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Int J Adv Manuf Technol (2009) 45:382–388DOI 10.1007/s00170-009-1959-1 ORIGINAL ARTICLEA method to exchange the demand of productsfor cost improvementSanjay SharmaReceived: 18 October 2007 / Accepted: 3 February 2009 / Published online: 24 February 2009# Springer-Verlag London Limited 2009Abstract In a multiproduct manufacturing environment, the facilities optimally. However, when most of the firmsactual demands of various products are either available, or achieve this level, there is loss of competitive edge, andthese are expected. There are situations when demand of a further cost reduction becomes necessary. In such a scenario,product can be substituted with that of another. In the context an examination of significant parameters is essential.of cyclic manufacture, all the items are produced in an optimal Demand management is critical nowadays, and therefore, acycle time, and the production facility runs at certain cost method is explored in the present paper to exchange thelevel. The total cost consists of the facility setup cost, demand of products for cost improvement in certain cases.inventory carrying costs, and the manufacturing time cost In a continuous production, single standard product isfor the basic case. The total cost is optimized. For the purpose manufactured in large quantities. Even if the type ofof total cost improvement, a method is presented in which the product is similar, it can be produced in a wide variety ofdemand of a product is exchanged with that of another item in sizes. For instance, in a tube or pipe manufacturingthe group. The basic model without backorders is analyzed industry, these are in different diameters/thicknesses. In afirst. Then, it is extended for an inclusion of shortages that are job shop/batch production also, several items are processedeither completely backlogged or partially. In addition to the in a cycle time. For example, if the cycle time is 3 monthscost components discussed before, shortage costs are included or 0.25 year, all items/product varieties are manufactured inin the total cost for this case. Finally, after a discussion of idle the cycle time. This is called as common cycle time. If thetime costs, these are also included briefly in the formulation of production rate of an item is, say 300 U per month, and thethe total cost. The proposed methods are useful for imple- demand rate is 100 U per month, the production time in amentation in a variety of industrial or business situations in the cycle time of 3 months will be 1 month, i.e., 3×(100/300).context of internal benchmarking or gradual improvement. Benefits can be achieved by synchronizing production activities sequentially in a cycle time [3]. A relevant costKeywords Multi-item cyclic manufacture . Demand rate . needs to be estimated/modeled for the concerning produc-Production time . Idle time costs tion environment. For example, if shortages are not allowed, the shortage costs will not become a component of the total relevant cost. After an optimization of the total1 Introduction relevant cost, a common cycle time is usually obtained in which all the items in a family are produced. A generalizedIn the manufacturing firms, one or more products are made production cost is used [1] including shop floor index, thein certain cycle time. In order to become competitive, the value of which lies in the range 0–1. The generalizedprogressive firms are expected to run their production production cost is obtained as the multiplication of fixed production cost and a factor that is an exponential order of the ratio of production rate to demand rate of an item.S. Sharma (*) In the context of modeling process, the rate of manufac-National Institute of Industrial Engineering (NITIE),Vihar Lake, Mumbai 400087, India ture and demand rate are among significant input parameters.e-mail: s_nsit@rediffmail.com Manufacturing rate is considered to be a decision variable
