An Inventory Model for Two Warehouses with Constant Deterioration and Quadratic Demand Rate under Inflation and Permissible Delay in Payments

Transcription

1 Ameican Jounal of Engineeing Reseach (AJER) 16 Ameican Jounal of Engineeing Reseach (AJER) e-issn: p-issn : Volume-5, Issue-6, pp Reseach Pape Open Access An Inventoy Model fo wo Waehouses with Constant Deteioation and Quadatic Demand Rate unde Inflation and Pemissible Delay in Payments U. B. Gothi 1, Pachi Saxena, Kitan Pama 3 1 Head & Asso. Pof., Dept. Of Statistics, St. Xavie s College (Autonomous), Ahmadabad, Gujaat, India. Reseach Schola, Dept. Of Statistics, St. Xavie s College (Autonomous), Ahmadabad, Gujaat, India. 3 Adhyapak Sahayak, Dept. Of Statistics, St. Xavie s College (Autonomous), Ahmadabad, Gujaat, India. ABSRAC: In this pape, we have analysed a two-waehouse inventoy model fo deteioating items with quadatic demand with time vaying holding cost. he effect of pemissible delay in payments is also consideed, which is usual pactice in most of the businesses i.e. puchases ae allowed a peiod to pay back fo the goods bought without paying any inteest. o make it moe suitable to the pesent envionment the effect of inflation is also consideed. Ou objective is to minimize the aveage total cost pe time unit unde the influence of inflation. Numeical examples ae povided to illustate the model and sensitivity analysis is also caied out fo the paametes. Keywods: Inventoy model, wo-waehouse, Deteioation, Quadatic demand. I. INRODUCION he main poblem in an inventoy management is to decide whee to stock the goods. Geneally, when the poducts ae seasonal o the supplies povide discounts on bulk puchase, the etailes puchase moe goods than the capacity of thei owned waehouse (OW). heefoe, the excess units ove the fixed capacity w of the owned waehouse ae stoed in ented waehouse (RW). Usually, the unit holding chage is highe in ented waehouse than the owned waehouse, as the ented waehouse povides a bette peseving facility esulting in a lowe ate of deteioation in the goods than the owned waehouse. And thus, the fim stoes goods in owned waehouse befoe ented waehouse, but cleas the stocks in ented waehouse befoe owned waehouse. Inventoy models fo deteioating items wee widely studied in the past but the two-waehouse inventoy issue has eceived consideable attention in ecent yeas. Hatley [1] was the fist peson to develop the basic two-waehouse inventoy model. Chung and Huang [5] poposed a two-waehouse inventoy model fo deteioating items unde pemissible delay in payments, but they assumed that the deteioating ate of two waehouses wee the same. An inventoy model with infinite ate of eplenishment with two-waehouse was consideed by Sama [1]. An optimization inventoy policy fo a deteioating items with impecise lead-time, patially/fully backlogged shotages and pice dependent demand unde two-waehouse system was developed by Rong et al. [18]. Lee and Hsu [13] investigated a two-waehouse poduction model fo deteioating items with time dependent demand ate ove a finite planning hoizon. Ealie, in Economic Ode Quantity (EOQ), it was usually assumed that the etaile must pay to the supplie fo the items puchased as soon as the items wee eceived. In the last two decades, the influence of pemissible delay in payments on optimal inventoy management has attacted attention of many eseaches. Goyal [9] fist consideed a single item EOQ model unde pemissible delay in payments. Aggawal and Jaggi [1] extended Goyal s [9] model to the case with deteioating items. Aggawal and Jaggi s [1] model was futhe extended by Jamal et al. [] to conside shotages. Chung and Huang [7] futhe extended Goyal s [9] model to the case that the units ae eplenished at a finite ate unde delay in payments and developed an easy solution pocedue to detemine the etaile s optimal odeing policy. A liteatue eview on inventoy model unde tade cedit is given by Chang et al. [8]. eng et al. [19] developed the optimal picing and lot sizing unde pemissible delay in payments by consideing the diffeence between the selling pice and the puchase cost and also the demand is a function of pice. Fo the elevant papes elated to pemissible delay in payments see Chung and Liao [6], Liao ([14], [15]), Huang and Liao [11]. Recently, Kitan Pama and U. B. Gothi [16] have developed ode level inventoy model fo deteioating items unde time vaying demand condition. Devyani Chatteji and U. B. Gothi [4] have developed w w w. a j e. o g Page 6

