Determining optimal retail price and lot size under day-terms supplier credit

Computers ind. Engng Vol. 33, Nos 3-4, pp. 717-720, 1997 O 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved

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Determining Optimal Retail Price and Lot Size Under Day-terms Supplier Credit S.W. Shinn Departmentof BusinessAdministration,SemyungUniversity Jecheon, KOREA Abstract This paper deals with the problem of determining the retailer's optimal price and lot size simultaneously under the condition of permissible delay in payments. It is assumed that the customer's demand rate is represented by a constant price elasticity function which is a decreasing function of retail price. Investigation of the properties of an optimal solution allows us to develop an algorithm whose validity is illustrated using an example problem. Also, the effect of credit period on the retailer's pricing and lot sizing policy is examined through sensitivity analysis. © 1997 Elsevier Science Ltd Keywords : Inventory, Credit period

capital losses incurred during the credit period. The positive effects of credit period on the product demand can be integrated into the EOQ model through the consideration of retailing situations where the demand rate is a function of the selling price the retailer sets for the product. The availability of the credit period from the supplier enables the retailer to choose the selling price from a wider range of option. Since the retailer's lot size is affected by the demand rate of the product, the problems of determining the retail price and the lot-size are interdependent and must be solved simultaneously (we will call the RPLS problem). Kunreuther and Richard[9], Kunreuther and Schrage[10], Ladany and Sternlieb[11], and Shah and Jha[14] dealt with the RPLS problem when the demand is a decreasing function of retail price. Abad[1] and Lee[12] dealt with the RPLS problem assuming that the supplier offers all-unit quantity discounts and the demand for the product is a decreasing function of price. Abad[2] also extended his model to the case of incremental quantity discounts. More recently, Kim et al.[7] introduced the RPLS problem under the condition of permissible delay in payments. They assumed that the retailer's borrowing and lending rates of capital are equal, which seems quite restrictive from the practical point of view.

1. Introduction

In deriving the economic order quantity(EOQ) formula, it is tacitly assumed that the retailer(buyer) must pay for the items as soon as he receives them from a supplier. However, in practice, a supplier will allow a certain fixed period(credit period) for settling the amount the retailer owes to him for the items supplied. Trade credit would play an important role in the conduct of business for many reasons. For a supplier who offers trade credit, it is an effective means of price discrimination which circumvents anti-trust measures and is also an efficient method to stimulate the demand of the product. For a retailer, it is an efficient method of bonding a supplier when the retailer is at the risk of receiving inferior quality goods or service and is also an effective means of reducing the cost of holding stocks. Recently, Haley and Higgins[6], Kingsman[8], Chapman et al.[3], Goyal[5], and Ward and Chapman[15] examined the effects of trade credit on the optimal inventory policy. With the average cost approach, they reported that the EOQ is invariant to the length of credit period. This is not consistent with our expectation since the availability of opportunity to delay the payments effectively reduces the cost of holding inventories, and thus is likely to result in larger order quantity, ceteris paribus. This inconsistency is resulted by the assumption commonly held by the previous research works in which the demand for the product is treated as a given constant. Consequently, they disregarded the effects of credit period on the quantity demanded. As implicitly stated by Mehta[13], a major reason for the supplier to offer a credit period to the retailers is to stimulate the demand for the product he produces. Also, Fewings[4] stated that the advantage of trade credit for the supplier is substantial in terms of influence on the retailer's purchasing and marketing decisions. The supplier usually expects that the profit increases due to rising sales volume can compensate the

Relaxing the above assumption, we allow the retailer's borrowing rate to be larger than or equal to his lending rate. This paper evaluates an inventory system when the supplier permits delay in payments for an order of a product whose demand rate is represented by a constant price elasticity function. 2. Model Formulation

The assumptions of this paper are essentially the same as the EOQ model except for the condition of permissible delay in payments. 717

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The following assumptions and notations are used: (1) Replenishments are instantaneous with a known and constant lead time. (2) The demand rate is represented by a constant price elasticity function of retail price. (3) No shortages are allowed. (4) The inventory system involves only one item. (5) The supplier proposes a certain credit period and sales revenue generated during the credit period is deposited in an interest bearing account with rate L At the end of the period, the credit is settled and the retailer starts paying the capital opportunity cost for the items in stock with rate R(R > 1). C S tc H

: unit purchase cost. : ordering cost. : credit period set by the supplier. :inventory carrying cost, excluding the capital opportunity cost. R : capital opportunity cost(as a percentage). I : earned interest rate(as a percentage). Q : order size. T : replenishment cycle time. P : unit retail price, p < p= where p, is a given upper limit of retail price. K : scaling factor( > 0). ,/] : index of price elasticity( > 0). D

: annual demand rate, as a function of retail price,

D = KP -~. The retailer's objective is to maximize the annual net profit I I ( P , T ) from the sales of the products. The annual net profit consists of the following five elements as stated by Goyal[5]. (1) Annual sales revenue = DP. (2) Annual purchasing cost = DE'.

