Benkő, János

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Investment allocation model at ABC Gelderland 1Investment allocation model at ABC GelderlandJ. BENKŐDepartment of Logistics, University of Agricultural Sciences, H-2103 Gödöllő(Hungary)KEY WORDS: investment allocation, transportation problem, Hungarian method.ABSTRACTThe author shows a decision making model for an investment allocation problem in Hollandwhich was developed from the base model of transportation problem. Former the problem inthe case study in general was solved by mixed integer programming (MIP). The suggestednew model has some essential advantages against the MIP regarding the memory demand andcomputation time.INTRODUCTIONScope of ABC GelderlandABC Gelderland is a centre for Dutch agrarian interest. Its activities comprise the supply of anextensive package of products and services attuned to modem farming. Marketing is done inHolland and abroad.The major activity is the production and sale of mixed feeds. For this ABC Gerderland com-mands four modem and well-equipped production companies. Total mixed feed productionand sales amounts to more than 725 000 tones annually and consists of 30% beef feed, 58%pig feed, 12% poultry feed. More than 85% of these feed are produced in complete form, 95%of delivery is in bulk.Production involves a large number of mixed feeds fine tuned to specific use in practice. Abroad assortment makes it possible to choose the right mixed feed for any type of animal un-der different operating conditions. Figures from practice show that these mixed feeds produceoutstanding results. Outstanding quality coupled to attractive prices means that mixed feedsfrom ABC Gelderland make an important contribution to achieving a maximum profit at live-stock management company.As was indicated above ABC Gelderland has four plants located in Lochem, Doetinchem,Aalten and Winterswijk. The last is specialized on one product line, while the others producethe whole assortment. The assortment of the products is very wide included 225 products.PROBLEM AND METHODSABC Gelderland is planning to expand its production capacity to satisfy the increasing de-mand of clients. Another reason of the investment is that the plant located in Doetinchem hasobsolete production line. The question is, what is the most advantageous allocation of the in-vestment. The possible places are Lochem and Doetinchem because both of them can bereached by ship.Management of ABC Gelderland needs to make decision on which place to choose. The basisfor the decision is the minimal total logistical and production costs. Since the production costis independent of the location (the fixed and variable cost of the production are the same onboth of places) it is sufficient to consider only the logistical cost.This problem can be described in very simple way as a transportation problem. The advan-tages of this method will be shown in the next part of the report.2 Bulletin of the University of Agricultural Sciences, Volume 1, Gödöllő, 1991-1992.2Transportation problemProbably the most important special type of linear programming problem is the so calledtransportation problem. Special solution procedure can be used for it which is a kind ofstreamlining of simplex method and also can be obtained by exploiting the special structure inthe problem.To describe the general model for transportation problem, we need to use terms. In particular,the general transportation problem is concerned with distributing any commodity from anygroup of supply centers, called sources, to any group of receiving centers, called destination,in such a way as to minimize the total distribution cost.Thus, in general source i ( i= 1, 2,...,n) has a supply of si units to distribute to destination, anddestination j (j= 1, 2,..., m) has demand for dj units to be received from the sources. A basicassumption is that the cost of distributing unit from source i to destination j is directly propor-tional to the number distributed, where c ij denotes the cost per unit distributed. These data canbe summarized conveniently in the cost and requirements table shown in Table 1.Table 1.Cost and requirements tableDestination1 2 … m Supply1 c11 c12 … c1m s1Source 2 c12 c22 … c12 s2… … … … … …n cn1 cn1 cnm snDemand d1 d2 dmLetting z be total distribution cost and xij (i= 1, 2, ... ,n;j= 1,2, .... ; m) be the number of unitsto be distributed from source i to destination j, the linear programming formulation of thisproblem becomes: i jijij zxc