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European Journal of Operational Research 40 (1989) 117-119 117 North-Holland

Short Communication

Some comments on "Resource allocation in a large decentralized enterprise"

Cornelis van de PANNE Department of Economics, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta T2N 1N4, Canada

In a recent article Gazi [3] has proposed a method for decentralized decision making in an enterprise consisting of a center controlling re- sources used by divisions for their productive op- erations. As this article proposes a new approach to this problem, it is useful to compare it with alternatives that can be found in the existing literature which were ignored by Gazi.

The classical approach to decentralized deci- sion making in linear programming models has been that of the Dantzig-Wolfe decomposition method [1], which has been the basis of extensive literature on the subject, see, for example, Dirickx and Jennergren [2]. For a more recent method for these types of models, see [5]. The references of the Gazi article consist of a previous paper and the linear programming book by Kreko [4], which devotes a chapter to Dantzig-Wolfe decomposi- tion, of which Gazi seems unaware.

Since the model used by Gazi is different from that of the Dantzig-Wolfe decomposition, it is useful to first state the differences. The latter model is formulated in matrix notation as follows:

Maximize Y'~ ek'x k (1) k

subject to

EAkx*

118 C. van de Panne / Some comments on "'Resource allocation in a large decentralized enterprise"

divisions according to its own perceptions. The second set of terms reflect the costs of distributing the resources over the divisions. If ~k= c k, and d k = 0, both models are comparable at least in this respect. The Gazi model has therefore a slightly more general objective function.

The constraints in Gazi's divisional problems only concern common resources; there are no special divisional constraints as in (3). In this respect Gazi's model is substantially less general. Divisional constraints may be included in the common constraints, but this will usually make the number of common constraints very large, and, moreover, centralize divisional decisions to a large extent.

The main constraints of the center problem are given by (5). The resources of the center given by the vector f may be different from the resources distributed to the divisions, but if they are the same, the E-matrices are unit matrices and the corresponding constraints may be given by

~.,ak

C van de Panne / Some comments on "'Resource allocation in a large decentralized enterprise" 119

constraints are present and in which the Gazi's minimal set happens to contain all variables in- cluded in the overall optimal solution, the require- ment of transferring the entire inverse of the sub- problems will make the method less attractive than decomposit ion methods which require transfer of just prices or quantities, both from a computational point of view and also in terms of interpretation as a decentralization.

[2] Dirickx, Y.M.I., and Jennergren, L.P., System Analysis by Multilevel Methods, Wiley, New York, 1979.

[3] Gazi, D.C., "Resource allocation in a large decentralized enterprise", European Journal of Operational Research 30 (1987) 339-343.

[4] Kreko, B., Linear Programming, American Elsevier, New York, 1968.

[5] van de Panne, C., "Local decomposition methods for linear programming", European Journal of Operational Research 28 (1987) 369-381.

References

[1] Dantzig, G.B., and Wolfe, P., "The decomposition al- gorithm for linear programs", Econometrica 29 (1961) 767-778.