Determination of the optimal procurement policy under integrating working capital.

INTRODUCTION

In view of the interrelationships among inventory, procurement, cash discounts, accounts payable and accounts receivable policies, both academicians and practitioners appear to have recognized the need to integrate them. Until now, there have been few studies on integrating working

Recently, Arcelus and Srinivasan [1] considered the problem of integrating the main components of working capital decisions within a discounted cash flow framework to study the interrelationships among inventory procurement, cash discounts, accounts payable and accounts receivable policies. This paper discusses the same model as that of Arcelus and Srinivasan [1]. The purpose of this paper is threefold: first, this paper will prove the discounted infinite horizon present value function is concave with respect to the cycle time; second, bounds for the optimal cycle time will be derived; third, a simple algorithm to locate the optimal cycle time will be developed. Finally, numerical examples are given to illustrate the algorithm.

THE MODEL

The following notation will be used. Let

U = demand rate per unit time;

T = cycle time;

[T.sup.*] = optimal cycle time;

P = sales price per unit;

C = purchasing costs;

h = holding cost per unit per unit time;

k = discount rate per unit time;

d = cash discount (percent of P) offered by wholesaler for early payment;

[Mathematical Expression Omitted] = cash discount (percent of C) offered to wholesaler for early payment;

M = cash discount period offered by wholesaler for early payment;

[Mathematical Expression Omitted] = cash discount period offered to wholesaler for early payment;

L = net credit period offered by wholesaler for full payment;

[Mathematical Expression Omitted] = net credit period offered to wholesaler for full payment;

r = average percent of sales with cash discount

= probability of paying within cash discount date, M;

p = average percent of sales on credit

= probability of paying at L;

E = fixed ordering costs per cycle;

s = selling unit cost;

N = discounted infinite horizon present value;

H = r(1 -d)[e.sup.-kM] + [pe.sup.-kL];

[Mathematical Expression Omitted],

where W = 1, if the cash discount is taken; = 0, otherwise.

The model considers a wholesaler firm that purchases its merchandise at the beginning of each ordering cycle, T, on credit, i.e., on the usual ([Mathematical Expression Omitted], [Mathematical Expression Omitted]) basis. Hence, its procurement plan calls for a discount, [Mathematical Expression Omitted], off the purchase price, C, if payment is made on or before time [Mathematical Expression Omitted], otherwise, the full cost of the purchase is due at [Mathematical Expression Omitted]. Sales of the merchandise occur uniformly throughout T, on similar (d/M, n/L) credit terms. This behavioral assumption of the wholesaler receiving and giving credit is consistent with the hypothesis in Schwartz [6] whereby larger firms whose size provides them with better access to financial markets may profitably transfer some of the benefits on to their own smaller customers. based on the above notation and assumption, Arcelus and Srinivasan [1] demonstrated that the present value of the cash flows over a discounted infinite horizon is given by

[Mathematical Expression Omitted] (1)

where [I.sub.T] = [kT - (1 - [e.sup.-kT])]/[k.sup.2].

The wholesaler problem is to select the credit policy, i.e., the values of d, M and L and its procurement policy, i.e., the value of T, so as to maximize the net present value N of the infinite stream of cash flows from revenues, purchasing, inventory holding and selling costs. Usually, the values of d, M and L are determined by top management. Hence, after the values of d, M and L are determined, the discounted infinite horizon present value function N will be the function of T only. Under the assumption that the values of d, M and L are fixed, the main purpose of this paper is to present a simple algorithm to locate the optimal procurement policy, i.e., the optimal value of T.

BOUNDS FOR THE OPTIMAL PROCUREMENT POLICY

In Arcelus and Srinivasan [1], the property of the discounted infinite horizon present value function N has remained unexplored. In this section, we shall first show that N is concave with respect to T.

THEOREM 1: N = N(T) is concave with respect to T.

Before proving Theorem 1, we need the following Lemma 1 (Theorem 2 in Chung and Lin [3]).

LEMMA 1: (a) [e.sup.x] [greater than] 1 + x + [x.sup.2]/2 [greater than] 0 for x [greater than] 0.

(b) [e.sup.-x] [greater than] 2-x/2+x for x [greater than] 0.

PROOF: From Equation (1) and Lemma 1, we see

[Mathematical Expression Omitted] (2)

and

[Mathematical Expression Omitted].

Hence N[inches](T) [less than] 0 and N(T) is concave with respect to T [greater than] 0.

