![Goal Programming Format
In general, the goal programming format is as follows:
Minimise = {P1 [f1 (d1 , d1 )] + P2 [ f2 (d2 , d2 )] +……
+Pm {fm (dm ,dm)]}
Subject to aj1 Xj C1
aj2 Xj C2
ajk Xj Ck
bj1 Xj + d1 – d1 = G1
bj2 Xj + d2 – d2 = G2
bjm Xj + dm – dm = Gm
Xj , di , di 0
The goal programming model is formulated in terms of three principal components: (i)
the objective function,(ii) the economic constraints, and (iii) the goal constraints.
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Chapter15 multipleprojectsandconstraints
- 1. CHAPTER 15 MULTIPLE PROJECTS AND CONSTRAINTS
- 2. OUTLINE • Constraints • Method of Ranking • Mathematical programming approach • Linear programming model • Integer linear programming model • Goal programming model
- 3. Constraints • Project dependence Mutual exclusiveness Negative economic dependency Positive economic dependency • Capital rationing • Project indivisibility
- 4. Methods of Ranking Because of economic dependency, capital rationing, or project indivisibility, a need arises for comparing projects in order to accept some and reject others. What approaches are available for determining which projects to accept and which projects to reject? Basically,two approaches are available: (i) the method of ranking, and (ii) the method of mathematical programming. Fairly simple, the method of ranking consists of two steps (i) Rank all projects in a decreasing order according to their individual NPVs IRRs or BCRs (iii) Accept projects in that order until the capital budget is exhausted. The method of ranking, originally proposed by Joel Dean is seriously impaired by two problems: (i) conflict in ranking as per discounted cash flow criteria, and (ii) project indivisibility
- 5. Feasible Combination Approach The following procedure may be used for selecting the set of investments under capital rationing. 1. Define all contributions of projects which are feasible, given the capital budget restriction and project interdependencies 2. Choose the feasible combination that has the highest NPV
- 6. Mathematical Programming Approach A mathematical programming model is formulated in terms of two broad categories of equations: (i) the objective function, and (ii) the constraint equations. The objective function represents the goal or objective the decision maker seeks to achieve. Constraint equations represent restrictions –arising out of limitations of resources, environmental restrictions, and managerial policies – which have to be observed. The mathematical model seeks to optimise the objective function subject to various constraints The objective function and constraint equations are defined in terms of parameters and decision variables. Parameters represent the characteristics of the decision environment which are given. Decision variables represent what is amenable to control by the decisions makers. Out of the wide variety of mathematical programming models, we shall discuss three types: Linear programming model Integer programming model Goal programming model
- 7. Linear Programming Model The most popular mathematical programming model, the linear programming model is based on the following assumptions • The objective function and the constraint equations are linear • All the coefficients in the objective function and constraint equations are defined with certainty. • The objective function is unidimensional • The decision variables are considered to be continuous • Resources are homogenous. This means that if 100 hours of direct labour are available, each of these hours is equally productive.
- 8. Linear Programming Model of a Capital Rationing Problem The general formulation of a linear programming model for a capital rationing problem is : Maximise NPVj Xj Subject to CFjt Xj ≤ Kt (t = 0,1,….m) 0 ≤ Xj ≤ 1 Where NPVj = net present value of project j Xj = amount of project j accepted CFjt = cash outflow required for project j in period t n n j =1 j =1
- 9. Kt = capital budget available in period t The following features may be noted. 1. All the input parameters – NPVj, CFjt , Kt – are assumed to be known with certainty. 2. The Xj decision variables are assumed to be continuous but limited by a lower restriction (0) and an upper restriction (1) 3. The NPV calculation is based on a cost of capital figure which is known with certainty.
- 10. Integer Linear Programming Model The principal motivation for the use of integer linear programming approach are: (i) It overcomes the problem of partial projects which besets the linear programming model because it permits only 0 or 1 value for the decision variables (ii) It is capable of handling virtually any kind of project interdependency. The basic integer linear programming model for capital budgeting under capital rationing is as follows: Maximise Xj NPVj Subject to CFjt Xj ≤ Kt (t = 0,1,….m) n j =1 n j =1
- 11. Xj = (0,1) It may be noted that the only difference between this integer linear programming model and the basic linear programming model discussed earlier is that the integer linear programming model ensures that a project is either completely accepted (Xj =1) or completely rejected (Xj= 0).
