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Towards a more economic production and just-in-time delivery system
international journal of
production economics
ELSEVIER
Towards
Int. J. Production
Economics
a more economic
36 (1994) 307-313
production system
and just-in-time
delivery
Li Zhuang Department
of Industrial
and Systems Engineering, National
Received
1 March
Unioersity
1994; accepted
qf Singapore.
10 Kent Ridge Crescent. Singaporr 051 I, Singapore
in revised form
1 July 1994
Abstract
We study a system in which a batch production supplier participates in just-in-time (JIT) delivery of its products to a JIT buyer firm. We demonstrate that under certain conditions a small adjustment of previously agreed product selling price can result in a more economic production and JIT delivery system with reduced delivery quantity and hence increased delivery frequencyln the case when both buyer and supplier share equal profit margin, optimal levels of adjustment for price and delivery frequency can be obtained that maximize the profit margin. A numerical example is used to illustrate the results.
1. Introduction
The concept of just-in-time (JIT) production and supply has recently received considerable attention from both academics and industries. An increasing number of empirical and theoretical studies have appeared in the literature that justify the economy of adopting the JIT concept by both buyer firms and their suppliers [l-5]. In essence, JIT is an approach for production management and productivity improvement that seeks to produce and deliver the right products with the right quantities at the right time, thereby reducing overall inventories (raw materials, work-in-process and finished products). Successful implementation of JIT requires a long-term relationship between a JIT firm and its suppliers. Under such a relationship the JIT buyer firm offers a long-term supply contract to a supplier who agrees to deliver quality products on a just-in0925-5273/94/$07.00 0 SSDI
1994 Elsevier
0925-5273(94)00055-7
Science B.V. All rights reserved
time basis. In some cases, the buyer provides a forecast of demand and shares market information with its supplier. In return the supplier shares cost information with the buyer and agrees to quality and cost improvement over time [6]. Furthermore, if a long-term commitment is established between the two parties, the JIT firm may share its production information continuously with the supplier. Hence there is no formal release to the purchase order. The supplier simply delivers the parts as they are required - weekly, daily or hourly [6]. Essentially, the relationship will ensure delivery of supplies that is synchronous with production so that the buyer does not need to keep high level inventory to smooth its production. Although it is not necessary for a supplier to become a JIT user, the supplier is strongly encouraged to implement the JIT approach to further reduce costs and to become more responsive to customer demands. A batch production supplier
can switch to a lot-for-lot production economically if setup-time and setup-cost have been reduced to certain target levels [a]. Even if reduction of setuptime and setup-cost is not enough to economically operate the system on a lot-for-lot basis, it is still beneficial for the supplier to participate in a JIT delivery system as the finished product inventory is found to be proportional to delivery quantity [3,7]. An important feature of a JIT delivery system is frequent and reliable delivery of quality parts in small lots. As a small delivery quantity means a lower average inventory to both parties, the resulting benefits such as a shorter lead time, improved visibility on the shop floor, earlier identification of defective lots (parts), and faster response to market demands can be enjoyed by both parties. In this paper, we consider a system in which a JIT firm schedules and receives inbound supplies from a batch production supplier on a just-in-time basis. We assume that both parties have agreed on a purchase price and they all commit to JIT production and delivery. It is, therefore, natural that each party makes its most economic decisions (production lot size and delivery quantity) according to its own objectives. Our purpose is to investigate how pricing decision affects the overall economies of the firm and its supplier. Particularly, we demonstrate that through proper adjustment (price discount in the case of delivery by the buyer firm) of a previously agreed purchase price, a more economic production and JIT delivery system with reduced delivery quantity and hence increased delivery frequency can be achieved. Traditionally, the problems of determining economic ordering and production quantities have been treated separately and often involved a quantity discount schedule. Typical issues addressed include how a buyer should react to a supplier’s quantity discount [S] and how a supplier dictates a buyer’s ordering pattern through manipulating different forms of quantity discount [S, 91. In both cases, the objective of the decision maker is to choose a policy that minimizes cost or maximizes profit according to its own cost structure. Recently, due to the advent of the JIT concept an increasing number of companies have realized the importance of improving relationships with their customers and suppliers. As a result, more
attention has been focused on the studies of interactive relationship between buyers and suppliers. Usually the studies are carried out through the use of an integrated model and the objective is to establish a policy (normally by the supplier) that benefits both parties [l&16]. For example, it has been shown that under certain conditions one party’s gain exceeds the other party’s loss. Thus, the net benefit can be shared equally by both parties through an appropriate price adjustment [lo, 171. However, little work has been done to study the effect of pricing decision within a JIT environment. Although some related papers assume a lot-for-lot production [lo, 151, they do not study how delivery schedule would affect the results. The organization of the remaining paper is as follows. Section 2 describes models and the associated optimal solutions that are used by the JIT firm and its supplier. These models are then used in Section 3 to demonstrate that both parties can be better off if the supplier agrees to offer a price discount to compensate for the buyer firm’s increased delivery expense. An optimum price discount and delivery frequency adjustment are obtained that maximize the equalized profit margin for both parties. A simple example illustrates the results. The paper ends in Section 4 with a discussion and conclusions.