Consider the following equation:

N[prime](T) = 0 (3)

Since N is concave, N[prime](T) is strictly decreasing for T [greater than] 0. From

[Mathematical Expression Omitted],

there exists a unique solution [T.sup.*] such that Equation (3) holds. The concavity of N implies that [T.sup.*] is the only optimal procurement policy. Equation (3) holds if and only if

[Mathematical Expression Omitted]. (4)

Let

[Mathematical Expression Omitted]

[Mathematical Expression Omitted].

Then

[Mathematical Expression Omitted] and

[Mathematical Expression Omitted].

Both [f.sub.1][prime](T) and [f.sub.2][prime](T) are less than 0. Therefore, both [f.sub.1](T) and [f.sub.2](T) are strictly decreasing on (0, [infinity]). Since [f.sub.1] (T) = [e.sup.kT][f.sub.2] (T) = k[(1 - [e.sup.-kT]).sup.2] N[prime](T), Equation (3) holds if and only if

[f.sub.1] (T) = 0 (5)

if and only if

[f.sub.2] (T) = 0. (6)

Consequently [T.sup.*] is the unique common root of Equations (5) and (6). Let

[Mathematical Expression Omitted] and

[Mathematical Expression Omitted].

Then we have the following result.

THEOREM 2: [Mathematical Expression Omitted].

PROOF: By Lemma 1, we see

[Mathematical Expression Omitted].

Since [f.sub.1] is strictly decreasing,

[Mathematical Expression Omitted]. (7)

On the other hand, we have

[Mathematical Expression Omitted].

Since [f.sub.2] is strictly decreasing,

[Mathematical Expression Omitted]. (8)

Combining Equations (7) and (8), we complete the proof.

THE ALGORITHM

Theorem 1 reveals that N is concave with respect to T. Therefore it is appropriate to use the Newton-Raphson method to locate the optimal cycle time. However, the Newton-Raphson method involves the first derivative. The whole process of Newton-Raphson is still very complicated. Consequently, with Theorem 2, the bisection algorithm can locate the root of Equation (5). The bisection algorithm is based on the intermediate value theorem (Thomas and Finney [7]).

COMPARISONS

Arcelus and Srinivasan [1] present three approximations [T.sub.i] (i = 1,2,3) to the optimal cycle time [T.sup.*]. These near-optimal policies [T.sub.i] (i = 1,2,3) are based on assumption (A).

ASSUMPTION (A): kT is generally very small, with k in the 10% to 20% range and T usually no more than 3 months or 0.25 years.

Obviously, Assumption (A) limits the applications of [T.sub.i] (i = 1,2,3). Sometimes it will yield a significant penalty as examples 1 and 2 show. Example 2 reveals that [kT.sub.i] [greater than] 0.85 and [T.sub.i] [greater than] 0.25 (i = 1,2,3). So, Assumption (A) does not hold in general. However, the above algorithm is free from Assumption (A). Hence, the above algorithm overcomes the shortcomings of the three approximations [T.sub.i] (i = 1,2,3).

With Assumption (A), let

exp(-kT) [approximately equal to] 1 - kT (9)

or

exp(-kT) [approximately equal to] 1 - kT + [k.sup.2][T.sup.2]/2 (10)

Arcelus and Srinivasan [1] indicate

[T.sub.1] = ([-square root of [k.sup.2] [E.sup.2] + 2EX - KE])/X, (11)

[Mathematical Expression Omitted] (12)

and

[Mathematical Expression Omitted] (13)

where [Mathematical Expression Omitted]. (14)

After recomputing, we find that the real situation should be

[Mathematical Expression Omitted]. (15)

Hence, Equation (11) should be in error. Consequently, later, Equation (11) is replaced by Equation (15). The following examples compare [T.sup.*], [T.sub.1], [T.sub.2] and [T.sub.3].

EXAMPLE 1: Let C = 3, P = 4.5, k = 0.2, h = 0.2, E = 1000, s = 0.05, M = 0.318, [Mathematical Expression Omitted], L = 0.36, [Mathematical Expression Omitted], p = 0.477, r = 0.402, d = -0.164, [Mathematical Expression Omitted] and U = 60000.