- 12. Incorporating Project Interdependencies in the Model By constraining the decision variables to 0 or 1, the integer linear programming model can handle almost any kind of project interdependency. To illustrate, let us see how the following kinds of project interdependence are incorporated in the integer linear programming model: • Mutual exclusiveness • Contingency • Complementariness
- 13. Goal Programming Model Throughout this text we have assumed that the principal goal of financial management is to maximise the wealth of shareholders, which, under conditions of perfect capital market, can be realised by selecting the set of capital projects that maximise net present value. However, in the real world, capital market imperfections (like capital rationing, differences in lending and borrowing rates, etc.) exist. Further, empirical observation show that managers pursue a multiple goal structure which includes, inter alia, the following: Therefore, a realistic representation of real life situations should reflect the multiple goals pursued by the management. The goal programming approach, a kind of mathematical programming approach, provides a methodology for solving an optimisation problem that involves multiple goals • Growth in sales and market share • Growth and stability of reported earnings • Growth and stability of dividends
- 14. This approach, originally proposed by Charnes and Cooper in 1961, has been extended by Ijiri, Ignizio, and others. To use the goal programming model, the decision maker must: 1. State an absolute priority order among his goals 2. Provide a target value for each of his goals. The goal programming methodology seeks to solve the programming problem by minimising the absolute deviations from the specific goals in order of the priority structure established. Goals at priority level one are sought to be optimised first. Only when this is done will the goals at priority level two be considered; so on and so forth. At a given priority level, the relative importance of two or more goals is reflected in the weights assigned to them.
- 15. Goal Programming Format In general, the goal programming format is as follows: Minimise = {P1 [f1 (d1 , d1 )] + P2 [ f2 (d2 , d2 )] +…… +Pm {fm (dm ,dm)]} Subject to aj1 Xj C1 aj2 Xj C2 ajk Xj Ck bj1 Xj + d1 – d1 = G1 bj2 Xj + d2 – d2 = G2 bjm Xj + dm – dm = Gm Xj , di , di 0 The goal programming model is formulated in terms of three principal components: (i) the objective function,(ii) the economic constraints, and (iii) the goal constraints. + + + - - - - - - + + + - +
- 16. SUMMARY Because of constraints like project dependencies, capital rationing, and project indivisibility, investment projects cannot be viewed in and project indivisibility, investment projects cannot be viewed in location. Two projects are said to be economically dependent if the acceptance or rejection of one changes the cash flow stream of the other or affects the acceptance or rejection of the other. The types of economic dependency found among projects are: mutual exclusiveness, negative dependency, and positive dependency (complementariness). Capital rationing exists when funds available for investment are inadequate to undertake all projects which are otherwise acceptable. Capital rationing may arise because of an internal limitation or an external constraint. Capital projects are generally indivisible. This means that a capital project has to be accepted or rejected in total – a project cannot be accepted partially. Because of economic dependency, capital rationing, or project indivisibility, a need arises for comparing projects in order to accept some and reject others. Two approaches are available for this purpose: the method of ranking, and the method of mathematical programming.
- 17. Fairly simple, the method of ranking consists of two steps: (i) Rank all projects in decreasing order according to their individual NPVs, IRRs, or BCRs. (ii) Accept projects in that order until the capital budget is exhausted. The method of ranking is seriously impaired by two problems: (i) conflict in ranking as per discounted cash flow criteria, and (ii) project indivisibility. In a given set of projects, preference ranking tends to differ from one criterion to another. Ranking conflicts are traceable to differing assumptions about the rate of return at which intermediate cash flows are reinvested. To select a set of projects, one may define all combinations of projects which are feasible given the capital rationing constraint and project dependencies, and then choose that combination which has the highest NPV associated with it. This procedure may be called as the feasible combinations procedure. A mathematical programming model is formulated in terms of two broad categories of equations: (i) the objective function, and (ii) constraint equations. Out of the wide variety of mathematical programming models, three types have greater applicability to the capital budgeting. problem: linear programming model, integer linear programming model, and goal programming model.