2. Modeling of a JIT production and delivery system The following notations are used: = the customer’s annual demand (units/yr) = the current delivered unit price paid by the P customer (S/unit) Cd = the customer’s aggregate cost per delivery ($/delivery) h, = the customer’s annual inventory carrying cost per unit, expressed as a percentage of the purchase price (%/yr) current annual number of m = the customer’s deliveries (deliveries/yr) C, = the supplier’s production setup cost per batch ($/batch) D
L. Zhuungllnt.
H,
J. Production
= the supplier’s annual inventory carrying cost per unit based on unit production cost (S/unit yr) Q = the supplier’s current production lot size (units/batch) P = the supplier’s annual production rate (units/yr) [x] = the nearest integer rounded from x. We assume that the buyer’s annual demand is not affected by the change in the purchase price. Similarly, the supplier’s annual inventory carrying cost H, is assumed not to fluctuate with the selling price as H, is normally based on unit production cost which is mainly determined by production process. As discussed earlier, the JIT concept calls for frequent deliveries of small quantity, which means that m is much larger than what it would be in traditional batch production systems. This is achievable in a JIT system because delivery cost can be reduced substantially through continuing effort in improving productivity and eliminating non-value added processes together with mixed loading and freight consolidation [4, 5, 181. In addition, the cost of inventory carrying, another factor that affects delivery frequency, is also expected to be rationalized. In fact, JIT challenges the appropriateness of traditional accounting practices and calls for using the process costing method instead of the currently used product costing method [19]. Jones [20] has recently identified relevant costs in a JIT environment that should be included in evaluating inventory carrying cost. These are the costs that are associated with facilities, operations and administration and accounting. It should be noted that with the establishment of a long-term supply contract, the reduced ordering cost may not have as much impact on the ordering quantity as it does in traditional supply systems. Thus, we do not consider ordering cost explicitly in our model. For those minor ordering costs that are associated with each delivery, we assume that they are included in the delivery cost Cd. In general the determination of delivery frequency in most organizations is based on analysis that tends to minimize the sum of all incremental costs associated with or influenced by material deliveries. The analysis is, however, usually based on
Economics
309
36 (1994) 307-313
certain assumptions which may vary in different organizations depending on the type of purchasing systems and interfaces that a particular company uses. For example, Hewlett-Packard makes its weekly delivery schedule using a model that assumes the following [l]: The only significant incremental costs associated with delivery frequency are freight and inventory holding costs. Buying costs, as well as related materials management overhead and direct labor costs, do not vary significantly with delivery frequency. Based on the above discussion, we assume that a JIT firm determines its optimal annual delivery frequency by minimizing the following cost function: l
l
TRC,(m) = 2
h,p + mCd.
Thus, an optimal
m* =
integer
solution
[Jl
Dhcp
is given by
(2)
?Ey
It should be noted that a reduction in incoming parts inventory has many benefits (see Section 1). These are mostly intangible benefits that are strategically important to an organization’s competitiveness. However, as these benefits are difficult to estimate, they are excluded from the model formulation. A cost structure similar to (1) has also been used by Hewlett-Packard [l]. On the other hand, with an average delivery quantity of D/m*, the supplier makes its production planning. We assume that the supplier’s production lot size Q is an integer multiple of the delivery quantity. This is reasonable as it has been proven that adoption of the integer multiple policy is more beneficial to the supplier [16]. In fact, this policy also proves to be optimal if supplier and customer jointly determine their production lot size and delivery quantity [2]. Under this condition, it can be shown [2,3,7] that the average inventory level of a supplier’s finished product is
Qavg= $1 - 4Q + +
- I),
where, CI= D/P < 1 is the ratio of customer’s annual demand to supplier’s annual production capacity.
We assume that the supplier uses the following criteria to determine its production lot size: TRC,(Q) = 7
+ f(1 - z)QHs + &(2,
- l)H, (4)
3.1. E.uisfence qf’ a more economic
s)xtern
Suppose that the supplier agrees to have a price discount of Ap > 0 for a delivery increment of Am > 0. Then after some simple algebra the following expression for the increased profit margin to the JIT buyer can be derived:
which gives AP,(Am, Ap) = ApD - AmCd +
Dh,(pAm
2m(m + Am)
(5) where the integer multiple policy described above is taken into account. Note that the remark made earlier regarding the exclusion of inventory reduction benefits from the cost structure is also applicable to the supplier. Note that when r > 0.5 the supplier’s cost is proportional to delivery quantity, which means that increasing delivery frequency is also beneficial to the supplier although its optimal lot size Q* may be changed accordingly. Thus, we assume cx > 0.5. This may be the case when the JIT firm is a major consumer of its supplier’s product. This agrees with JIT purchase and supply philosophy of keeping fewer, ideally sole, sources of supplies.
