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

Short Communication

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

Cornel is van de P A N N E 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

E A k x * <~a, (2) k

B , x k <~b*, x k>~O, k = l . . . . . N. (3)

Received March 1988; revised May 1988

The constraints of (2) represent the common constraints for shared resources, while those of (3) give the constraints applying to the N divisions separately. Given this overall model, the master problem for the center and the subproblems for the divisions are created.

Gazi does not have an overall problem, but gives a center problem and divisional problems. The divisional problems are, in a notation similar to the one used above.

Maximize Y" c k'x k k

subject to

~_~Akx k <~a k, x k>l-O, k

where a k is the ink-vector of common resources allocated by the center to the k-th division. The problem for the center is

Maximize ~_,~k'xk-- ~_,dk'a k (4) k k

subject to

~.E~a ~ ~ f , a m >1 o, (5),(6) k

g, <~ y" Gkx k <~ g2. (7) k

The first set of terms in (4) is the same as the corresponding terms in the objective functions of the divisions, except that gk may differ from c k because the center may adjust revenues from the

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118 C. van de Panne / S o m e 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 <~ f . (8) k

The need for the E-matrices being non-unit is not discussed by Gazi.

If the constraints in (7) are deleted, the Gazi formulation is similar to a reformulation of the decomposition model used for resource-directive decomposition methods:

Maximize ~_,ck'x k (9) k

subject to

Y'~a k ~ a, (10) k

A k x k <~a k, k = l . . . . . N , (11)

B k x k <~b k, k = l . . . . . N , (12)

xk>~O, k = l . . . . . N,

where a is used instead of f . Note that the divi- sional constraints (12) are absent in Gazi's formu- lation.

The compatibility constraints (7) are basically the same as the common constraints with 0 right- hand sides.

Gazi's method is based on certain assumptions summarized in the two lemma's. The main as- sumption is that the same variables remain basic

in optimal solutions of the divisional problems as the center varies the allocations a k of the common resources. The optimal bases of the divisional problems are supposed to be generated by a pro- cedure that is independent of the available total resource quantities. A counterexample can easily be given. For the problem

Maximize

f - - x 1 + 1.5X 2 q- 0 .5X 3

subject to

x I + 2x 2 + x 3 ~< 11,

x I + x 2 + 0 . 1 x 3 ~ < 1 ,

x I , x 2, x 3 ~ 0 ,

the optimal solution with normalized constraints has x~ and x 2 as basic variables, whereas the optimal solution has x3 as basic variable. It is easy to show that this leads to a suboptimal solution for Gazi's procedure.

Compared with both price- and resource direc- tive decomposition methods the information transfer requirements of Gazi's method are very high, because the inverse of basic matrix of divi- sional solutions must be given to the center. In resource-directive decomposition divisions trans- mit just the prices for the resources to the center.

Whereas both price- and resource directed de- composition methods iterate towards the optimal solution, Gazi's method consists of just two steps, in the first of which the divisions find their opti- mal bases. In the second step the center uses the supposedly optimal bases to find the optimal al- location of common resources to the divisions.

In a comment Gazi suggests that the center may determine for the divisions the prices of the resources instead of the quantities. This would lead in the direction of the original Dantzig-Wolfe decomposition method which is also based on resource prices.

In another comment Gazi writes about intro- ducing iterations, which would also make the method more similar to the classical decomposi- tion methods.

The conclusion is that compared with classical decomposition, Gazi's model formulation is very limited because it leaves out divisional constraints, that its information transfer requirements are very high, while it does not necessarily finds the opti- mal solution. Even in cases in which no divisional

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 decompos i t 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.