The classical economic order quantity model has two types of costs: ordering and inventory holding costs. In this paper we try to investigate the effect of purchasing activity on cash flow of a firm. In the examinations we use a cash flow identity similar to that of in inventory modeling. In our approach we analyze the purchasing and ordering process with discounted costs. The cost function of the model consists of linear cash holding, linear opportunity cost of spending cash, and linear interest costs. We show the optimal solution of the proposed model. The optimal solutions will be presented by numerical examples

Dobos, Imre

Vörösmarty, Gyöngyi

English

Periodica Polytechnica (Budapest University of Technology and Economics)

Ŕ periodica polytechnicaSocial and Management Sciences21/2 (2013) 67–70doi: 10.3311/PPso.7085http://periodicapolytechnica.org/soCreative Commons AttributionRESEARCH ARTICLEAnalysis of purchasing activity withdiscounted cash flow inventory modelsGyöngyi Vörösmarty / Imre DobosReceived 2013-02-08AbstractThe classical economic order quantity model has two types ofcosts: ordering and inventory holding costs. In this paper wetry to investigate the effect of purchasing activity on cash flow ofa firm. In the examinations we use a cash flow identity similarto that of in inventory modeling. In our approach we analyzethe purchasing and ordering process with discounted costs. Thecost function of the model consists of linear cash holding, linearopportunity cost of spending cash, and linear interest costs. Weshow the optimal solution of the proposed model. The optimalsolutions will be presented by numerical examples.KeywordsNet present value · discounted cash flow · cash balance prob-lem · inventory models · purchasingAcknowledgementImre Dobos gratefully acknowledges the financial supports byTÁMOP-4.2.2.A-11/1/KONV-2012-0051 research program andthe Deutscher Akademischer Austauschdienst (DAAD).Gyöngyi VörösmartyDepartment of Logistics and Supply Chain Management, Corvinus Universityof Budapest, Fővám tér 8., 1093 Budapest, Hungarye-mail: gyongyi.vorosmarty@uni-corvinus.huImre DobosDepartment of Logistics and Supply Chain Management, Corvinus Universityof Budapest, Fővám tér 8., 1093 Budapest, Hungarye-mail: imre.dobos@uni-corvinus.hu1 IntroductionThe cash balance model was first analyzed by Baumol [1].He has applied the classical EOQ inventory model to investigatethe demand of a firm for cash. This model is a deterministiccash model. Other deterministic cash balance problems was ex-amined by Mensching et al. [15]. They have analyzed a simplecash balance problem similar that of aggregate production plan-ning problems. The analysis is seeking of an optimal solutionand the planning horizons. Chand and Morton [2] have supplieda further examination of the deterministic cash balance problemin direction of the planning horizons. Sethi and Thompson [24]have modeled a two-asset dynamic cash balance problem. Theyhave solved the problem with the help of Pontryagin’s maximumprinciple. The solution is a bang-bang optimal control. An in-troductory cash management chapter is presented in textbook ofRoss and Westerfield [22].The first stochastic cash balance model was initiated by Millerand Orr [16]. This basic model was generalized in several direc-tion, e.g. Eppen and Fama [6–8], Girgis [9], Neave [18], Por-teus [19], Heyman [12], Kamin [14], Constantinides [3], In-derfurth and Schneeweiss [13], Constantinides and Richard [4].These stochastic models are solved with dynamic programmingand under different cost structure. The mentioned models arecritically examined by Daellenbach [5] and asked, whether thesemodels fit the practice.Other papers investigate the cash management practices in thecontext of cash flow, e.g. Gitman, Moses and White [10], Mor-ris [17], Sartoris and Hill [23], Vickson [27], and Premachan-dra [20]. A good introduction in the stochastic cash manage-ment problems are supplied by Tapiero [25], who presents thebasic models with solution propositions.Thorstenson [26] has examined the capital costs in the inven-tory models. In his work Thorstenson has investigated the effectof discounting in cash flow inventory models. Our approach issimilar that of Thorstenson. The aim of the paper is to ana-lyze the allocation of cash in purchasing activity. The proposedmodel is a dynamic cash flow model with discounted costs. Thecash flow identity consists of cash transfer, credit and