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Int J Adv Manuf Technol (2009) 45:382–388 383[8]. Shortages are included in the production system. These With the purpose of an internal benchmarking/improve-may be backordered completely/partially. Various cases are ment activities, it seems reasonable to consider an appro-analyzed [5–7, 9] for single/multi-item scenario. The priate item whose demand is to be interchanged by anydemand rate per year or an annual demand needs to be other remaining item in the group. The present paper isadjusted in order to incorporate partial or fractional back- divided into nine sections. Assumptions and notations areordering situation. For a single product case, the demand provided in the “Assumptions” section, followed byincrease is included in different context [2, 4] considering methodology in the “Methodology” section. Mathematicaldemand function with respect to time. As it will be discussed formulation for the basic problem is dealt with in thelater, a quite different approach is presented in this paper in “Mathematical formulation” section followed by an illus-the context of multiproduct manufacturing environment. trative numerical example in the “Illustrative example”This is expected to be useful in certain situations of business section. Shortages are included in the “Extension forwhen more or less stable product demands exist. shortages” section with the assumption that all the shortage In the traditional production/manufacturing setup, the quantities will be backordered completely. This assumptiondemand is analyzed solely as an input parameter. In the is relaxed in the “Partial backlogging” section. An idle timepresent paper, the demands are being viewed in an uncon- cost is introduced in the “Incorporating an idle time cost”ventional manner. For instance, several production lines run in section for this approach, and finally, the concludingparallel in the pharmaceutical industries. Whether it is remarks are provided in the “Concluding remarks” section.multiple or single production line, a batch production isusually adopted. After certain development or value addition,the management wishes to promote the improved product 2 Assumptions(which may be patented in a different name) at the cost ofsimilar (more or less for medicinal purpose) matured product. An industrial organization is engaged in the production ofHowever, the improved product is at least presently in lower multiple items in a common cycle time. The manufacturingdemand because of either the availability of a familiar matured facility is being run conventionally in an optimum manner. Itproduct at higher demand level or lack of awareness. This may is often difficult to obtain information for benchmarkingalso be due to purely psychological or emotional reasons purpose particularly at the production facility level. With theattached to a familiar product. As the aggregate demand is aim of a gradual improvement, an intentional search is mademore or less uniform for similar types of products, the to exchange the demand of an item (strategically selected byproduction strategy may be based on a conscious anticipated the management) with another appropriate item in the familydemand swapping. Further, there should be a strong justifica- for any potential cost reduction. A business environment oftion if it yields into the total relevant cost reduction. stable demand exists in general. The proposed method In oligopoly, few firms dominate the market. While in considers an exact interchange of the demand level of twothe monopolistic competition, many firms are active in items because it is in the interest of the organization tosatisfying the market demands. Whether it is monopolistic maintain a similar aggregate demand for the whole family ofcompetition or oligopoly, each progressive firm in the items.industrial sector would run their production operations at a In addition to the above, the following assumptions arecertain optimum level. There is continuous pressure to also made:adopt a kind of internal benchmarking and improve the 1. The facility is set-up for a family of items, andproduction/operational cost further. In a planning period, it therefore, the facility setup cost is included in theis possible to substitute the demand of an item by