An optimal policy for a single-vendor single-buyer integrated production}inventory system with capacity constraint of the transport equipment

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1 Int. J. Production Economics 65 (2000) 305}315 Technical Note An optimal policy f a single-vend single-buyer integrated production}inventy system with capacity constraint of the transpt equipment M.A. Hoque, S.K. Goyal* Department of Mathematics, Jahangirnagar University, Savar Dhaka, Bangladesh Department of Decision Sciences & M. I. S., Faculty of Commerce and Administration, Concdia University, 1455 de Maisonneuve Blvd.,-West Montreal, Quebec, Canada H3G 1M8 Received 29 December 1998; accepted 14 June 1999 Abstract This paper deals with the development of an optimal policy f the single-vend single-buyer integrated production}inventy system. The successive batches of a lot are transferred to the buyer in a "nite number of unequal and equal sizes. The successive unequal batch sizes increase by a "xed fact. The capacity of the transpt equipment used to transfer batches from the vend to the buyer is limited. The objective is to minimize the total joint annual costs incurred by the vend and the buyer Elsevier Science B.V. All rights reserved. Keywds: Production; Inventy; Capacity constraint 1. Introduction An interesting optimization problem is encountered whenever a single product needs to besupplied by a vend to a buyer over an in"nite time hizon. It has been established that by integrating the vend'sas well as the buyer's production/inventy/transptation problem the total of all the costs incurred by the vend and the buyer can be reduced signi"cantly. Goyal [1] considered an integrated inventy model f the single-supplier single-customer problem. Banerjee [2] investigated the lot f lot policy in which the vend manufactures a lot at a "nite rate of production. Goyal [3] suggested equal sized shipments to the buyer only after "nishing the entire production lot. Based on equal sized shipments to the buyer Lu [4] considered heuristics f the single-vend single-buyer problem. Goyal [5] suggested an alternative policy in which successive shipments of a lot increase by a fact equal to the ratio of the production rate to the demand rate. Hill [6] introduced a me general class of policy f determining optimal total cost by increasing the successive batch size by a "xed fact ranging from 1 to the production rate divided by the demand rate. By combining Goyal's [5] policy * Cresponding auth /00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S ( 9 9 )

2 306 M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315 and an equal shipment size policy, Hill [7] derived a globally optimal batching and shipping policy f the single-vend single-buyer integrated production}inventy problem. This policy gives a lower total cost as compared to the previous policies. In this paper we develop an optimal solution procedure f the single-vend single-buyer production}inventy system with unequal and equal sized shipments from the vend to the buyer and under the capacity constraint of the transpt equipment. 2. Assumptions and notation 2.1. Assumptions (i) The demand rate f the product is deterministic and constant over an in"nite time hizon. (ii) The entire production lot can be shipped in unequal and/ equal sized batches. A "xed transptation cost is incurred f each shipment. (iii) Shtages are not allowed. (iv) Set-up and transptation times are insigni"cant and hence igned. (v) The unit holding cost represents the cost of carrying one unit of physical inventy of the product. (vi) Manufacturing set-up cost, unit inventy holding cost f the vend and the buyer, the cost of a shipment from the vend to the buyer and the transpt capacity are known. (vii) Time hizon is in"nite Notation A cost of a production set-up A cost of a shipment from the vend to the buyer h stock holding cost per unit of time f the vend h stock holding cost per unit of time f the buyer D demand per year P the production rate k the ratio between the rates of production and demand g capacity of the transpt equipment z the smallest batch size Q lot size m total number of batches (m is a positive integer) in which a lot is transpted to the buyer e number of unequal sized batches (e is a positive integer) C the annual cost of the integrated system. The vend transpts the entire lot, Q, inm shipments in which e!1 are unequal and m#1!e are equal. It is assumed that P'D and h 'h. 3. Development of the model 3.1. Stock holding cost In the model one production cycle is the time during which Q satis"es the demand rate D. Thus the length of each cycle is Q/D and the number of cycles per unit time is D/Q. Following the policy adopted by Goyal [5] the batch sizes are z, kz, kz, 2, kz. Here all the batches are unequal and each of them increases to the

