Two New Uncertainty Programming Models of Inventory with Uncertain Costs

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1 Journal of Informaion & Compuaional Science 8: 2 (211) Available a hp:// Two New Uncerainy Programming Models of Invenory wih Uncerain Coss Lixia Rong Compuer Science and Technology Deparmen, Dezhou Universiy, Dezhou, Shandong 25323, China Absrac This paper provides wo new models of Economic Order Quaniy (EOQ) for invenory based on uncerain heory. In he models, he holding cos, shorage cos and ordering cos per uni are assumed o be uncerain variables. Taking advanages of some properies of uncerainy heory, he models can be ransformed ino deerminisic form and solved by 99-mehod. In he end, wo numerical examples are provided o illusrae he effeciveness of he models. Keywords: Uncerain Variable; Economic Order Quaniy; Expeced Value Model; Chace-consrained Programming 1 Inroducion Economic order quaniy (EOQ) model is one of fixed order quaniy models of invenory problem. In EOQ, we need o deermine he opimal selling period and order quaniy. EOQ is also known as ha shorage is no permied and producion ime is very shor invenory model. The EOQ model was firs proposed by Harris [7] in 1915, which gave he order quaniy so as o mee cusomer service levels while minimizing he oal invenory cos. In he classical EOQ models, he cusomer demand, ordering cos and holding cos were assumed o consan number. Cheng [5] proposed an EOQ model wih demand-dependen uni cos and formulae he opimizaion problem as a geomeric program. Goyal [19] and Teng [8] developed an economic order quaniy under condiions of permissible delay in paymens. Exended economic quaniy model under cash discoun and paymen delay were sudied by Chang [2]. Teng e al. [9] discussed EOQ model for deerioraing iems wih ime-varying demand and parial backlogging. However, owing o some objecive and subjecive facors, he parameers of EOQ are assumed o be random or fuzziness. Parler [18] Proposed wo differen invenory models under yield randomness. Liberaore [11] sudied he EOQ model where he uncerainies in he lead ime was represened sochasically. Chen [4] developed an EOQ model under random demand. The effecs Projec suppored by he Naional Naure Science Foundaion of Shandong Province (No. ZR21BL9), Shandong Provincial Scienific and Technological Research Plan Projec (No. 29GG2129). Corresponding auhor. address: lx rong@163.com (Lixia Rong) / Copyrigh 211 Binary Informaion Press February 211

2 281 L. Rong /Journal of Informaion & Compuaional Science 8: 2 (211) of inflaion and ime-value of money in an economic order quaniy model wih a random produc life cycle were sudied by Moon [15].Roy and Maii [2] exended he classical EOQ model by inroducing fuzziness boh in he objecive funcion and consrains of sorage. Vujosevic [23] and Park [17] provided a fuzzy EOQ model by he fuzziness of ordering cos and holding cos. Chen and Wang [3] developed an EOQ model where he demand, ordering cos, holding cos and backorder cos were represened by rapezoidal fuzzy numbers. Producion invenory problems were sudied by Lee an Yao [1] and Lin and Yao [12] where he demand and order/producion quaniy were represened by fuzzy ses. Yao and Chang [27] and Wang e al. [24] discussed invenory problems wihou backorder, where Yao and Chang represened he order quaniy and oal demand quaniy by rapezoidal fuzzy numbers. The fuzzy EOQ model wih backorder was presened by Chang and Yao [1], where he backorder quaniy was fuzzified. Ouyang and Yao [16] proposed a mixed invenory model wih variable lead ime, where demand is fuzzy variables. On he oher hand, in some pracical applicaions, he parameers ake on he randomness and fuzziness simulaneously, and he decision maker needs o consider hem in a formal framework. Generally, here are wo approaches o deal wih he combinaion of randomness and fuzziness. One is fuzzy random, he oher is random fuzzy. Halim e al. [6] presened a fuzzy economic order quaniy model for perishable iems wih sochasic demand. Vijayan e al. [22] sudied fuzzy economoic order ime models where demand was random. Wang [25] [26] proposed a fuzzy random and random fuzzy EOQ model wih imperfec qualiy iems. A producion invenory model wih fuzzy random demand and wih flexibiliy and reliabiliy consideraions was sudied by Soumen e al. [21]. Yu e al. [28] Sudied he exended newsboy problem based on fuzzy random demand. The above lieraures sudy he problem in a sochasic or fuzzy environmen. In many cases, he uncerainy behaves neiher like randomness nor like fuzziness. This fac provides a moivaion o sudy he behavior of uncerain phenomena. In order o model uncerainy, uncerainy heory was founded by Liu [13] and refined by Liu [14] in 21. Nowadays uncerainy heory has become a branch of mahemaics based on normaliy, monooniciy, self-dualiy, counable subaddiiviy, and produc measure axioms. Since hen, uncerainy heory has been developed seadily and applied widely. In his paper, he EOQ models for invenory are provided when he holding cos, shorage cos and ordering cos per uni are assumed o be uncerain variables. In order o solve he models, 99-mehod was employed. The res of he paper is organized as follows. We will inroduce some definiions and resuls abou uncerain variable and 99-mehod in Secion 2. In Secion 3, he expeced value EOQ model and chance-consrained programming model of invenory problem under uncerainy environmens are formulaed. To illusrae he effeciveness of he models, wo numerical examples are provided in secion 4. In he end, some conclusions are lised in Secion 5. 2 Preliminaries Le Γ be a nonempy se, and le A be a σ-algebra over Γ. Each Λ A is called an even. In order o provide an axiomaic definiion of uncerain measure, i is necessary o assign o each even Λ a number M{Λ} which indicaes he level ha Λ will occur. In order o ensure ha he number M{Λ} has uncerain mahemaical properies, Liu [13] proposed he following four axioms: Axiom 1 (Normaliy) M{Γ} 1.

