Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol65.Issue3.May2000:
Int. J. Production Economics 65 (2000) 305}315
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 production}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
* Corresponding author.
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 8 2 - 1
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 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
A
2
h
1
h
2
D
P
k
g
z
Q
m
e
C
cost of a production set-up
cost of a shipment from the vendor to the buyer
stock holding cost per unit of time for the vendor
stock holding cost per unit of time for the buyer
demand per year
the production rate
the ratio between the rates of production and demand
capacity of the transport equipment
the smallest batch size
lot size
total number of batches (m is a positive integer) in which a lot is transported to the buyer
number of unequal sized batches (e is a positive integer)
the annual cost of the integrated system.
The vendor transports the entire lot, Q, in m shipments in which e!1 are unequal and m#1!e are equal.
It is assumed that P'D and h 'h .
2
1
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, k2z, 2, km~1z. Here all the batches are unequal and each of them increases to the
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
307
next by a factor k"P/D. The entire production lot of size Q is transported in e!1 unequal batches followed
by (m#1!e) equal batches. The unequal batches are z, kz, k2z, 2 km~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:
C
D
kz
k2z
ke~1z
ke~1z
1 z
z #kz #k2z #2#ke~1z
.
#(m!e)ke~1z
D
D
D
D
2 D
Since z"Q/f (m, e) and f (m, e)"(m!e)ke~1# +e~1kr, the inventory cost per lot for the buyer is given by
r/0
h
Q2
k2e!1
2
#(m!e)k2(e~1) .
2D M f (m, e)N2 k2!1
D
C
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
A
B
C
D
Q2 1
1
Q2
Qzh
h
k2e!1
1#
! h ! 1
#(m!e)k2(e~1) .
2 D P 1 2D M f (m, e)N2 k2!1
P
Therefore, the total inventory cost per lot of the vendor}buyer system is evaluated as follows:
A
B
C
D
Q2 1
1
Qzh
Q2
k2e!1
1#
! h #
#(m!e)k2(e~1) (h !h ).
2
1
1
2 D P
P
2DM f (m, e)N2 k2!1
As there are D/Q cycles per unit of time, the total inventory cost per unit of time is evaluated from the
following:
A
B
QDh
QDh 1
1
Q (k2e!1)/(k2!1)#(m!e)k2(e~1)
1 #
1
! #
(h !h ).
1
Pf (m, e)
2
2 M(ke!1)/(k!1)#(m!e)k(e~1)N2 2
D P
3.2. Set-up and shipment cost
Cost of set-up is (D/Q)A and cost of shipment is (D/Q)mA .
1
2
Therefore, the total annual cost, C of the vendor}buyer system can be expressed as
C
D
B mb
a
h !h (k2e!1)/(k2!1)#(m!e)k2(e~1)
1
,
C" # #AQ#Q
# 2
Q
M(ke!1)/(k!1)#(m!e)k(e~1)N2
2
Q
f (m, e)
(1)
where B"DA , b"DA , A"(Dh /2) (1/D! 1/ P), and a"Dh /P.
1
2
1
1
3.3. The constraint
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~1
Q
e! + k~r)m! .
g
r/0
(2)
308
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
Therefore, the single-vendor single-buyer problem can be expressed as
Minimize
C
h !h (k2e!1)/(k2!1)#(m!e)k2(e~1)
a
B mb
1
# #AQ#Q
# 2
2
Q
f (m, e)
Q
M(ke!1)/(k!1)#(m!e)k(e~1)N2
D
e~1
Q
e! + k~r)m! .
g
r/0
Note that e)m.
subject to
4. Solution of the model
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 of m, e and so for given Q
Q
h !h (k2e!1)/(k2!1)#(m!e)k2(e~1)
1
# 2
f (m, e)
M(ke!1)/(k!1)#(m!e)k(e~1)N2
2
is a non-increasing function of m and e. The solution algorithm starts with the determination of the minimum
value of the partial cost function
C
a
h !h (k2e!1)/(k2!1)#(m!e)k2(e~1)
mb
1
# 2
H(Q, m, e)" #Q
M(ke!1)/(k!1)#(m!e)k(e~1)N2
2
f (m, e)
Q
D
for 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]. For 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). For e"m all batches are unequal and the constraint given by (2)
reduces to
m~1
Q
+ k~r* .
g
r/0
For 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 for the set (m, e) to form a basic feasible solution
is that it satis"es
A
B
B A
Q
e~1
1!k~e' m! ! e! + k~r *0.
