J. Opl Res. Soc. Vol. 40, No. 9, pp. 805-813, 1989 0160-5682/89 $3.00 + 0.10 Printed in Great Britain. All rights reserved Copyright ? 1989 Operational Research Society Ltd

Effect of Delayed Payments (Trade Credit) on Order Quantities

RAM RACHAMADUGU School of Business Administration, University of Michigan, Ann Arbor, Michigan, USA

We address the problem of determining the economic order quantity when the vendor permits delay in payment. Some early researchers argued that the best order quantity is invariant with respect to the trade credit. Others argued that the order quantity should increase as the delay in payment increases. We analyse this problem using the discounted cash-flow approach, and provide clarification on the inconsis- tencies between these approaches. First, we show that the best order quantity is an increasing function of the permitted delay in payment. We also show that an approach suggested by Chand and Ward not only yields an upper bound on the optimum, but also provides robust results. Though the classical square- root formula disregards trade-credit information, under some circumstances, it will yield better results than the formulation taking into account that information. We illustrate this anomaly with an example, and provide analytical explanation for it. We also discuss some potential conceptual pitfalls in using the average cost analysis as an approximation to the discounted cash-flow approach.

INTRODUCTION

It is well known that the square root formula provides optimal order quantities in a deterministic demand environment when backlogging or shortages are not permitted. The decision criterion used in that analysis is minimization of the average cost per period. However, a common practice in industry is to provide a specific delay period for the payment after the goods are delivered. Some earlier researchers1'2 analysed this problem. Their analysis leads to the conclusion that when the cost of funds is the same as the return on opportunities available to the firm, the order quantity is invariant with respect to the payment delay period. Further, the order quantity accord- ing to their analysis is equal to the quantity derived using the square root formula. Intuitively, it is clear that the availability of opportunity to delay the payment effectively reduces the cost of holding inventories, and thus is likely to result in larger order quantities. Chand and Ward3 analysed the same problem and suggested an order quantity which increases as the delay time for payment increases. Their approach takes into consideration delay in payments through dis- counting, and superimposes this effect on the square root formula. Thus we have two schools of thought one arguing that the order quantity is invariant with respect to the delay time for payment, and the other arguing for an increase in order quantity as the delay time for payment increases.

In this paper, we analyse the trade-credit problem using the discounted cash-flow approach. In the next section, we briefly review approaches used by earlier researchers. Later, we present our approach based on the discounted cash-flow method. Further we show that the approach sug- gested by Chand and Ward yields an upper bound on the optimum order quantity. Next, we illustrate the procedure using a numerical example. Further, we shall show why the square root formula, in spite of disregarding the delay in payment information, may provide better results than the Chand and Ward approach under some circumstances. However, we shall also point out why the formulation given by Chand and Ward is robust. In the final section, we conclude with a discussion on the appropriateness of using average cost analysis, and provide future research directions.

PRIOR RESEARCH In this paper, we assume that the demand for the product is deterministic and constant over an

infinite horizon. Further, it is assumed that no backlogging or shortages are permitted. The vendor permits some time delay in payment for the goods after their receipt by the customer. Also, it is assumed that the rate of return earned on funds equals the cost of funds. The notation used in the paper is defined in Table 1.

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TABLE 1. Notation

D= Demand rate h = Holding cost per unit per period

(exclusive of the capital charges) p = Price of the item r = Discount rate or interest rate S = Set-up cost or order cost t = Delay time permitted in paying for the

goods after their receipt by the customer p' = pl(1 + rt) (for continuous discounting, it is given by pe ")

Q(EOQ) = Order quantity as determined by the square root formula = /2SD/(h + pr)

Q(CW) = Order quantity determined using the Chand and Ward approach = 2SD/h + r1 (for discrete discounting) =- S + prert) (for continuous discounting)

Q(OPT) = Optimum order quantity determined using discounted cash-flow analysis

T(EOQ), T(CW), T(OPT) = Reorder intervals corresponding to Q(EOQ), Q(CW) and Q(OPT)