3. A more economic JIT production and delivery system We assume that the system has been operational for some time under the JIT delivery contract. Suppose that in its continuing effort in reducing inventory, the JIT buyer firm wants to increase delivery frequency. However, by doing so the firm will certainly incur additional cost. One way to offset this additional cost is to ask for a price discount from the supplier. As we noted earlier, this may be acceptable to the supplier because a decrease in delivery quantity will reduce the finished product inventory and eventually reduces its production cost although production lot size may change accordingly. In this section, we show that there may exist a purchase price under which both parties will be better off economically. For brevity, we will drop superscript from the current delivery frequency m*.
+ mAp) ’
(6) where the last two terms correspond to the cost saving. It can be easily proved that AP,(Am, Ap) is a monotonically decreasing and increasing function of delivery increment Am and price discount Ap, respectively. For a given Am, the buyer will not be worse off, i.e., P,(Am, Ap) > 0 if the amount of price discount offered by the supplier satisfies the following condition: (2C,/D)m(m AP ’
+ Am) - h,p Am
2(m + Am) + h,
%-Cd D
m
kP Am, 2m(m + Am)
(7)
where the approximate equality is obtained by assuming that 2(m + Am) + h, z 2m(m + Am) as normally 2(m + Am) $ h,. Similarly, the supplier’s increased profit margin is given by AP,(Am, Ap) = - ApD + ATRCf(Am) + (2~ - 1) DH,Am 2m(m + Am)
’
(8)
where ATRCf(Am)
= C,D +
t(l - Co(Q* - Q)Hs.
(9)
Then from supplier’s viewpoint, for a given Am, the supplier will not be worse off, i.e., AP,(Am, Ap) > 0 if the price discount offered satisfies the following condition: AP<
ATRCt(Am) D
+ (2~ - 1) H,Am 2m(m + Am) ’
(10)
L. Zhuang/Int.
J. Production
Therefore, under given system parameters, a more economic production and supply system is achievable if there exists a Am > 0 such that the following condition is satisfied: C _ ATRC,O(Am) < [(2cr - l)Hs + h,p]D d 2m(m + Am) . Am
(11)
The condition is obtained by setting the right-hand side of Eq. (10) to be greater than the right-hand side of Eq. (7).
3.2. Optimal delivery ,frequency
and price discount
Suppose there exists an alternative production and JIT delivery system that outperforms the current one. Furthermore, we assume that both parties agree to improve system performance by sharing an equal amount of achievable profit margin. Then we can obtain an optimal adjustment for delivery frequency and price discount that maximizes this profit margin. It should be noted that for such an agreement to take place, both parties should be willing to share sensitive cost information. In the traditional supply environment, suppliers may be reluctant to give out such information for the fear that it could be used against them in subsequent contract negotiations. However, in a JIT supply environment where there is a mutual dependence and sharing of benefit and risk, the sharing of such sensitive cost information is not impossible [6]. The sharing of net profit is also assumed in [lo, 173. Let AP,(Am, Ap) = AP,(Am, Ap) and make use of Eqs. (6) and (8); we can easily derive the following expression for Ap as a function of Am: AP z &(CdAm
+
+ ATRCz(Am))
(2a - 1ws - kPAm 4m(m + Am)
’
Economics
36 11994)
AP(Am)
C(2a- 1)K + bIDAm 4m(m + Am) - tCdAm
+ iATRCf(Am).
(13)
Note that the profit function becomes negative at some point and goes to infinity as Am is increased to infinity. Our concern is, however, whether the profit function is positive (determined by condition (11)) and if yes, where the positive profit is maximized. A straightforward method to find the optimal Am is through enumeration that calculates profit for each integer Am > 0. Such an enumeration method is expected to be efficient as nomally the segment of profit function with positive value is small. However, as we will see later, a simple heuristic formula can be obtained to estimate the optimal Am with good accuracy. In fact, for all the numerical examples we studied the formula always gives an optimal solution. Take the first derivative of function AP(Am) with respect to Am; we have
AP’(Am)= C(2g- l)ffs + kPlD 4(m + Am)’ -$c,++
dATRCz(Am) dAm
(14)
’
As ATRCz(Am) is an implicit and discontinuous function of Am, its derivative cannot be obtained. However, as ATRCg(Am) is a function of Q which does not change much with Am (see Eq. (5)) we assume that its derivative to Am is negligible. (For instance, in the example described below, the numerical value of the last term in (14) is - 0.25 at the optimum point.) Thus, the following heuristic formula for optimal Am can be obtained by setting the derivative in (14) to be zero:
(12)
where the same approximation described earlier is used. Eq. (12) states that for a given Am both buyer and supplier will gain equal profit margin if the price discount is accepted. To maximize the profit margin, substituting Ap from Eq. (12) into Eq. (8) and after some simple algebra, we obtain the following equation:
=
311
307-313
Am* z
C(2a - l)N, + h,PlD 2Cd
1-
m.