2 Ameican Jounal Of Engineeing Reseach (AJER) 16 an integated inventoy model with exponential amelioation and two paamete Weibull deteioation. Ankit Bhojak and U. B. Gothi [3] have developed inventoy models fo amelioating and deteioating items with time dependent demand and inventoy holding cost. Paekh R.U. and Patel R.D. [17] have developed a two-waehouse inventoy model in which they assumed that the demand is linea function of time t. hey took diffeent deteioation ates and diffeent inventoy holding costs in both OW and RW unde inflation and pemissible delay in payments. In this pape, we have tied to develop a two-waehouse inventoy model unde time vaying holding cost and quadatic demand unde inflation and pemissible delay in payments. In the pesent wok we have consideed same deteioation ate and same linea holding cost thoughout the peiod [, ]. In this model t and ae taken as decision vaiables. Numeical examples ae povided to illustate the model and sensitivity analysis of the optimal solutions fo majo paametes is also caied out. he pupose of this study is to make the model moe elevant and applicable in pactice. II. NOAIONS 1. I (t) : Inventoy level fo the ented waehouse (RW) at time t.. I o (t) : Inventoy level fo the owned waehouse (OW) at time t. 3. w : he capacity of the owned waehouse. 4. D(t) : Demand ate. 5. θ(t) : Rate of deteioation pe unit time. 6. R : Inflation ate. 7. A : Odeing cost pe ode duing the cycle peiod. 8. C d : Deteioation cost pe unit pe unit time. 9. C h : Inventoy holding cost pe unit pe unit time. 1. Q : Ode quantity in one cycle. 11. k : Puchase cost pe unit. 1. p : Selling pice pe unit. 13. Ie : Inteest eaned pe yea 14. Ip : Inteest chage pe yea. 15. M : Pemissible peiod of delay in settling the accounts with the supplie 16. t : time at which the inventoy level eaches zeo in RW in two waehouse system. 17. : he length of cycle time. 18. Ci : otal cost pe unit time in the i th case. (i = 1,, 3) III. ASSUMPIONS 1. he demand ate of the poduct is D (t) = a + bt + ct (whee a, b, c > ).. Holding cost is a linea function of time and it is C h = h+t (h, > ) fo both OW and RW 3. Shotages ae not allowed. 4. Replenishment ate is infinite and instantaneous. 5. Repai o eplacement of the deteioated items does not take place duing a given cycle. 6. OW has a fixed capacity W units and the RW has unlimited capacity. 7. Fist the units kept in RW ae used and then of OW. 8. he inventoy costs pe unit in the RW ae highe than those in the OW. IV. MAHEMAICAL MODEL AND ANALYSIS At time t = the inventoy level is S units. Fom these w units ae kept in owned waehouse (OW) and est in the ented waehouse (RW). he units kept in ented waehouse (RW) ae consumed fist and then of owned waehouse (OW). Due to the maket demand and deteioation of the items, the inventoy level deceases duing the peiod [, t ] and the inventoy in RW eaches to zeo. Again with the same effects, the inventoy level deceases duing the peiod [t, ] and the inventoy in OW will also become zeo at t =. he pictoial pesentation is shown in the Figue 1. w w w. a j e. o g Page 63

3 Ameican Jounal Of Engineeing Reseach (AJER) 16 he diffeential equations which descibe the instantaneous state of inventoy at time t ove the peiod [, ] ae given by d I t θ I t a b t c t t t (1) dt d I t θ I t ( t t ) () dt d I ( t ) θ I t a b t ct ( t t ) (3) dt Unde the bounday conditions I (t ) =, I o () = w, and I o () =, solutions of equations (1) to (3) ae given by t t t t t t t t t t t t I t a t t b a θ c b θ cθ aθ t t t b θ cθ (4) θt (5) I t w e t t t t t t t I t a t b a θ c b θ cθ aθ t (6) t b θ cθ V. COSS COMPONENS he total cost pe eplenishment cycle consists of the following cost components. 1) Odeing Cost he opeating cost (OC) ove the peiod [, ] is OC = A (7) ) Deteioation Cost he deteioation cost (DC) ove the peiod [, ] is t t R t R t R t D C C d θ I ( t ) e d t θ I ( t ) e d t θ I ( t ) e d t w w w. a j e. o g Page 64

