Multi-Item EOQ Model with Varying Holding Cost: A Geometric Programming Approach

Transcription

1 Intenational Mathematical Foum, Vol. 6, 0, no. 3, Multi-Item EOQ Model with Vaying Holding Cost: A Geometic Pogamming Appoach K. A. M. Kotb () and Hala A. Fegany () () Depatment of Mathematics and Statistics, Faculty of Science Taif Univesity, Taif, Saudi Aabia () Depatment of Mathematical Statistics, Faculty of Science Tanta Univesity, Tanta, Egypt Abstact: A multi-item inventoy model with deceasing holding cost, subject to linea and non-linea constaints is consideed in this pape. The vaying holding cost is consideed to be a continuous function of poduction quantity. An analytical solution of the economic poduction un size is deived using simple technique called the geometic pogamming appoach. Some special cases ae deduced, a numeical example is pesented to illustate how the closed-fom optimal solution of a given model is deived. Keywods: Economic poduction quantity, geometic pogamming method, vaying holding cost, numbe of odes, ate of poduction.. Intoduction The constained multi-item inventoy model had been teated by many eseaches. Maloney and Klein [6] discussed constained multi-item inventoy systems: an implicit appoach using algoithms method. Abou El-Ata and Kotb [] developed a cisp inventoy model unde two estictions. The pionee wok began by Cheng [], who studied an EOQ model with demand-dependent unit cost of single-item using geometic pogamming

2 36 K. A. M. Kotb and H. A. Fegany appoach. Othe elated studies wee witten by Juneau and Coates [4], Jung and Klein [5], Teng and Yang [8] and Mandal et. al. [7]. This pape examines the n-item inventoy model with deceasing vaying holding cost unde linea and non-linea constaints which ae assumed binding. The objective is to minimize the total cost function based on the values of demand ate, ate of poduction, unit cost, inventoy caying cost and ode quantity ove a long peiod of time. Also, the optimal ode quantity of each item is deived. A geometic pogamming appoach, which is moe poweful than the Lagangian multiplies method is used to solve the esulting models. Then some special cases can be deduced. A numeical example is solved to illustate the model.. Model and assumptions Fo ou model, we define the following notation: K the limitation on the total numbe of odes, K the limitation on the total holding cost, D the annual demand ate fo the th item, d the annual ate of poduction fo the th item, C p the puchase (poduction) cost fo the th item, C o the setup cost fo the th item, n numbe of diffeent items caied in inventoy, Q the poduction quantity (a decision vaiable) fo the th item, C h ( Q ) the vaying holding cost fo the th item, and ( Q ),,3, L TC the aveage annual total cost, n. In addition, the following basic assumptions ae adopted fo developing the mathematical model of ou inventoy poblem:

3 Multi-item EOQ model with vaying holding cost 37. Demand ate is unifom ove time.. Shotages ae not allowed. 3. The ate of poduction fo each poduct is finite and constant. 4. The holding cost fo th item is a deceasing continuous function of the poduction quantity C h Q and takes the fom: ( Q ) + β Q,,,3, L n α () whee α > 0 and β > 0 ae eal constants selected to povide the best fit of the estimated cost function, since C h ( Q ) must be non-negative. The objective is to minimize the annual elevant total cost (i. e. the sum of poduction, setup and inventoy caying costs) which, accoding to the basic assumptions of the EOQ model, is: D D TC(Q ) DCp + Co + ( )QCh (Q ) () Q d n Substituting () into () yields: n n D Q D p Co + ( )( + βq ) Q d TC(Q ) D C + α (3) The constaint set can be stated as follows: n D Q K and n ( D d )Q C h K, (4) whee K and K set limits on total numbe of odes and total holding cost espectively. n D The tem D Cp + β ( ) is constant and hence can be ignoed. d To solve this pimal function which is a convex pogamming poblem, let us wite it in the following simplified vesion of equation (3):

4 38 K. A. M. Kotb and H. A. Fegany n D mintc Co + α D Q, (5) Q Subject to: n D n h KQ and C Q D K, D d D (6) Applying the geometic pogamming technique to elations (5) and (6), the enlaged pedual function could be witten as: n D ( ) Co G Q n D Co α D Q α D D KQ3 D K3 3 3 C hqd K4 C hd K4 + + Q 3 4 (7) 4 4 whee the dual vaiable vecto j, 0 j < <, j,,3,4,,,3,...,n is abitay and can be chosen accoding to convenience subject to the nomality condition: + (8) e choose such that the exponent of Q is zeo, thus making the ight hand side of (7) independent of the decision vaiable. To do this we equie: (9) 3 4 This is called the othogonality condition, which togethe with (8) ae two linea equations in the fou unknowns having infinite numbe of solutions. Howeve the poblem is to select the optimal solution of the weights j. Solving equations (8) and (9), we get: ( ) and ( ) (0) Substituting and in equation (7), then the dual function is given by:

