Division of Research School of Business Administration November 1987 Revised June 1988 Revised August 1988 ERROR BUNDS FR EOQ Working Paper #527 R2 Ram Rachamadugu The University of Michigan

Abstract In this paper we explore the properties of the discounted total cost function for the economic order quantity. We show that it is convex. Further, it is shown that the Classical Economic Order Quantity (based on Wilson's formula) is not less than the true optimum value based on discounting. Bounds for the discounted reorder interval (or order quantity) based on average cost analysis are also provided. Further, we analytically show that larger the ratio of noncapital related holding charges to the total holding charges, more adverse is the effect on the accuracy of the average cost analysis.

ERROR BOUNDS FOR EOQ Introduction It is well known that the classical Economic Order Quantity, which is based on average cost analysis, is an approximation for the conceptually more rigorous discounted cost version. It is clear that the objective of maximizing the net present value of shareholder wealth leads to the use of net present value as the criterion for decision making. For a more detailed exposition, see the arguments presented by Grubbstrom [1]. In prior analyses (Hadley and Whitin [2]), it was shown that the total cost function based on average cost analysis is convex, and using that property, the optimal order quantity was determined. However, properties of the discounted total cost function have remained unexplored. Many researchers recognize that the classical economic order quantity is an excellent first order approximation to the discounted cost version. Hadley and Whitin [3] discussed this in a problem form in their classical book on analysis of inventory systems. This was further verified by Hadley [2] in an extensive computational study. Trippi and Lewin [5] also addressed the problem of determining optimal lot sizes using discounting. However, they implicitly assumed that there are no noncapital related holding charges such as material handling, insurance and warehousing etc. In this paper, we analyze the general case. We show that the discounted total cost function for the general case is convex. We further show that, for the general case, the reorder interval (or the order quantity) derived using average cost analysis is never less than the true optimum based on discounting. Relative error bounds for the optimum reorder interval (or order quantity) are provided based on average cost analysis. These bounds are useful in search methods used to determine the optimum reorder interval (or order quantity) for the discounted model. Further, we derive an expression for the relative total cost

-2 - error if the classical reorder interval (or order quantity) based on the averrage cost analysis is used. It is shown that the accuracy of the average cost analysis is dependent, among other factors, on the ratio of noncapital related holding charges such as material handling and warehousing to the total holding charges. Thus using the analysis provided in this paper, the decision maker can verify if the average cost analysis has provided acceptable approximate solution. If not, search methods can be used to determine more accurate order quantities. Notation T: Reorder interval D: Demand rate r: Discount rate p: Price per unit S: Set up cost or Order cost h: Inventory carrying cost, exclusive of capital charges (in dollars per unit per period) NPV(T): Net present value if the reorder interval is T ANN(T): Annualized cost per period if the reorder interval is T AVC(T) = Cost per period based on the average cost analysis if the reorder interval is T TE = Best reorder interval using the classical average cost analysis = /2S/D(h + pr) T = Reorder interval based on the discounted cost approach (true optimum). Analysis Since the assumptions made in determining the classical economic order quantity (or reorder interval) are well known, they are not repeated here. The reader is refered to Hax and Candea [4].

-3 - If we discount all future costs at a discount rate r, T NPV(T) = S + f hD(T - k)er dI + DTp + erT NPV(T) (1) 0 sdgTp 1 () T -rT f hD(T - Z)e t dt + DTp (2) 1 -e 1 -e 0 1- e Equation (2) indicates the net present value of all cash outflows if the reorder interval is T. In order to compare this with conventional average cost analysis, we use ANN(T), an alternate, but equivalent measure. ANN(T) represents the equivalent uniform cash flow stream that generates the same NPV(T). Since we are considering an infinite horizon (from the basics of discounting), ANN(T) = r * NPV(T) _ Sr + r T -rR DTpr 1 _-rT 1 -rT -f hD(T - z)e r d + D Tp (3) 1-e 1 - e 0 1-e -rT Sr hD rT - 1 + e DprT ( 1-e 1 -e 1 -e For small values of rT, the above expression can be approximated as ANN(T) S + hT + Dp + D T 2 2 s + 7 (h + rp)DT + Dp (5) However, based on average cost analysis, s 1 AVC(T) = T + (h + rp)DT + Dp (6) Dp is a constant and independent of the decision variable T. In most discussions of average cost analysis, it is not included in the cost expression