(3)
g
r/0
For 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. For m , the
0
0
resulting initial value of e represented by e will be the largest integer satisfying constraint (2). Let the
0
di!erence between the right-hand and the left-hand sides of the inequality in (2) for (m , e ) be denoted by
0 0
e ranging between 0 and 1. That is,
0
e0 ~1
Q
e " m ! ! e ! + k~r *0.
0
0
0
g
r/0
A
B A
B
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
309
If R"Int(e #1/(k!1)ke0 ~1), then by repeated application of the inequalities in (2) and (3) it can be shown
0
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)
0
0
for all n)N ,
1
(m #n, e #n#r)
0
0
for all n such that N #1)n)N
and 1)r)R!1,
r
r`1
(m #n, e #n#R) for all n*N #1,
0
0
r
lnM1!(r!e )(ke0 !ke0 ~1N
0
where N )!
)N #1.
r
r
ln k
If e "m , all basic feasible solutions are of the form (m #n, e #n) for all n*0. It can be shown that
0
0
0
0
the partial cost function is a convex function of m or n for each of the individual ranges
(m #n, e #n#r), where (m #n)!(e # n#r), is always a constant. For that set of basic feasible
0
0
0
0
solution, maximum of the partial cost function, (H, m, e) is at N #1 or N
or the value of n satisfying
r
r`1
ke0 ~1Q2a
[k#(m !e )(k!1)]
g(n#1)g(n)kn*
0
0
b
*g(n)g(n!1)k1~n,
where a"h D/P and
1
g(n)"f (m #n, e #n)
0
0
"(m !e )ke0 `n~1#+e0`n~1kr.
r/0
0
0
Hence for given Q, 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 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 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 on Q as Q0"J[B#b]/[A#a#(h !h )/2] obtained by setting
2
1
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 algorithm 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"Q1.
Step 2. Q"Q1 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 or below the initial value Q0 or 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, > for
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 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 j such that
1(j(k, and let the total inventory cost per unit time corresponding to j is less than that of the same
310
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
corresponding to k. That is,
A
A
B
B
QDh
Q (j2e!1)/(j2!1)#(m!e)j2(e~1)
QDh 1
1
1 #
1
! #
(h !h )
2
1
Pf (m, e, j)
D P
2
Mf (m, e, jN2
2
QDh 1
1
Q (k2e!1)/(k2!1)#(m!e)k2(e~1)
QDh
1
1 #
! #
(h !h ),
2
1
2
2
D P
M f (m, e, k)N2
Pf (m, e, k)
where
f (m, e, k)"(m!e)ke~1#+e~1kr,
r/0
f (m, e, j) "(m!e)je~1#+e~1jr
r/0
or
C
C
2Dh
(j2e!1)/(j2!1)#(m!e)j2(e~1)
(k2e!1)/(k2!1)#(m!e)k2(e~1)
1 (
!
P(h !h )
M f (m, e, j)N2
M f (m, e, k)N2
2
1
DN
D
1
1
.
!
f (m, ej) f (m, e, k)
It is shown in Appendix B that M(k2e!1)/ (k2!1)#(m!e)k2(e~1)N/M f (m, e, k)N2 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 j. So after "nding the minimum total cost at k"P/D, a simple interval search
procedure over j is carried out for determining the minimum total cost. This search procedure starts with the
value of j very close to k.
5. Numerical example
We solve the problem considered in Goyal [5]. The data for this problem is A "400, A "25,
1
2
h "4, h "5, P"3200, D"1000. Here k"P/ D"3.2.