Chapman, Ward, Cooper and Page1 (CWWP) and Goyal2 used average cost analysis for ana- lysing the trade-credit problem. One of the cases discussed by CWWP conforms to the assump- tions we have stated here. They found that the order quantity is independent of the trade-credit period or the time delay permitted in paying for the goods after they are received. Goyal2 also analysed this problem, although from a slightly different perspective. He developed a generic model, where he assumes that in a deterministic environment, opportunities available to the firm may yield a return different from the cost of funds (or borrowing) for the firm. When these returns are equal, the problem he described is the same as the case discussed here. Goyal also used average cost analysis for solving the problem. His conclusion also was that when the returns are equal, the order quantity is independent of the trade credit or the time permitted by the vendor to pay for the goods. Chand and Ward3 also studied the trade-credit problem. Superimposing the principle of discounting on average cost analysis, they provided a formulation for the order quantity.

Some researchers recognized the need to analyse the inventory problems using the net present- value concept or discounted cash-flow analysis. Hadley4 found, through extensive computational study, that the results provided by average cost analysis are a good approximation to net present- value analysis for the classical formulation. His analysis was limited to the case where payment for the goods was instantaneous. Trippi and Lewin,5 Grubbstrom6 and Rachamadugu7 also elabo- rated on this issue. In the next section, we discuss formally the motivation to use the net present- value (or discounted cash-flow) approach for the trade-credit problem, and derive some necessary results using this approach. Throughout this paper, the terms net present value and discounted cash-flow analysis are used synonymously.

DISCOUNTED CASH-FLOW APPROACH In this section, we analyse the problem using discounted cash-flow analysis. The basic objective

of the firm is to maximize the net present value of the shareholder wealth. Given that the revenue inflow stream is generated by the constant demand rate, we maximize the net present value of the shareholder wealth by minimizing the net present value of the cash outflows resulting from the inventory decisions. Hence we shall use the discounted cash-flow approach to analyse this problem. The net present value is estimated by discounting cash outflows originating at different times using the time value of money. For further elaboration on the need to analyse inventory problems using discounting, the reader is directed to the article by Grubbstrom.6

Suppose that the reorder interval is T. The net present value [NPV(T)] of all future costs is given by

NPV(T) = S + { hD(T - l)e-rl dl + DTpe-rt + e-rT NPV (T). (1)

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The first term in the above expression relates to set-up or order costs. The second term relates to the non-capital-related holding charges such as warehousing etc. The third item relates to the price paid for the items. The fourth term is the recursive term accounting for all cash outflows beyond the first cycle. Since we are using discounting, there is no need to state the capital-related holding charges explicitly. The above expression can be rewritten as

S hD rT - 1 + erT Dpe-&tT NPV(T) = -T+ r 1-- + 1--(2) 1e-r.T r le-rT l -e-rT

For our further analysis, we use annualized cost, ANN(T), instead of NPV(T). ANN(T) is that uniform cash stream over infinite time whose net present value is NPV(T). It is a surrogate for NPV(T). From the basic principles of discounting,

ANN(T) - r* NPV(T) Sr hD rT - 1 + e rT Drpe-rtT

1-erT 1-e-rT +1- e-rT (

It can easily be seen that for small values of rT, the above expression can be approximated as

S hDT +pe eD prtDTr(4 T 2 pe D + 2 (4)

- + - (h + rpert)DT + pertD. T 2

Now consider the case where no delay in payment is permitted-i.e. t = 0. The above expres- sion (4) reduces to

Si1 -+ - (h + rp)DT + Dp. (5) T 2

(5) is the same as the average cost per period. It is expression (5) that is generally used for inventory analysis. Thus, average cost analysis is an approximation to the annualized cost approach or the discounted cash-flow analysis. It is well known that the average cost per period (5) is a convex function. This property is generally used to derive the classical square-root formula. However, since the net present value or annualized cost approach is conceptually more rigorous and directly relates to the concept of maximizing the net present value of current shareholder wealth, we shall make use of ANN(T) for our subsequent analysis. First, we shall establish that ANN(T) (3), just like the expression for the average cost per period (5), is also convex. We make use of this property later in our paper.