minsubject to  jiij sx for i = 1, 2,..., n ijij dx for j = 1, 2,..., m,and 0ijx for all i and j.Moreover, a necessary and sufficient condition for transportation problem to have any feasiblesolution is that i jji dsInvestment allocation model at ABC Gelderland 3For many applications, the supply and demand quantities in the model (the si and dj) have in-teger values, and implementation will require that the distributed quantities (the x ij) also haveinteger value. Fortunately, because of the special structure the problems have the followingproperty: for transportation problem where every si and dj has and integer value, all the basicvariables (allocations) in every basic feasible solution also have integer values. Therefore, it isunnecessary to add a constraint to the model that the x ij must have integer value.Remarkable, in almost every case can be reached by multiplication si and dj to be integer, andeven, in this way it can be obtained c ij to be integer also. As a result, we can save dramaticcomputational time and also memory place.Mathematical model of allocation problemThe allocation problem can be drafted: Given a company with j plants which produce i sortsof products for k number of customers. The question is how to split the products between theplants in such a way that demand of the customers will be satisfied and the total logistical costwill be minimal.At the same time, we have to insure that two kinds of options can be available:permit product splitting, where the same product is produced in more than one places,prohibit product splitting.This problem can be described as a so-called more steps transportation problem.To understand the model, we need to introduce some new terms.To obtain the most advantageous product splitting, we have to use two types of capacity:CAPACITYij the potential capacity of jth plant from ith product,CAPACITYj the total real capacity of jth plant.With CAPACITYij we can reach that ith product will be prohibited or restricted at jth plant. IfCAPACITYij=0, it means the ith product is prohibited at jth plant. If CAPACITYij -≤CAPAC-ITYj, it means the ith product is restricted.In this way, in the model i jjij CAPACITYCAPACITY ,therefore we have to provide that the difference i jjijj CAPACITYCAPACITYFDEMANDwill be contracted, to avoid overloading of jth plant. FDEMANDj is fictitious quantity. To geta feasible solution for the problem, necessary and sufficient condition is that  DEMANDSOURCE .This condition can be obtained with introducing.SURPLUS capacity surplus,LACK lack of capacity, what can be calculated the next way. Let it be.  i i kikjij DEMANDCAPACITY4 Bulletin of the University of Agricultural Sciences, Volume 1, Gödöllő, 1991-1992.4than j i kikij DEMANDCAPACITYSURPLUS 0if,0if,0 i k jijik CAPACITYDEMANDLACK 0if,0if,0After this introduction let see the model (Table 1 and Table 2).Variables:xijk production from ith product at ith product at jth plant for kth c1ient,yij fictitious production from ith product for jth fictitious c1ient,ij surplus of capacity from ith product at jth plant,βik lack of capacity from ith product for kth c1ient.Coefficients:cijk cost matrixc ijk=(average supply cost)ij+(average distribution cost)ijkCAPACITYij potential production capacity from ith product at jth plant,CAPACITYj total real production capacity of jth plant,DEMANDik demand of kth client from ith product,FDEMANDj fictitious demand of jth fictitious client,SURPLUS capacity surplus,LACK lack of capacity.Object function: i j kijkijk xc minConditions:x ijk≥0; yij≥0; ij; βik≥0 kijijijijk CAPACITYyx  for i=1,2,…I and j=1,2,…,J i kik LACK for k=1,2,…K, where i k jijik CAPACITYDEMANDLACK 0if,0if,0, and  i i kikjij DEMANDCAPACITYInvestment allocation model at ABC Gelderland 5COST AND REQUIREMENTS TABLE OF INVESTMENT ALLOCATION PROBLEMPRODUCT(1) PRODUCT(i) PRODUCT(I)CLIENT CLIENT CLIENTFICTITIOUS CLIENT(1) (k) (K) (1) (k) (K) (1) (k) (K) (1) (j) (J)FICT.CLI-ENT SUPPLY(1) c(1,1,1) c(1,1,k) c(1,1,K) M M M M M M 0 M M 0 CAPACITY(1,1)(j) c(1, j,1) c(1,j,k) c (1,j, K) M M M M M M M 0 M 0 CAPACITY(1,j)PRO-DUCT(1)PLANT(J) c (1, J, 1) c (1,J,k) c (1, J, K) M M M M M M M M 0 0 CAPACITY(1,J)(1) M M M c(i,1,1) c(i,1,k) c(i,1,K) M M M 0 M M 0 CAPACITY(i,1)(j) M M M c(i,j,1) c(i,j,k) c(i,j,K) M M M M 0 M 0 CAPACITY(i,i)PRO-DUCT(i)PLANT (J) M M M c(i,J,l) c(i,J,k) c(i,J,K) M M M M M 0 0CAPACITY(i,J)(1) M M M M M M c(I,1,1) c(I,1, k) c(I,1,K) 0 M M 0 CAPACITY(I,1)(j) M M M M M M c(I,j,1) c(I,j,k) c(I,j ,K) M 0 M 0 CAPACITY(I,j)PRO-DUCT(I)PLANT (J) M M M M M M c(I,J,l) c(I,J,k) c(I,J,K) M M 0 0 CAPACITY(I,J)FICT. PRODUCT 0 0 0 0 0 0 0 0 0 M M M M LACKDEMAND Dem(1,1) Dem(1,k) Dem(1,K) Dem(i,1) Dem(i,k) Dem(i,K) Dem(I,1) Dem(I,k) Dem(I,K) Fd(1) Fd(j) Fd(J) SURP6 Bulletin of the University of Agricultural Sciences, Volume 1, Gödöllő, 1991-1992.6Table 3SOLUTION TABLE OF INVESTMENT ALLOCATION PROBLEMPRODUCT (1) PRODUCT (i) PRODUCT (I)CLIENT CLIENT CLIENTFICTITIOUS CLIENT(1) (k) (K) (1) (k) (K) (1) (k) (K) (1) (j) (J)FICT.CLI-ENT SUPPLY(1) x(1,1,1) x(1,1,k) x(1,1,K) M M M M M M y(1,1) M M (1,1) CAPACITY(1,1)(j) x(1, j,1) x(1,j,k) x(1,j, K) M M M M M M M y(1,j) M (1,j) CAPACITY(1,j)PRO-DUCT(1)PLANT(J) x(1, J, 1) x(1,J,k) x(1, J, K) M M M M M M M M y(1,J)0(1,J) CAPACITY(1,J)(1) M M M x(i,1,1) x(i,1,k) x(i,1,K) M M M y(i,1) M M (i,1) CAPACITY(i,1)(j) M M M x(i,j,1) x(i,j,k) x(i,j,K) M M M M y(i,j) M (i,j) CAPACITY(i,i)PRO-DUCT(i)PLANT (J) M M M x(i,J,l) x(i,J,k) x(i,J,K) M M M M M y(i,J) (i,J) CAPACITY(i,J)(1) M M M M M M x(I,1,1) x(I,1,k) x(I,1,K) y(I,1) M M (I,J) CAPACITY(I,1)(j) M M M M M M x(I,j,1) x(I,j,k) x(I,j ,K) M y(I,j) M (I,j) CAPACITY(I,j)PRO-DUCT(I)PLANT (J) M M M M M M x(I,J,1) x(I,J,k) x(I,J,K) M M y(I,J) (I,J) CAPACITY(I,J)FICT. PRODUCT β(1,1) β(1,k) β(i,K) β(i,1) β(i,k) β(i,K) β(I,1) β(I,k) β(I,K) M M M M LACKDEMAND Dem(1,1) Dem(1,k) Dem(1,K) Dem(i,1) Dem(i,k) Dem(i,K) Dem(I,1) Dem(I,k) Dem(I,K) Fd(1) Fd(j) Fd(J) SURPInvestment allocation model at ABC Gelderland 7 jikikijk DEMANDx  for i=1,2,…I and k=1,2,…,Kjjik FDEMAND for j=1,2,…J, where i jjijj CAPACITYCAPACITYFDEMAND i jij SURPLUS , for i=1,2,…I and j=1,2,…,J, where j i kikij DEMANDCAPACITYSURPLUS 0if,0if,0SURPLUSFDEMANDDEMANDLACKCAPACITYjji kiki jij  DISCUSSIONClustering Of products and clientsTo decrease the number of variables, the clustering of products and clients is recommended.Since the large number of products and clients may result into too large effectiveness matrix.Therefore the products and clients were aggregated into clusters. This operation resulted into 9products clusters including more than 200 animal feeds and 20 clients clusters with more than5000 customers. The clients clusters are assumed as average customer who is located in thecenter of the geographical cluster.Complements to usage of the modelIn order to get answer for our question, as to what is the most advantageous allocation or allo-cations of the investment, we need to run the model three times with different boundary condi-tions. The next possible varieties of allocation have to be analyzed:a) plants: Lochem with expanded capacityAaltenWinterswijkb) plants: LochemDoetinchem with expanded capacityAaltenWinterswijkc) plants: Lochem with expanded capacityDoetinchem with expanded capacityAaltenWinterswijkThe value of the object function will give the answer of which is the most beneficial solution.At the same time, the values of variables (x ijk,ij andβik) indicate the following:8 Bulletin of the University of Agricultural Sciences, Volume 1, Gödöllő, 1991-1992.8x ijk shows satisfaction in demand of the kth client for ith product from jth plant, ij orβik in-forms us on, how much capacity of jth plant is under utilized or how much the lack of capac-ity of jth plant.To apply the model we need to use special algorithm and software. Besides the simplexmethod there are also two special methods which give exact solution of transportation prob-lem: the transportation simplex and the Hungarian methods. Both methods can be used but inour case, where the model consists of a lot of M (infinite) elements the Hungarian method ismore beneficial. The Hungarian method does not require the storage the infinite elementssince during the iterative solution procedure these elements have never been allocated. But inthis case special matrix handling method have to be used.REFERENCESBENKŐJ. 1987. Transport Organization in the Grain Processing Industry. PhD thesis, Hun-garian Academy of Sciences, Budapest (in Hungarian)BOWERSOX D. J. CLOSS D. J. HELFERICH O. K. 1986. Logistical Management, A Sys-tem Integration of Physical Distribution, Manufacturing Support and Material Procurement,Macmillan Publishing Co., New York.HILLIER F. S., LIBERMAN G. J. 1990. Introduction to operations Research, Mc Graw-HillPubl. Co., NewYork.Information on ABC Gelderland, The Netherlands.Publikálva: Bulletin of the University of Agricultural Sciences, Volume 1, Gödöllő, 1991/1992. 103-111 p.

Investment Allocation Model at ABC Gelderland

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