quantity model Economic production model Quantity discount model How Much to Order: EOQ Models Skip 13-14 Inventory Management Basic EOQ Model The basic EOQ model is used to find a fixed order quantity that will minimize total annual inventory costs
â Inventory models o Economic Lot Size Models with Constant Demands Economic Order Quantity, EOQ() Economic Production Quantity, EPQ() o Economic Lot Size Models with Varying Demands, Wagner-Whitin algorithm, WW() o Stochastics Inventory Models Newsvendor model, Newsvendor() â Safety Stocks, SS() â Inventory and Supply Chain Management (SCM)
Table of contentsI 1 Introduction 2 Inventory Management 3 Inventory models 4 Economic Order Quantity (EOQ) EOQ model When-to-order? 5 Economic Production Quantity (EPQ): model description EPQ model 6 The Newsboy Problem-Unknown demand (probabilistic model) The newsvendor model 7 Multiple-period stochastic model: model description 8 Managing inventory in the supply chain
inventory level. In order to help them reduce their stock outs, a forecasting model was provided along with an economic order quantity. Finally, the economic order quantity is, optimized the order quantity for each product when an order is placed, reducing the companies product stock out issue.
Types of Inventory Control Policies Fixed order quantity policies The order quantity is always the same but the time between the orders will vary depending on demand and the current inventory levels Inventory levels are conti nuously monitored and an order is placed whenever the inventory level drops below a prespecified reorder point.
CPS.PAC.QTY QUANTITY R 1 (10) Number of articles in the package. C186 QUANTITY DETAILS M 6063 Quantity qualifier M an.3 Value: 52 Quantity per pack 6060 Quantity M n.10 (an.15) Quantity of articles in specified package type 6411 Measure unit qualifier O an.3
Balance sheet Inventory Cost / Unit Inventory Value x Holding Cost Inventory Turns Inventory Value Inventory Turns Wal Mart Stores Inc. Kmart Corp. . Restaurant; High Tech; Inventory decisions 1 Christmas Tree Problem 100 8 15 22 29 2 9 16 23 30 3 10 17 24 31 4 11 18 25 5 12 19 26 6 13 20 27 7 14 21 28
additif alimentaire ainsi que d’une nouvelle utilisation pour un additif alimentaire déjà permis. Les dispositions réglementaires pour les additifs alimentaires figurent à la partie B du titre 16 du RAD. L’article B.16.001 énumère les exigences relatives à l’étiquetage des additifs alimentaires. En particulier, l’article B.16.002 énumère la liste des critères qui doivent .