(15)
Observe that in order to have a feasible solution for Am > 0, 51must be greater than 0.5. This condition is consistent with what we described earlier about the system (see (2)). That is, the system must be such that a decrease in delivery quantity will result in a decrease in the supplier’s average inventory level.
312
Note that as a result of the optimal adjustment, the buyer’s delivery frequency becomes m* + Am* which is greater than m*. This means that the analysis described above cannot be simply carried out by substituting the optimal m* and Q* into TRC, and TRC,, respectively, for a given price p, To demonstrate how the method can be used in practice, we consider a simple example. Assume that the following parameters are given: D = 10000 units, p = $50.0/unit, H, = $lS/unit yr., h, = 0.3, Cd = $200/dehvery, C, = $500/setup and x = 0.8. Then the current optimal policy for the buyer and its supplier are, respectively, m* z 19 and Q* = 3 x 526 = 1578. Now suppose that the buyer wants to increase delivery frequency to further reduce its inventory. Then in order not to incur additional cost, the buyer wants to negotiate with the supplier for a price discount. The buyer demonstrates that by obtaining as little as Ap = $0.035 price discount per unit, both parties will be better off with an increased profit margin of AP = $179. As a result of this policy adjustment, the buyer’s delivery frequency will be increased to 24 times per annum (i.e. Am * = 5) and the supplier’s production lot size becomes Q* = 4 x 417 = 1668. Furthermore, the JIT buyer firm’s average inventory is reduced by Am/(m + Am) = 20.8%.
the associated adjustment for delivery frequency and production lot size can be obtained that maximize the profit margin. To implement the proposed approach, a JIT firm needs to know a supplier’s inventory carrying cost H,. This may not be possible in traditional supply systems in which each party only works towards its own advantage. However, in a just-in-time supply and delivery system, a JIT firm and its supplier will have an established long-term relationship under which there is a mutual dependence and sharing of benefit and risk. Furthermore, an adjustment of price and delivery schedule is also economically beneficial to the supplier. Thus, it may not be difficult to carry out the analysis. The method can be extended in a number of ways. For example, the supplier’s inventory carrying cost H, can be dependent on the selling price. It can also be extended to the case when the supplier is responsible for the delivery. In this case, however, if the buyer wants to reduce his inventory level by asking for an increase in delivery frequency, the supplier may have to increase its selling price to compensate for the increased delivery cost. It is expected that there may still exist an alternative system in which both parties will be better off.
4. Discussion
Acknowledgment
and conclusions
We studied a system in which a JIT firm schedules and delivers its inbound freight from a batch production supplier. We demonstrated that when the JIT firm is a major buyer of its supplier’s product, a more economic production and supply system can be achieved if the supplier is willing to offer a small discount on its previous selling price. The price discount is in exchange with buyer’s willingness to increase delivery frequency for reduced on hand inventory and to compensate the buyer for his increased delivery expense. The enticement to the supplier for offering a price discount is the achievement of lower finished product inventory. In fact, it is possible that both parties can be better off economically if an alternative production and supply policy is adopted. In this case both parties share an equal profit margin. An optimal price discount and
The author and comments
thanks a referee for his suggestions to improve the presentation.
References [l] Ansari. A. and Modarress. [2]
[3]
[4]
[S]
B., 1990. Just-in-Time Purchasing. The Free Press. New York. Hahm. J. and Yano. C.A., 1992. The economic lot and delivery scheduling problem: The single item case. Int. J. Prod. Econom., 28: 235~ 252. Golhar, D.Y. and Sarker, B.R., 1992. Economic manufdcturing quantity in a just-in-time delivery system. Int. J. Prod. Res., 30: 961-972. Schonberger, R.J., 1982. Japanese Manufacturing Techniques: Nine Hidden Lessons in Simplicity. The Free Press, New York. Suzaki, K.. 1987. The New Manufacturing Challenge: Techniques for Continuous Improvement. The Free Press, New York.
L. Zhuang/Inr.
J. Production
[6] Harding, M., 1990. Profitable Purchasing: An Implementation Handbook for Just-in-Time. Industrial Press Inc., New York. [7J Zhuang, L.. 1993. Comments on economic manufacturing quantity in a just-in-time delivery system. Int. J. Prod. Res., 31: 2747-2748. [SJ Silver, E.A. and Peterson, R., 1985. Decision Systems for Inventory Management and Production Planning. Wiley, New York. [9] Dolan, R.J., 1987. Quantity discount: Managerial issues and research opportunities. Marketing Sci., 6: l-22. [IO] Banerjee, A., 1986. A joint economic-lot-size model for purchaser and vendor. Dec. Sci., 17: 292-3 1 I. [l I] Goyal, S.K., 1987. Determination of a supplier’s economic ordering policy. J. Oper. Res. Sot., 38: 8533857. [IZ] Joglekar, P.N., 1988. Comments on a quantity discount pricing model to increase vendor profits. Mgmt. Sci., 34: 1391l1398. [13J La], R. and Staelin, R., 1984. An approach developing an optimal discount pricing policy. Mgmt. Sci., 30: 1524-l 539. [l4J Lee, H.L. and Rosenblatt, M.J., 1986. A generalized quantity discount pricing model to increase supplier’s profits. Mgmt. Sci., 32: 1177-l 185.