4 Ameican Jounal Of Engineeing Reseach (AJER) 16 D C C d R 6 a t 6 a t 6 b t 4 b t 4 c t 3c t 6 R 6 a 6 a t 3 b t c t 4 R a b 1 R b c 4 c Rt 3 1 e 3 R a b t c t R a b b t c t c t R b c c t c 5 1 R R 6 a 6 b 4 b 4 c 3c 1 6 R 6 a 6 a 3 b c 4 R a b 1 R b c 4 c R 3 1 e 3 R a b c R a b b c c R b c c c t R w e 1 R 3) Inventoy Holding Cost he inventoy holding cost (IHC) ove the peiod [, t ] is (8) IHC = Holding cost duing the cycle peiod in RW [HC(RW)] + Holding cost duing the cycle peiod in RW [HC(OW)] (9) whee, t Rt H C ( R W ) ( h t ) I ( t ) e d t 1 H C ( R W ) 1 R t h 4 8 R a b t c t 3 6 R a b b t c t c t 4 R b c c t 4 c R Rt e 4 8 R a t b t c t 3 6 R a a t 3 b t b t 4 c t c t 4 R 3a 3 b 4 b t 8 c t 5 c t 4 R b 4 c 5 c t 1 c R 1 a t 6 a t 6 b t 4 b t 4 c t 3c t h R 6 a 6 a t c t 3 b t 3 6 R a b 4 R b c 4 c R R 1 a t 6 a t 6 b t 4 b t 4 c t 3c t R 7 R a 4 R c t 3 6 R b t 7 R a t 7 R a b 4 8 R b c 1 c R t t R t H C (O W ) ( h t ) I ( t ) e d t ( h t ) I ( t ) e d t (1) w w w. a j e. o g Page 65

5 Ameican Jounal Of Engineeing Reseach (AJER) 16 1 t ( R ) e ( R t t 1) h ( R ) h ( R ) w ( R ) 3 4 c R R 4 b 4 8 c ( 1 t ) R 3 6 a 3 6 b ( 1 t ) 3 6 c ( t t ) h R 4 8 a ( 1 t ) 4 b ( t t ) 1 6 c ( 3 t t ) 3 3 a ( t t t ) 1 b ( 3 t t 3 3 t 5 R ) 5 c ( 4 t t 4 4 t 3 ) Rt e R 4 8 b 4 c ( 4 5 t 5 ) R 7 a 4 b ( 3 4 t ) 4 c ( 8 t 5 t ) 6 1 R R 3 6 a ( 3 t ) 3 6 b ( 3 t t ) 1 c ( 1 t 5 t ) a ( 4 t 3 t 4 t ) 8 b ( 9 t 4 t 3 6 t ) 4 R c ( 1 6 t 5 t 4 8 t 3 ) 3 3 a ( t t t t t ) R 1 b ( 3 t t 3 t 3 t t ) 5 c ( 4 t t 4 t 4 t 4 3 t ) 1c 4 c R 4 R b c (1 ) h R a b (1 ) c ( ) 4 8 R (a b c ) R e 4 R ( b 4 c ) 4 R 3a b (3 4 ) c (8 5 ) 6 1 R R a ( ) b (3 ) c ( 4 c ) R (a b c ) 1 c (11) 4) Inteest Eaned: hee ae two cases Case 1: (M ) In this case, inteest eaned is: 1 e M R t IE p I (a b t ct ) t e d t p I R R e R M IE e R b R a 6 c 1 R R M a M b M c 6 c b 3 M c a M b 3 M c (1) Case : (M > ) In this case, inteest eaned is: e R t IE p I [ (a b t ct ) t e d t (a b c ) ( M )] w w w. a j e. o g Page 66

6 Ameican Jounal Of Engineeing Reseach (AJER) 16 1 R e R b 3c R a b 3c 3 R a b c 6 c ar b R 6 c} IE p I R e (a b c ) ( M ) (13) 5) Inteest Payable: hee ae thee cases descibed in figue-1 Case 1: (M t ) In this case, annual inteest payable is: IP t t R t R t R t IP k I 1 p I ( t ) e d t I ( t ) e d t I ( t ) e d t M M t 1 R b c ( 1 M ) 4 R a b ( 1 M ) c ( M M ) R 6 a ( 1 t M ) 3 b ( t M ) c ( t 3 M M ) 4 a t t M M t M RM e R b 1 t 8 t 1 M t 1 M 9 M 4 M 5 1 R c 4 t 3 t 4 M t 4 M M 4 c 1 R b c (1 t ) 4 R a b (1 t ) c ( t t ) Rt e R ( a b t c t ) 4 c M ( R ) t ( R ) w e e k I R 1 R b c ( 1) 4 R a b (1 ) c ( ) R e R ( a b c ) 1 1 R b c ( 1 t ) 4 R a b ( 1 t ) c ( t t ) 5 1 R Rt e 6 R 6 a ( 1 t ) 3 b ( t t ) c ( 3 t t ) a ( t t t ) b ( 3 t t 3 3 t ) 4 4R c ( 4 t t 4 4 t 3 ) 1 p (14) Case : (t M ) In this case, inteest payable is: p M Rt IP k I I ( t ) e d t w w w. a j e. o g Page 67