Proposition I ANN(T) is convex in T.

Proof The proof is similar to Rachamadugu.7 The second derivative of ANN(T) in expression (3)

yields

ANN"(T) Sr 3e-rT(1 + e-rT) (1 - e rT)3

+ (hpe)rT {(2 + rT)erT - (2 - rT)}. (6)

It is clear that ANN"(T) is non-negative if the term in curled parentheses is non-negative.

(2 +rT)e-rT-(2-rT) = -4ijZn(rT) ( 2}.

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Since ANN"(T) > 0, ANN(T) is convex in T. The convexity property of ANN(T) will be useful in rigorously analysing the alternative

approaches to order-size determination for the trade-credit problem suggested by earlier researchers. In the next section we shall analyse the approach to this problem suggested by some researchers based on the average analysis.

ANALYSIS OF THE AVERAGE COST APPROACH

As mentioned earlier, Chapman et al.' and Goyal3 analysed the trade-credit problem using the average cost analysis. For the case described here (i.e. constant demand, no shortages or back- logging, cost of funds equals return available to the firm, and the payment to be made to the vendor no later than t periods after delivery of the order), their arguments, based on the average cost analysis, lead to the following order quantity:

Optimum order quantity = + p (7)

Average cost per period = vl/2SD(h + pr) - Dprt (8) The reader will notice that expression (7) is the same as the classical economic order quantity,

popularly known as the square root formula. It is clear that this order quantity is independent of the credit period t. This happens for the following reasons. If the order quantity is large, then we may get large benefits due to free use of funds (based on delay in payment), but the number of occasions on which such benefits are available decreases on a per period basis. The reverse is true when order quantities are small. Overall, the benefit per period due to the delay in payment remains the same. This shows up as a constant negative term in (8). However, order quantity does not depend on t. The reader is directed to the original sources for further details as to how (7) is derived.

The above result is somewhat counterintuitive. Suppose that the vendor permits an increase in the delay to pay for the order. Effectively, this reduces financing costs, and hence we would expect an increase in the order quantity. We rigorously establish this through the proposition below.

Proposition 2 The optimum order quantity and the optimum reorder interval are increasing functions of the

delay. time:

Proof Earlier [proposition (1)] we showed that ANN(T) is convex. Hence T(OPT) can be found by

setting the first derivative of ANN(T) equal to zero.

ANN'(T) = - Sr er 1-ee -rTr r er (1 - eT + (Dper2Tr + hD) (1 - erT)2

ANN'(T) = 0 => Sr2

erT(OPT) -rT(OPT) = D(h - rt) + 1. (9)

Using (9), dT(OPT) Spr. ert 1

dt D h + prert erT(OPT)-

Note that the right-hand side of the above expression is always positive. Hence T(OPT) and Q(OPT) are increasing functions of the delay time, t.

The above proposition contradicts the conclusions drawn by earlier researchers that when the cost of funds is the same as the returns available to the firm, the order quantity is equal to the classical order quantity and remains invariant with respect to the delay time. This does not imply

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R. Rachamadugu Effect of Delayed Payments on Order Quantities

that their analysis is at fault. On the contrary, their conclusions are perfectly consistent with the criterion they use (average cost). We wish merely to point out that though the average cost analysis may be adequate for most practical purposes, it can lead to erroneous conceptual conclu- sions. Next, we discuss and analyse an alternative approach to the trade-credit problem suggested by Chand and Ward.3

AN ALTERNATIVE APPROACH

Chand and Ward3 suggested an alternative approach to the trade-credit problem. Recognizing the fact that the effect of the trade-credit is to reduce the inventory financing costs, they dis- counted the item price to the start of the order cycle. Substituting this in the classical square-root formula,

optimum order 2SD quantity - Q(CW) { h + r* (discounted value of the item at

the start of the order cycle) / 2SD (for discrete (10) h + pr discounting) h 1 + rt

| 2SD (for continuous (11) h + pre-rt discounting).

