Financing Terms in the EOQ Model


 Ambrose Poole
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1 Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 August 6, Intoduction This note discusses two tems that ae often omitted fom the standad economicodequantity (EOQ) model when the holding cost is based on financing costs. Both of these tems wee fist identified in Poteus (1985). One was identified explicitly, and the othe was implicit in the calculation of an appoximation bound. This note was witten fo anyone teaching the EOQ model to gaduate business students. The motivation fo this note is the need to econcile the EOQ model with what students may have leaned about woking capital in an accounting couse. To eview, total cost in the EOQ model is assumed to be the sum of puchase costs, odeing costs, and holding costs, namely cλ + λ q k + q h, whee λ is a constant demand pe unit of time, c is a constant, peunit puchase cost, k is a fixed cost pe ode, and h is a constant, peunit holding cost. The decision vaiable, q, is the ode size. (Fo an ealy histoy of the model, see Elenkotte (1990). Fo a moden teatment, see, fo example, Poteus (00) o Zipkin (000).) When the holding cost is based on financing costs, the usual appoach is to set h = c, whee is the cost of money, typically an inteest ate. This gives a total cost of C(q) =cλ + λ q k + q c and an optimal ode quantity of q = λk λk h = c. Fo students who undestand woking capital, it appeas that the total cost is undestated. (The analysis that follows is called the conventional appoach in Appendix C of Poteus (00).) Since the cost of an ode is, thecostpeunitis k q + c. This implies that the aveage woking capital equied by the cyclestock inventoy is q k q + c = 1 (), 1
2 and so the cost of this woking capital is This suggests a total cost of (). C 1 (q) = cλ + λ q k + q c + k = C(q)+ k. The exta tem can be intepeted as the cost of the woking capital needed fo the fixed cost potion of the cycle stock inventoy. The exta tem is not elevant to the ode size decision, so the usual appoach of setting h = c yields the optimal decision. Since the EOQ model was designed to answe only the ode size question, using C(q) instead of C 1 (q) does no ham. But if the EOQ model is used to evaluate diffeent vendos, the exta tem in C 1 (q) can be elevant. Conside the following example. Example Demand λ,000 Cost of Money 0% Vendo A chages a fixed cost of $4,000 and a peunit cost of $0. Vendo B chages a fixed cost of $1,000 and a peunit cost of $0.50. With Vendo A, q =8, 000, soletq EOQ =8, 000. With Vendo B, q =, 951, soletq EOQ =4, 000. The best choice of vendo depends upon whethe o not the k/ tem is included in the analysis: Vendo A Vendo B C(q EOQ ) $67, 000 $67, 00 k/ $400 $100 C 1 (q EOQ ) $67, 400 $67, 00 Pesent Value Analysis At the close of the pevious section, we indulged in a (often employed) slip of logic: we took a model designed fo detemining an optimal ode size and used it fo a diffeent pupose, namely evaluating diffeent vendos. To see if this slip can be justified, we conside the tue economic cost associated with a paticula choice of vendo. In this simple (and classic) vesion of the EOQ model, it suffices to conside the pesent value of the payments. (See, fo example, Poteus (00, Appendix C.1) o Zipkin (000, Section.7).) The pesent value of the payments is PV(q) = ()+ e q λ = 1 e q/λ, + e q λ +... whee q/λ is the elevant inteest ate, and the facto e q/λ epesents continuous discounting. The question now is whethe we can find a elationship between PV(q) and C 1 (q). We can. Conside the
3 following expansion of PV(q): PV(q) = 1 e q/λ = q λ q + q... = q q λ 6λ λ (1 λ + q q ±...) 6λ 4λ λ() = 1+ q q λ + q 1λ 4 q 4 70λ = 1 cλ + λ q k + q c + k q() + q () 1λ 70λ +... (1) Fom equation (1), it follows that lim PV(q) C 1(q) =0, () 0 showing that the k/ tem should be included in the EOQ model. Befoe poceeding, it is useful to be specific about ou pefomance citeion. Let k A and c A be the cost paametes associated with Vendo A. Define PV A (q) =PV(q; k A,c A,). Define PV B similaly. We want to find a function C(q; k, c, ) such that PV A (q) PV B (q) C A (q) C B (q). (*) Ideally, the ode size q would be chosen to optimize the elevant function, but fo pactical easons, we will use ode sizes that ae easy to compute and close to the optimal q. Given ou pefomance citeion, it tuns out that equation () is misleading. (The autho thanks Paul Zipkin fo this obsevation.) Since the optimal q depends upon, we need to conside equation (1) with q = q. PV(q ) = C 1(q ) = C 1(q ) = C 1(q ) + q ( )... 1λ + k λk 1λ c + c 1λ k + 7λc + k 6... λk... c Thus, lim PV(q ) C 1(q ) = k 0 6. This suggests that if q is chosen optimally, the total cost in the EOQ model will be bette epesented by C (q) =C 1 (q)+ k 6. The additional k/6 facto can be intepeted as incemental inteest due to compounding. The fist ode appoximation of this incemental inteest is epesented by the q()/1λ tem in equation (1). With optimal ode sizes, the incemental inteest on the k/ pat of the cycle stock cost, namely q k/1λ, goestozeoas goes to zeo. But the incemental inteest on the cq/ pat of the cycle stock cost does not; c (q ) /1λ equals k/6 fo all.