3 L. Rong /Journal of Informaion & Compuaional Science 8: 2 (211) Axiom 2 (Monooniciy) M{Λ 1 } M{Λ 2 } whenever Λ 1 Λ 2. Axiom 3 (Self-Dualiy) M{Λ} + M{Λ c } 1 for any even Λ. Axiom 4 (Counable Subaddiiviy) For every counable sequence of evens {Λ i }, we have M{ Λ i } M{Λ i } (1) i1 Definiion 1 (Liu [13]) The se funcion M is called an uncerain measure if i saisfies he normaliy, monooniciy, self-dualiy, and counable subaddiiviy axioms. Definiion 2 (Liu [13]) An uncerain variable is measurable funcion ξ from an uncerainy space (Γ, A, M) o he se of real numbers, i.e., for any Borel se B of real numbers, he se is an even. i1 {ξ B} {γ Γ ξ(γ) B} (2) Definiion 3 (Liu [13]) The uncerainy disribuion Φ : R [, 1] of an uncerain variable ξ is defined by Φ(x) M{ξ x} (3) Definiion 4 (Liu [13]) Le ξ be an uncerain variable. Then he expeced value of ξ is defined by E[ξ] M{ξ r}dr Definiion 5 (Liu [13]) Le ξ be an uncerain variable, and (, 1]. Then is called he -pessimisic value o ξ. M{ξ r}dr (4) ξ inf () inf{ r M{ξ r} } (5) Theorem 1 (Liu [14]) Le ξ be an uncerain variable wih uncerainy disribuion Φ. If he expeced value exiss, hen E[ξ] Φ 1 ()d (6) Theorem 2 (Liu [14]) Le ξ be an uncerain variable wih uncerainy disribuion Φ. Then is -pessimisic value is ξ inf () Φ 1 () (7) Theorem 3 (Liu [14]) Le ξ 1, ξ 2,, ξ n be independen uncerain variables wih uncerainy disribuions Φ 1, Φ 2,, Φ n, respecively. If f is a sricly increasing funcion. Then is an uncerain variable wih inverse uncerainy disribuion ξ f(ξ 1, ξ 2,, ξ n ) (8) Ψ 1 () f(φ 1 1 (), Φ 1 2 (),, Φ 1 n ()) (9)