3 next by a fact k"p/d. The entire production lot of size Q is transpted in e!1 unequal batches followed by (m#1!e) equal batches. The unequal batches are z, kz, kz, 2 kz. Following the policy developed by Goyal and Szendrovits [8], the general expression f the time weighted inventy is QZ/P#(Q/2)(1/D!1/P). This includes the inventies of both the vend and the buyer. The inventy per lot f the buyer can be evaluated as follows: 1 2 z z D #kzkz D #kzkz D #2#kzkz D #(m!e)kzkz D. Since z"q/f (m, e) and f (m, e)"(m!e)k# k, the inventy cost per lot f the buyer is given by h Q 2D f (m, e) k!1 k!1 #(m!e)k. The inventy f the vend can be determined by subtracting the inventy f the buyer from the total inventy of the system, so the inventy cost f the vend per lot is Qzh P #Q 2 1 D!1 P h!h 2D Q f (m, e) k!1 k!1 #(m!e)k. Therefe, the total inventy cost per lot of the vend}buyer system is evaluated as follows: Qzh P #Q 2 1 D!1 P h # Q 2D f (m, e) k!1 k!1 #(m!e)k (h!h ). As there are D/Q cycles per unit of time, the total inventy cost per unit of time is evaluated from the following: QDh Pf (m, e) #QDh 2 1 D!1 P #Q Set-up and shipment cost (k!1)/(k!1)#(m!e)k (k!1)/(k!1)#(m!e)k (h!h ). Cost of set-up is (D/Q)A and cost of shipment is (D/Q)mA. Therefe, the total annual cost, C of the vend}buyer system can be expressed as C" B Q #mb Q #AQ#Q a f (m, e) #h!h (k!1)/(k!1)#(m!e)k 2 (k!1)/(k!1)#(m!e)k, (1) where B"DA, b"da, A"(Dh /2) (1/D! 1/ P), and a"dh /P The constraint M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305} When the capacity of the equipment used f transpting batches from the vend to the buyer is limited, the largest batch size must not exceed the capacity of the transpt equipment. So the largest batch size must be equal (based on Goyal and Szendrovits [8]) to less than g. Hence the following constraint must be satis"ed: e! k)m! Q g. (2)

4 308 M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315 Therefe, the single-vend single-buyer problem can be expressed as B Minimize Q #mb Q #AQ#Q a f (m, e) #h!h (k!1)/(k!1)#(m!e)k 2 (k!1)/(k!1)#(m!e)k subject to e! Note that e)m. k)m! Q g. 4. Solution of the model It is shown in Appendix A that (k!1)/(k!1)#(m!e)k (k!1)(k!1)#(m!e)k is a non-increasing function of m, e and so f given Q Q f (m, e) #h!h (k!1)/(k!1)#(m!e)k 2 (k!1)/(k!1)#(m!e)k is a non-increasing function of m and e. The solution algithm starts with the determination of the minimum value of the partial cost function H(Q, m, e)" mb Q #Q a f (m, e) #h!h (k!1)/(k!1)#(m!e)k 2 (k!1)/(k!1)#(m!e)k f given Q, considering the integer nature of m and e. The non-convex nature of the cost function in Q leads us to carry out a directed search procedure over Q in the next step. The analysis presented here is given in Hoque and Kingsman [9]. F given Q and m, note that the minimum of the partial cost function, H(Q, m, e) is where f (m, e) has its greatest value. It can be shown that this minimum is where e is the largest integer satisfying the constraint given by (2). F e"m all batches are unequal and the constraint given by (2) reduces to k* Q g. F given Q, a set of value (m, e) such that (m, e) satis"es constraint (2) but (m, e#1) does not, is de"ned to be a basic feasible solution. A necessary and su$cient condition f the set (m, e) to fm a basic feasible solution is that it satis"es 1!k' g m!q! e! k *0. (3) F given Q, the smallest integer greater than Q/g can always be taken as the initial value of m. Let it be represented by m. Obviously, the right-hand side of constraint (2) is always non-negative. F m, the resulting initial value of e represented by e will be the largest integer satisfying constraint (2). Let the di!erence between the right-hand and the left-hand sides of the inequality in (2) f (m, e ) be denoted by ε ranging between 0 and 1. That is, ε " m!q g! e! k *0.