Following the bisection algorithm, we get [T.sup.*] = 0.1668, [T.sub.1] = 0.1650, [T.sub.2] = 0.2340, [T.sub.3] = 0.2340, N([T.sup.*]) = 19988.5, N([T.sub.1]) = 19985.1, N([T.sub.2]) = 16496.0, N([T.sub.3]) = 16496.0 for W = 1 and [T.sup.*] = 0.1664, [T.sub.1] = 0.1646, [T.sub.2] = 0.2329, [T.sub.3] = 0.2330, N([T.sup.*]) = 11706.0, N([T.sub.1]) = 11702.6, N([T.sub.2]) = 8250.7, N([T.sub.3]) = 8248.9 for W = 0.

The above results show that [T.sup.*] and [T.sub.i](i = 1,2,3) are less than 0.25. Hence, [T.sup.*] is better than all [T.sub.i](i = 1,2,3). Basically, Assumption (A) holds. However, using [T.sub.i](i = 1,2,3) instead of [T.sup.*] yields penalties of N: 0.02%, 17.47%, 17.48% for W = 1, and 0.03%, 29.51%, 29.53% for W = 0, respectively. Although Assumption (A) holds, the penalties of N cannot be ignored. Hence [T.sub.i] (i = 1,2,3) may not be good approximations for [T.sup.*] in general.

EXAMPLE 2: Let C = 1, P = 7, k = -0.2, h = 0.2, E = 1000, s = 0.05, M = 0.318, [Mathematical Expression Omitted], L = 0.36, [Mathematical Expression Omitted], p = 0.477, r = 0.402, d = 0.164, [Mathematical Expression Omitted] and U = 140.

Following the bisection algorithm, we get [T.sup.*] = 5.0150, [T.sub.1] = 4.2948, [T.sub.2] = 7.5510, [T.sub.3] = 8.3903, N([T.sup.*]) = 406.55, N([T.sub.1]) = 380.44, N([T.sub.2]) = 212.50, N([T.sub.3]) = 93.50 for W = 1, and [T.sup.*] = 5.0054, [T.sub.1] = 4.2861, [T.sub.2] = 7.5238, [T.sub.3] = 8.3531, N([T.sup.*]) = 396.52, N([T.sub.1]) = 370.31, N([T.sub.2]) = 203.77, N([T.sub.3]) = 86.11 for W = 0.

The above results show that using [T.sub.i](i = 1,2,3) instead of [T.sup.*] yields penalties of N: 6.42%, 47.72%, 77.00% for W = 1, and 6.60%, 48.61%, 78.28% for W = 0, respectively. Hence [T.sup.*] is better than all [T.sub.i] (i = 1,2,3). We also find:

(1) [kT.sup.*] and [kT.sub.i](i = 1,2,3) are more than 0.85,

and

(2) [T.sup.*] and [T.sub.i](i = 1,2,3) are more than 0.25.

Therefore, Assumption (A) does not hold. Consequently, approximations (9) and (10) are not appropriate for this example. This explains why using [T.sub.i](i = 1,2,3) instead of [T.sup.*] yields significant penalties. The observations of Examples 1 and 2 justify the establishment of the bisection algorithm to locate the optimal cycle time [T.sup.*].

REFERENCES

[1] Arcelus, F.J. and G. Srinivasan, "Integrating Working Capital Decisions," The Engineering Economist, Vol. 39, 1993, pp. 1-15.

[2] Chapman, C.B., S.C. Ward, D.F. Cooper and M.J. Page, "Credit Policy and Inventory Control," journal of the Operational Research Society, Vol. 35, 1984, pp. 1055-1065.

[3] Chung, K.J. and S.D. Lin, "A Note on the Optimal Cycle Length with a Random Planning Horizon," The Engineering Economist, Vol. 40, No. 4, 1995, pp. 385-392.

[4] Kim, Y.H. and K.H. Chung, "An Integrated Evaluation of Investment in Inventory and Credit: A Cash Flow Approach," Journal of Business Finance and Accounting, Vol. 17, 1990, pp. 381-390.

[5] Sartoris, W.L. and N.C. Hill, "A Generalized Cash Flow Approach to Short-term Financial Decisions," Journal of Finance, Vol. 38, 1983, pp. 349-360.

[6] Schwartz, R.A., "An Economic Model of Trade Credit," Journal of Financial and Quantitative Analysis, Vol. 9, 1974, pp. 643-657.

[7] Thomas, C.B. and R.L. Finney, Calculus and Analytic Geometry, 9th Edition, Addison-Wesley Publishing, 1996.