International Journal of Mathematical Archive-3(12), 2012, Available online through ISSN

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1 Intenational Jounal of Mathematical Achive-3(), 0, Available online though ISSN STATISTICAL QUALITY CONTROL OF MULTI-ITEM EOQ MOEL WITH VARYING LEAING TIME VIA LAGRANGE METHO K. A. M. Kotb, Sayed A. Zaki, Zenab M. Elakkad and S. E. Albenday epatment of Mathematics and Statistics, Faculty of Science, Taif Univesity, Taif, Saudi Aabia. (Received on: 4--; Revised & Accepted on: 3--) ABSTRACT The goal of this pape is to confim that the poduction pocess is in contol using statistical quality contol pocess with subgoup anges. The analytical solution of the economic ode quantity model of multiple items with vaying leading time using Lagangian multiplies is deived. The vaying leading time cashing cost is consideed to be continuous function of leading time. The model is esticted to the budget inventoy investment. The optimal ode quantity is deduced as a decision vaiable. Finally the model is illustating by applied example and the aveage of the subgoup anges appoach is used to confim that the poduction pocess is in contol. Keywods: Inventoy model, Lagangian multiplies, economic ode quantity (EOQ), multiple items, leading time, statistical quality contol, budget inventoy investment.. INTROUCTION Many eseaches have consideed the single item economic ode quantity inventoy models unde one o two constaints such as the maximum budget inventoy investment, the maximum waehouse space o the aveage inventoy level. They solved these poblems eithe by the classical Lagangian multiplies method o the method of "equal ode inteval fo each item" as suggested by Page and Paul [5]. They poposed a heuistic appoach fo allocating n poducts ove m goups. In fact thei method is always significantly good except fo lage numbe of poducts as given in Goyal [4, 5, ]. Shawky and Abou-El-Ata [] solved a constained poduction lot size model with tade policy by Geometic Pogamming and Lagangian methods. Othe elated woks wee witten by Kun-Chan [], Juneau and Eyle [7], as and Maiti [], Mandal and Pal [3], Mehta and Shah [4] and Teng [7]. The poblem of inventoy models involving lead time as a decision vaiable have been succinctly descibed by Ben- aya and Abdul Raouf []. Recently, Kotb and Fegany [9, 0] deived the analytical solution of the economic ode quantity model of multiple items with demand-dependent unit cost, leading time and vaying holding cost using geometic pogamming appoach. The statistical quality contol fo pobabilistic and deteministic models is studied by Kotb [8], El-Wakeel [3] and Kotb et al []. The aim of this pape is to discuss statistical quality contol and inventoy policy of deteministic multi-item economic ode quantity model with vaying leading time unde linea constaint which is assumed bending. The optimal ode quantity of each item was obtained as decision vaiable using Lagangian method. In the final the aveage of the subgoup anges appoach was used to confim that the poduction pocess is in contol.. BASIC NOTATIONS AN ASSUMPTIONS To constuct the model of this poblem, we define the following vaiables: C h Unit holding (inventoy caying) cost pe item pe unit time, Fo the th item. C o Odeing cost, Fo the th item. C p Unit puchase (poduction) cost, Fo the th item. CL Contol limit. Annual demand ate, Fo the th item. K Limitation of budget inventoy investment. L Leading ate time, Fo the th item. Coesponding autho: K. A. M. Kotb epatment of Mathematics and Statistics, Faculty of Science, Taif Univesity, Taif, Saudi Aabia. Intenational Jounal of Mathematical Achive- 3 (), ec