5 If R"Int(ε #1/(k!1)k), then by repeated application of the inequalities in (2) and (3) it can be shown that all the possible alternative relative values of m and e which can give the minimum of H(Q, m, e) are given by discontinuous ranges: (m #n, e #n) f all n)n, (m #n, e #n#r) f all n such that N #1)n)N and 1)r)R!1, (m #n, e #n#r) f all n*n #1, where N )! ln1!(r!ε )(k!k )N #1. ln k If e "m, all basic feasible solutions are of the fm (m #n, e #n) f all n*0. It can be shown that the partial cost function is a convex function of m n f each of the individual ranges (m #n, e #n#r), where (m #n)!(e # n#r), is always a constant. F that set of basic feasible solution, maximum of the partial cost function, (H, m, e) isatn #1N the value of n satisfying g(n#1)g(n)k* kqa [k#(m!e )(k!1)] b where a"h D/P and *g(n)g(n!1)k, g(n)"f (m #n, e #n) M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305} "(m!e )k#k. Hence f given Q, the maximum of the partial cost function is at one of the number of possible alternative values f the total number of batches and the number of the unequal sized batches. All these values can be derived to "nd out the lowest one f that value of Q. Now the total cost function is of non-convex nature. So its minimum is attained by a simple interval search procedure. The search procedure used to "nd out the economic production quantity is described below. Hence f given Q, the maximum of the partial cost function is at one of the number of possible alternative values f the total. Step 1. Starting with the lower bound on Q as Q"[B#b]/[A#a#(h!h )/2] obtained by setting m"1, e"1 in the objective function and then equating the di!erential coe$cient of it with respect to Q to zero. Q is incremented in increasing steps, say, x until a (local) minimum is obtained. The algithm starts with a low value of x, then doubles at each step until it "rst exceeds some preset value, whence the steps are kept at this preset value. Let this local minimum be at Q"Q. Step 2. Q"Q is then decreased at each step, by, say, X to obtain converged local minimum. The process of decreasing the production quantity continues until Q falls to below the initial value Q exceeds the maximum preset number of steps, say, S pre-speci"ed at the beginning of the procedure. Let QH and CH be the economic production quantity and the relevant cost, respectively. Step 3. Having obtained a converged local minimum, lot quantity is incremented by the step size, say, > f a pre-speci"ed number of steps, S. Step 4. Whenever a lower cost is found during the "xed search, the value replaces C* and the cresponding value f Q replaces Q*. Steps 3 and 4 are repeated until there is no reduction in the total cost f a pre-speci"ed number of steps. F k"p/d, the total inventy cost per unit time is given in Section 3.1. Now consider λ such that 1(λ(k, and let the total inventy cost per unit time cresponding to λ is less than that of the same

6 310 M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315 cresponding to k. That is, QDh Pf (m, e, λ) #QDh 2 1 D!1 P #Q (λ!1)/(λ!1)#(m!e)λ (h!h ) 2 f (m, e, λ QDh Pf (m, e, k) #QDh 2 1 D!1 P #Q (k!1)/(k!1)#(m!e)k (h!h ), 2 f (m, e, k) where f (m, e, k)"(m!e)k# k, f (m, e, λ) "(m!e)λ# λ 2Dh P(h!h ) ( (k!1)/(k!1)#(m!e)k! (λ!1)/(λ!1)#(m!e)λ f (m, e, k) f (m, e, λ) 1 f (m, eλ)! 1 f (m, e, k). It is shown in Appendix B that (k!1)/ (k!1)#(m!e)k/ f (m, e, k) is an increasing function of k. Applying this property, it is proved in Appendix C that the right-hand side of the above inequality is an increasing function of λ. So after "nding the minimum total cost at k"p/d, a simple interval search procedure over λ is carried out f determining the minimum total cost. This search procedure starts with the value of λ very close to k. 5. Numerical example We solve the problem considered in Goyal [5]. The data f this problem is A "400, A "25, h "4, h "5, P"3200, D"1000. Here k"p/ D"3.2. In this example x and X are set at 1 and 5, respectively, and S is set at 50. Whenever production quantity is changed, the total cost is calculated by "nding out the total number of batches and the number of unequal sized batches. F g"380, the total cost is obtained as 1972 and the relevant production quantity is 560. The batch sizes are 23, 73, 232, 232. The total cost found by the method in this study is the same as the cost found by Hill [7]. Table 1 Value of h Value of g Method Batch sizes Lot Total annual cost 5 Goyal [5] 36, 116, Lu [4] 111, 111, 111, 111, Hill [6] 31, 68, 142, Hill [7] 24, 76, 229, This paper 23, 73, 232, This paper 23, 73, 232, Goyal [5] 32, 101, Lu [4] 91, 91, 91, 91, 91, Hill [6] 54, 72, 99, 131, Hill [7] 31, 99, 137, 137, This paper 40, 128, 128, 128, This paper 20, 64, 205,

