African Journal of Mathematics and Computer Science Research Vol. 4 (7), pp. 266-269 July 2011 Available online at http://www.academicjournals.org/AJMCSR ISSN 2006-9731 ©2011 Academic Journals Full Length Research Paper A Branch and Bound approach to a state government capital budgeting N. P. Akpan 1 , E. H. Etuk 2 and I. D. Essi 2 * 1 Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Rivers State, Nigeria. 2 Department of Mathematics and Computer Science Rivers State University of Science and Technology, Port Harcourt, Rivers State, Nigeria. Accepted 18 April, 2011 The capital budgeting problem is a multi-constraint Knapsack problem. Hence this paper provides a solution to a capital budgeting problem of Akwa Ibom State, Nigeria using a Branch and Bound approach. The capital projects are summarized in four broad groups: The Economic sector ( ) 1 x , The Social Service sector ( ) 2 x , The Environmental/Regional Development sector ( ) 3 x , and The Administration sector ( ) 4 x . Using Tora software (version 11) for the analysis, it was observed that the optimal solution was found in node 60 while the upper bound and the lower bound were 154.46 and 129.76 respectively. The result shows that the second and the third sector will be selected ( 1 , ; 0 , 3 2 4 1 = = x x x x ). Key words: Capital budgeting, branch and bound, Knapsack problem, capital projects. INTRODUCTION Branch and Bound method is one of the deterministic techniques of solving a Knapsack problem. It is seen as the most efficient tool among other deterministic methods as it is based on a restriction of the search tree growth; (Sedova and Seda, 2008). They also observed that avoiding much enumeration depends on the precise bounds (the lower and the upper bounds, the faster the finding of the solution is). Lawler and Wood (1966) observed Branch and Bound as algorithm for finding optimal solutions to combinatorial problems. The method produces convergent lower and upper bound for the optimal solution using an implicit enumeration scheme. Branch and bound method does not go through iterative partial solution until an optimal solution is found like the case of dynamic programming neither does it enumerate all feasible solutions which may lead to 2 n different solutions and select the one with the highest objective function value; brute force approach. The concept of Branch and Bound is based on an intelligent complete *Corresponding author. E-mail: ddmetra_utibe@yahoo.com. enumeration of the solution space since in many cases, only small subsets of the feasible solutions are enumerated explicitly. It is however guaranteed that the parts of the solution space which were not considered explicitly cannot contain the optimal solution (Kellerer et al., 2004). In this work, we will apply this method in a capital budgeting problem in order to maximize the value of some selected sectors among the other sectors. We are using a state budget; Akwa Ibom State, Nigeria, as our case study for the period of ten years. We are considering the entire Capital budget in terms of their sectors grouping. Each sector has various areas in it. The sectors are, the Economic Sector, the Social Services Sector, the Environment/Regional Development sector and the General Administrative sector. The problem is a multidimensional Knapsack Problem and the model will be stated as we progress. Assumptions (I)All the projects are independent of each other. (ii) All the sectors have equal importance to the