(3) Annual ordering cost = S__. T (4) Annual inventory carrying cost = DTH 2 (5) Annual capital opportunity cost

= DC(R-I)tc 2 + DTRC - DCRtc if tc < T, 2T 2 = ___DTIC DCltc ff tc > T. 2 The annual net profit H(P,T) can be expressed as H(P, T) = Sales revenue - Purchasingcost - Ordering cost - Inventory carrying cost - Capital opportunity cost. Depending on the relative size of tc to T, I-[(P,T) has two different expressions as follows:

1. Case 1 (tc<_T)

n,( P,T) = D P - D C - ~ - DTH T

2

( D C ( R - I ) tc2 ~DTRC_DCRtc),2T 2

(1)

2. Case 2 (tc > T )

I'I2(P'T)=DP-DcDTH2

TS(DTI2C-DCItc)"

(2)

3. Determination of Optimal Policy The problem is to find an optimal retail price P" and an optimal replenishment cycle time T" which maximizes H ( P , T ) . Also, an optimal lot size Q' can be obtained by the relation, Q = D T . For a fixed P, H ( P , T ) is a concave function of T, and there exists a unique value Ti, which maximizes

[Ii(P,T ) as follows:

= 2~s,,

(3)

T1 y OH,

where S, = S + DC(R- I)tc 2 and H, = H +CR ,

2 T~=

2S

DH~

, where H2 = H + C I .

(4)

Note that since the demand rate D is a function of P, each Ti can be represented by a real valued function of P; that is, T~ = T~(P). Now, from equation (3) and (4), we have a following useful property for a fixed P. Property 1 T~(P)>_tc if and only if T2(P)>tc. If T l ( P ) 2 t c, then ]]2 (P,T) is increasing in T over T < tc. If Tz(p ) < tc, then

T~(P) _ tc. Proof From equation (7), T~(P) _>tc can be rewritten as

~[2S + DC(R ~t~ ~ >_tc"

(P1)

D(H + CR) Squaring both side of equation (P1) and rearranging,

2S + DC(R - l)tc 2 > D(H + CR)tc 2, 2S

>_tc"

(P2) (P3)

D(H +CI) Equation (P3) implies that T2(p)>tc and T2(p ) is a maximum point of H2(P,T ), which is a concave function. Thus, I-]2(p,T ) is increasing in T over T < tc. Also, from equations (P1) and (P3), T~(P) tc. Q.E.D.

Proceedings of 1996 ICC&IC Property 1 states that for a fixed P, the optimal replenishment cycle time which maximizes rI(P,T) is known to be either Tt(p ) or T2(p) because the annual net profit function is continuous at T = tc. Moreover, if Tt(P) > tc, then Tt ( p ) becomes an optimal replenishment cycle time T'(P)- And if Tt(P) tc. Since the demand rate D is also a function of P, the inequality can be rewritten as

T,(P) =

[25 + DC(R- l)tc 2 ~

(5)

>tc"

and fz(p)=IH2~sD 2)28 • For l~ 0 and l-l°d(p) is an increasing function of P. Thus an optimal value Pi of I'I°l(P) occurs at the maximum point of the price interval(p/~7) corresponding to T/(P). For ,fl> 1, it can be shown that both f l ( p ) and f2(p ) become positive and the second order condition is satisfied if p < ([3+l)c(l_Rtc).

Note

(,8-1)

that, given the second order assumption, p~ is the one which has the minimum absolute value of rl0~(p)• on the price interval(P/T). Based on the above properties, we develop the following solution procedure to determine an optimal retail price and lot size.

Rearranging equation (5), |

Solution algorithm

Let !

Po . . . . .

(7)

It is self evident that for any p ~ P/~t' P/T1 = {PIP ;~ Po}' the inequality Tt(P)>tc holds. So, we conclude that Tt(p ) determined by P value which satisfies the inequality (6) becomes an optimal replenishment cycle time T'(P). Similarly, T2(p) becomes an optimal replenishment cycle time T'(P) only if P is an element of price interval PlT2 = {pIp
i-[o~(p)=D{P_C(l_Rtc)}_~,

(8)

1-I%(P) = D{P - C(1- tic)} - 2VI-~2DS. (9) Therefore, an optimal solution (P',T') which maximizes I[(P,T) is found by searching over 1-I°~(p). Note that Fl°~(p) is valid only on the interval PEPFI~. Now, we are going to investigate the characteristics of l-[°~(p). Using the ,classical optimization technique, the first derivative of l-[°t(p) becomes n°,(P)'

719

=

'

1-I°2(P)' = D+ D ' I P - C(1- ltc)}- D'. H ~2S" '!' 2/9

(lo) (11 )

Also, the second order condition for concavity is Flo,(p)- =BDp-z{(/j_l)p_(O+l)C(l_Rtc)_ft(p)}< O, (12) ilo,(p)-= ODp-2{(O_ l)p_(/$+ l)C(l_ /tc)_ fz(p)}< O, (13) where ft (P) = ~ t [ ( f l + 2)S 2 + (fl + 1)DCrcZ(R-1){3S + DCtc~(R- 1)}] ~SDSt 3

Step 1: Determine P0 from equation (7). Step 2: If p, < P0' then go to step 4. Otherwise, go to step 3. Step 3: This step determines the optimal retail price Pt and replenishment cycle time T=(pt) for Case 1. 3.1 Determine Pl, which maximizes [Iol(p) on the following price interval: p E P/~t and p _
4. Numerical Example and Sensitivity Analysis To illustrate the solution algorithms, the following problem is considered. 8 = $ 50, K = 250000, C= $ 3, H = $ 0.1, R = 0.15( = 15%), ] = 0.1( = 10%), tc = 0.3 and ~ = 2.5.