4 283 L. Rong /Journal of Informaion & Compuaional Science 8: 2 (211) mehod 1 (Liu [14]) I is suggesed ha an uncerain variable ξ wih uncerainy disribuion Φ is represened by a 99-able as Table 1. Where.1,.2,,.99 in he firs row are he values of uncerainy disribuion Φ, and x 1, x 2,, x 99 in he second row are he corresponding values of Φ 1 (.1), Φ 1 (.2),, Φ 1 (.99). Essenially, he 99-able is a discree represenaion of uncerainy disribuion Φ. Then for any sricly increasing funcion f(x), he uncerain variable f(ξ) has a 99-able as Table mehod 2 (Liu [14]) Assume ξ 1, ξ 2,, ξ n are uncerain variables, and each ξ i is represened by a 99-able as Table 3. Then for any sricly increasing funcion f(x 1, x 2,, x 3 ), he uncerain variable f(ξ 1, ξ 2,, ξ n ) has a 99-able as Table 4. Table 1: The 99-able of ξ x x 1 x 2 x 3 x 99 Table 2: The 99-able of f(ξ) f(x) f(x 1 ) f(x 2 ) f(x 3 ) f(x 99 ) Table 3: The 99-able of ξ i x i x i 1 x i 2 x i 3 x i 99 Table 4: The 99-able of f(ξ 1, ξ 2,, ξ n ) f(x 1, x 2,, x n ) f(x 1 1, x 2 1,, x n 1 ) f(x 1 2, x 2 2,, x n 2 ) f(x 1 99, x 2 99,, x n 99) 3 Model Formulaion Economic order quaniy model is concerned wih giving he order period and order quaniy so as o mee cusomer service levels while minimizing he oal invenory cos. Due o he uncerainy of informaion, his secion presens wo uncerain programming models for EOQ in which he coss are assumed o be uncerain variables wih known uncerainy disribuion. Before inroducing he economic order quaniy model wih uncerain coss, we make some assumpions: (a) he ime of supplemen is long, and (b) shorage is permied. In order o describe and analyze he problem, we inroduce he following indices and parameers:

5 L. Rong /Journal of Informaion & Compuaional Science 8: 2 (211) R: demand per uni ime; P : produc number per uni ime; : selling period, a decision variable; 1 : he ime of produc saring; 2 : he ime of shorage is zero; 3 : he ime of produc ending; K: purchasing cos per uni of goods; ξ 1 : holding cos per uni, an uncerain variable; ξ 2 : shorage cos per uni, an uncerain variable; ξ 3 : ordering cos per uni, an uncerain variable; Φ i : uncerainy disribuion of ξ i, i 1, 2, Toal Cos The oal cos per period includes holding cos, shorage cos and ordering cos. Le ξ i (i 1, 2, 3) be independen uncerain variables wih uncerainy disribuion Φ i (i 1, 2, 3). If ξ i (i 1, 2, 3) are holding cos, shorage cos and ordering cos per uni, and he oal cos per uni ime is represen by f(ξ 1, ξ 2, ξ 3,, 2 ), hen f(ξ 1, ξ 2, ξ 3,, 2 ) R(P R) [ ( 2 ) 2 ] ξ ξ 2 + ξ 3 Then he oal cos per uni ime f(ξ 1, ξ 2, ξ 3,, 2 ) is an uncerain variable wih inverse uncerainy disribuion [ ] Ψ 1 R(P R) ( 2 ) 2 () Φ 1 1 () Φ 1 2 () + 1 Φ 1 3 () (11) If each ξ i (i 1, 2, 3) is represened by a 99-able as Table 5, hen he f(ξ 1, ξ 2, ξ 3,, 2 ) has a 99-able as Table 6. (1) Table 5: The 99-able of ξ i x i x i 1 x i 2 x i 3 x i 99 R(P R) Table 6: The 99-able of f(ξ 1, ξ 2, ξ 3,, 2 ).1.99 [ ( 2 ) 2 ] x x [ R(P R) ( 2 ) 2 x3 1 x x2 3 ] + 1 x3 3