Economics
[15] [163
[l7]
[lSJ
[19J
[20] 1213
[22]
36 (1994) 307-313
313
Monahan, J.P., 1984. A quantity discount pricing model to increase vendor’s profits. Mgmt. Sci., 30: 720-726. Rosenblatt, M.J. and Lee, H.L., 1985. Improving profitability with quantity discounts under fixed demand. IIE Trans., 17: 388-395. Zusman, P. and Etgar, M., 1981. The marketing channel as an equilibrium set of contracts. Mgmt. Sci., 27: 284-302. Schorr, J.E., 1990. Managing transportation costs in an MRP lI/JIT environment. 33rd Internat. Conf. Proc. American Production and Inventory Control Society, New Orleans, USA, pp. 662-663. Golhar, D.Y. and Stamm, CL., 1991. The just-in-time philosophy: A literature review. Int. J. Prod. Res., 29: 6577676. Jones, D.J., 1991. JIT and the EOQ model. Mgmt. Accounting, February, 54-57. Goyal, SK. and Gupta, Y.P., 1989. Integrated inventory models: The buyer-vendor coordination. Eur. J. Oper. Res., 41: 261-269. Ramasesh, R.V., 1990. Recasting the traditional inventory model to implement just-in-time purchasing. Prod. Inv. Mgmt., 31: 71-75.
production economics
ELSEVIER
Towards
Int. J. Production
Economics
a more economic
36 (1994) 307-313
production system
and just-in-time
delivery
Li Zhuang Department
of Industrial
and Systems Engineering, National
Received
1 March
Unioersity
1994; accepted
qf Singapore.
10 Kent Ridge Crescent. Singaporr 051 I, Singapore
in revised form
1 July 1994
Abstract
We study a system in which a batch production supplier participates in just-in-time (JIT) delivery of its products to a JIT buyer firm. We demonstrate that under certain conditions a small adjustment of previously agreed product selling price can result in a more economic production and JIT delivery system with reduced delivery quantity and hence increased delivery frequencyln the case when both buyer and supplier share equal profit margin, optimal levels of adjustment for price and delivery frequency can be obtained that maximize the profit margin. A numerical example is used to illustrate the results.
1. Introduction
The concept of just-in-time (JIT) production and supply has recently received considerable attention from both academics and industries. An increasing number of empirical and theoretical studies have appeared in the literature that justify the economy of adopting the JIT concept by both buyer firms and their suppliers [l-5]. In essence, JIT is an approach for production management and productivity improvement that seeks to produce and deliver the right products with the right quantities at the right time, thereby reducing overall inventories (raw materials, work-in-process and finished products). Successful implementation of JIT requires a long-term relationship between a JIT firm and its suppliers. Under such a relationship the JIT buyer firm offers a long-term supply contract to a supplier who agrees to deliver quality products on a just-in0925-5273/94/$07.00 0 SSDI
1994 Elsevier
0925-5273(94)00055-7
Science B.V. All rights reserved
time basis. In some cases, the buyer provides a forecast of demand and shares market information with its supplier. In return the supplier shares cost information with the buyer and agrees to quality and cost improvement over time [6]. Furthermore, if a long-term commitment is established between the two parties, the JIT firm may share its production information continuously with the supplier. Hence there is no formal release to the purchase order. The supplier simply delivers the parts as they are required - weekly, daily or hourly [6]. Essentially, the relationship will ensure delivery of supplies that is synchronous with production so that the buyer does not need to keep high level inventory to smooth its production. Although it is not necessary for a supplier to become a JIT user, the supplier is strongly encouraged to implement the JIT approach to further reduce costs and to become more responsive to customer demands. A batch production supplier
can switch to a lot-for-lot production economically if setup-time and setup-cost have been reduced to certain target levels [a]. Even if reduction of setuptime and setup-cost is not enough to economically operate the system on a lot-for-lot basis, it is still beneficial for the supplier to participate in a JIT delivery system as the finished product inventory is found to be proportional to delivery quantity [3,7]. An important feature of a JIT delivery system is frequent and reliable delivery of quality parts in small lots. As a small delivery quantity means a lower average inventory to both parties, the resulting benefits such as a shorter lead time, improved visibility on the shop floor, earlier identification of defective lots (parts), and faster response to market demands can be enjoyed by both parties. In this paper, we consider a system in which a JIT firm schedules and receives inbound supplies from a batch production supplier on a just-in-time basis. We assume that both parties have agreed on a purchase price and they all commit to JIT production and delivery. It is, therefore, natural that each party makes its most economic decisions (production lot size and delivery quantity) according to its own objectives. Our purpose is to investigate how pricing decision affects the overall economies of the firm and its supplier. Particularly, we demonstrate that through proper adjustment (price discount in the case of delivery by the buyer firm) of a previously agreed purchase price, a more economic production and JIT delivery system with reduced delivery quantity and hence increased delivery frequency can be achieved. Traditionally, the problems of determining economic ordering and production quantities have been treated separately and often involved a quantity discount schedule. Typical issues addressed include how a buyer should react to a supplier’s quantity discount [S] and how a supplier dictates a buyer’s ordering pattern through manipulating different forms of quantity discount [S, 91. In both cases, the objective of the decision maker is to choose a policy that minimizes cost or maximizes profit according to its own cost structure. Recently, due to the advent of the JIT concept an increasing number of companies have realized the importance of improving relationships with their customers and suppliers. As a result, more