-4 - that is to be minimized. It is included here in (6) to show that the total cost per period based on the average cost analysis is approximately the same as ANN(T). It is well known that (6) is convex. The classical Economic Order Quantity or reorder interval (TE) are determined using the convexity properties of (6). However, properties of ANN(T) have remained unexplored. We show that ANN(T) is also convex. Remark 1: ANN(T) is convex. Proof: Rewriting (4), Sr T Dh ANN(T) = +(Dpr +hD) (7) -rT -rT r 1 - e 1 - e 2 -rT -rT -rT Sr e 1 -e rTe ANN'(T) = + (Dpr + hD) (8) -rT -rT (1 - e ) ( - e ) r3 -rT + e-rT ANN'(T) Sr e (1 + e ) -rT (1 - e ) + (DPP + hD)re - (rT + ^ -rT + (Dpr + hD)re {(2 + rT)er - (2 - rT)} (9) -rT (1 -e ) ANN(T) is convex if and only if ANN'(T) > 0 It can easily be seen that all terms in (9) except the term in curled parenthesis are positive. Consider the term in curled parenthesis. (2 + r rTe - (2 - rT) {(2 + rT) - (2 - rT)e } ~~~~(2 + rT)e ~rT e e { rT(1 + erT) + 2(1 - e)} e

-5 - c1 nTn eT n{ n! (n - 2)}. e n=3 > 0 Hence ANN"(T) > 0 and ANN(T) is convex. Since ANN(T) is proportionate to NPV(T), NPV(T) is also convex. We next explore the optimality of the classical reorder interval which is based on the average cost analysis. Remark 2: The reorder interval (or order quantity) based on average cost analysis is never less than the optimum value derived using discounting. Proof: It is well known that T 2S (10) E D(h + pr) where T is reorder interval based on average cost analysis. Let T be E o the true optimum based on discounted cost analysis. Since ANN(T) is convex, T is determined by setting ANN'(T) zero. Reconsider (8). Sr2 rT ANN'(T) = 0 => r =e - - rT D(h + pr) o r2T 2 r3T 3 r4T 4 0 + o + o 2 6 24 3 2 4 rT r T 2S =T + 0o o (11) D(h + pr) o 3 12 Using (10) in (11),

-6 - rT 3 rT 4 2 2 o o T2 = T +. + 0. E o 3 12 Since T > 0, the above expression can be rewritten as, 2 2 T - T > 0 E - TE > To It may be noted that Remarks 1 and 2 are generalizations of the results derived by Trippi and Lewin [51. As noted earlier, their analysis is a special case where it is assumed that the noncapital inventory related cash outflows such as material handling, insurance, taxes and warehouse leasing do not exist. Error Bounds From equation (11) we can derive an error bound for the classical reorder interval. Equation (11) can be rewritten as T T rT r T r T E ~ 1 0+ + / + 6-I (12a) T 3 12 60....a o Since T > T, the above expression can be rewritten as - O-T T rT rT TE < E + E + E T 3 12 60 rTE < 2(e " - rT - 1) r E -1 (12b) rTE Hence, equation (12) provides ex-post relative error bound on the optimal reorder interval (T ). Using Remark 2 and expression (12), we bound T as given below.

-7 - rTE < rT < rT - o - E (13) Table 1 shows the ex-post relative error for the reorder interval ((TE - To)/To) based on the value of TE and compares it with the exact value of the relative error. It can be seen that for all practical purposes, the ex-post relative error is a good approximation of the exact relative error.. TABLE 1 RELATIVE ERROR BOUNDS FOR rTE Exact value of relative Ex-post relative error in percent rTE error for TE in percent T - T = expression (12b) * 100 T 100,_____ ___________ -0. o 0.05 0.85 0.84 0.10 1.72 1.69 0.15 2.63 2.56 0.21 3.57 3.45 0.26 4.54 4.35 0.32 5.55 5.26 0.37 6.60 6.19 0.43 7.68 7.14 0.49 8.81 8.10 0.55 9.98 9.08 0.61 11.20 10.07 0.67 12.46 11.09 0.73 13.78 12.12 0.79 15.15 13.16 0.86 16.57 14.23 0.92 18.06 15.32 0.99 19.61 16.42 1.06 21.23 17.55 1.13 22.92 18.69 1.20 24.69 19.86 1,. _ 1 -_ _ _ _ _ 1 l _,_,_,_l Note: Computations are shown upto a value of rTE = 1.2. For example, if rTE equals 1.2 and r is 20% per year, TE is 6 years!