It is clear from the above analysis that the order size determined using the Chand and Ward approach is always more than the order size derived using the classical square-root formula.

Remark I

Q(CW) > Q(EOQ). Next we establish an interesting property of the approach suggested by Chand and Ward.3 The

order quantity derived using their approach is an upper bound on the true optimum.

Remark 2 The Chand and Ward approach (11) yields an order quantity which is an upper bound on the

true optimum; i.e. Q(CW) > Q(OPT). Proof

ANN(T) - Sr hD rT -1 + errT DrxT (12) 1- e-+ r 1er'T - e'T~ where x is the discounted price of the item. Rachamadugu7 has shown that an order quantity equal to /2SD/(h + xr) is an upper bound on the true optimum for expression (12). Substituting pe -rt for x,

2SD

Dath + prert > Q(OPT).

Details are omitted here for the sake of brevity. The interested reader is referred to Rachamadugu.7

DISCUSSION In this section, we discuss the conceptual and computational analysis of the alternative

approaches to the trade-credit problem. First, it is clear that the Chand and Ward approach is conceptually consistent with our expectations in that the optimal order quantity increases as t increases. Also, note that the approach suggested by them has only one source of approximation using average cost analysis to determine the effects of set-up and holding costs. However, the other approach, where the average cost approach is used to take into account the

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effects of trade-credit and the normal set-up and holding costs, has two sources of approximation- effects of trade-credit and other inventory-related costs. Thus, we would expect the approach suggested by Chand and Ward to yield more accurate results. In fact, it can be shown that the Chand and Ward approach yields robust results. The analysis is similar to the one given in Rachamadugu.7 Based on an extreme scenario (25% capital charge and reorder interval of one year), for all practical purposes, the relative error resulting from the use of the Chand and Ward approach can be no more than 3.75%. These details are shown in the Appendix.

The above discussion need not be construed as implying that the approach suggested by Chand and Ward dominates the other approach under all circumstances. Note that the other approach totally discards trade-credit information. However, under some circumstances it may yield better results than the Chand and Ward approach. We illustrate this using a numerical example.

Let S = $200, r = 20% per period,

p = $4 per unit, h = $0.6 per unit per period, D = 200 units per period.

Figure 1 illustrates the analytical relationship between the optimum order quantity, the order size found using the square-root formula, and the Chand and Ward approach. Results for t 0,

t R

z~~~~

6 = Q(OPT) 0 | I b= Q(EOQ)

i c = Q(CW) z l l l

a b c Order quantity

FIG. 1.

1/6 and 4 periods are shown in Figures 2, 3 and 4 respectively and also in Table 2. When t equals zero (Figure 2), both the square root formula and the Chand and Ward approach yield the same order quantity. This is shown in Figure 2. Arrows in Figure 2 indicate the direction of change for various quantities as the payment-delay period increases.

Figure 3 indicates the result for t = 1/6. Notice that in this case, the Chand and Ward approach yields an order quantity greater than the classical EOQ. Further, classical EOQ is greater than the true optimum. Thus, even though the square root formula disregards the delay-payment informa- tion, it results in less annualized cost than the CW approach. This may appear to be somewhat

TABLE 2.

t=0 t= 1/6 t=4

Percentage Percentage Percentage Reorder above Reorder above Reorder above interval optimum interval optimum interval optimum

Square-root 1.195 0.2724 1.195 0.158 1.195 3.042 formula approach

Chand and 1.195 0.2724 1.206 0.271 1.444 0.335 Ward approach

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R. Rachamadugu Effect of Delayed Payments on Order Quantities Delay time = 0 periods

1.16235-

1.1623 -

1.16225 -

1.1622 -

C,) v 1.16215- c 0 T(EOQ) C,) :3 1.1621_ - T(CW) 0 -C

1.16205 - F-/ T(CW) z z 1.162-

1.16195 -

1.1619-

1.16815- T(OPT)

1.1618- 1 . . .. ....... , 1.13 1.138 1.146 1.154 1.162 1.17 1.178 1.186 1.194 1.202

Reorder interval (T) FIG. 2.