attention has been focused on the studies of interactive relationship between buyers and suppliers. Usually the studies are carried out through the use of an integrated model and the objective is to establish a policy (normally by the supplier) that benefits both parties [l&16]. For example, it has been shown that under certain conditions one party’s gain exceeds the other party’s loss. Thus, the net benefit can be shared equally by both parties through an appropriate price adjustment [lo, 171. However, little work has been done to study the effect of pricing decision within a JIT environment. Although some related papers assume a lot-for-lot production [lo, 151, they do not study how delivery schedule would affect the results. The organization of the remaining paper is as follows. Section 2 describes models and the associated optimal solutions that are used by the JIT firm and its supplier. These models are then used in Section 3 to demonstrate that both parties can be better off if the supplier agrees to offer a price discount to compensate for the buyer firm’s increased delivery expense. An optimum price discount and delivery frequency adjustment are obtained that maximize the equalized profit margin for both parties. A simple example illustrates the results. The paper ends in Section 4 with a discussion and conclusions.
2. Modeling of a JIT production and delivery system The following notations are used: = the customer’s annual demand (units/yr) = the current delivered unit price paid by the P customer (S/unit) Cd = the customer’s aggregate cost per delivery ($/delivery) h, = the customer’s annual inventory carrying cost per unit, expressed as a percentage of the purchase price (%/yr) current annual number of m = the customer’s deliveries (deliveries/yr) C, = the supplier’s production setup cost per batch ($/batch) D
L. Zhuungllnt.
H,
J. Production
= the supplier’s annual inventory carrying cost per unit based on unit production cost (S/unit yr) Q = the supplier’s current production lot size (units/batch) P = the supplier’s annual production rate (units/yr) [x] = the nearest integer rounded from x. We assume that the buyer’s annual demand is not affected by the change in the purchase price. Similarly, the supplier’s annual inventory carrying cost H, is assumed not to fluctuate with the selling price as H, is normally based on unit production cost which is mainly determined by production process. As discussed earlier, the JIT concept calls for frequent deliveries of small quantity, which means that m is much larger than what it would be in traditional batch production systems. This is achievable in a JIT system because delivery cost can be reduced substantially through continuing effort in improving productivity and eliminating non-value added processes together with mixed loading and freight consolidation [4, 5, 181. In addition, the cost of inventory carrying, another factor that affects delivery frequency, is also expected to be rationalized. In fact, JIT challenges the appropriateness of traditional accounting practices and calls for using the process costing method instead of the currently used product costing method [19]. Jones [20] has recently identified relevant costs in a JIT environment that should be included in evaluating inventory carrying cost. These are the costs that are associated with facilities, operations and administration and accounting. It should be noted that with the establishment of a long-term supply contract, the reduced ordering cost may not have as much impact on the ordering quantity as it does in traditional supply systems. Thus, we do not consider ordering cost explicitly in our model. For those minor ordering costs that are associated with each delivery, we assume that they are included in the delivery cost Cd. In general the determination of delivery frequency in most organizations is based on analysis that tends to minimize the sum of all incremental costs associated with or influenced by material deliveries. The analysis is, however, usually based on
Economics
309
36 (1994) 307-313
certain assumptions which may vary in different organizations depending on the type of purchasing systems and interfaces that a particular company uses. For example, Hewlett-Packard makes its weekly delivery schedule using a model that assumes the following [l]: The only significant incremental costs associated with delivery frequency are freight and inventory holding costs. Buying costs, as well as related materials management overhead and direct labor costs, do not vary significantly with delivery frequency. Based on the above discussion, we assume that a JIT firm determines its optimal annual delivery frequency by minimizing the following cost function: l
l
TRC,(m) = 2
h,p + mCd.