2 K. A. M. Kotb, Sayed A. Zaki, Zenab M. Elakkad and S. E. Albenday/ STATISTICAL QUALITY CONTROL OF MULTI-ITEM EOQ MOEL WITH VARYING LEAING TIME VIA LAGRANGE METHO /IJMA- 3(), ec.-0. LCL Lowe contol limit. n Numbe of diffeent items caied in inventoy. Q Poduction (ode) quantity batch (a decision vaiable), Fo the th item. Q Optimal ode quantity, Fo the th item. R i Subgoup anges. R Aveage of subgoup anges (CL). S R Vaiance of subgoup anges. SS k σ L Safety stock, whee k is known as safety facto and σ is standad deviation. TC ( Q ) Aveage annual total cost. Fo the th item. UCL Uppe contol limit. In addition, the following basic assumptions about the model ae made: () emand ate is unifom ove time. () Time hoizon is finite. (3) Shotages ae not allowed. (4) Lead time cashing cost is elated to the lead time by a function of the fom: ( ) α,,,3,, R L L n () > whee α 0 is eal constant selected to povide the best fit of the estimated cost function. (5) Ou objective is to confim that the poduction pocess is in contol. 3. MATHEMATICAL MOEL: The annual elevant total cost was composed of fou components (poduction, ode, inventoy caying and lead time cashing costs) accoding to the basic notations and assumptions of the EOQ model is: n Q TC (Q ) Cp + Co + + kσ L Ch + R( L ) () Q Q To detemine the optimal values of Q we Substitute () in () obtaining: n + Q + TC (Q + σ + ) Cp Co k L Ch α L (3) Q Q The following constaint can be stated as: n C p Q K, Q > 0 fo all (4) whee K is the limitation of budget inventoy investment. In ode to solve this pimal function which was a convex pogamming poblem, it can be ewitten in the following fom: n + Q + Min TC(Q + σ + ) Cp Co k L Ch α L (5) Q Q n subject to Cp Q K This is a minimization poblem in n vaiables with single constaint. Hence it can be solved by the Lagangian multiplie appoach. The Lagangian function associated with poblem (5) can be estated as: n n Q L (Q, λ) Cp + Co + + kσ L Ch + α L + λ K CpQ () Q Q whee λ is non-positive Lagangian multiplie. 0, IJMA. All Rights Reseved 480

3 K. A. M. Kotb, Sayed A. Zaki, Zenab M. Elakkad and S. E. Albenday/ STATISTICAL QUALITY CONTROL OF MULTI-ITEM EOQ MOEL WITH VARYING LEAING TIME VIA LAGRANGE METHO /IJMA- 3(), ec.-0. Taking the patial deivative of Lagange function with espect to Q at Q Q, and then setting equal to zeo, gives: L Co + Ch α L λ Cp 0 (7) Q Q Q Q Q Simplifying equation (7), we obtain the optimal ode quantity: ( Co + α L ) Q (8) Ch λ Cp We shall veify that the Q minimize TC (Q ). This is done by showing that the second deivative of Lagange function with espect to Q is always positive: L Co + α L Q 3 3 Q Q Q Q 0 (9) > Substituting the values of Q in elation (3), we deduce the minimum total cost as: ( )( ) ( ) + α n + + α λ + Co L Min TC (Q ) + σ Cp Co L Ch Cp k L C (0) h ( ) C + h Cp As special case, we assume L 0 R(L) 0 and K. This is the unconstained classical multi-item inventoy model. 4. STATISTICAL QUALITY CONTROL The decision vaiables Q should be computed hose values ae to be detemined to minimize the total cost and to confim that the poduction pocess is in contol fo thee items (n 3). The paametes of the model ae shown in TABLE : L C o C p C h α 5 Units 8 Weak $ 00 $ 0 $ 0.8 $ 5 Units 4 Weak $ 40 $ 08 $ 0.5 $ 3 5 Units Weak $ 00 $ 05 $ 0.3 $ 3 TABLE Assume that the standad deviation σ units/weak and k. The optimal esults of the poduction batch quantity Q, the minimum annual total cost and subgoup anges ae given in TABLE fo each values of λ : λ Q Q Q Q Min TC R i TABLE 0, IJMA. All Rights Reseved 4803