5 Multi-item EOQ model with vaying holding cost 39 g( n 3,4) To find D Co 3 + ( 3+ 4) (+ 3 4) αd 4 ( C 4 hd K 4 D K () 3 3 and 4 which maximize g ( 3, 4 ),taking the logaithm of both sides of (), then taking the patial deivatives with espect to 4 espectively, and setting each equal to zeo and simplifying gives: α D D C o D 3 K e and () and D C o D C h D K e α Multiplying elation () by elation (3), we have: 3 4 D D C K h K e Substituting 4 and 3 into elations () and (3) espectively, we get: (3) ( A B ) B A B 0, j 3, f(j ) j + j j j j j j (4) whee: A α DD KK e, B 3 α D D DC o Ke and B4 DCo D Ch D K e α Now we have: f(0) A B j < 0, j 3,4, and f() A ( + B j ) > 0, A <<, B j <<, j 3,4 4 This means that thee exists a oot j ( 0, ), j 3, 4. Any method, such as the tial and eo, could be used to calculate these oots. Howeve, we shall fist veify that any oot j, j 3, 4 calculated fom equations (4) maximizes

6 40 K. A. M. Kotb and H. A. Fegany g( j ), j 3,4. This is done by applying the conditions of the second deivative as: lng( 3, 4 ) lng( 3, 4 ) lng( < 0, < 0 and 3,4) > Hence Δ lng(, ) lng(, ) lng( 3, 4 ) 3 4 < Thus, the oots 3 and 4 calculated fom (4) maximize the dual function g(3, 4 ). Hence the optimal solution is j, j,,3,4, whee, ae the solutions of (4) and 3 4, ae calculated by substituting the values of 3 and 4 in (0). To find the optimal economic poduction un size and Peteson s theoem [3] of geometic pogamming as: D Q C o α D Q g(, ) and g(, Solving these elations, the optimal poduction un size is: ) Q, we apply Duffin D C Q o, α D,, 3,..., n (5) Substituting the values of Q in (3) we get: min TC n α D + C o D C p D D + β (6) 3. Special cases Some inventoy models can be obtained as special cases: Case ()

7 Multi-item EOQ model with vaying holding cost Let K 0 ( A and B 0 ). This is the multi-item inventoy model with vaying holding cost unde one estiction which is non-linea. Case () 3 3 Let K 0( A and B 0 ). this is the multi-item inventoy model with vaying holding cost unde one estiction which is linea. Case (3) Let K and K 3 and 4 0 ( A B3 B4 0 ). This is unconstained multi-item inventoy model with vaying holding cost. Case (4) Let β C ( Q ) α 0 h cons tan t model with constant holding cost unde two estictions. Case (5) C Let (Q. This is the multi-item inventoy K,K and β 0 ( 3 4 0, ), and ) α constant. This is the classical economic poduction un size model. h 4. Numeical illustation e shall compute the decision vaiable Q whose values ae to be detemined to minimize the annual elevant total cost fo thee items (n 3) and diffeent values of β. The paametes of the model ae shown in Table : D d C o C p α 00 Units 300 Units $ 00 $ 0 $ 070 Units 00 Units $ 40 $ 08 $ Units 00 Units $ 00 $ 05 $ Table Assume the total set numbe of odes each yea and the total holding cost ae given by K and $ ft K espectively.

8 4 K. A. M. Kotb and H. A. Fegany It follows that the optimal values of poduction batch quantity Q, the vaying holding cost and the minimum annual total cost ae given in Table fo each value of β : b Q Q Q 3 Ch ( Q ) C h ( Q ) C h 3 ( Q 3 ) min TC Table The above esults ae veified gaphically in Figues, and 3. mintc β Figue

9 Multi-item EOQ model with vaying holding cost 43 C (Q) h C (Q) C (Q) C (Q3) h h h β Figue C (Q) h C (Q) C (Q) C (Q3) h h h Figue 3 mintc 5. Conclusion In this pape, the authos obtained the analytical solution of multi-item economic poduction un size inventoy model with vaying holding cost unde linea and non-linea constaints which ae assumed binding. The optimal ode quantity Q is evaluated and the minimum annual total cost min TC is deduced. Five inventoy models ae obtained as special cases, a numeical example is solved.

10 44 K. A. M. Kotb and H. A. Fegany Refeences [] M. O. Abou-EL-Ata, and K. A. M. Kotb, Multi-Item EQO Inventoy Model with Vaying Holding Cost unde two Restictions: A Geometic Pogamming Appoach, Poduction Planning & Contol, 6 (997), [] T. C. E. Cheng, An Economic Ode Quantity Model with Demand-Dependent Unit Cost, Euopean Jounal Of Opeational Reseach, 40(989), [3] R. J. Duffin, E. L. Peteson and C. Zene, Geometic Pogamming Theoy and Application, John iley, New Yok, 967 [4] J. Juneau and E. R. Coates, An Economic Ode Quality Model Fo Time- Vaying Demand, Jounal of Moden Engineeing, (00) [5] H. Jung and C. M. Klein, Optimal Inventoy Policies unde Deceasing Cost Functions via Geometic Pogamming, Euopean Jounal of Opeational Reseach, 3 (00), [6] B. M. Maloney and C. M. Klein, Constained Multi-Item Inventoy Systems: An Implicit Appoach, Computes Ops. Res. 6.(993), [7] N. K. Mandal, T. K. Roy and M. Maiti, Inventoy Model of Deteioated Items with a Constaints: A Geometic Pogamming Appoach, Euopean Jounal of Opeational Reseach, 73 (006),99-0. [8] J. T. Teng and H. L. Yang, Deteministic Inventoy Lot-Size Models with Time-Vaying Demand and Cost unde Genealized Holding Costs, Infomation and Management Sciences, (007), 3-5. Received: Octobe, 00