Thus, an optimal
m* =
integer
solution
[Jl
Dhcp
is given by
(2)
?Ey
It should be noted that a reduction in incoming parts inventory has many benefits (see Section 1). These are mostly intangible benefits that are strategically important to an organization’s competitiveness. However, as these benefits are difficult to estimate, they are excluded from the model formulation. A cost structure similar to (1) has also been used by Hewlett-Packard [l]. On the other hand, with an average delivery quantity of D/m*, the supplier makes its production planning. We assume that the supplier’s production lot size Q is an integer multiple of the delivery quantity. This is reasonable as it has been proven that adoption of the integer multiple policy is more beneficial to the supplier [16]. In fact, this policy also proves to be optimal if supplier and customer jointly determine their production lot size and delivery quantity [2]. Under this condition, it can be shown [2,3,7] that the average inventory level of a supplier’s finished product is
Qavg= $1 - 4Q + +
- I),
where, CI= D/P < 1 is the ratio of customer’s annual demand to supplier’s annual production capacity.
We assume that the supplier uses the following criteria to determine its production lot size: TRC,(Q) = 7
+ f(1 - z)QHs + &(2,
- l)H, (4)
3.1. E.uisfence qf’ a more economic
s)xtern
Suppose that the supplier agrees to have a price discount of Ap > 0 for a delivery increment of Am > 0. Then after some simple algebra the following expression for the increased profit margin to the JIT buyer can be derived:
which gives AP,(Am, Ap) = ApD - AmCd +
Dh,(pAm
2m(m + Am)
(5) where the integer multiple policy described above is taken into account. Note that the remark made earlier regarding the exclusion of inventory reduction benefits from the cost structure is also applicable to the supplier. Note that when r > 0.5 the supplier’s cost is proportional to delivery quantity, which means that increasing delivery frequency is also beneficial to the supplier although its optimal lot size Q* may be changed accordingly. Thus, we assume cx > 0.5. This may be the case when the JIT firm is a major consumer of its supplier’s product. This agrees with JIT purchase and supply philosophy of keeping fewer, ideally sole, sources of supplies.
3. A more economic JIT production and delivery system We assume that the system has been operational for some time under the JIT delivery contract. Suppose that in its continuing effort in reducing inventory, the JIT buyer firm wants to increase delivery frequency. However, by doing so the firm will certainly incur additional cost. One way to offset this additional cost is to ask for a price discount from the supplier. As we noted earlier, this may be acceptable to the supplier because a decrease in delivery quantity will reduce the finished product inventory and eventually reduces its production cost although production lot size may change accordingly. In this section, we show that there may exist a purchase price under which both parties will be better off economically. For brevity, we will drop superscript from the current delivery frequency m*.
+ mAp) ’
(6) where the last two terms correspond to the cost saving. It can be easily proved that AP,(Am, Ap) is a monotonically decreasing and increasing function of delivery increment Am and price discount Ap, respectively. For a given Am, the buyer will not be worse off, i.e., P,(Am, Ap) > 0 if the amount of price discount offered by the supplier satisfies the following condition: (2C,/D)m(m AP ’
+ Am) - h,p Am
2(m + Am) + h,
%-Cd D
m
kP Am, 2m(m + Am)
(7)
where the approximate equality is obtained by assuming that 2(m + Am) + h, z 2m(m + Am) as normally 2(m + Am) $ h,. Similarly, the supplier’s increased profit margin is given by AP,(Am, Ap) = - ApD + ATRCf(Am) + (2~ - 1) DH,Am 2m(m + Am)
’
(8)
where ATRCf(Am)
= C,D +
t(l - Co(Q* - Q)Hs.
(9)
Then from supplier’s viewpoint, for a given Am, the supplier will not be worse off, i.e., AP,(Am, Ap) > 0 if the price discount offered satisfies the following condition: AP<
ATRCt(Am) D
+ (2~ - 1) H,Am 2m(m + Am) ’
(10)
L. Zhuang/Int.
J. Production
Therefore, under given system parameters, a more economic production and supply system is achievable if there exists a Am > 0 such that the following condition is satisfied: C _ ATRC,O(Am) < [(2cr - l)Hs + h,p]D d 2m(m + Am) . Am
(11)
The condition is obtained by setting the right-hand side of Eq. (10) to be greater than the right-hand side of Eq. (7).
3.2. Optimal delivery ,frequency
and price discount
Suppose there exists an alternative production and JIT delivery system that outperforms the current one. Furthermore, we assume that both parties agree to improve system performance by sharing an equal amount of achievable profit margin. Then we can obtain an optimal adjustment for delivery frequency and price discount that maximizes this profit margin. It should be noted that for such an agreement to take place, both parties should be willing to share sensitive cost information. In the traditional supply environment, suppliers may be reluctant to give out such information for the fear that it could be used against them in subsequent contract negotiations. However, in a JIT supply environment where there is a mutual dependence and sharing of benefit and risk, the sharing of such sensitive cost information is not impossible [6]. The sharing of net profit is also assumed in [lo, 173. Let AP,(Am, Ap) = AP,(Am, Ap) and make use of Eqs. (6) and (8); we can easily derive the following expression for Ap as a function of Am: AP z &(CdAm
+
+ ATRCz(Am))
(2a - 1ws - kPAm 4m(m + Am)
’
Economics
36 11994)
AP(Am)
C(2a- 1)K + bIDAm 4m(m + Am) - tCdAm
+ iATRCf(Am).