4 Since lim PV(q ) C (q ) =0, 0 the function C satisfies ou pefomance citeion (*) in the limit. Retuning to the Example, the table below lists the costs associated with the diffeent models. In the pesent value model, the optimal quantities fo Vendo A and Vendo B ae 7, 94 and, 95, espectively. Vendo A Vendo B C(q EOQ ) $67, 000 $67, 00 C 1 (q EOQ ) $67, 400 $67, 00 C (q EOQ ) $67, 5 $67, P V (q EOQ ) $67, 57 $67, 5 P V (q opt ) $67, 56 $67, Discounted Aveage Value Poteus (1985) povides the fist fomal teatment of the analysis in the peceding section by using the discounted aveage value appoach. Poteus (00) descibes the discounted aveage value... ove a specific time inteval... [as] the cost ate that, if incued continuously ove that inteval, yields the same NPV as the actual steam. The goal is to find a function DAV (q) such that PV(q) = DAV (q)+ = DAV (q) 1 e. DAV (q) e + DAV (q) e +... Theoem 1 of Poteus (1985), fo the paametes given in the fist section above, states that λ() DAV (q) = 1+ q q λ + q 1λ + O( ) = C 1 (q)+ q() + O( ). 1λ Fo q = q, DAV (q ) = C 1 (q )+ k λk 1λ c + c 1λ λk + O( ) c k = C (q ) λc + O( ). Theoem 1 implies that lim (DAV (q) C 1(q)) = 0 0 and lim (DAV 0 (q ) C (q )) = 0. () As mentioned in the vey beginning, the k/6 tem in C (q ) is implicit in Poteus (1985), but it is not specifically identified. And it is discussed vebally as inteest on... inteest... chaged. Finally, note that since 1/ (1 e ) does not depend on q, DAV (q) satisfies the pefomance citeion (*) in the limit. 4
5 4 Compound Inteest Effect We have just shown how adding two tems to the taditional EOQ model leads to a bette appoximation of the pesent value model. The intuitive explanation fo the fist of these tems, namely k/, is staightfowad. In fact, it was the motivation fo this note. In this section, we povide some intuition fo the second tem, namely k/6. In the two analyses above, we have descibed it as inteest on inteest. In this section, we will ty to isolate the souce of this tem. We stat by asking whee, in the pesent value model, is the cycle stock? In the taditional EOQ model, the cycle stock cost can actually be seen. Each ode cycle, units aive and ae dawn down. Consequently, we see the inventoy value cycle in the same way. But in the pesent value model, whee is the cycling? We have an inceasing step function, in which the total costs incued jump with each ode. Futhe, the evenue steam does not depend on the ode policy, so it can t be used to explain the missing cycle stock. To answe this question, conside Figue 1 below. It depicts the balance ove time. (λ is denoted by l.) kl/q + lc 0 q/l q/l... 1 Figue 1 To find the missing cycle stock, conside the aveage balance in Figue 1: 1 k ()+1 λq + cλ. The fist pat of the expession is the missing cycle stock. It is the amount by which the balance exceeds the dotted, diagonal line. As the ode fequency inceases, this aea deceases. Loosely speaking, one can think of this as money being spent on units befoe the money is needed to be spent. This paallels nicely with thinking of cycle stock as units onhand befoe they ae needed. To demonstate that k/6 is due to a compound inteest effect, fist conside what the simple inteest would be on the balances in Figue 1. This would be just the inteest ate times the aveage balance, namely k ()+ λq + cλ. (4) To intoduce a compound inteest effect, we next detemine how the simple inteest in equation (4) accumulates ove time. This is depicted in Figue below. (λ is denoted by l, is denoted by a, and n = λ/q.) Note that the value at t =1educes to the value in equation (4). 5
6 (aq/l) ()* (1/)(n )* (n+1) 6(aq/l)() (aq/l)() (aq/l)() 0 q/l q/l... 1 Figue To appoximate the effects of compound inteest, we compute inteest on the inteest. Let i denote the ode inteval in the above gaph, i.e. i anges fom 1 to n = λ/q. The aveage simple inteest in inteval i is ³ q i(i 1) SI i (q) = () + λ i ³ q i = (). λ The inteest on the simple inteest in inteval i is then ³ q II i (q) = SI i (q) λ ³ = q i (). λ Summing ove the intevals yields nx ³ II(q) = II i (q) = q X n i () λ i=1 ³ i=1 = q n () λ 6 + n 4 + n 1 λ = () 6q q. 1λ At q = q, II(q ) lim = c(q ) = k 0 1λ 6. Thus, we see why c (q ) /1λ can be thought of as a compound inteest effect on the cq pat of the cycle stock. 6
7 Refeences [1] Elenkotte, Donald, Fod Whitman Hais and the Economic Ode Quantity Model, Opeations Reseach, 1990, 8:6, [] Poteus, Evan L., Undiscounted Appoximations of Discounted Regeneative Models, Opeations Reseach Lettes, 1985,, [] Poteus, Evan L., Foundations of Stochastic Inventoy Theoy. Stanfod: Stanfod Univesity Pess, 00. [4] Zipkin, Paul H., Foundations of Inventoy Management. New Yok: McGawHill,
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