7 M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305} If h changes from 5 to 7 and g"128, then the following policy is obtained: QH"552, CH"1942 and batch sizes are 40, 128, 128, 128, 128. The total cost found by the method in this research is 3 units me than the total cost found by Hill [7]. This is due to the integer nature of the shipment sizes obtained by him. The results obtained by various methods f h "5 and 7 are given in Table Conclusion This paper extends the idea of producing a single product in a multistage serial production system with equal and unequal sized batch shipments between stages, iginally presented by Goyal and Szendrovits [8] and modi"ed by Hoque and Kingsman [9], to the single-vend single-buyer production}inventy system. A number of properties that the optimal solution must satisfy have been established. With the help of these properties an algithm f determining the optimal policy has been developed. Appendix A Show that (k!1)/(k!1)#(m!e)k (k!1)/(k!1)#(m!e)k is a non-increasing function of m and e whether e increases by 1 2 as m increases by 1. Proof. where (k!1)/(k!1)#(m!e)k (k!1)/(k!1)#(m!e)k "(k!k)/(k!1)#(m!e)k (k!k)/(k!1)#(m!e)k " (k!1)/(k!1)#(m!e)k!1 (k!1)/(k!1)#(m!e)k!1 " a!1 (b!1), a" k!1 #(m!e)k, k!1 b" k!1 k!1 #(m!e)k. Now let (a!1)/(b!1)(a/b a(2b!1)(b ab(b (as b(2b!1 b(b (as a'b) which is a contradiction. So (a!1)/(b!1)*a/b. Substituting the values of a, b it becomes (k!1)/(k!1)#(m!e)k (k!1)/(k!1)#(m!e)k *(k!1)/(k!1)#(m!e)k (k!1)/(k!1)#(m!e)k. Thus, if e increases by 1 as m increases by 1, then (k!1)/(k!1)#(m!e)k (k!1)/(k!1)#(m!e)k is a non-increasing function of m and e.

8 312 M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315 Now consider the case when e increases by 2 as m increases by 1. In this case (k!1)/(k!1)#(m!e)k (k!1)/(k!1)#(m!e)k "(k!k)/(k!1)#(m!e)k (k!k)/(k!1)#(m!e)k where " a#k!(k#1) b#k!(k#1), a" k!1 #(m!e!1)k, k!1 k"1#k!1, k"1#k!1, b" k!1 k!1 #(m!e!1)k. Now let a#k!(k#1) b#k!(k#1) ( a b. Then ab!b(k#1)(ab#a(k#1)!2ab (k#1) #ka!b!2ak(k#1)!b. Since a(b and b'k#1, ab!b(k#1)( ab#a(k#1)!2ab(k#1) a(k#1)2b! (k#1)(b(k#1) [if 2b!(k#1)(b, then b(k#1, a contradiction] a(k#1)b( b(k#1) which implies a/b( (1#k)/(1#k) 1#k#k#2#k#(m!e!1)k ( 1#k 1#k#k#2#k#(m!e!1)k 1#k, a contradiction. So a#k!(k#1) b#k!(k#1) * a b. Therefe (k!1)/(k!1)#(m!e)k (k!1)/(k!1)#(m!e)k is a non-increasing function of m and e in this case also. Thus the theem is proved. Appendix B Show that (k!1)/(k!1)#(m!e)k (k!1)/(k!1)#(m!e)k is an increasing function of k when e*2.

9 Proof. Consider the case when e"2. M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305} Let 1#k#(m!e)k 1#k#(m!e)k * 1#(1#k)#(m!e)(1#k) 1#(1#k)#(m!e)(1#k) 1#(m#1!e)k 1#(m#1!e)k * 1#(m#1!e)(1#k) 1#(m#1!e)(1#k) which after simpli"cation implies 1!2k* (m# 1!e)(2k!1). This is a contradiction because 1#k#(m!e)k 1#k#(m!e)k ( 1#(k#1)#(m!e)(k#1) 1#(k#1)#(m!e)(k#1). Now assume that 1#k#k#2#k#(m!e)k 1#k#k#2#k#(m!e)k ( 1#(k#1)#(k#1)#2#(k#1)#(m!e)(k#1) 1#(k#1)#(k#1)#2#(k#1)#(m!e)(k#1) and let 1#ka (1#kb) * 1#(k#1)c 1#(k#1)d, where a"1#k#k#2#k#(m!e)k, b"1#k#k#2#k#(m!e)k, c"1#(k#1)#(k#1)#2#(k#1)#(m!e)(k#1), d"1#(k#1)#(k#1)#2#(k#1)#(m!e)(k#1). Then 1#2(k#1)d#(k#1)d#ka#2k(k#1)ad#k(k#1)ad 1#2kb#kb#(k#1)c#2k(k#1)bc#k(k#1)bc. From the assumed relation ad(bc. Hence 2(k#1)d#(k#1)d#ka#2k(k#1)ad*2kb#kb# (k#1)c#2k(k#1)bc#2k(k#1) bc [since 2k(k#1)bc"2k(k#1)bc#2k (k#1)bc]. Now let ad*bc. This means 1*(bc)/(ad)'(bc)/ (ad)(b/d) (as d'b) 1*bc/ad'1 (as a/b( c/d). Whatever may be the case, we have a contradiction. So bc'ad. Therefe, 2(k#1)d# (k#1) bc#ka*2kb#kb#(k#1)c#2k(k#1)bc.