1
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. For 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
5
5
5
5
5
5
7
7
7
7
7
7
2
Value of g
380
250
128
250
Method
Batch sizes
Lot
Total annual cost
Goyal [5]
Lu [4]
Hill [6]
Hill [7]
This paper
This paper
Goyal [5]
Lu [4]
Hill [6]
Hill [7]
This paper
This paper
36, 116, 370
111, 111, 111, 111, 111
31, 68, 142, 310
24, 76, 229, 229
23, 73, 232, 232
23, 73, 232, 232
32, 101, 322
91, 91, 91, 91, 91, 91
54, 72, 99, 131, 177
31, 99, 137, 137, 137
40, 128, 128, 128, 128
20, 64, 205, 205
552
555
551
258
560
560
455
546
533
541
552
494
1818
1903
1814
1793
1792
1792
2089
2008
1972
1939
1942
1985
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
311
If h changes from 5 to 7 and g"128, then the following policy is obtained: QH"552, CH"1942 and
2
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 for h "5 and 7 are given in Table 1.
2
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 of m and e whether e increases by 1 or 2 as m increases by 1.
Proof.
(k2e!1)/(k2!1)#(m!e)k2(e~1) (k2(e`1)!k2)/(k2!1)#(m!e)k2e
"
M(ke!1)/(k!1)#(m!e)ke~1N2
M(ke`1!k)/(k!1)#(m!e)keN2
(k2(e`1)!1)/(k2!1)#(m!e)k2e!1
a!1
"
"
,
M(ke`1!1)/(k!1)#(m!e)ke!1N2
(b!1)2
where
k2(e`1)!1
#(m!e)k2e,
a"
k2!1
ke`1!1
b"
#(m!e)ke.
k!1
Now let (a!1)/(b!1)2(a/b2 or a(2b!1)(b2 or ab(b2 (as b(2b!1 or b2(b2 (as a'b) which is
a contradiction.
So (a!1)/(b!1)2*a/b2. Substituting the values of a, b it becomes
(k2e!1)/(k2!1)#(m!e)k2(e~1) (k2(e`1)!1)/(k2!1)#(m!e)k2e
*
.
M(ke!1)/(k!1)#(m!e)ke~1N2
M(ke`1!1)/(k!1)#(m!e)keN2
Thus, if e increases by 1 as m increases by 1, then
(k2e!1)/(k2!1)#(m!e)k2(e~1)
M(ke!1)/(k!1)#(m!e)ke~1N2
is a non-increasing function of m and e.
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
(k2e!1)/(k2!1)#(m!e)k2(e~1) (k2(e`2)!k4)/(k2!1)#(m!e)k2(e`1)
"
M(ke`2!k2)/(k!1)#(m!e)ke`1N2
M(ke!1)/(k!1)#(m!e)ke~1N2
a#k2(e`1)!(k2#1)
"
, k4"1#k4!1, k2"1#k2!1,
Mb#ke`1!(k#1)N2
where
k2(e`2)!1
a"
#(m!e!1)k2(e`1),
k2!1
ke`2!1
#(m!e!1)ke`1.
b"
k!1
Now let
a#k2(e`1)!(k2#1)
a
( .
Mb#ke`1!(k#1)N2 b2
Then
ab2!b2(k2#1)(ab2#a(k#1)2!2ab (k#1) #k2(e`1)Ma!b2N!2ake`1M(k#1)!bN.
Since a(b2 and b'k#1,
ab2!b2(k2#1)( ab2#a(k#1)!2ab(k#1)
or
a(k#1)M2b! (k#1)N(b2(k2#1)N
[if 2b!(k#1)(b, then b(k#1, a contradiction] or a(k#1)b( b2(k2#1) which implies a/b(
(1#k2)/(1#k) or
1#k2#k4#2#k2(e`1)#(m!e!1)k2(e`1) 1#k2
(
,
1#k#k2#2#ke`1#(m!e!1)ke`1
1#k
a contradiction. So
a
a#k2(e`1)!(k2#1)
* .
Mb#ke`1!(k#1)N2 b2
Therefore
(k2e!1)/(k2!1)#(m!e)k2(e~1)
M(ke!1)/(k!1)#(m!e)ke~1N2
is a non-increasing function of m and e in 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
is an increasing function of k when e*2.
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
313
Proof. Consider the case when e"2.
Let
1#k2#(m!e)k2
1#(1#k)2#(m!e)(1#k)2
*
M1#k#(m!e)kN2 M1#(1#k)#(m!e)(1#k)N2
or
1#(m#1!e)(1#k)2
1#(m#1!e)k2
*
M1#(m#1!e)kN2 M1#(m#1!e)(1#k)N2
which after simpli"cation implies
1!2k* (m# 1!e)(2k2!1).