6 285 L. Rong /Journal of Informaion & Compuaional Science 8: 2 (211) Expeced Value Model Assume ha he expeced value crierion is rue. In order o obain a decision we need o find he opimal order period o minimize he expeced value of he oal cos per uni ime. We have he following expeced value model, min E[f(ξ 1, ξ 2, ξ 3,, 2 )] subjec o: (12) > 2 > where is a decision variable. For decision-makers, i is necessary o find he opimal value of, say, which minimizes E[f(ξ 1, ξ 2, ξ 3,, 2 )]. In he model, E[f(ξ 1, ξ 2, ξ 3,, 2 )] Ψ 1 ()d [ ( ) R(P R) ( 2 ) 2 Φ 1 1 () Φ 1 2 () + 1 ] Φ 1 3 () d R(P R) ( 2 ) 2 [ R(P R) ( 2 ) 2 Φ 1 1 ()d + R(P R) (1 Φ 1 (x))dx Φ 1 Φ 1 2 ()d + 1 ] (1 Φ 2 (x))dx + 1 So he model (12) is equivalen o [ ] R(P R) ( 2 ) 2 min (1 Φ 1 (x))dx (1 Φ 2 (x))dx + 1 subjec o: 3 ()d (1 Φ 3 (x))dx (1 Φ 3 (x))dx > 2 > (13) 3.3 Chance-consrained Programming Model Assume ha he pessimisic value crierion is rue. If we wan o minimize he pessimisic value subjec o some chance consrains, he chance-consrained programming is as follows, min f subjec o: M{f(ξ 1, ξ 2, ξ 3,, 2 ) f} > 2 > (14)

7 L. Rong /Journal of Informaion & Compuaional Science 8: 2 (211) where is specified confidence level, and min f is he -pessimisic reurn. In he model, ξ inf () Ψ 1 () [ ] R(P R) ( 2 ) 2 Φ 1 1 () Φ 1 2 () + 1 Φ 1 3 () So he model (14) is equivalen o [ ] R(P R) ( 2 ) 2 min Φ 1 1 () Φ 1 2 () + 1 Φ 1 3 () subjec o: > 2 > (15) 4 Numerical Example Example 1 Assume ha he holding cos, shorage cos and ordering cos are linear uncerain variables, ξ i L(a i, b i ), i 1, 2, 3 he oal cos per period f(ξ 1, ξ 2, ξ 3, ) is linear uncerain variable. Case 1 If using model (13), we can ge he following opimal selling period and order quaniy. 2(a 3 + b 3 ) 2(a 1 + b 1 + a 2 + b 2 ) P R(a 1 + b 1 ) a 2 + b 2 P R Q R 2R(a 3 + b 3 ) (a 1 + b 1 ) 2(a 1 + b 1 + a 2 + b 2 ) a 2 + b 2 P P R Case 2 By using model (15), he opimal selling period and order quaniy are as follows. 2[(1 )a 3 + b 3 ] 2[(1 )(a 1 + a 2 ) + (b 1 + b 2 )] P R[(1 )a 1 + b 1 ] (1 )a 2 + b 2 P R Q R 2R[(1 )a 3 + b 3 ] (1 )a 1 + b 1 2[(1 )(a 1 + a 2 ) + (b 1 + b 2 )] (1 )a 2 + b 2 P P R Example 2 Suppose ha he holding cos, shorage cos and ordering cos are normal wih normal uncerainy disribuion ( ( )) 1 π(e x) Φ(x) 1 + exp, x R. 3σ Where e and σ are real number wih σ >. ξ i N(e i, σ i ), i 1, 2, 3

8 287 L. Rong /Journal of Informaion & Compuaional Science 8: 2 (211) Case 1 By using model (13), he opimal selling period and order quaniy are as follows. 2e1 Re 3 Q 2Re1 R Case 2 If using model (15), we can ge he following opimal selling period and order quaniy. 2(e 3 + σ 3 3ln /π) 1 e 1 + e 2 + (σ 1 + σ 2 ) 3ln /π 1 P R(e 1 + σ 1 3ln /π) e σ 2 3ln /π P R 1 Q R 2R(e 3 + σ 3 3ln e 1 + σ 1 3ln /π 1 5 Conclusions 1 /π) e 3 e 1 + e 2 + (σ 1 + σ 2 ) 3ln e 2 + σ 2 3ln /π 1 1 /π P P R In his paper, he expeced value model and chance-consrained programming model of economic order quaniy for invenory are provided when he holding cos, shorage cos and holding cos are assumed o be uncerain variables. Based on uncerainy heory, he models can be ransformed ino a deerminisic form, and we ge he soluion by 99-mehod. Finally, wo numerical examples are given o illusrae he effeciveness of he models. References [1] Chang S C, Yao J S, Lee H M. Economic reorder poin for fuzzy backorder quaniy. European Journal of Operaional Research. 19 (1998) [2] Chang C T. Exended economic quaniy model under cash discoun and paymen delay, Informaion and Managemen Science. 13(3) (22) [3] Chen S H, Wang C C. Backorder fuzzy invenory model under funcional principle. Informaion Sciences. 95 (1996) [4] Chen Y C. A probabilisic approach for radiional EOQ model, Journal of Informaion Opimizaion Sciences. 24(2) (23) [5] Cheng. An economic order quaniy model wih demand-dependen uni cos, European Journal of Operaional Research. 4(2) (199) [6] Halim K A, Giri B C, Chaudhuri K S. Fuzzy economic order quaniy model for perishable iems wih sochasic demand, parial backlogging and fuzzy deerioraion rae, Inernaional Journal of Operaional Research. 3 (28) [7] Harris, F. W., Operaion and coss, The librarg of Facory Managemen, Vol. V. A. W shaw company, Chicago, 1915, PP [8] Teng J T. On he economic order quaniy under condiions of permissible delay in paymens, Operaional Research Sociey. 53 (22)