(13)
Note that the profit function becomes negative at some point and goes to infinity as Am is increased to infinity. Our concern is, however, whether the profit function is positive (determined by condition (11)) and if yes, where the positive profit is maximized. A straightforward method to find the optimal Am is through enumeration that calculates profit for each integer Am > 0. Such an enumeration method is expected to be efficient as nomally the segment of profit function with positive value is small. However, as we will see later, a simple heuristic formula can be obtained to estimate the optimal Am with good accuracy. In fact, for all the numerical examples we studied the formula always gives an optimal solution. Take the first derivative of function AP(Am) with respect to Am; we have
AP’(Am)= C(2g- l)ffs + kPlD 4(m + Am)’ -$c,++
dATRCz(Am) dAm
(14)
’
As ATRCz(Am) is an implicit and discontinuous function of Am, its derivative cannot be obtained. However, as ATRCg(Am) is a function of Q which does not change much with Am (see Eq. (5)) we assume that its derivative to Am is negligible. (For instance, in the example described below, the numerical value of the last term in (14) is - 0.25 at the optimum point.) Thus, the following heuristic formula for optimal Am can be obtained by setting the derivative in (14) to be zero:
(12)
where the same approximation described earlier is used. Eq. (12) states that for a given Am both buyer and supplier will gain equal profit margin if the price discount is accepted. To maximize the profit margin, substituting Ap from Eq. (12) into Eq. (8) and after some simple algebra, we obtain the following equation:
=
311
307-313
Am* z
C(2a - l)N, + h,PlD 2Cd
1-
m.
(15)
Observe that in order to have a feasible solution for Am > 0, 51must be greater than 0.5. This condition is consistent with what we described earlier about the system (see (2)). That is, the system must be such that a decrease in delivery quantity will result in a decrease in the supplier’s average inventory level.
312
Note that as a result of the optimal adjustment, the buyer’s delivery frequency becomes m* + Am* which is greater than m*. This means that the analysis described above cannot be simply carried out by substituting the optimal m* and Q* into TRC, and TRC,, respectively, for a given price p, To demonstrate how the method can be used in practice, we consider a simple example. Assume that the following parameters are given: D = 10000 units, p = $50.0/unit, H, = $lS/unit yr., h, = 0.3, Cd = $200/dehvery, C, = $500/setup and x = 0.8. Then the current optimal policy for the buyer and its supplier are, respectively, m* z 19 and Q* = 3 x 526 = 1578. Now suppose that the buyer wants to increase delivery frequency to further reduce its inventory. Then in order not to incur additional cost, the buyer wants to negotiate with the supplier for a price discount. The buyer demonstrates that by obtaining as little as Ap = $0.035 price discount per unit, both parties will be better off with an increased profit margin of AP = $179. As a result of this policy adjustment, the buyer’s delivery frequency will be increased to 24 times per annum (i.e. Am * = 5) and the supplier’s production lot size becomes Q* = 4 x 417 = 1668. Furthermore, the JIT buyer firm’s average inventory is reduced by Am/(m + Am) = 20.8%.
the associated adjustment for delivery frequency and production lot size can be obtained that maximize the profit margin. To implement the proposed approach, a JIT firm needs to know a supplier’s inventory carrying cost H,. This may not be possible in traditional supply systems in which each party only works towards its own advantage. However, in a just-in-time supply and delivery system, a JIT firm and its supplier will have an established long-term relationship under which there is a mutual dependence and sharing of benefit and risk. Furthermore, an adjustment of price and delivery schedule is also economically beneficial to the supplier. Thus, it may not be difficult to carry out the analysis. The method can be extended in a number of ways. For example, the supplier’s inventory carrying cost H, can be dependent on the selling price. It can also be extended to the case when the supplier is responsible for the delivery. In this case, however, if the buyer wants to reduce his inventory level by asking for an increase in delivery frequency, the supplier may have to increase its selling price to compensate for the increased delivery cost. It is expected that there may still exist an alternative system in which both parties will be better off.
4. Discussion
Acknowledgment
and conclusions
We studied a system in which a JIT firm schedules and delivers its inbound freight from a batch production supplier. We demonstrated that when the JIT firm is a major buyer of its supplier’s product, a more economic production and supply system can be achieved if the supplier is willing to offer a small discount on its previous selling price. The price discount is in exchange with buyer’s willingness to increase delivery frequency for reduced on hand inventory and to compensate the buyer for his increased delivery expense. The enticement to the supplier for offering a price discount is the achievement of lower finished product inventory. In fact, it is possible that both parties can be better off economically if an alternative production and supply policy is adopted. In this case both parties share an equal profit margin. An optimal price discount and
The author and comments
thanks a referee for his suggestions to improve the presentation.
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