cash re-quirements. The goal function has three linear elements cashAnalysis of purchasing activity with discounted cash flow inventory models 672013 21 2holding, cash transfer, and interest costs. We solve this problemwith linear programming.The second part of the paper shows parameters, variables andgoal function of the model. The next chapter characterizes someproperties of the optimal solution. The fourth part presents somenumerical examples, and last we summarize the results of thepaper.2 The modelThe parameters of the model:r discount rateT length of the planning horizonI0 initial cash levelM available quantity of money for purchasing in the plan-ning horizonptiprice of the ith product or service in the tth periodxtirequired quantity of the ith product or service in the tthperiodXt the cash requirements in the tth periodh the holding cost of cashc opportunity cost of spending cash to buy products andservicesi interest rate of creditThe variables of the model:It available cash level in time t, nonnegativeYt cash transfer paid for products and services in period t,nonnegativeCt credit requirements in period t, nonnegativeLet us assume that cash requirements of the firm are given forthe purchasing department. If the prices (pti) and the requiredquantities (xti) of products and services are known in any period,then these requirements can be calculated as follows:Xt =n∑i=1pti · xti,where number n is the number of the sum of products and ser-vices.In this model we assume that the cash flow of the purchasingdepartment can be modeled as an inventory balance equation.The inventory balance equation consists of the sum of spent cashby purchasing department (Yt) and used credit (Ct) reduced bythe cash requirements (Xt) defined above:It = It−1 + Yt + Ct − Xt t = 1, 2, . . . ,T, (1)I0 is given and equal to zero.Let us assume that the purchasing department disposes of agiven quantity of money:T∑t=1Yt ≤ M. (2)We assume that the variables of the model are nonnegative:It ≥ 0, Yt ≥ 0, Ct ≥ 0 (t = 1, 2, . . . ,T ). (3)The goal function of the model is the sum of the discountedperiod costs:T∑t=11(1 + r)t−1· (h · It + c · Yt + i ·Ct)→ min . (4)We will analyze the optimal solution of model (1)-(4).3 Properties of the optimal solution of the model (1)-(4)Model (1)-(4) is a linear programming problem. Before wesolve this problem we give some properties of the model. Weinvestigate the dependence of the model on the parameters. Firstwe examine the dependence of optimal solution on the cost pa-rameters c and i. Then we analyze the solution in dependenceon available cash (M) and the sum of the required cash(∑Tt=1 Xt)for purchasing.Let us assume that the optimal solution of model (1)-(4) is{I0t , Y0t , C0t}T1.The next property shows that the optimal inven-tory level is zero along the planning horizon.Property 1 In the optimal solution the cash levels are zero inthe planning horizon: I0t = 0 (t = 1, 2, . . . ,T ).Proof (a) It is assumed that c > i. Let us reformulate the goalfunction as follows:T∑t=11(1 + r)t−1· (h · It + c · Yt + i ·Ct)=T∑t=11(1 + r)t−1· [h · It + i · (Yt + Ct) + (c − i) · Yt].In this formula using (1) we can writeYt + Ct = It − It−1 + Xt. (5)Substituting (5) in the reformulated goal function we haveT∑t=11(1 + r)t−1· [h · It + i · (Yt + Ct) + (c − i) · Yt]=T−1∑t=11(1 + r)t−1·(h +r1 + r· i)· It +h + i(1 + r)T−1· IT+T∑t=1c − i(1 + r)t−1· Yt + i ·T∑t=1Xt(1 + r)t−1−i · I0.The goal function has a lower bound: i ·∑Tt=1Xt(1+r)t−1−i · I0,because the cash levels and cash transfers are not lower thenzero. This lower bound is achieved if values I0t and Y0t (t =1, 2, . . . ,T ) are equal to zero. This means in this case that it isbetter to borrow from a bank the required money and to investPer. Pol. Soc. and Man. Sci.68 Gyöngyi Vörösmarty / Imre Dobosthe available cash. The optimal solution then: I0t = 0, Y0t = 0,C0t = Xt (t = 1, 2, . . . ,T ).