7 Ameican Jounal Of Engineeing Reseach (AJER) 16 IP k I 1 R p 5 R 3 e 1 R b c 1 4 R a b 1 c 3 6 R b c 1 4 c 1 R b c 1 M 4 R a b 1 M c M M R M e 6 R 6 a 1 M 3 b M M c 3 M M a M M M b 3 3M 3M M 4 4R 4c c M M 4 M (1 5 ) Case 3: (M ) In this case, no inteest chages ae paid fo the item and so IP 3 = (16) Substituting values fom equations (7) to (11) and equations (1) to (16) in equations (17) to (19), the etaile s total cost duing a cycle in thee cases will be as unde: 1 C [ A H C (O W ) H C ( R W ) D C IP IE ] (17) C [ A H C (O W ) H C ( R W ) D C IP IE ] (18) 1 1 C [ A H C (O W ) H C ( R W ) D C IP IE ] (19) 3 3 Ou objective is to detemine the optimum values t * and * of t and espectively so that C i is minimum. Note that t * and * can be obtained by solving the equations C t i C i a n d (i = 1,, 3) () C C C i i i t t * * t t, C i t * * t t, (1) he optimum solution of the equations () can be obtained by using appopiate softwae. he above developed model is illustated by the means of the following numeical example. Numeical Example 1 o illustate the poposed model, an inventoy system with the following hypothetical values is consideed. By taking A = 15, w = 1, a = 8, b =.5, c =., k = 1, p = 15, θ =., h = 1, =.5, R =.6, M = 1, C d = 4, Ip =.15 and Ie =.1 (with appopiate units). he optimal values of t and ae t * = , * = units and the optimal total cost pe unit time C = units. Numeical Example By taking A = 15, w = 1, a = 8, b =.5, c =., k = 1, p = 15, θ =., h = 1, =.5, R =.6, M = 16, C d = 4, Ip =.15 and Ie =.1 (with appopiate units). he optimal values of t and ae t * = , * = units and the optimal total cost pe unit time C = units. Numeical Example 3 By taking A = 15, w = 1, a = 8, b =.5, c =., k = 1, p = 15, θ =., h = 1, =.5, R =.6, M = 5, C d = 4, Ip =.15 and Ie =. (with appopiate units). he optimal values of t and ae t * = , * = units and the optimal total cost pe unit time C = units. w w w. a j e. o g Page 68

8 Ameican Jounal Of Engineeing Reseach (AJER) 16 VI. SENSIIVIY ANALYSIS Sensitivity analysis depicts the extent to which the optimal solution of the model is affected by the changes in its input paamete values. Hee, we study the sensitivity fo total cost pe time unit C with espect to the changes in the values of the paametes A, w, a, b, k, p, θ, h,, R, M, C d, Ip and Ie. he sensitivity analysis is pefomed by consideing vaiation in each one of the above paametes keeping all othe emaining paametes as fixed. able 1: Patial Sensitivity Analysis Based Numeical Example 1 Paamete % t C A w a b c k p h M θ C d Ip Ie w w w. a j e. o g Page 69

9 Ameican Jounal Of Engineeing Reseach (AJER) 16 able : Patial Sensitivity Analysis Based Numeical Example Paamete % t C A w a b c k p h M θ C d Ip Ie w w w. a j e. o g Page 7

10 Ameican Jounal Of Engineeing Reseach (AJER) 16 able 3: Patial Sensitivity Analysis Based Numeical Example 3 Paamete % t C A w a b c p h M θ C d Ie VII. GRAPHICAL PRESENAION Figue w w w. a j e. o g Page 71

11 Ameican Jounal Of Engineeing Reseach (AJER) 16 Figue 3 Figue 4 VIII. CONCLUSIONS Fom the able 1, we obseve that as the values of the paametes a, b, h, and θ incease the aveage total cost also inceases and fo the values of the paametes A, w, c, k, p, M, C d, Ip and I e the aveage total cost deceases. able shows that as the values of the paametes a, b, k, h,, θ, Ip and M incease the aveage total cost also inceases and fo the values of the paametes A, w, c, p, C d and I e the aveage total cost deceases. Fom the able 3, we note that as the values of the paametes a, b, p, h,, M, θ and Ie incease the aveage total cost also inceases and fo the values of the paametes A, w, c and C d the aveage total cost deceases. Fom the Figue, we obseve that the total cost pe time unit is highly sensitive to changes in the values of c, C d, modeately sensitive to changes in the values of b, and less sensitive to changes in the values of A, w, a, k, p, h, M, C d, Ip, I e. Fom the Figue 3 we note that the total cost pe time unit is highly sensitive to changes in the values of c, b,, C d, modeately sensitive to changes in the values of a, h and less sensitive to changes in the values of A, w, p, M, k, Ip, I e. Figue 4 shows that the total cost pe time unit is highly sensitive to changes in the values of b, c, M,, C d, modeately sensitive to changes in the values of a, h and less sensitive to changes in the values of A, w, p, I e. w w w. a j e. o g Page 7