another formulation. As the individual item setup time is notsuitable item in the product family. The firm may have relevant in the present context, it is ignored.invested in product development activities. It would like to 2. All the items are manufactured in a common cycle time.exchange the lower demand of new product with higher 3. Shortages may or may not be allowed.demand of an old matured product, and the firm manage- 4. In case shortages are allowed, these may be backorderedment is confident of getting it consumed as a substitute in completely/partially depending on the situation.the market. In yet another situation, a factory may face 5. An idle time exists usually in a common cycle time. If thequality problems related to the input item of a product, and idle time costs are significant, these may be incorporatedit wants to exchange the demand of such a product with in the modeling process depending on the case.another in the short-run. In many cases, contribution perunit is almost similar for the products in a family. It is an Based on these assumptions, a formulation is first madeinteresting approach to explore the possibility concerning for the basic production situation. Then the shortages arethe exchange of demand of items and examine the effects incorporated with complete backordering. This is extendedon total relevant cost. for a fractional backordering case. The idle time cost is
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384 Int J Adv Manuf Technol (2009) 45:382–388further discussed briefly with its inclusion in the suggested Compute the existing cost, Emethod.2.1 Notation Select Dk from the set Di , i≠j∝ Shop floor index lying usually in the range (0≤ ∝ <1). No Exchange Dj with DkAi Setup cost for item i.bi A faction of shortage quantity which is not backordered for product i. ∑(Di/Pi)< 1c Fixed production cost per year.c1 Idle time cost per year. YesDi Annual demand for item i. Compute the revised costE Total relevant cost.E1 Total cost after exchange of the demand rate of two Retain the minimum cost along with corresponding exchange and implement items.Hi Inventory carrying cost for an item i per unit-year.j An item whose demand rate is desired to be Fig. 2 An iterative process of demand exchange exchanged with another appropriate item.Ji Shortage quantities for a product i.k Selected another appropriate item whose demand rate All the remaining items can be considered one at a time. would be exchanged with that of item j. However, the conditions are developed next in order toKi Annual shortage cost per unit for a product i. have a small subset of items to make the search proceduren Number of items in the group. convenient.Pi Production rate per year for item i.T Common cycle time in year. 4 Mathematical formulation A generalized production cost is cðPi =Di Þa per year, and as the manufacturing time for an item i is (Di/Pi), the annual manufacturing time cost for an item i is cðDi =Pi Þ1Àa . With3 Methodology the inclusion of this cost component, a total relevant cost for the basic model without shortages,From a family of n items, an item j is selected by themanagement whose demand rate is to be exchanged by that X n 1X n TX n E¼c ðDi =Pi Þ1Àa þ Ai þ Di Hi ð1 À Di =Pi Þof another appropriate item k among the remaining items. T i¼1 2 i¼1 i¼1Figure 1 represents the process of exchange of demand rates. Pn ð1Þ The production time is T ðDi =Pi Þ in a cycle time T, and i¼1in order to have a feasible schedule, the production time The optimal cycle time can be obtained by differentiating Pnshould be less than T, i.e., ðDi =Pi Þ < 1. In the iterative Eq. 1 with respect to T and equating to 0. The optimal i¼1process of exchange (Fig. 2), Dj is exchanged by Dk such values (T* and subsequently E*) can easily be obtained as,that the constraint on total production time is satisfied. vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u P n u 2 Ai u u T * ¼ uP i¼1Fig. 1 Exchanging the demand D1 ð2Þrate t n D2 Di Hi ð1 À Di =Pi Þ i¼1 . . Dj . X n Dk and E * ¼ c ðDi =Pi Þ1Àa ð3Þ . i¼1 . vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ . u " n #" # Dn u X X n þ t2 Ai Di Hi ð1 À Di =Pi Þ Di, i≠j i¼1 i¼1