This is a contradiction because
1#(k#1)2#(m!e)(k#1)2
1#k2#(m!e)k2
(
.
M1#k#(m!e)kN2 M1#(k#1)#(m!e)(k#1)N2
Now assume that
1#k2#k4#2#k2(e~1)#(m!e)k2(e~1)
M1#k#k2#2#k(e~1)#(m!e)ke~1N2
1#(k#1)2#(k#1)4#2#(k#1)2(e~1)#(m!e)(k#1)2(e~1)
(
M1#(k#1)#(k#1)2#2#(k#1)e~1#(m!e)(k#1)e~1N2
and let
1#(k#1)2c
1#k2a
*
,
(1#kb)2 M1#(k#1)dN2
where
a"1#k2#k4#2#k2(e~1)#(m!e)k2(e~1),
b"1#k#k2#2#ke~1#(m!e)ke~1,
c"1#(k#1)2#(k#1)4#2#(k#1)2(e~1)#(m!e)(k#1)2(e~1),
d"1#(k#1)#(k#1)2#2#(k#1)e~1#(m!e)(k#1)e~1.
Then
1#2(k#1)d#(k#1)2d2#k2a#2k2(k#1)ad#k2(k#1)2ad2
1#2kb#k2b2#(k#1)2c#2k(k#1)2bc#k2(k#1)2b2c.
From the assumed relation ad2(b2c. Hence 2(k#1)d#(k#1)2d2#k2a#2k2(k#1)ad*2kb#k2b2#
(k#1)2c#2k2(k#1)bc#2k(k#1) bc [since 2k(k#1)2bc"2k2(k#1)bc#2k (k#1)bc].
Now let ad*bc. This means
1*(bc)/(ad)'(bc)/ (ad)(b/d) (as d'b)
or
1*b2c/ad2'1 (as a/b2( c/d2).
Whatever may be the case, we have a contradiction. So bc'ad. Therefore,
2(k#1)d# (k#1)2 bc#k2a*2kb#k2b2#(k#1)2c#2k(k#1)bc.
314
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
Now let a/b2(c/d2. This means
d2(b2c/a"(b/a)bc (as a'b)
or
d2(bc.
Therefore,
2(k#1)d# (k#1)2bc#k2a'2kb#k2b2#(k#1)2c# 2k(k#1)bc.
Thus
2(k#1)d#k2a*2kb#k2b2#(k#1)2c
[as (k#1)2bc(2k(k#1)bc (otherwise, there is a contradiction that 1*k)]
or
k2a'2kb#k2b2
[as 2(k#1)d((k#1)2 (otherwise, there is a contradiction that 1*k.)]
or k2a'2kb#k2a (as b2'a) or 0'kb. This is a contradiction. So,
1#(k#1)2c
1#k2a
(
.
(1#kb)2 M1#(k#1)dN2
That is
1#k2#k4#2#k2e#(m!e)k2e 1#(k#1)2#(k#1)4#2#(k#1)2e#(m!e)(k#1)2e
(
.
M1#k#k2#2#ke#(m!e)keN2 M1#(k#1)#(k#1)2#2#(k#1)e#(m!e)(k#1)eN2
Thus, if the relation is true for e!1, then it is true for e!1#1"e. It is shown that the relation is true for
e"2, so it is true for 3. Again, since the relation is true for e"3, so it is true for e"4. Thus the above
relation is true for any value of e. 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 of k when e*2.
Appendix C
Show that
1#k2#k4#2#k2(e~1)#(m!e)k2(e~1) 1#j2#j4#2#j2(e~1)#(m!e)j2(e~1)
!
M1#k#k2#2#k(e~1)#(m!e)ke~1N2
M1#j#j2#2#je~1#(m!e)je~1N2
1
1
!
1#j#j2#2#je~1#(m!e)je~1 1#k#k2#2#ke~1#(m!e)ke~1
is an increasing function of j.
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
315
Proof. Let
r/t2!a/b2 r/t2!c/d2
*
,
1/b!1/t
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)
(because ad(bc; otherwise, 1'b2c/ad2'1, since a/b2(c/d2)
which implies r/t)c/d. Since r/t2*c/d2 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. 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.