4 K. A. M. Kotb, Sayed A. Zaki, Zenab M. Elakkad and S. E. Albenday/ STATISTICAL QUALITY CONTROL OF MULTI-ITEM EOQ MOEL WITH VARYING LEAING TIME VIA LAGRANGE METHO /IJMA- 3(), ec.-0. In ode to study the statistical quality contol of the model, applying contol limit (CL) appoach when σ is unknown as: The aveage of the subgoup anges (CL) is: R i i R The standad deviation of the subgoup anges is: ( R i R) i SR The lowe contol limit is LCL R 3SR , the aveage of the subgoup anges is R and the uppe contol limit is UCL R + 3SR It is clea that LCL < R < UCL. Theefoe the poduction pocess is in contol. 5. CONCLUSION This pape is devoted to study statistical quality contol fo multi-item inventoy model that consides ode quantity as decision vaiable. An analytical solution of the EOQ model with vaying leading time and one estiction is deived using Lagangian appoach. Finally, we used the optimal ode quantity of each of the 3 items and subgoup anges method to investigate quality contol of the poduction pocess. REFERENCES [] M. Ben-aya and Abdul Raouf, Inventoy Models Involving Lead Time as a ecision Vaiable, J. Opl. Res. Soc., Vol. 45, No. 5(994), pp [] K. as and M. Maiti, Inventoy of a diffeential item sold fom two shops unde single management with shotages and vaiable demand, Applied Mathematical Modelling, 77(003), pp [3] M. F. El-Wakeel, Quality contol fo pobabilistic (Q, ) inventoy model with vaying ode cost and continuous lead time demand unde the annual holding constaint, Intenational Jounal of Mathematical Achive, (7) (0), [4] S. K. Goyal, Optimal odeing policy fo a multi-item single supplie system, Opeational Reseach Quately, Vol. 5(975), pp [5] S. K. Goyal, A note on multi-poduct inventoy situations with one estiction, Jounal of the Opeational Reseach Society, Vol. 9, No. 3(978), pp [] S. K. Goyal, Economic ode quantity unde conditions of pemissible delay in payment, Jounal of the Opeational Reseach Society, Vol. 3, No. 4(985), pp [7] J. Juneau and R. C. Eyle, An economic ode quantity model fo time-vaying demand, The Intenational Jounal of Moden Engineeing, (00), pp. -0. [8] K. A. M. Kotb, Statistical quality contol fo constained multi-item inventoy lot-size model with inceasing vaying holding cost via geometic pogamming, Intenational Jounal of Mathematical Achive, (4) (0), [9] K. A. M. Kotb and H. A. Fegany, Multi-item EOQ model with vaying holding cost: A geometic pogamming appoach, Int. Math. Foum, (0), [0] K. A. M. Kotb and H. A. Fegany, Multi-item EOQ model with both demand-dependent unit cost and vaying leading time via geometic pogamming, Applied Mathematics, (0), [] K. A. M. Kotb, H. M. Genedi and S. A. Zaki, Quality contol fo pobabilistic single-item EOQ model with zeo lead time unde two estictions: A geometic pogamming appoach, Int. Math. Foum, (0), [] Kun-Chan Wu, eteministic inventoy model fo items with time vaying demand, Weibull distibution deteioation and shotages, Yugoslav Jounal of Opeational Reseach,, No. (00), pp. -7. [3] B. Mandal and A. K. Pal, Ode level inventoy system with amp type demand ate fo deteioating items, Jounal of Intenational Mathematics, (998), pp [4] N. Mehta and N. Shah, An Inventoy model fo deteioating items with exponentially inceasing demand and shotages unde inflation and time discounting, Investigacao Opeational, 3(003), pp [5] E. Page and R. Paul, Multi-poduct inventoy situations with one estiction, Opeational Reseach Quately, Vol. 7(97), pp , IJMA. All Rights Reseved 4804

5 K. A. M. Kotb, Sayed A. Zaki, Zenab M. Elakkad and S. E. Albenday/ STATISTICAL QUALITY CONTROL OF MULTI-ITEM EOQ MOEL WITH VARYING LEAING TIME VIA LAGRANGE METHO /IJMA- 3(), ec.-0. [] Shawky A. I. and M. O. Abou-El-Ata, Constained poduction lot-size model with tade cedit policy: a compaison geometic pogamming appoach via Lagange, Poduction Planning and contol, (7) (00), [7] J. T. Teng, A deteministic inventoy eplenishment model with a linea tend in demand, Opeat. Res. Lett. 9(99), pp Souce of suppot: Nil, Conflict of inteest: None eclaed 0, IJMA. All Rights Reseved 4805