10 314 M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315 Now let a/b(c/d. This means d(bc/a"(b/a)bc (as a'b) d(bc. Therefe, 2(k#1)d# (k#1)bc#ka'2kb#kb#(k#1)c# 2k(k#1)bc. Thus 2(k#1)d#ka*2kb#kb#(k#1)c [as (k#1)bc(2k(k#1)bc (otherwise, there is a contradiction that 1*k)] ka'2kb#kb [as 2(k#1)d((k#1) (otherwise, there is a contradiction that 1*k.)] ka'2kb#ka (as b'a) 0'kb. This is a contradiction. So, That is 1#ka (1#kb) ( 1#(k#1)c 1#(k#1)d. 1#k#k#2#k#(m!e)k 1#k#k#2#k#(m!e)k (1#(k#1)#(k#1)#2#(k#1)#(m!e)(k#1) 1#(k#1)#(k#1)#2#(k#1)#(m!e)(k#1). Thus, if the relation is true f e!1, then it is true f e!1#1"e. It is shown that the relation is true f e"2, so it is true f 3. Again, since the relation is true f e"3, so it is true f e"4. Thus the above relation is true f any value of e. Therefe, 1#k#k#2#k#(m!e)k 1#k#k#2#k#(m!e)k is an increasing function of k when e*2. Appendix C Show that 1#k#k#2#k#(m!e)k 1#k#k#2#k#(m!e)k!1#λ#λ#2#λ#(m!e)λ 1#λ#λ#2#λ#(m!e)λ 1 1#λ#λ#2#λ#(m!e)λ! 1 1#k#k#2#k#(m!e)k is an increasing function of λ.

11 M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305} Proof. Let r/t!a/b 1/b!1/t *r/t!c/d 1/d!1/t, where r"1#k#k#2#k#(m!e)k, t"1#k#k#2#k#(m!e)k, a"1#λ#λ#2#λ#(m!e)λ, b"1#λ#λ#2#λ#(m!e)λ, c"1#(λ#1)#(λ#1)#2#(λ#1)#(m!e)(λ#1), Then d"1#(λ#1)#(λ#1)#2#(λ#1)#(m!e)(λ#1). bct(t!b)#(bct(d!t)'rbd(d!b) (because ad(bc; otherwise, 1'bc/ad'1, since a/b(c/d) which implies r/t)c/d. Since r/t*c/d we have r/t*(c/d)(t/d)'c/d (as t/d'1). So r/t)c/d is a contradiction. Hence the proof. References [1] S.K. Goyal, Determination of optimum production quantity f a two-stage production system, Operational Research Quarterly 28 (1977) 865}870. [2] A. Banerjee, A joint economic lot size model f purchaser and vend, Decision Sciences 17 (1985) 292}311. [3] S.K. Goyal, A joint economic lot size model f purchaser and vend: A comment, Decision Sciences 19 (1988) 236}241. [4] L. Lu, A one-vend multi-buyer integrated inventy model, European Journal of Operational Research 81 (1995) 312}323. [5] S.K. Goyal, A one-vend multi-buyer integrated inventy model: A comment, European Journal of Operational Research 82 (1995) 209}210. [6] R.M. Hill, The single-vend single-buyer integrated production inventy model with a generalized policy, European Journal of Operational Research 97 (1997) 493}499. [7] R.M. Hill, The optimal production and shipment policy f the single-vend single-buyer integrated production inventy problem, International Journal of Production Research 37 (1999) 2463}2475. [8] S.K. Goyal, A.Z. Szendrovits, A constant lot size model with equal and unequal sized batch shipments between production stages, Engineering Costs and Production Economics 10 (1986) 203}210. [9] M.A. Hoque, B.G. Kingsman, An optimal solution algithm f the constant lot size model with equal and unequal sized batch shipments f the single product multistage production system, International Journal of Production Economics 42 (1995) 161}174.

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