[9] M.A. Hoque, B.G. Kingsman, An optimal solution algorithm for the constant lot size model with equal and unequal sized batch
shipments for the single product multistage production system, International Journal of Production Economics 42 (1995) 161}174.
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 production}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
* Corresponding author.
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 8 2 - 1
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 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
A
2
h
1
h
2
D
P
k
g
z
Q
m
e
C
cost of a production set-up
cost of a shipment from the vendor to the buyer
stock holding cost per unit of time for the vendor
stock holding cost per unit of time for the buyer
demand per year
the production rate
the ratio between the rates of production and demand
capacity of the transport equipment
the smallest batch size
lot size
total number of batches (m is a positive integer) in which a lot is transported to the buyer
number of unequal sized batches (e is a positive integer)
the annual cost of the integrated system.
The vendor transports the entire lot, Q, in m shipments in which e!1 are unequal and m#1!e are equal.
It is assumed that P'D and h 'h .
2
1
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, k2z, 2, km~1z. Here all the batches are unequal and each of them increases to the
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
307
next by a factor k"P/D. The entire production lot of size Q is transported in e!1 unequal batches followed
by (m#1!e) equal batches. The unequal batches are z, kz, k2z, 2 km~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:
C
D
kz
k2z
ke~1z
ke~1z
1 z
z #kz #k2z #2#ke~1z
.
#(m!e)ke~1z
D
D
D
D
2 D
Since z"Q/f (m, e) and f (m, e)"(m!e)ke~1# +e~1kr, the inventory cost per lot for the buyer is given by
r/0
h
Q2
k2e!1
2
#(m!e)k2(e~1) .
2D M f (m, e)N2 k2!1
D
C
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
A
B
C
D
Q2 1
1
Q2
Qzh
h
k2e!1
1#
! h ! 1
#(m!e)k2(e~1) .
2 D P 1 2D M f (m, e)N2 k2!1
P
Therefore, the total inventory cost per lot of the vendor}buyer system is evaluated as follows:
A
B
C
D
Q2 1
1
Qzh
Q2
k2e!1
1#
! h #
#(m!e)k2(e~1) (h !h ).
2
1
1
2 D P
P
2DM f (m, e)N2 k2!1
As there are D/Q cycles per unit of time, the total inventory cost per unit of time is evaluated from the
following:
A
B
QDh
QDh 1
1
Q (k2e!1)/(k2!1)#(m!e)k2(e~1)
1 #
1
! #
(h !h ).
1
Pf (m, e)
2
2 M(ke!1)/(k!1)#(m!e)k(e~1)N2 2
D P
3.2. Set-up and shipment cost
Cost of set-up is (D/Q)A and cost of shipment is (D/Q)mA .
1
2
Therefore, the total annual cost, C of the vendor}buyer system can be expressed as
C
D
B mb
a
h !h (k2e!1)/(k2!1)#(m!e)k2(e~1)
1
,
C" # #AQ#Q
# 2
Q
M(ke!1)/(k!1)#(m!e)k(e~1)N2
2
Q
f (m, e)
(1)
where B"DA , b"DA , A"(Dh /2) (1/D! 1/ P), and a"Dh /P.
1
2
1
1
3.3. The constraint
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~1
Q
e! + k~r)m! .
g
r/0
(2)
308
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
Therefore, the single-vendor single-buyer problem can be expressed as
Minimize
C
h !h (k2e!1)/(k2!1)#(m!e)k2(e~1)
a
B mb
1
# #AQ#Q
# 2
2
Q
f (m, e)
Q
M(ke!1)/(k!1)#(m!e)k(e~1)N2
D
e~1
Q
e! + k~r)m! .
g
r/0
Note that e)m.
subject to
4. Solution of the model
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 of m, e and so for given Q
Q
h !h (k2e!1)/(k2!1)#(m!e)k2(e~1)
1
# 2
f (m, e)
M(ke!1)/(k!1)#(m!e)k(e~1)N2
2
is a non-increasing function of m and e. The solution algorithm starts with the determination of the minimum
value of the partial cost function
C
a
h !h (k2e!1)/(k2!1)#(m!e)k2(e~1)
mb
1
# 2
H(Q, m, e)" #Q
M(ke!1)/(k!1)#(m!e)k(e~1)N2
2
f (m, e)
Q
D
for 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]. For 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). For e"m all batches are unequal and the constraint given by (2)
reduces to
m~1
Q
+ k~r* .
g
r/0
For 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 for the set (m, e) to form a basic feasible solution
is that it satis"es
A
B
B A
Q
e~1
1!k~e' m! ! e! + k~r *0.