12 Ameican Jounal Of Engineeing Reseach (AJER) 16 REFERENCES [1]. Aggawal S. P., and Jaggi C. K., Odeing Policies of Deteioating Items unde Pemissible Delay in Payments, Jounal of Opeational Reseach Society, 1995, 46, []. A.M.M. Jamal, B.R. Saka. and S. Wang, An odeing policy fo deteioating items with allowable shotages and pemissible delay in payment, Jounal of Opeational Reseach Society, 1997, 48, [3]. Bhojak A. and Gothi U. B., Inventoy Models fo Amelioating and Deteioating Items with ime Dependent Demand and IHC. Intenational Jounal of Management and Social Science Reseach Review, 15, Vol. 1, Issue, [4]. Chatteji, D. A., Gothi U. B. and Pama K. C., An Integated Inventoy Model with Exponential Amelioation and wo Paamete Weibull Deteioation, Jounal of Statistics and Mathematical Engineeing, MA Jounals, 16, Vol., No.. [5]. Chung K. J., and Huang Y. F., he Optimal Cycle ime fo EPQ Inventoy Model Unde Pemissible Delay in Payments, Intenational Jounal of Poduction Economics, 3, 84, [6]. Chung K. J., and Liao J. J., Lot-Sizing Decisions Unde ade Cedit Depending on the Odeing Quantity, Computes and Opeations Reseach, 4, 31, [7]. Chung K. J., and Huang. S., he Optimal Retaile s Odeing Policies fo Deteioating Items with Limited Stoage Capacity Unde ade Cedit Financing, Intenational Jounal of Poduction Economics, 7, 16, [8]. C.. Chang, J.. eng and S.K. Goyal, Inventoy lot sizing models unde tade cedits, Asia Pacific Jounal Opeational Reseach, 8, 5, [9]. Goyal S. K., Economic Ode Quantity unde Conditions of Pemissible Delay in Payments, Jounal of the Opeational Reseach Society, 1985, 36, [1]. Hatley V. R., Opeations Reseach A Manageial Emphasis, 1976, Goodyea, Santa Monica. [11]. Huang K. N., and Liao J. J., A Simple Method to Locate the Optimal Solution fo Exponentially Deteioating Items Unde ade Cedit Financing, Computes and Mathematics with Applications, 8, 56, [1]. K.V.S. Sama, A deteministic inventoy model fo deteioating items with two stoage facilities; Euopean Jounal of Opeational Reseach, 1987, 9, 7 7. [13]. Lee C. C., and Hsu S. L., A wo-waehouse Poduction Model fo Deteioating Inventoy Items with ime-dependent Demands, Euopean Jounal of Opeational Reseach, 9, 194, [14]. Liao J. J., On an EPQ Model fo Deteioating Items Unde Pemissible Delay in Payments, Applied Mathematical Modelling, 7, 31, [15]. Liao J. J., An Inventoy Contol System Unde Defeable Delivey Conditions, Mathematical and Compute Modelling, 8, 47, [16]. Pama, K., Gothi U. B., Ode level inventoy model fo deteioating items unde time vaying demand condition, Sankhaya Vignan, 15, NSV 11, No. 1, 9. [17]. Paekh R. U. and Patel R. D., Deteioating items inventoy models fo two waehouses with linea demand, time vaying holding cost unde inflation and pemissible delay in payments, Intenational Jounal of Mathematics and Statistics Invention, 14, Vol., Issue 5, pp [18]. Rong M., Mahapata N. K., and Maiti M., A wo Waehouse Inventoy Model fo a Deteioating Item with Patially/Fully Backlogged Shotage and Fuzzy Lead ime, Euopean Jounal of Opeational Reseach, 8, 189, [19]. eng J.., Change C.., and Goyal S. K., Optimal Picing and Odeing Policy Unde Pemissible Delay in Payments, Intenational Jounal of Poduction Economic, 5, 97, w w w. a j e. o g Page 73