2.3 multipleprojectsandconstraints

8/9/2019 2.3 MultipleProjectsandConstraints

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Outline

Constraints

Method of Ranking

Mathematical programming approach

Linear programming model

Integer linear programming model

Goal programming model


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Constraints

Project dependence

Mutual exclusiveness

Negative economic dependency

Positive economic dependency

Capital rationing

Project indivisibility


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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 availablefor 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


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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


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Mathematical Programming ApproachA 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 restrictionsarising out of limitations of resources, environmental restrictions, and

managerial policieswhich 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


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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.


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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

CFjtXjKt (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


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Kt = capital budget available in period t

The following features may be noted.

1. All the input parametersNPVj, CFjt, Ktare

assumed to be known with certainty.

2. The Xjdecision variables are assumed to be continuous

but limited by a lower restriction (0) and an upper

restriction (1)

3. The NPVcalculation is based on a cost of capital

figure which is known with certainty.


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Integer Linear Programming Model

The principal motivation for the use of integer linear

programming approach are: (i) It overcomes the problem ofpartial 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

XjNPVj

Subject to

CFjtXjKt (t = 0,1,.m)

n

j =1

n

j =1


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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).


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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 incorporatedin the integer linear programming model:

Mutual exclusiveness

Contingency Complementariness


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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 selectingthe 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 multiplegoal 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 involvesmulti le oals

Growth in sales and market share

Growth and stability of reported earnings

Growth and stability of dividends


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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 theprogramming 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.

2.3 multipleprojectsandconstraints

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