(3)
g
r/0
For 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. For m , the
0
0
resulting initial value of e represented by e will be the largest integer satisfying constraint (2). Let the
0
di!erence between the right-hand and the left-hand sides of the inequality in (2) for (m , e ) be denoted by
0 0
e ranging between 0 and 1. That is,
0
e0 ~1
Q
e " m ! ! e ! + k~r *0.
0
0
0
g
r/0
A
B A
B
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
309
If R"Int(e #1/(k!1)ke0 ~1), then by repeated application of the inequalities in (2) and (3) it can be shown
0
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)
0
0
for all n)N ,
1
(m #n, e #n#r)
0
0
for all n such that N #1)n)N
and 1)r)R!1,
r
r`1
(m #n, e #n#R) for all n*N #1,
0
0
r
lnM1!(r!e )(ke0 !ke0 ~1N
0
where N )!
)N #1.
r
r
ln k
If e "m , all basic feasible solutions are of the form (m #n, e #n) for all n*0. It can be shown that
0
0
0
0
the partial cost function is a convex function of m or n for each of the individual ranges
(m #n, e #n#r), where (m #n)!(e # n#r), is always a constant. For that set of basic feasible
0
0
0
0
solution, maximum of the partial cost function, (H, m, e) is at N #1 or N
or the value of n satisfying
r
r`1
ke0 ~1Q2a
[k#(m !e )(k!1)]
g(n#1)g(n)kn*
0
0
b
*g(n)g(n!1)k1~n,
where a"h D/P and
1
g(n)"f (m #n, e #n)
0
0
"(m !e )ke0 `n~1#+e0`n~1kr.
r/0
0
0
Hence for given Q, 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 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 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 on Q as Q0"J[B#b]/[A#a#(h !h )/2] obtained by setting
2
1
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 algorithm 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"Q1.
Step 2. Q"Q1 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 or below the initial value Q0 or 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, > for
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 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 j such that
1(j(k, and let the total inventory cost per unit time corresponding to j is less than that of the same
310
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
corresponding to k. That is,
A
A
B
B
QDh
Q (j2e!1)/(j2!1)#(m!e)j2(e~1)
QDh 1
1
1 #
1
! #
(h !h )
2
1
Pf (m, e, j)
D P
2
Mf (m, e, jN2
2
QDh 1
1
Q (k2e!1)/(k2!1)#(m!e)k2(e~1)
QDh
1
1 #
! #
(h !h ),
2
1
2
2
D P
M f (m, e, k)N2
Pf (m, e, k)
where
f (m, e, k)"(m!e)ke~1#+e~1kr,
r/0
f (m, e, j) "(m!e)je~1#+e~1jr
r/0
or
C
C
2Dh
(j2e!1)/(j2!1)#(m!e)j2(e~1)
(k2e!1)/(k2!1)#(m!e)k2(e~1)
1 (
!
P(h !h )
M f (m, e, j)N2
M f (m, e, k)N2
2
1
DN
D
1
1
.
!
f (m, ej) f (m, e, k)
It is shown in Appendix B that M(k2e!1)/ (k2!1)#(m!e)k2(e~1)N/M f (m, e, k)N2 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 j. So after "nding the minimum total cost at k"P/D, a simple interval search
procedure over j is carried out for determining the minimum total cost. This search procedure starts with the
value of j very close to k.
5. Numerical example
We solve the problem considered in Goyal [5]. The data for this problem is A "400, A "25,
1
2
h "4, h "5, P"3200, D"1000. Here k"P/ D"3.2.