Delay time = 1/6 periods 1.1335- 1.1334- 1.1333-

c 1.1332- 1.1331-

0 . 1.1330-

R 1.1329- z 1.1328- < 1.1327- T(CW)

1.1326-

1.1325 T(OPT) T(EOQ) 1.1324

1.148 1.164 1.18 1.196 1.212 1.228 1.244 1.14 1.156 1.172 1.188 1.204 1.22 1.236

Reorder interval (T) FIG. 3.

Delay time = 4 periods 667.5 -

667 - T(EOQ) 666.5 -

666 -

Z 665.5- z

665 -

664.5 - T(CW) 664 TOT) 663.5 -. ............ .......

1.204 1.252 1.3 1.348 1.396 1.444 1 .1 8 1 .228 1 .276 1 .324 1 .372 1 .42

Reorder interval (T) FIG. 4.

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anamolous, but can be explained as follows. The classical EOQ is an upper bound on the true optimum at t = 0. As t increases, the true optimum increases, and the classical EOQ remains the same. However, the Chand and Ward approach yields an order quantity greater than the classical EOQ for all positive values of t. Since the net present-value function is convex, for small values of t, the classical EOQ under some circumstances yields better results than the Chand and Ward approach. It can be verified that, in this example, in the range t = 0 and 0.734 periods, as the value of t increases, relative error associated with the use of the classical economic order quantity decreases. At t = 0.734 periods, the classical EOQ yields an optimal solution! (Details of the exact value of t at which the classical EOQ provides an optimal solution are provided in the Appendix.) However, as t further increases, the true optimum will be greater than the classical EOQ. Further increases in t result in deteriorating performance of the classical EOQ. This is shown in Figure 4 for the case of t = 4. It is clear from this numerical example and the Appendix that for small values of t, the classical EOQ may yield better results than the Chand and Ward approach. However, the reader should note that the Chand and Ward approach is quite robust. In fact, an error bound on the annualized cost resulting from the use of the Chand and Ward approach can be derived on exactly the same lines as given in Rachamadugu.7 As mentioned earlier using the numerical illustration, for all practical purposes, the use of the Chand and Ward approach results in no more than 3.75%. However, the error resulting from the use of the classical square-root formula can be arbitrarily bad.

CONCLUSION In this paper, we analysed two approaches to order-size determination when delay in payment

is permitted. One of them results in the order size being invariant with respect to delay time. The other approach increases the order size as the delay time increases. The former approach is based on the average cost analysis. The latter approach is a hybrid using the average cost analysis and discounting. Since the average cost analysis is only an approximation to the conceptually rigorous discounted cash-flow approach, we analysed both procedures using the discounted cash-flow approach. First, we showed that, contrary to the claims made by some earlier authors, the optimal order quantity increases as payment delay-time increases. Next, we showed that the Chand and Ward approach yields an order quantity which is an upper bound on the true optimum. This, together with the fact that the net present value function is convex, is helpful in determining the optimum order quantity should the need arise to determine the same more accurately. The hybrid procedure suggested by Chand and Ward is robust. Our analysis here also has both research and pedagogical implications. Even if the average cost analysis is computationally adequate for most practical purposes, it can inadvertently lead to conceptually anomalous conclusions. It is essential to be aware of the fact that the average cost analysis is only an approximation and a surrogate for the basic objective of shareholder wealth maximization.