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Int J Adv Manuf Technol (2009) 45:382–388 385 With reference to Eq. 3, the components concerning itemj and item k are separated from the remaining items. Afterexchanging Dj and Dk, the total optimal cost, 2 3 6Xn À Á1Àa À Á1Àa 7E1 ¼ c6 Ã 4 ðDi =Pi Þ1Àa þ Dk Pj þ Dj Pk 7 5 ð4Þ i6¼j i6¼k vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 3 u u u X #6X n n À Á À Á7 u þ u2 Ai 6 4 fDi Hi ð1 À Di =Pi Þg þ Dk Hj 1 À Dk Pj þ Dj Hk 1 À Dj Pk 7 5 t i¼1 i6¼j i6¼kSubtracting Eqs. (4) from (3), any potential cost improvement, hÀ Á À Á1Àa À Á1Àa iE Ã À E1 ¼ c Dj Pj Ã 1Àa þ ðDk =Pk Þ1Àa À Dk Pj À Dj Pk ð5Þ 2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 Pn sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ6 fDi Hi ð1 À Di =Pi Þg 7 X 6n i¼1 7 6 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 þ 2 A i 6 uP 7 6 Àu fD H ð1 À D =P Þg þ D H À1 À D P Á þ D H À1 À D P Á 7 n i¼1 4 t i i i i k j k j j k j k 5 i6¼j i6¼kEquation (5) has two components, the first component is The second component is certain to be positive if,certain to be positive if,À Á1Àa À Á1Àa À Á1Àa Dj Pj þðDk =Pk Þ1Àa Dk Pj þ Dj Pk ð6Þ À Á À Á À ÁDj Hj 1 À Dj hPj þ Dk Hk ð1 Dk =PÞ Dk Hj 1 À Dk Pj þ Dj Hk 1 À Dj Pk À k À Á À Á H À Ái ð7Þor Dj À Dk Hj À Hk þ Hkk À Pjj Dj þ Dk 0 PThere is a guaranteed cost improvement if the conditions 6and 7 are satisfied. The entire feasible remaining itemdemand rate can be exchanged if it is difficult to draw any Table 1 Input parametersconclusion with the use of conditions 6 and 7. Item 1 25 Illustrative example Annual demand Di 400 300Table 1 shows the input parameters concerning two items. Annual production rate Pi 720 750As it is a simple numerical example for illustration purpose, Setup cost, Ai ($) 100 150Pn P n Di Hi ð1 À Di =Pi Þ ¼ 0 and ðDi =Pi Þ1Àa ¼ 0. Annual carrying cost Hi ($ per unit) 13 5i6¼j i6¼ji6¼k i6¼k Annual shortage cost Ki ($ per unit) 120 80 Using the relevant parameters for the basic case, i.e., Fraction bi Pn 0.2 0.3without shortages, ðDi =Pi Þ ¼ 0:955 1, and the feasible i¼1data are ensured. c=$9,000 per year; α=0.2
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386 Int J Adv Manuf Technol (2009) 45:382–388 From Eq. 3, the total relevant cost, E* =$11,214.88. Now, let j=1 and k=2. After exchanging Dj with Dk,Pn ðDi =Pi Þ ¼ 0:95, and the feasibility is ensured. Vii¼1 Production From condition 6, 1.1051.101. inventory Pi – D i Di From condition 7, 2.780. As the both conditions are satisfied, there is a guaranteed 0cost improvement with the implementation of the proposed Timemethod. With the use of Eq. 4, a reduced total relevant cost after Jidemand exchange, E1* =$11,177.19. T Fig. 3 The production cycle with shortages6 Extension for shortages TDi Hi ð1 À Di =Pi Þ Substituting optimal Ji ¼ ð11ÞQuite often, the shortages are included in a manufacturing ðHi þ Ki Þsystem. These are assumed to be completely backordered atpresent. Figure 3 shows this kind of environment. X n 1 Xn T X Di Hi Ki ð1 À Di =Pi Þ n Since the shortages exist for a period Ji =ðPi À Di Þþ E¼c ðDi =Pi Þ1Àa þ Ai þ T i¼1 2 iÀ1 ðHi þ Ki ÞðJi =Di Þ, the annual shortage cost for an item i, i¼1 h i ð12Þ¼ Ji ðPi ÀDi Þ þ Dii Ki 2 Ji J T The optimal values can be obtained as, P Ki Ji2 nand the total annual shortage cost ¼ 2T1 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Di ð1ÀDi =Pi Þ u P n i¼1 u 2 Ai u ð8Þ u T * ¼ uP i¼1 ð13Þ Now, the maximum inventory level, Vi ¼ ðPi ÀiDi ÞTDi =Pi À Ji h t n ½Di Hi Ki ð1 À Di =Pi Þ=ðHi þ Ki ÞŠand the annual carrying cost ¼ Vi T À ðPi ÀDi Þ À Dii Hi Substitut- 2 Ji J T i¼1ing Vi, the total annual carrying cost, T Xn Xn 1 Xn Hi Ji2¼ Di Hi ð1 À Di =Pi Þ À Hi J i þ 2 i¼1 i¼1 2T i¼1 Di ð1 À Di =Pi Þ P n ð9Þ and E * ¼ c ðDi =Pi Þ1Àa i¼1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! n !Adding the Eqs. 8, 9, and the remaining cost components, P n P þ 2 Ai Di Hi Ki ð1 À Di =Pi Þ=ðHi þ Ki Þ X n 1X n 1 X ðHi þ Ki ÞJi2 n i¼1 i¼1E¼c ðDi =Pi Þ1Àa þ Ai þ i¼1 T i¼1 2T i¼1 Di ð1 À Di =Pi Þ ð14Þ T X n X n þ Di Hi ð1 À Di =Pi Þ À Hi Ji With the swapping of Dj and Dk, 2 i¼1 i¼1 ð10Þ 2 3 * 6Xn À Á1Àa À Á1Àa 7E1 ¼ c 6 4 ðDi =Pi Þ1Àa þ Dk Pj þ D j Pk 7 5 i6¼j i6¼k vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 3 u u u X #6X n n À ÁÀ Á À Á 7 u þ u2 Ai 6 4 fDi Hi Ki ð1 À Di =Pi Þ=ðHi þ Ki Þg þ Dk Hj Kj 1 À Dk Pj Hj þ Kj þ Dj Hk Kk 1 À Dj Pk ðHk þ Kk Þ7 5 t i¼1 i6¼j i6¼k ð15Þ
