*Corresponding author.
Technical Note
An optimal policy for a single-vendor single-buyer integrated
production
}
inventory system with capacity constraint of the
transport 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, Concordia 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 for the single-vendor single-buyer integrated produc-tion}inventory 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 factor. The capacity of the transport equipment used to transfer batches from the vendor to the buyer is limited. The objective is to minimize the total joint annual costs incurred by the vendor and the buyer. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Production; Inventory; Capacity constraint
1. Introduction
An interesting optimization problem is encountered whenever a single product needs to besupplied by a vendor to a buyer over an in"nite time horizon. It has been established that by integrating the vendor's as
well as the buyer's production/inventory/transportation problem the total of all the costs incurred by the
vendor and the buyer can be reduced signi"cantly.
Goyal [1] considered an integrated inventory model for the single-supplier single-customer problem. Banerjee [2] investigated the lot for lot policy in which the vendor 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 for the single-vendor single-buyer problem. Goyal [5] suggested an alternative policy in which successive shipments of a lot increase by a factor equal to the ratio of the production rate to the demand rate. Hill [6] introduced a more general class of policy for determining optimal total cost by increasing the successive batch size by a"xed
factor ranging from 1 to the production rate divided by the demand rate. By combining Goyal's [5] policy
and an equal shipment size policy, Hill [7] derived a globally optimal batching and shipping policy for the
single-vendor single-buyer integrated production}inventory problem. This policy gives a lower total cost as
compared to the previous policies. In this paper we develop an optimal solution procedure for the
single-vendor single-buyer production}inventory system with unequal and equal sized shipments from the
vendor to the buyer and under the capacity constraint of the transport equipment.
2. Assumptions and notation
2.1. Assumptions
(i) The demand rate for the product is deterministic and constant over an in"nite time horizon.
(ii) The entire production lot can be shipped in unequal and/or equal sized batches. A"xed transportation cost is incurred for each shipment.
(iii) Shortages are not allowed.
(iv) Set-up and transportation times are insigni"cant and hence ignored.
(v) The unit holding cost represents the cost of carrying one unit of physical inventory of the product. (vi) Manufacturing set-up cost, unit inventory holding cost for the vendor and the buyer, the cost of
a shipment from the vendor to the buyer and the transport capacity are known. (vii) Time horizon is in"nite.
2.2. Notation
A
1 cost of a production set-up
A
2 cost of a shipment from the vendor to the buyer
h
1 stock holding cost per unit of time for the vendor
h
2 stock holding cost per unit of time for the buyer
D demand per year
P the production rate
k the ratio between the rates of production and demand
g capacity of the transport equipment
z the smallest batch size
Q lot size
m total number of batches (mis a positive integer) in which a lot is transported to the buyer
e number of unequal sized batches (eis a positive integer)
C the annual cost of the integrated system.
The vendor transports the entire lot,Q, inmshipments in whiche!1 are unequal andm#1!eare equal.
It is assumed that P'Dandh
2'h1.
3. Development of the model
3.1. Stock holding cost
In the model one production cycle is the time during whichQsatis"es the demand rateD. Thus the length
of each cycle isQ/Dand the number of cycles per unit time isD/Q. Following the policy adopted by Goyal [5]
next by a factork"P/D. The entire production lot of sizeQis transported ine!1 unequal batches followed
by (m#1!e) equal batches. The unequal batches arez,kz,k2z,2km~1z. Following the policy developed
by Goyal and Szendrovits [8], the general expression for the time weighted inventory is
QZ/P#(Q2/2)(1/D!1/P).