APPENDIX

Relative Error of Chand and Ward Approach The relative error resulting from the use of the Chand and Ward approach is given by

ANN(T(CW)) - ANN(T(OPT)) AM ANN(T(OPT)) (1)

At the optimum, ANN'(T(OPT)) = 0. This implies Sr2 = (Dpre-rt + hD)(erT(OPT) - rT(OPT)). A(2)

Using (11) and A(2), with a little algebraic manipulation, A(1) can be rewritten as

r(T(CW) -T(OPT)) -(e -rT(OPT) - e T(CW)) A(3) (1 -er'T(cw)){erT(OPT) -(oh + pre-rT)}

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Numerical evaluation of A(3) was done by Rachamadugu7 for different values of rT(CW) and (h/h + pre-rt). Details can be found in Rachamadugu7 (Figure 1). For example, consider an extreme scenario-capital charges are 25% per year (excluding material handling, insurance etc.), and the reorder interval is 1 year. It can easily be verified that as delay time and price tend to infinity, the relative error resulting from the use of the classical economic order quantity can be arbitrarily bad. The relative error for the Chand and Ward approach can be no more than 3.75%. Thus the Chand and Ward approach provides robust results for practical purposes.

Value oft at which Classical EOQ Yields Optimal Solution From (9) it is clear that, at the optimum,

Sr2 = erT(OPT) 1 - rT(OPT) A(4)

D(h + pre - r) r2 2S h + pr - eTOPT - 1-rT(OPT) 2 D(h + pr) h + pre

If the classical EOQ should equal the optimum order quantity, then (rT(EOQ))2 h + pr - erT(EOQ)

-

1 - rT(EOQ). A(5) 2 h +preT

The value of t at which A(5) holds good determines the delay period for which the square root formula yields the optimal solution. For the numerical illustration in the paper, t equals 0.734 periods.

Acknowledgements-I am grateful to anonymous referees for their critical and constructive comments on an earlier version of this paper.

REFERENCES 1. C. B. CHAPMAN, S. C. WARD, D. F. COOPER and M. J. PAGE (1984) Credit policy and inventory control, J. Opl Res. Soc.

35, 1055-1065. 2. S. K. GOYAL (1985) Economic order quantity under conditions of permissible delay in payments. J. Opl Res. Soc. 36,

335-338. 3. S. CHAND and J. WARD (1987) A note on economic order quantity under conditions of permissible delay in payments.

J. Opl Res. Soc. 38, 83-84. 4. G. HADLEY (1964) A comparison of order quantities computed using the average annual cost and the discounted cost.

Mgmt Sci. 10, 472-476. 5. R. RPTRIPPI and D. E. LEWIN (1974) A present value formulation of the classical EOQ problem. Decis. Sci. 5, 30-35. 6. R. W. GRUBBSTROM (1980) A principle for determining the correct capital costs of work-in-progress and inventory. Int.

J. Prod. Res. 18, 259-271. 7. R. RACHAMADUGU (1988) Error bounds for EOQ. Nav. Res. Logist. Q. 35, 419-425.

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Article Contentsp. 805p. 806p. 807p. 808p. 809p. 810p. 811p. 812p. 813

Issue Table of ContentsThe Journal of the Operational Research Society, Vol. 40, No. 9 (Sep., 1989), pp. i-iii+737-837Front Matter [pp. i - iii]General PapersStrategies for Backward-Area Development: A Systems Approach [pp. 737 - 751]OR as Technology: Some Issues and Implications [pp. 753 - 759]A Language for Teaching Discrete-Event Simulation [pp. 761 - 770]

Case-Oriented PapersA Successful Implementation of Locating Passive Cooling Facilities: The Case of Northern Thailand [pp. 771 - 778]Dynamic Programming in Action [pp. 779 - 787]

Theoretical PapersThe Effects of Varying Production Rates on Inventory Control [pp. 789 - 796]A Numerical Approach to the Repairman Problem with Two Different Types of Machine [pp. 797 - 803]Effect of Delayed Payments (Trade Credit) on Order Quantities [pp. 805 - 813]Maximum Entropy Analysis of Multiple-Server Queueing Systems [pp. 815 - 825]

Technical NoteOn a Heuristic Inventory-Replenishment Rule for Items with a Linearly Increasing Demand Incorporating Shortages [pp. 827 - 830]

Book Selectionuntitled [p. 831]untitled [pp. 831 - 832]untitled [pp. 832 - 834]untitled [p. 834]

Letters and Viewpoints [pp. 835 - 836]Corrigendum: Simple Inspection Scheme for Two Types of Defect [p. 837]Back Matter