1
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. For 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
5
5
5
5
5
5
7
7
7
7
7
7
2
Value of g
380
250
128
250
Method
Batch sizes
Lot
Total annual cost
Goyal [5]
Lu [4]
Hill [6]
Hill [7]
This paper
This paper
Goyal [5]
Lu [4]
Hill [6]
Hill [7]
This paper
This paper
36, 116, 370
111, 111, 111, 111, 111
31, 68, 142, 310
24, 76, 229, 229
23, 73, 232, 232
23, 73, 232, 232
32, 101, 322
91, 91, 91, 91, 91, 91
54, 72, 99, 131, 177
31, 99, 137, 137, 137
40, 128, 128, 128, 128
20, 64, 205, 205
552
555
551
258
560
560
455
546
533
541
552
494
1818
1903
1814
1793
1792
1792
2089
2008
1972
1939
1942
1985
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
311
If h changes from 5 to 7 and g"128, then the following policy is obtained: QH"552, CH"1942 and
2
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 for h "5 and 7 are given in Table 1.
2
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 of m and e whether e increases by 1 or 2 as m increases by 1.
Proof.
(k2e!1)/(k2!1)#(m!e)k2(e~1) (k2(e`1)!k2)/(k2!1)#(m!e)k2e
"
M(ke!1)/(k!1)#(m!e)ke~1N2
M(ke`1!k)/(k!1)#(m!e)keN2
(k2(e`1)!1)/(k2!1)#(m!e)k2e!1
a!1
"
"
,
M(ke`1!1)/(k!1)#(m!e)ke!1N2
(b!1)2
where
k2(e`1)!1
#(m!e)k2e,
a"
k2!1
ke`1!1
b"
#(m!e)ke.
k!1
Now let (a!1)/(b!1)2(a/b2 or a(2b!1)(b2 or ab(b2 (as b(2b!1 or b2(b2 (as a'b) which is
a contradiction.
So (a!1)/(b!1)2*a/b2. Substituting the values of a, b it becomes
(k2e!1)/(k2!1)#(m!e)k2(e~1) (k2(e`1)!1)/(k2!1)#(m!e)k2e
*
.
M(ke!1)/(k!1)#(m!e)ke~1N2
M(ke`1!1)/(k!1)#(m!e)keN2
Thus, if e increases by 1 as m increases by 1, then
(k2e!1)/(k2!1)#(m!e)k2(e~1)
M(ke!1)/(k!1)#(m!e)ke~1N2
is a non-increasing function of m and e.
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
(k2e!1)/(k2!1)#(m!e)k2(e~1) (k2(e`2)!k4)/(k2!1)#(m!e)k2(e`1)
"
M(ke`2!k2)/(k!1)#(m!e)ke`1N2
M(ke!1)/(k!1)#(m!e)ke~1N2
a#k2(e`1)!(k2#1)
"
, k4"1#k4!1, k2"1#k2!1,
Mb#ke`1!(k#1)N2
where
k2(e`2)!1
a"
#(m!e!1)k2(e`1),
k2!1
ke`2!1
#(m!e!1)ke`1.
b"
k!1
Now let
a#k2(e`1)!(k2#1)
a
( .
Mb#ke`1!(k#1)N2 b2
Then
ab2!b2(k2#1)(ab2#a(k#1)2!2ab (k#1) #k2(e`1)Ma!b2N!2ake`1M(k#1)!bN.
Since a(b2 and b'k#1,
ab2!b2(k2#1)( ab2#a(k#1)!2ab(k#1)
or
a(k#1)M2b! (k#1)N(b2(k2#1)N
[if 2b!(k#1)(b, then b(k#1, a contradiction] or a(k#1)b( b2(k2#1) which implies a/b(
(1#k2)/(1#k) or
1#k2#k4#2#k2(e`1)#(m!e!1)k2(e`1) 1#k2
(
,
1#k#k2#2#ke`1#(m!e!1)ke`1
1#k
a contradiction. So
a
a#k2(e`1)!(k2#1)
* .
Mb#ke`1!(k#1)N2 b2
Therefore
(k2e!1)/(k2!1)#(m!e)k2(e~1)
M(ke!1)/(k!1)#(m!e)ke~1N2
is a non-increasing function of m and e in 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
is an increasing function of k when e*2.
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
313
Proof. Consider the case when e"2.
Let
1#k2#(m!e)k2
1#(1#k)2#(m!e)(1#k)2
*
M1#k#(m!e)kN2 M1#(1#k)#(m!e)(1#k)N2
or
1#(m#1!e)(1#k)2
1#(m#1!e)k2
*
M1#(m#1!e)kN2 M1#(m#1!e)(1#k)N2
which after simpli"cation implies
1!2k* (m# 1!e)(2k2!1).
