A multilevel programming approach to decentralized (or hierarchical) re-source allocation systems

Semu Mitiku1,∗

1 Department of Mathematics, Addis Ababa University, P.O.Box 1176, Addis Ababa, Ethiopia.

In many decision processes there is a hierarchy of decision-makers and decisions are taken at different levels in this hierarchy.In business (and many other practical activities) decision making has changed over the last decades. From a single person(the boss!) and a single criterion (e.g. profit), decision environments have developed increasingly to become multi-personand multi-criteria and even multi-level (or hierarchical) situations. In organization with hierarchical decision systems, thesequential and preemptive nature of the decision process makes the problem of selecting an optimum strategy and actionvery different from the usual operations research methods. Therefore, a multilevel programming approach is considered inmodeling such problems. In particular a three-level mathematical programming model has been proposed for an optimalresource allocation problem in Ethiopian universities.

1 Introduction

Many resource allocation or planning problems require compromises among the objectives of several interacting individualsor agencies, most of the time, arranged in hierarchical administrative structure and can have independent even sometimesconflicting objectives. A planner at one level of the hierarchy may have its objective function determined partly by variablescontrolled at other levels. Assuming that the decision process has a preemptive nature and having r levels of hierarchy, weconsider the decision maker at level r to be the leader and those at lower levels to be followers. Therefore, a multilevelprogramming approach is considered in modeling such problems.

A multilevel programming problem partitions control of the decision variables among several decision makers, each actingin a sequence to maximize his/her own objective function. A University resource allocation procedure and the problem ofoptimal allocation of the meager resources it has is studied and modeled with three-level linear programming approach.

2 The Model and Methods in Three Level Programming

Resource allocation for universities by the federal government (in Ethiopia) is a three level decision problem. The threedecision makers are the federal government, the University central administration and the Faculty dean’s offices. For instance,if the federal government has a total budget of T to be distributed among (or allocated to) N Universities in the country, howcan the government body allocate its resources in terms of the effectiveness of the use of these resources by the universities, andhow can the universities utilize the allocated resources effectively to maximize their benefits at their faculty levels? Assumethat a university labeled by i has ri faculties and there are some k departments and units that utilize resources and carry outplanned activities.

The above problem is a three level problem and resource allocations can usually be modeled using linear functional rela-tionships. Similar problem was studied by Cassidy, et. al. [4]. However, at the lower level of the hierarchy, it is assumed thatprojects are chosen to be funded or not. This will be a kind of 0-1 problem at the lower level. At a department of a university,however, activities could be undertaken possibly with a lower scale but may not be omitted. Therefore, we assumed a certainpercentage of an activity will be carried out in each faculty at each university. We adopted and modified the “relative regrate”function defined by Kirby and Raike [5] to state the objective functions at each level to minimize the dissatisfaction of thelower level budget units.

Linear three level programming problems are especially formulated as ( [2], [3])

maxx3

(f3(x) = c31x1 + c32x2 + c33x3)

where x2 solves

maxx2

(f2(x) = c21x1 + c22x2 + c23x3)

∗ Semu Mitiku E-mail: smtk@math.aau.edu.et,

PAMM · Proc. Appl. Math. Mech. 7, 2060003–2060004 (2007) / DOI 10.1002/pamm.200700127

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

where x1 solves

maxx1

(f1(x) = c31x1 + c32x2 + c33x3)

subject to A1x1 + A2x2 + A3x3≤ b.

The problem has been modeled and formulated in similar form. There are a few varying approaches yet known in solvingsuch a mathematical problem. We used the simplex-cutting plane algorithm introduced by Bard [2] and improved by White [6]which is based on the Kuhn-Tucker conditions.

One of the main assumptions in the above class of problems is that there is a single decision maker at each level. In realitythere are several competing decision makers who like to get the maximum share as possible. To entertain this fact and to comeup with a single compromising decision at level, we used Willick’s power index approach [7] to propose a coalition amongthe players within each level of the hierarchy.

References

[1] Bialas, W. F., and Karwan M. H. Mathematical Methods for Multilevel Planning, Technical report No. 79-2, Department of IndustrialEngineering, State University of Newyork at Buffalo, New York (1979).

[2] Bard, J. F., An Investigation of the Linear Three Level Programming Problem, IEEE Transactions on Systems, Man, and Cybernetics,14, 711-717 (1984).

[3] Bard, J. F., Geometric and Algorithmic Developments for a Hierarchical Planning Problem, European J. of OR 19, 372-383 (1985).[4] Cassidy, R. G., Kirby, M. J. L., and Raike, W. M., Efficient Distribution of Resources Through Three Levels of Government, Manage-

ment Science, 17, No. 8, B462 - B473 (1971).[5] Kirby, M. J. L., and Raike, W. M., Priority Planning in a University Computation Center CBA Working Paper, 69-114, Univ. of Texas

(1968).[6] White, D. J., Penalty Function Approach to Linear Trilevel Programming JOTA, 93, No. 1, 183-197 (1997).[7] Willick, W. J., A Power Index for Cooperative Games with Applications to Hierarchical Organizations, Ph.D Thesis, Dept. of Industrial

Eng., Sunny at Buffalo, New York, (1995).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Contributed Papers 2060004