This includes the inventories of both the vendor and the buyer. The inventory per lot for the buyer can be evaluated as follows:
The inventory for the vendor can be determined by subtracting the inventory for the buyer from the total inventory of the system, so the inventory cost for the vendor per lot is
Qzh
Therefore, the total inventory cost per lot of the vendor}buyer system is evaluated as follows:
Qzh
As there are D/Qcycles per unit of time, the total inventory cost per unit of time is evaluated from the
following:
Therefore, the total annual cost,Cof the vendor}buyer system can be expressed as
C"B
When the capacity of the equipment used for transporting batches from the vendor to the buyer is limited, the largest batch size must not exceed the capacity of the transport equipment. So the largest batch size must
be equal (based on Goyal and Szendrovits [8]) to or less than g. Hence the following constraint must be
satis"ed:
e!e~1+
r/0
k~r)m!Q
Therefore, the single-vendor single-buyer problem can be expressed as
It is shown in Appendix A that
(k2e!1)/(k2!1)#(m!e)k2(e~1)
M(ke!1)(k!1)#(m!e)k(e~1)N2
is a non-increasing function ofm,eand so for givenQ
Q
is a non-increasing function ofmande. The solution algorithm starts with the determination of the minimum
value of the partial cost function
H(Q,m,e)"mb
for givenQ, considering the integer nature ofmande. The non-convex nature of the cost function inQleads
us to carry out a directed search procedure overQin the next step. The analysis presented here is given in
Hoque and Kingsman [9]. For givenQandm, note that the minimum of the partial cost function,H(Q,m,e)
is wheref(m,e) has its greatest value. It can be shown that this minimum is whereeis the largest integer
satisfying the constraint given by (2). For e"m all batches are unequal and the constraint given by (2)
reduces to a basic feasible solution. A necessary and su$cient condition for the set (m,e) to form a basic feasible solution is that it satis"es
For givenQ, the smallest integer greater thanQ/gcan always be taken as the initial value ofm. Let it be
represented by m
0. Obviously, the right-hand side of constraint (2) is always non-negative. For m0, the
resulting initial value of e represented by e
0 will be the largest integer satisfying constraint (2). Let the
di!erence between the right-hand and the left-hand sides of the inequality in (2) for (m
0,e0) be denoted by
e0 ranging between 0 and 1. That is,
IfR"Int(e0#1/(k!1)ke0~1), then by repeated application of the inequalities in (2) and (3) it can be shown that all the possible alternative relative values ofmandewhich can give the minimum ofH(Q,m,e) are given by discontinuous ranges:
the partial cost function is a convex function of m or n for each of the individual ranges
(m
0#n,e0#n#r), where (m0#n)!(e0# n#r), is always a constant. For that set of basic feasible
solution, maximum of the partial cost function, (H,m,e) is atN
r#1 orNr`1or the value ofnsatisfying
Hence for givenQ, the maximum of the partial cost function is at one of the number of possible alternative
values for the total number of batches and the number of the unequal sized batches. All these values can be derived to"nd out the lowest one for that value ofQ. Now the total cost function is of non-convex nature. So
its minimum is attained by a simple interval search procedure. The search procedure used to"nd out the
economic production quantity is described below. Hence for given Q, the maximum of the partial cost
function is at one of the number of possible alternative values for the total. Step 1. Starting with the lower bound onQasQ0"J[B#b]/[A#a#(h
2!h1)/2] obtained by setting
m"1,e"1 in the objective function and then equating the di!erential coe$cient of it with respect toQto
zero.Qis incremented in increasing steps, say,xuntil a (local) minimum is obtained. The algorithm starts
with a low value ofx, 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 atQ"Q1.
Step 2.Q"Q1is then decreased at each step, by, say,Xto obtain converged local minimum. The process
of decreasing the production quantity continues untilQfalls to or below the initial valueQ0or exceeds the
maximum preset number of steps, say,Spre-speci"ed at the beginning of the procedure. LetQHandCHbe 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,>for
a pre-speci"ed number of steps,S.
Step 4. Whenever a lower cost is found during the"xed search, the value replacesC*and the corresponding
value for Q replaces Q*. Steps 3 and 4 are repeated until there is no reduction in the total cost for
a pre-speci"ed number of steps.
For k"P/D, the total inventory cost per unit time is given in Section 3.1. Now consider jsuch that
Table 1 Value ofh
2 Value ofg Method Batch sizes Lot Total annual cost
5 Goyal [5] 36, 116, 370 552 1818
5 Lu [4] 111, 111, 111, 111, 111 555 1903
5 Hill [6] 31, 68, 142, 310 551 1814
5 Hill [7] 24, 76, 229, 229 258 1793
5 380 This paper 23, 73, 232, 232 560 1792
5 250 This paper 23, 73, 232, 232 560 1792
7 Goyal [5] 32, 101, 322 455 2089
7 Lu [4] 91, 91, 91, 91, 91, 91 546 2008
7 Hill [6] 54, 72, 99, 131, 177 533 1972
7 Hill [7] 31, 99, 137, 137, 137 541 1939
7 128 This paper 40, 128, 128, 128, 128 552 1942
7 250 This paper 20, 64, 205, 205 494 1985
corresponding tok. That is, k. Applying this property, it is proved in Appendix C that the right-hand side of the above inequality is an increasing function of j. So after "nding the minimum total cost at k"P/D, a simple interval search
procedure overjis carried out for determining the minimum total cost. This search procedure starts with the
value ofjvery close tok.