This is a contradiction because
1#(k#1)2#(m!e)(k#1)2
1#k2#(m!e)k2
(
.
M1#k#(m!e)kN2 M1#(k#1)#(m!e)(k#1)N2
Now assume that
1#k2#k4#2#k2(e~1)#(m!e)k2(e~1)
M1#k#k2#2#k(e~1)#(m!e)ke~1N2
1#(k#1)2#(k#1)4#2#(k#1)2(e~1)#(m!e)(k#1)2(e~1)
(
M1#(k#1)#(k#1)2#2#(k#1)e~1#(m!e)(k#1)e~1N2
and let
1#(k#1)2c
1#k2a
*
,
(1#kb)2 M1#(k#1)dN2
where
a"1#k2#k4#2#k2(e~1)#(m!e)k2(e~1),
b"1#k#k2#2#ke~1#(m!e)ke~1,
c"1#(k#1)2#(k#1)4#2#(k#1)2(e~1)#(m!e)(k#1)2(e~1),
d"1#(k#1)#(k#1)2#2#(k#1)e~1#(m!e)(k#1)e~1.
Then
1#2(k#1)d#(k#1)2d2#k2a#2k2(k#1)ad#k2(k#1)2ad2
1#2kb#k2b2#(k#1)2c#2k(k#1)2bc#k2(k#1)2b2c.
From the assumed relation ad2(b2c. Hence 2(k#1)d#(k#1)2d2#k2a#2k2(k#1)ad*2kb#k2b2#
(k#1)2c#2k2(k#1)bc#2k(k#1) bc [since 2k(k#1)2bc"2k2(k#1)bc#2k (k#1)bc].
Now let ad*bc. This means
1*(bc)/(ad)'(bc)/ (ad)(b/d) (as d'b)
or
1*b2c/ad2'1 (as a/b2( c/d2).
Whatever may be the case, we have a contradiction. So bc'ad. Therefore,
2(k#1)d# (k#1)2 bc#k2a*2kb#k2b2#(k#1)2c#2k(k#1)bc.
314
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
Now let a/b2(c/d2. This means
d2(b2c/a"(b/a)bc (as a'b)
or
d2(bc.
Therefore,
2(k#1)d# (k#1)2bc#k2a'2kb#k2b2#(k#1)2c# 2k(k#1)bc.
Thus
2(k#1)d#k2a*2kb#k2b2#(k#1)2c
[as (k#1)2bc(2k(k#1)bc (otherwise, there is a contradiction that 1*k)]
or
k2a'2kb#k2b2
[as 2(k#1)d((k#1)2 (otherwise, there is a contradiction that 1*k.)]
or k2a'2kb#k2a (as b2'a) or 0'kb. This is a contradiction. So,
1#(k#1)2c
1#k2a
(
.
(1#kb)2 M1#(k#1)dN2
That is
1#k2#k4#2#k2e#(m!e)k2e 1#(k#1)2#(k#1)4#2#(k#1)2e#(m!e)(k#1)2e
(
.
M1#k#k2#2#ke#(m!e)keN2 M1#(k#1)#(k#1)2#2#(k#1)e#(m!e)(k#1)eN2
Thus, if the relation is true for e!1, then it is true for e!1#1"e. It is shown that the relation is true for
e"2, so it is true for 3. Again, since the relation is true for e"3, so it is true for e"4. Thus the above
relation is true for any value of e. 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 of k when e*2.
Appendix C
Show that
1#k2#k4#2#k2(e~1)#(m!e)k2(e~1) 1#j2#j4#2#j2(e~1)#(m!e)j2(e~1)
!
M1#k#k2#2#k(e~1)#(m!e)ke~1N2
M1#j#j2#2#je~1#(m!e)je~1N2
1
1
!
1#j#j2#2#je~1#(m!e)je~1 1#k#k2#2#ke~1#(m!e)ke~1
is an increasing function of j.
M.A. Hoque, S.K. Goyal / Int. J. Production Economics 65 (2000) 305}315
315
Proof. Let
r/t2!a/b2 r/t2!c/d2
*
,
1/b!1/t
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)
(because ad(bc; otherwise, 1'b2c/ad2'1, since a/b2(c/d2)
which implies r/t)c/d. Since r/t2*c/d2 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. h
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