-8 - Table 2 compares T with the ex-post lower bound and upper bounds for it based on TE These bounds are from the expression (13). It can be seen that the ex-post lower bound on T is much tighter than the upper bound. TABLE 2 BOUNDS FOR rTo BASED ON THE AVERAGE COST ANALYSIS I ' I - - - -I - I I Ex-post lower Upper bound bound on rTo Exact value of on rTo rTE from (13) rTo (=rTE) 0.05042 0.05000 0.05000 0.05042 0.10169 0.09997 0.10000 0.10169 0.15385 0.14990 0.15000 0.15385 0.20689 0.19976 0.20000 0.20689 0.26087 0.24953 0.25000 0.26087 0.31578 0.29917 0.30000 0.31578 0.37167 0.34866 0.35000 0.37167 0.42854 0.39797 0.40000 0.42854 0.48644 0.44706 0.45000 0.48644 0.54538 0.49589 0.50000 0.54538 0.60540 0.54444 0.55000 0.60540 0.66651 0.59266 0.60000 0.66651 0.72875 0.64052 0.65000 0.72875 0.79215 0.68796 0.70000 0.79215 0.85674 0.73494 0.75000 0.85674 0.92254 0.78141 0.80000 0.92254 0.98959 0.82733 0.85000 0.98959 1.05793 0.87264 0.90000 1.05793 1.12757 0.91729 0.95000 1.12757 1.19857 0.96122 1.00000 1.19857._ _I_ _,_ _l_ _, I. I Note: Computations are shown upto a value of rTE = 1.2. and r = 20% per year, TE is 6 years! For rTE = 1.2 and Similarly, we derive the relative error value for the discounted cost function. Exact value of the relative error for the discounted total cost function are obtained from:

-9 - ANN(TE) - ANN(T ) ANN(T ) = -rT r(TE - T) - (e E 0 T (1. - rTE K (1 e E) (e ~ (14) -rT -e ) h h + pr Details of the derivation are shown in the appendix. Note that TE and T are related by expression (12a). Thus how close an approximation TE is to T depends on r. h/(h + pr) is the ratio of non-capital related holding charges (such as warehousing and material handling) to the total handling charges. The relative error for the discounted total cost function depends on this ratio. This is shown in Figure 1. FIGURE 1 EFFECT OF h/(h + pr) ON RELATIVE DISCOUNTED TOTAL COST ERROR z 10 0 I L. < 10 -4-0 0 z _N z -asz r l (D z Legend * h/(h+pr)=0 D h/(h+pr)=.5 * h/(h+er)-= 0 0.5 1 1.5 r T

-10 - It is clear from Figure 1 that the ratio h/(h + pr) influences the relative error for the discounted total cost function. As the ratio of non-capital related holding charges (such as material handling and warehousing) to total holdcharges increases, the relative error for the discounted total cost function increases. Conclusion We have shown in this paper that the discounted total cost function is also convex. Further, we have established that the reorder interval (or order quantity) based on average cost analysis is not less than the true optimum based on the discounted cost version. We also provide ex-post lower and upper bounds for the reorder interval (or order quantity). Computational results show that the lower bound is tight. Relative error for the discounted total cost is influenced by the proportion of non-capital related holding charges such as material handling and warehousing in the total holding costs. When this ratio is high, the reorder interval (or order quantity) based on the average cost analysis may be inadequate. However, the bounds for T based on TE provided by us will be useful in search procedures for determining T. References [1] Grubbstrom, R. W., "A Principle for Determining the Correct Capital Costs of Work-in-Progress and Inventory," International Journal of Production Research, 18, 1980, pp. 259-271. [2] Hadley, G., "A Comparison of Order Quantities Computed Using the Average Annual Cost and the Discounted Cost," Management Science, 10, 1964, pp. 472-476. [3] Hadley, G., and Whitin, T. M., Analysis of Inventory Systems, PrenticeHall Inc., Englewood Cliffs, New Jersey, 1963, pp. 29-81. [4] Hax, A. C., and Candea, D., Production and Inventory Management, PrenticeHall Inc., Englewood Cliffs, New Jersey, 1984, pp. 133-134. [5] Trippi, R. R. and Lewin, D. E., "A Present Value Formulation of the Classical EOQ Problem," Decision Sciences, 5, 1974, pp. 30-35.

-11 - APPENDIX Derivation for the relative discounted total cost error. Using expression (7), (ANN(To) - ANN(TE ))/ANN(T ) o o 0 2 -rT -rT -rT -rT Sr (e - e ) + (Dpr + hD)r {TE(1 - e ) - T (1 - e )} -rT -rT (1 - e )Sr2 + (Dpr + hDr)T - hD(l - e ~)} At T, derivative of ANN(T) equals zero. This implies (using expression (11)), ~~2 ~rT Sr = (Dpr + hD)(e - 1 - rT) (II) 0 Substituting II in I, relative discounted total cost error can be rewritten as -rTE -rT rT -rT -rT (e - e )(e - 1- rT ) + r{T(1- e ) -T (- e rT rT h -rT (1 -e ){e 0- 1 -( pr(1 -e 0)} With a little algebraic manipulation, above expression can be rewritten as -rT -rTE r(TE - T ) - (e - e ) -rE rT (1 - e E)e - (h pr)}