Constrained Probabilistic Economic Order Quantity Model under Varying Order Cost and Zero Lead Time Via Geometric Programming

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1 Journal of Matematics and Statistics 7 (): -7, 0 ISSN Science Publications Constrained Probabilistic Economic Order Quantity Model under Varying Order Cost and Zero Lead Time Via Geometric Programming Kotb Abd-El-Hamid Mamoud Kotb and Huda Moamed Hamid Al-Sanbari Department of Matematics and Statistics, Faculty of Science, Taif University, Taif, Saudi Arabia Department of Statistics, Faculty of Science, Princess Nora University for Women, Riyad, Saudi Arabia Abstract: Problem statement: In tis study, we provide a simple metod to determine te inventory policy of probabilistic single-item Economic Order Quantity (EOQ) model, tat as varying order cost and zero lead time. Te model is restricted to te expected olding cost and te expected available limited storage space. Approac: Te annual expected total cost is composed of tree components (expected purcase cost, expected ordering cost and expected olding cost. Te problem is ten solved using a modified Geometric Programming metod (GP). Results: Using te annual expected total cost to determine te optimal solutions, number of periods, maximum inventory level and minimum expected total cost per period. A classical model is derived and numerical example is solved to confirm te model. Conclusion/Recommendations: Te results indicated te total cost decreased wit canges in optimal solutions. Possible future extension of tis model was include continuous decreasing ordering function of te number of periods and introducing expected annual demand rate as a decision variable. Key words: Inventory model, olding costs, storage area, lead time, geometric programming, Economic Order Quantity (EOQ), limited storage space, probabilistic single-item, varying order INTRODUCTION Te simple EOQ model is te most fundamental of all inventory models. It is assumed tat te expected order cost and demand rate are constants. Fabrycky and Banks (967) studied some probabilistic models of te case, were bot demand and procurement lead time are identically and independently rodmen variables distributed. Discussed a simple metod for determining order quantities in oint replenisment of deterministic demand. Unconstrained probabilistic inventory problem wit constant cost units as been treated. Ben-Daya et al. (006) presented integrated inventory control and inspection policies wit deterministic demand. Also, Abou-El-Ata and Kotb (997) developed a crisp inventory model under two restrictions. Teng and Yang (007) studied deterministic inventory lot-size models wit time-varying demand and cost under generalized olding costs. Oter related studies are presented by Hadly and Witin (96); Ceng (989); Jung and Klein (00); Das et al. (000) and Mandal et al. (006). An optimal inventory policy for items aving linear demand and variable deterioration rate wit trade credit as been discussed by Sarbit and Ra (00). Recently, EL-Sodany (0) presented periodic review probabilistic inventory system wit zero lead time under constraint and varying olding cost. Also, Kotb and Fergany (0) discussed multi-item EOQ model wit varying olding cost: a geometric programming approac. In tis study, we ave proposed constrained probabilistic single-item EOQ model wit varying order cost and zero lead time. Te varying order cost is continually increasing function of number of periods per inventory cycle. Te constraints are proposed to be te expected olding cost and te expected available limited storage space. Te optimal number of periods, te optimal maximum inventory level and te minimum expected total cost per period are obtained using a modified geometric programming metod. Finally, a numerical example is used to confirm te results. Assumptions and notations: Te following assumptions are made for developing te model: Corresponding Autor: Kotb Abd-El-Hamid Mamoud Kotb, Department of Matematics and Statistics, Faculty of Science, Taif University, Taif, Saudi Arabia Tel: Fax:

2 J. Mat. & Stat., 7 (): -7, 0 Demand rate is random variable aving a known probability distribution Lead time is zero Sortages are not allowed Review of te stock level is made every N period Ordering cost C 0(N) α + βn, α > 0, β 0 i is continuous increasing function of te number of periods. Were α and β are real constants selected to provide te best fit of te estimated cost function Te minimization of te expected total cost is te obective In addition, te following notation is adopted for developing te model: α C DN min TC C D C Dv N Subect to: p + + β + + () CDN and SDN () K K Te term Cp D + β + C Dv is constant and ence can be ignored. Applying Duffin et al. (967) results of geometric programming tecnique on () and (), te enlarged predual function can be written in te form Eq. 5: C C p C o Holding cost. Purcase cost. Ordering cost. C o (N) Varying order cost per period. D Expected annual demand rate. K Limitation on te expected olding cost. K Limitation on te storage area. N Number of periods. N* Optimal number of periods. Q m Maximum inventory level. Q* m Optimal maximum inventory level. S Available storage area. TC Expected total cost. Model formulation and analysis: Te annual expected total cost is composed of tree components (expected purcase cost, expected ordering cost and expected olding cost) according to te basic assumptions and notation of te EOQ model provided by Eq. Fabrycky and Banks (967): TC C (N) C D[v + N] N o Cp D + + () w w w w N W W W W α C DN C DN SDN G(W) NW W KW KW W W W W α C D C D SD W W KW KW Te restrictions on te expected olding cost and By solving Eq. 6, we ave Eq. 7: te expected storage area are te following two conditions Eq. : + W + W W and CDN (7) K and SDN K () W W W In order to solve tis primal function wic is a convex programming problem, it can be rewritten in te Substituting W and W in Eq. 5, ten te dual following form Eq. and : function is Eq. 8: (5) were, W W,J,,,(0 < W <,) are te weigts and could be easily deduced from Equation 5 troug te use of te following normal and ortogonal conditions Eq. 6: W + W and (6) W + W + W + W 0 Tese are two linear equations in four unknowns aving an infinite number of solutions. However, te problem is to select te optimal solution of te weigts W,0 < W <,J,,,.

3 J. Mat. & Stat., 7 (): -7, 0 W W + W + W + W + W α C D g(w ) + W + W W W C D SD KW KW (8) In order to find te optimal W and W wic maximize g(w, W ), te logaritm of bot side of Eq. 8 and te partial derivatives were taken relative to W and W, respectively. Setting eac of tem to equal zero and simplifying, we get Eq. 9 and 0: α W W C D C D W W ekw + + and: α W W SD C D W W ekw + + (9) (0) Multiplying relation (9) by te inverse of relation (0), we find Eq. : W W C K () SK Substituting W and W into relations (9) and (0), respectively, we ave Eq. : f (W ) W + C W + B W B C 0,,, i, () i i i i i Were: αc D αs D B, B e K e C K derivative to ln g(w, W ) wit respect to W and W ), respectively, wic is always negative. Tus, te roots W * and W * calculated from Eq. maximize te dual function g(w, W ). Hence, te optimal solutions are W * and W * of Eq., respectively. W * and W * are evaluated by substituting te value of W * and W * in expression (7). To find te optimal number of periods N * and te optimal maximum inventory level Q m *, we applied te results of Duffin et al. (967) for geometric programming as indicated blow: α C * DN * W g(w ) and Wg(W ) N By solving tese relations, te optimal number of periods is given by Eq. : * * αw α( W W ) N * C DW C D( + W + W ) () and te optimal maximum inventory level Q * m is Eq. : * α m + Q DN g(n ) Dv D( W W ) C ( + W + W ) () By substituting te value of N * in relation (), we get te minimum expected total cost as Eq. 5: αcd min TC β + (Cp + vc )D + (5) W W ) As a special case, we assume β 0 and k i C o (N), I,. Tis is te probabilistic single-item inventory model wit constant order cost and witout any restrictions, wic is consistent wit te results of Fabrycky and Banks (967). CK SK C C C K + SK C K + SK An illustrative example: Te decision variables (te optimal number of periods N * and te optimal maximum inventory level Q * m ) sould be computed to minimize te annual relevant expected total cost. Assume te parameters of te inventory model as: It is clear tat f i (0)<0 and f i ()>0, i,, wic means tat tere exists a root W ε(0,),,. Te trial and error approac can be used to find tese roots. However, we sall first verify any root W *,, C o $per procurement, calculated from Eq. to maximize f i (W ), I,,,, respectively. Tis was confirmed by te second C $0.05 per unit per period,c p $5 per unit 5