4.1. Numerical Example The optimal solution with p= = C(I - Rtc) ([3 + 1) = 6.68

([3-1) can be obtained through the following steps. Step 1: From equation (7), P0 = 6.05.

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Step 2: Since p, = 6.68 > Po' go to step 3. Step 3: Solving equation (10) numerically on the price interval, 6.05
solution (P',T') becomes (4.93, 0.233) with its maximum annual net profit $8927. 4.2. Sensitivity Analysis

An interesting question is how much effect the length of credit period has on the retail price, the lot size and the retailer's profit. Since the problem structure of equation (1) and (2) does not permit sensitivity analysis, the same example problem is solved to answer the above question. Five different values of tc are adopted, rc = 0.0, 0.05, 0.1, 0.15, 0.3. For each value of tc, three different values of /3 are tested. The results are shown in Table 1. and the following observations can be made whinh are consistent with our expectation. (i) As either tc or/~ increases, p" decreases. (ii) With j~ fixed, as tc increases, p" decreases while _rI(P',T') increases. (iii) With/~ fixed, as tc increases, Q" is increasing. 5. Conclusion This paper dealt with the joint price and lot size determination problem when the supplier offers a certain fixed credit period. Recognizing that a major reason for the supplier to offer a credit period to the retailers is to stimulate the demand of the product, we expressed the retail demand rate of the product with a constant price elasticity function. The function value depends on the retail price which in turn is affected by the length of credit period. Table 1. Sensitivityanalysiswi~ variousvaluesof tc and .fJ. tc

1.5

2.5

3.0

0.0 1285 9.12 54845

0.05 1314 9.06 55039

0.1 1384 9.00 55211

0.15 1492 8.96 55363

0.3 1544 8.83 55788

881 5.10 II(F,T') 8454 Q" 683 p4.60 H(F,T') 3734

895 5.06 8547 691 4.58 3790

927 5.02 8634 708 4.55 3844

969 4.99 8714 734 4.52 3895

1075 4.93 8927 843 4.45 4032

Q" P" I~(P',T" ) Q"

p-

After formulating the mathematical model, we proposed the solution procedure which leads to an optimal retailing policy. Also, through sensitivity analysis, the effects of the length of credit period and index of the price elasticity are examined on the retail price, the lot size and the retailer's profit. The re.suits indicated that the retailer's profit seems quite sensitive to the parameter values examined. References 1. P. L. Abad, 1988, Determining optimal selling price and lot size when the supplier offers all-unit quantity discounts, Decision Sci., 19:622-634 2. P. L. Abad, 1988, Joint price and lot-size determination when supplier offers incremental quantity discounts, J. Opl Res. Soc., 39:603-607 3. C. B. Chapman, S. C. Ward, D. F. Cooper and M. J. Page, 1985, Credit policy and inventory control, J. Opl Res. Soc., 35:1055-1065 4. D. R. Fewings, 1992, A credit limit decision model for inventory floor planning and other extended trade credit arrangement, Decision Sci., 23:200-220 5. S. K. Goyal, 1985, Economic order quantity under conditions of permissible delay in payments, J. Opl Res. Soc., 36:335-338 6. C.W. Haley and R. C. Higgins, 1973, Inventory policy and trade credit financing, Mgmt Sci., 20:464-471 7. J. S. Kim, H. Hwang and S. W. Shinn, 1995, An optimal credit policy to increase suppller's profits with price dependent demand functions, Prod. Planning and Control, 6:45-50 8. B.G. Kingsman, 1983, The effect of payment rule on ordering and stock holding in pumhasing, J. Opl Res. Soc., 34:1085-1098 9. H. Kunreuther and J. F. Richard, 1971, Optimal pricing and inventory decisions for non-seasonal items, Economstrica, 39:173-175 10. H. Kunreuther and L. Schrage, 1973, Joint pricing and inventory decisions for constant priced items, Mgmt Sci., 19" 732-738 11. S. Ladany and A. Sternliab, 1974, The interaction of Economic ordering quantities and marketing policies, AIIE Trans., 6:35-40 12.W.J. lee, 1993, Determining order quantity and selling price by geometric programming: optimal solution, bounds, and sensitivity, Decision Sci., 24: 76-87 13. D. Mehta, 1968, The formulation of credit policy models, Mgmt Sci., 15:B30-B50 14. Y. K. Shah and P. J. Jha, 1991, A single period stochastic inventory model under the influence of marketing policies, J. Opl Res. Soc., 42:173-176 15. S. C. Ward and C. B. Chapman, 1987, Inventory control and trade credit - A reply to Daellenbach, J. Opl Res. Soc., 38:1081-1084