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Int J Adv Manuf Technol (2009) 45:382–388 387Following the procedure discussed in the “Mathematical obtained. The first condition is similar to 6. The secondformulation” section, the relevant conditions can be condition is obtained as, ( )#À Á Hj Kj Hk Kk À Á Hk Kk Hj Kj Dj À Dk À ÁÀ þ Dj þ Dk À À Á 0 ð16Þ Hj þ Kj ðHk þ Kk Þ Pk ðHk þ Kk Þ Pj Hj þ KjWith the input parameters of Table 1, the condition 6 is advertising costs apportioned for unit product and loss ofalready satisfied. profit among other factors, an explicit computation for From condition 16, 1.2090. contribution of the lost units of product is not necessary. A As the both conditions (6) and (16) are satisfied, there is suitable parameter for relevant cost is assumed for all thecertain cost improvement using the proposed approach. shortage quantities whether these are backlogged or not. An From Eq. 14, E*=$ 11,158.62. annual demand needs to be adjusted in order to incorporate Ã The reduced relevant cost from Eq. 15, E1 ¼ $11; 121:22. the partial backordering. The corresponding costs are also lower than that From Eq. 8, the annual shortage quantity can be obtainedobtained in the previous section. This can be justified by as,observing Eqs. 3 and 14. As Ki =ðHi þ Ki Þ is less than 1, the X n Ji2relevant costs are lower with relaxation of the constraint ¼that the backorders would not be allowed. i¼1 2TDi ð1 À Di =Pi Þ A fraction bi of the shortage quantity is not backordered, and therefore, the annual manufacturing cost,7 Partial backlogging X 1 n !1Àa bi Ji2In a real-world situation, a portion of the shortage quantities ¼c Di À P1Àa i¼1 i 2TDi ð1 À Di =Pi Þmay not be backordered. A particular customer may switchover to another competitive firm in the industry. However, Equation 10 can now be adjusted as follows for thiswith the advertising among other efforts, a new customer situation,can replace the old one, at a later date. In case where theshortage costs are estimated to be a good representation of X 1 !1Àa n bi Ji2 1 Xn 1 X ðHi þ Ki ÞJi2 n T X n XnE¼c Di À þ Ai þ þ Di Hi ð1 À Di =Pi Þ À H i Ji ð17Þ P1Àa i¼1 i 2TDi ð1 À Di =Pi Þ T i¼1 2T i¼1 Di ð1 À Di =Pi Þ 2 i¼1 i¼1Mathematical/analytical procedure as discussed before, 7.1 Specific casecannot be followed for the optimization of Eq. 17. However,conventional search process such as univariate method can α=0 in a specific case, and the Eq. 17 can be written as,be implemented conveniently for any real data set. X n 1 Xn 1 X ðHi þ Ki À cbi =Pi ÞJi2 T X n n XnE¼c ðDi =Pi Þ þ Ai þ þ Di Hi ð1 À Di =Pi Þ À H i Ji ð18Þ i¼1 T i¼1 2T i¼1 Di ð1 À Di =Pi Þ 2 i¼1 i¼1The optimal relevant cost can be obtained as, vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u n # # X n u X X nE* ¼ c ðD =P Þ þ t2 i i A i D H ð1 À D =P ÞðK À cb =P Þ=ðH þ K À cb =P Þ i i i i i i i i i i i ð19Þ i¼1 i¼1 i¼1
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388 Int J Adv Manuf Technol (2009) 45:382–388As c/Pi is the unit production cost, and shortage costs are framework of organization along with several inputmuch greater than this in the real world, an optimality/ parameters. However, these are continuously striving forfeasibility condition, i.e., Ki c/Pi, is satisfied easily. the cost improvement. Internal benchmarking practices are In order to exchange the demand of products, Eq. 19 can also adopted where the standards are bound to vary withbe used as a reference equation. time. A method is proposed and analyzed in which the Consider the input data of Table 1. demand of a strategically selected item is exchanged with From Eq. 19, E*=$9,809.46 another suitable item in the group. Analysis is first made for After an exchange of the demands, the reduced relevant the basic case without shortages and conditions arecost is obtained