5. Numerical example
We solve the problem considered in Goyal [5]. The data for this problem is A
1"400,A2"25,
h
1"4,h2"5,P"3200,D"1000. Herek"P/D"3.2.
In this examplexandXare set at 1 and 5, respectively, andSis 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
Ifh
2changes from 5 to 7 andg"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 more 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 forh
2"5 and 7 are given in Table 1.
6. 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, originally presented by Goyal and Szendrovits [8]
and modi"ed by Hoque and Kingsman [9], to the single-vendor single-buyer production}inventory system.
A number of properties that the optimal solution must satisfy have been established. With the help of these properties an algorithm for determining the optimal policy has been developed.
Appendix A
Show that
(k2e!1)/(k2!1)#(m!e)k2(e~1)
M(ke!1)/(k!1)#(m!e)ke~1N2
is a non-increasing function ofmandewhethereincreases by 1 or 2 asmincreases by 1.
Proof.
Thus, ifeincreases by 1 asmincreases by 1, then
(k2e!1)/(k2!1)#(m!e)k2(e~1)
M(ke!1)/(k!1)#(m!e)ke~1N2
Now consider the case when eincreases by 2 asmincreases by 1. In this case
is a non-increasing function ofmandein this case also. Thus the theorem is proved. h
Appendix B
Show that
(k2e!1)/(k2!1)#(m!e)k2(e~1)
M(ke!1)/(k!1)#(m!e)ke~1N2
Proof. Consider the case whene"2.
which after simpli"cation implies
1!2k*(m#1!e)(2k2!1).
Whatever may be the case, we have a contradiction. Sobc'ad. Therefore,
Now leta/b2(c/d2. This means relation is true for any value ofe. Therefore,
1#k2#k4#2#k2(e~1)#(m!e)k2(e~1)
M1#k#k2#2#k(e~1)#(m!e)ke~1N2
is an increasing function ofkwhene*2.
Appendix C
Proof. Let
r/t2!a/b2
1/b!1/t *
r/t2!c/d2
1/d!1/t ,
where
r"1#k2#k4#2#k2(e~1)#(m!e)k2(e~1),
t"1#k#k2#2#ke~1#(m!e)ke~1,
a"1#j2#j4#2#j2(e~1)#(m!e)j2(e~1),
b"1#j#j2#2#je~1#(m!e)je~1,
c"1#(j#1)2#(j#1)4#2#(j#1)2(e~1)#(m!e)(j#1)2(e~1),
d"1#(j#1)#(j#1)2#2#(j#1)e~1#(m!e)(j#1)e~1.
Then
bct(t!b)#(bct(d!t)'rbd(d!b)
(becausead(bc; otherwise, 1'b2c/ad2'1, sincea/b2(c/d2)
which impliesr/t)c/d. Sincer/t2*c/d2we haver/t*(c/d)(t/d)'c/d(ast/d'1). Sor/t)c/dis a
contra-diction. Hence the proof. h
References
[1] S.K. Goyal, Determination of optimum production quantity for a two-stage production system, Operational Research Quarterly 28 (1977) 865}870.
[2] A. Banerjee, A joint economic lot size model for purchaser and vendor, Decision Sciences 17 (1985) 292}311.
[3] S.K. Goyal, A joint economic lot size model for purchaser and vendor: A comment, Decision Sciences 19 (1988) 236}241. [4] L. Lu, A one-vendor multi-buyer integrated inventory model, European Journal of Operational Research 81 (1995) 312}323. [5] S.K. Goyal, A one-vendor multi-buyer integrated inventory model: A comment, European Journal of Operational Research 82
(1995) 209}210.
[6] R.M. Hill, The single-vendor single-buyer integrated production inventory model with a generalized policy, European Journal of Operational Research 97 (1997) 493}499.
[7] R.M. Hill, The optimal production and shipment policy for the single-vendor single-buyer integrated production inventory 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.