9 L. Rong /Journal of Informaion & Compuaional Science 8: 2 (211) [9] Teng J T, Yang H L, Ouyang L Y. On an EOQ model for deerioraing iems wih ime-varying demand and parial backlogging [J]. Journal of he Operaional Research Sociey. 54 (23) [1] Lee H M, Yao J S. Economic producion quaniy for fuzzy demand quaniy and fuzzy produion quaniy. European Journal of Operaional Research. 19 (1998) [11] Liberaore M J. The EOQ model under sochasic lead ime, Operaions Research. 27 (1979) [12] Lin D C, Yao J S. Fuzzy economic producion for producion invenory. Fuzzy Ses and Sysem. 111 (2) [13] Liu B. Uncerainy heory, 2nd ediion, Springer-Verlag, Berlin, 27 [14] Liu B. Uncerainy heory: A branch of mahemaics fo modeling human uncerainy, Springer- Verlag, Berlin, 21 [15] I Moon, S Lee. The effecs of inflaion and ime-value of money on an economic order quaniy model wih a random produc life cycle, European Journal of Operaional Research. 125(3) (2) [16] Ouyang L Y, Yao J S. A minimax diribuion free procedure for mixed invenory model involving variable lead ime wih fuzzy demand. Compuers and Operaions Research. 29 (22) [17] Park K S. Fuzzy se heoriic inerpreaion of economic order quaniy. IEEE Transacions on Sysems, Man and Cybemieics. 17(6) (1987) [18] Parler M, Wang D. Diversificaion under yield randomness in invenory models, European journal of operaional research. 66(1) (1993) [19] S. K. Goyal. Economic Order Quaniy under Condiions of Permissible Delay in Paymens, Operaional Research Sociey. 36(4) (1985) [2] Roy T K, Maii M. A fuzzy EOQ model wih demand dependen uni cos under limied sorage capaciy, European Journal of Operaional Research. 99 (1997) [21] Soumen Bag, Debjani Chakrabory, A.R. Roy. A producion invenory model wih fuzzy random demand and wih flexibiliy and reliabiliy consideraions. Compuers and Indusrial Engineering. 56(1) (29) [22] Vijayan T, Kumaran M. Fuzzy economoic order ime models wih random demand, Inernaionl Journal of Approximae Reasoning. 5 (29) [23] Vujosevic M, Perovic D, Perovic R. EOQ foumula when invenory cos is fuzzy. Invenory Journal of Producion Ecnomics. 45 (1996) [24] Wang X, Tang W, Zhao R. Fuzzy economic order quaniy invenory models wihou backordering, Tsinghua Science and Technology. 21(1) (27) [25] Wang X, Tang W, Zhao R. Random fuzzy EOQ model wih imperfec qualiy iems, Fuzzy Opimal Decision Making. 6 (27) [26] Wang X, Tang W, Xue F. Fuzzy random EOQ model wih imperfec qualiy iems, Machine Learning and Cyberneics. 26 [27] Yao J S, Chang S C, Su J S. Fuzzy invenory wihou backorder for fuzzy order quaniy and fuzzy oal demand quaniy. Compuers and Operaions Research. 27 (2) [28] Yu C Y, Zhao X N e al. Exended newsboy problem based on fuzzy random demand, Sysems Engineering. 24(9) (26) 13-17

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