as, developed for convenience in the search of another suitable Ã item. The process is illustrated with the help of a numericalE1 ¼ $ 9; 759:21 example. Further extensions are concerning the inclusion of shortages that may be backlogged completely or partially. The costs are obtained at a lower level with the allowable8 Incorporating an idle time cost backorders. However, an annual shortage cost needs to be estimated with care considering the all relevant factors.In the cyclic manufacture, a production activity usually takes In a production cycle time, a certain period is usually idle.place for certain portion of the cycle time, and the remaining This idle time frequently repeats itself in case where the Pnportion is idle. With reference to Eq. 3, ðDi =Pi Þ is the associated manufacturing schedule is implemented. Idle time i¼1annual manufacturing time. After an exchange of demand, cost is introduced for the proposed method. With the inclusionthis parameter will vary. For instance, an annual manufac- of this cost, the reference equations are obtained which can beturing time has been reduced after the exchange of demand useful for an exchange of demand. In the presence of ain the illustrative example of the “Illustrative example” relevant situation, these are suitable for a trade-off concerningsection. This means that the idle time during the cycle has the production time and idle time among other factors.increased. In few cases, the problems are associated with an The possibilities for demand exchange can be conve-idle production facility such as the maintenance problems. niently explored, and depending on the business strategy, theConsistency in the quality of a product and skills of the proposed approach may be implemented in a short-run/long-human resources may also get affected up to some extent. run. In case of the various problems being faced by the firm,With the occurrence of this type of problems, it seems an alternate schedule is available on the basis of certainreasonable to introduce the idle time cost. methodology. This will help in incorporating flexibility in the industrial system and also in the decision-making processConsider an idle time cost per year ¼ c1 ðc1 cÞ in a variety of situations. # X nIdle time cost in a year ¼ c1 1 À ðDi =Pi Þ i¼1 ReferencesEquation 3 can now be transformed as follows: 1. Chowdhury MR, Sarker BR (2001) Manufacturing batch size and # Xn 1Àa X n ordering policy for products with shelf lives. Int J Prod Res 39E* ¼ c ðDi =Pi Þ þ c1 1 À ðDi =Pi Þ (7):1405–1426. doi:10.1080/00207540110052148 i¼1 i¼1 2. Giri BC, Jalan AK, Chaudhari KS (2005) An economic production vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ lot size model with increasing demand, shortages and partial u n # # backlogging. Int Trans Oper Res 12:235–245 u X X n þ t2 Ai Di Hi ð1 À Di =Pi Þ ð20Þ 3. Hall RW (1988) Cyclic scheduling for improvement. Int J Prod Res 26(3):457–472. doi:10.1080/00207548808947876 i¼1 i¼1 4. Hill RM (1995) Inventory models for increasing demand followed by level demand. J Opl Res Soc 46(10):1250–1259The above equation can be used as a reference equation for 5. Sharma S (2004) Optimal production policy with shelf lifethe exchange of demand. including shortages. J Opl Res Soc 55(8):902–909 Similarly an idle time cost can be added in Eq. 14 with 6. Sharma S (2006) Incorporating fractional backordering in the multi-the inclusion of shortages in a manufacturing system. product manufacturing situation with shelf lives. Proc IMechE, Part B: Journal of Engineering Manufacture 220:1151–1156 7. Sharma S, Sadiwala CM (1997) Effects of lost sales on composite lot sizing. Computers Ind Engng 32(3):671–6779 Concluding remarks 8. Silver EA (1990) Deliberately slowing down output in a family production context. Int J Prod Res 28(1):17–27 9. Viswanathan S, Goyal SK (2000) Incorporating planned backordersAlmost all competitive firms in an industrial/business sector in a family production context with shelf life considerations. Int Jare expected to perform in an optimal manner within the Prod Res 38(4):829–836
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