4 J. Mat. & Stat., 7 (): -7, 0 Table : Te optimal results of different values of α and β C o(n*) min TC β α N Q m D unit per period, S 50 cubic unit per item, v K $000 per unit and K 00 cubic units of space Te Optimal results of different values of α and β are sown in Table. CONCLUSION Tis work investigated ow ordering cost function, two constraints and geometric programming approac affect te probabilistic EOQ model. Ordering cost function was assumed to depend on number of periods. In addition, te constraints were expected olding cost and expected available limited storage space. A geometric programming approac was devised to determine te optimal solution for probabilistic EOQ, number of periods, maximum inventory level and minimum expected total cost per period instead of te traditional Lagrangian metod. Finally, a classical model is derived and numerical example is solved to confirm te model. Te results indicated tat te total cost decreased wit canges in α,β,n * and Q m *. Possible future extension of tis work was include continuous decreasing ordering function of te number of periods and introducing expected annual demand rate as a decision variable. REFERENCES Abou-EL-Ata, M.O. and K.A.M. Kotb, 997. Multiitem EOQ inventory model wit varying olding cost under two restrictions: A geometric programming approac. Product. Plann. Control, 8: Ben-Daya, M., S.M. Noman and M. Hariga, 006. Integrated inventory control and inspection policies wit deterministic demand. Comput. Operat. Res., : DOI: 0.06/.cor Ceng, T.C.E., 989. An economic order quantity model wit demand-dependent unit cost. Eur. J. Operat. Res., 0: DOI: 0.06/077-7(89)90- Das, K., T.K. Roy and M. Maiti, 000. Multi-item inventory model wit quantity-dependent inventory costs and demand-dependent unit cost under imprecise obective and restrictions: A geometric programming approac. Product. Plann. Control, : DOI: 0.080/ Duffin, R.J., E.L. Peterson and C.M. Zener, 967. Geometric Programming: Teory and Application. st Edn., Jon Wiley and Sons, New York, ISBN: 07700, pp: 78. EL-Sodany, N. H., 0. Periodic review probabilistic multi-item inventory system wit zero lead time under constraint and varying olding cost. J. Mat. Stat., 7: -9. DOI: 0.8/mssp.0..9 Fabrycky, W.J. and J. Banks, 967. Procurement and Inventory Systems: Teory and Analysis. st Edn., Reinold Publising Corporation, USA., ISBN-0: , pp: 56. Hadly, G. and T.M. Witin, 96. Analysis of Inventory Systems. st Edn., Prentice-Hall, Englewood Cliffs, New Jersey, ISBN-0: 0095, pp: 58. Jung, H. and C.M. Klein, 00. Optimal inventory policies under decreasing cost functions via geometric programming. Eur. J. Operat. Res., : DOI: 0.06/S077-7(00) Kotb, K.A.M. and H.A. Fergany, 0. Multi-Item EOQ model wit varying olding cost: A geometric programming approac. Int. Mat. Forum, 6: 5-. Mandal, N.K. T.K. Roy and M. Maiti, 006. Inventory model of deteriorated items wit a constraint: A geometric programming approac. Eur. J. Operat. Res., 7: DOI: 0.06/.eor

5 J. Mat. & Stat., 7 (): -7, 0 Sarbit, S. and S.S. Ra, 00. A stock dependent economic order quantity model for perisable items under inflationary conditions. Am. J. Econ. Bus. Admin., : 7-. DOI: 0.8/aebasp Teng, J.T. and H.L. Yang, 007. Deterministic inventory lot-size models wit time-varying demand and cost under generalized olding costs. Inform. Manage. Sci., 8: -5. 7

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