(b) Now we can assume that i ≥ c. We write the goal functionasT∑t=11(1 + r)t−1· (h · It + c · Yt + i ·Ct)=T∑t=11(1 + r)t−1· [h · It + c · (Yt + Ct) + (i − c) ·Ct].In this formula substituting (5) the goal function can be writtenin the following way:T∑t=11(1 + r)t−1· [h · It + c · (Yt + Ct) + (i − c) ·Ct]=T−1∑t=11(1 + r)t−1·(h +r1 + r· c)· It +h + c(1 + r)T−1· IT+T∑t=1i − c(1 + r)t−1·Ct + c ·T∑t=1Xt(1 + r)t−1−c · I0.The lower bound of this cost function is c ·∑Tt=1Xt(1+r)t−1−c · I0.This lower bound is achieved if I0t = 0, Y0t = Xt, C0t = 0 (t =1, 2, . . . ,T ), i.e. we have attained the optimal solution. This op-timal solution holds if∑Tt=1 Xt ≤ M. If∑Tt=1 Xt > M, then theoptimal solution has the next form: I0t = 0 (t = 1, 2, . . . ,T ) andY0t =Xt t = 1, 2, . . . , t′0 t = t′ + 1, . . . ,T,C0t =0 t = 1, 2, . . . , t′Xt t = t′ + 1, . . . ,T.Time period t′ is defined as∑t′t=1 Xt = M. With these calcula-tions we have proven the property. Using the proof of Property1., we have two additional properties. Property 2 If c > i, then in the optimal solution Y0t = 0, C0t =Xt (t = 1, 2, . . . ,T ).The meaning of this property is that if the unit cost of cashtransfer is greater than the interest rate, then it is better to usebank loan, and not to spend the asset of the firm.Tab. 1. Optimal solution of the model in dependence of parametersc > i i ≥ cT∑t=1Xt ≤ MI0t = 0 I0t = 0Y0t = 0 (t = 1, 2, . . . ,T ) Y0t = Xt (t = 1, 2, . . . ,T )C0t = Xt C0t = 0T∑t=1Xt > M(t′∑t=1Xt = M)I0t = 0 I0t = 0Y0t = 0 (t = 1, 2, . . . ,T ) Y0t =Xt t = 1, 2, . . . , t′0 t = t′ + 1, . . . ,TC0t = Xt C0t =0 t = 1, 2, . . . , t′Xt t = t′ + 1, . . . ,TProperty 3 If i ≥ c, then in the optimal solution(a) ifT∑t=1Xt ≤ M, I0t = 0, Y0t = Xt , C0t = 0 (t = 1, 2, . . . ,T ),and(b) ifT∑t=1Xt > M,Y0t =Xt t = 1, 2, . . . , t′0 t = t′ + 1, . . . ,Tand C0t =0 t = 1, 2, . . . , t′Xt t = t′ + 1, . . . ,Twheret′∑t=1Xt = M and t′ < T.This third property shows that it is more rational to spend theasset of the firm if the interest rate is not smaller than the oppor-tunity cost of spending cash. The results can be summarized inTable 1.4 Numerical examplesTable 1 presents three different cases of the optimal solutions.We construct problems to demonstrate the solutions with thehelp of data from the Table 2.Case 1.In this case the opportunity cost of cash transfer (c = 2) isgreater than the interest rate (i = 0.5). The solution for this typeof models is borrowing from a bank and using the asset of thefirm for other investments. The optimal solution then I0t = 0,Y0t = 0, C0t = 10, (t = 1, 2, . . . , 12). The minimal costs are$46.532.Case 2.The opportunity costs (c = 0.5) are lower than then interestrate (i = 1). In this model it is better to spend the available cashfor purchasing. The sum of the required cash(∑12t=1 Xt = 120)islower than the available asset (M = 130). The optimal solutionthen I0t = 0, Y0t = 10, C0t = 0 (t = 1, 2, . . . , 12). The minimalcosts are $46.532.Case 3.The opportunity costs (c = 1) are lower than then interestrate (i = 1.5). In this model it is better to spend the avail-able cash for purchasing, as it was in case 2. The sum of therequired cash(∑12t=1 Xt = 177)is lower than the available asset(M = 130). It means that the purchasing department must bor-row some money from a bank. The sum of the borrowed cashis equal to $43 =∑12t=1 Xt − M. It is known in this model that125 =∑10t=1 Xt < M = 130, i.e. in the 10th period there areuse of asset and borrowing. The optimal solution then I0t = 0(t = 1, 2, . . . , 12), Y0t = {10, 15, 13, 20, 11, 8, 14, 16, 18, 5, 0, 0},and C0t = {0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 22, 10}. The minimal costsare $149.429.Analysis of purchasing activity with discounted cash flow inventory models 692013 21 2Tab. 2. Parameters of the modelsT r I0 h c i {Xt}12112∑t=1Xt MCase 1. 12 0.05 0 2 1 0.5 {10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10} 120 100Case 2. 12 0.05 0 0.1 0.5 1 {10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10} 120 130Case 3. 12 0.05 0 0.1 1 1.5 {10, 15, 13, 20, 11, 8, 14, 16, 18, 20, 22, 10} 177 1305 ConclusionsIn this paper we have investigated a discounted cash flow pur-chasing model. In the optimal solution of the model the cash lev-els are equal to zero in the planning horizon. The cash transferis equal to zero in the model if the opportunity costs of transferare higher than that of interest rate. If the interest rate is higherthan the transfer costs, then it is optimal to spend all availablecash, and if it is necessary to borrow some money from a bank.This basic model can be generalized in several ways. A pos-sible generalization is to take into account the date of payment.In this model form we have not examined the net present valuerepresentation of the cash flow identity. 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Pol. Soc. and Man. Sci.70 Gyöngyi Vörösmarty / Imre Dobos

Analysis of purchasing activity with discounted cash flow inventory models

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