2. a) Understand the constraints in the selection of
new projects.
b) Explain the techniques of mathematical
programming that may be applied in project
management.
3. When investment projects are considered
individually, any of the discounted cash flow technique
may be applied for obtaining a correct accept or reject
criteria.
In an existing organization, however, capital
investment projects often cannot be considered
individually or in isolation.
This is because the pre-conditions for viewing
projects individually- project independence, lack of
capital rationing, and project divisibility are rarely, if
ever, fulfilled.
Under the constraints obtained in the real world, the
so called rational criteria per se may not necessarily
signal the correct decision.
4. CONSTRAINTS
Project Dependence : Project A and B are
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
most conspicuous kind of economic dependency
occurs when projects are mutually exclusive.
If two or more projects are mutually exclusive,
acceptance of any one project out of the set of
mutually exclusive project automatically precludes
the acceptance of all other projects in the set. From
an economic point of view, mutually exclusive projects
are substitutes for each other.
For example, the alternative possible uses of a
building represent a set of mutually exclusive projects.
Clearly if the building is put to one use, it cannot be
put to any other use.
5. Economic dependency also exists when projects,
even though not mutually exclusive, negatively
influence each other’s cash flows if they are
accepted together.
Illustration of this kind of economic dependency:
a project for building a toll bridge and a project for
operating a toll ferry.
These two project are such that when they are
undertaken together, the revenues of one will be
negatively influenced by the other.
6. Further, the projects are said to have positive
when there is complementarity between projects.
If undertaking a project influences favourably the
cash flows of another project, the two projects are
complementary projects.
Complementarity may be of two types:
asymmetric complementarity and symmetric
complementarity. In asymmetric complementarity,
the favourable effect extends only in one
direction.
7. Capital Rationing:
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.
Internal capital rationing is caused by a decision taken by
the management to set a limit to its capital expenditure
outlays; or, it may be caused by a choice of hurdle rate
higher than the cost of capital of the firm.
Internal capital rationing, in either case, results in rejection
of some investment projects which otherwise are
acceptable.
External capital rationing arises out of the inability of the
firm to raise sufficient amounts of funds at a given cost of
capital. In a perfect market, a firm can obtain all its funds
8. Project Indivisibility
: Capital projects are considered indivisible, i.e. a capital project
has to be accepted or rejected in toto - a project cannot be
accepted partially.
Given the indivisibility of capital projects and the existence of
capital rationing, the need arises for comparing projects. To
illustrate this point, consider an example. A firm is evaluating
three projects A, B, and C which involve an outlays of Rs. 0.5
million, Rs. 0.4 million, and Rs. 0.3 million respectively. The net
present value of these projects are Rs. 0.2 million, Rs. 0.15
million, Rs. 0.1 million respectively.
The funds available to the firm for investment are Rs. 0.7 million.
In this situation, acceptance of project A (project with the highest
net present value) which yields a net present value of Rs. 0.2
million results in the rejection of projects B and C which together
yield a combined net present value of Rs. 0.25 million. Hence,
because of the indivisibility of projects, there is a need for the
comparison of projects before the acceptance/rejection decisions
9. METHOD OF RANKING
Two approaches are available for determining
which project to accept and which projects to
reject : (i) the method of ranking, and (ii) the
method of mathematical programming.
10. The Method of ranking
The method of ranking consists of two steps :
(i) Rank all projects in a decreasing order
according to their individual NPV’s, IRR’s or
BCR’s.
(ii) Accept project 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.
11. Conflict in Ranking
In a given set of projects, preference ranking
tends to differ from one criterion to another. For
example, NPV and IRR criteria may yield different
preference rankings.
Likewise, there may be a discrepancy between
the preference rankings of NPV and BCR (benefit
cost ratio) criteria.
When preference rankings differ, the set of
projects selected as per one criterion tends to
differ from the set of projects selected as per
some other criterion. This may be illustrated by an
example.
12. Consider a set of five projects, A, B, C, D, and E,
for which the investment outlay, expected annual
cash flow, and project life are as shown:
13.
14. The NPV, IRR and BCR for the five projects and
the ranking along these dimensions are shown in
Exhibit on next slide:
15.
16. It is clear that in the above case the three criteria
rank the projects differently.
If there is no capital rationing, all the projects
would be accepted under all the three criteria
though internal ranking may differ across criteria.
However, if the funds available are limited, the
set of projects accepted would depend on the
criterion adopted.
17. Project Indivisibility
A problem in choosing the capital budget on the
basis of individual ranking arises because of
indivisibility of capital expenditure projects. To
illustrate, consider the following set of projects
(ranked according to their NPV) being evaluated
by a firm which has a capital budget constraint of
Rs. 2,500, 000.
18.
19. If the selection is based on individual NPV
ranking, projects A and B would be included in the
capital budget- these projects exhaust the capital
budget.
A cursory examination, however, would suggest
that it is more desirable to select projects B, C,
and D. These three projects can be
accommodated within the capital budget of Rs.
2,500,000, and have a combined NPV of Rs.
850,000, which is greater than the combined NPV
of projects A and B.
20. Feasible Combinations Approach
The above example suggests that the following
procedure may be used for selecting the set of
investments under capital rationing.
1. Define all combinations of projects which are
feasible, given the capital budget restriction and
project interdependencies.
2. Choose the feasible combination that has the
highest NPV.
21.
22.
23. The most desirable feasible combination consists
of projects B, D and E as it has the highest NPV.
24. MATHEMATICAL PROGRAMMING
APPROACH
The ranking procedure described above becomes
cumbersome as the number of projects increases
and as the number of years in the planning
horizon increases. To cope with a problem of this
kind, it is helpful to use mathematical
programming models. The advantage of
mathematical programming models is that they
help in determining the optimal solution without
explicitly evaluating all feasible combinations.
25. 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
optimize the objective function subject to various
constraints.
26. Though a wide variety of mathematical
programming models is available, but we will
cover only one type:
Linear programming model
27. LINEAR PROGRAMMING
MODEL
The linear programming model is based on the
following assumptions :
• The objective functions 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 homogeneous. This means that if 100
hours of direct labour are available, each of these
hours is equally productive.
28. Linear Programming Model of a
Capital Rationing Problem
The general formulation of a linear programming
model for a capital rationing problem is:
29.
30.
31.
32.
33.
34. The basic variables (variables which take a
positive value in the optimal solution) are X1, X3,
X4, X6, X7, X9, S4, S7, S8, S9, and S10.
Their values are shown in the last column of the
tableau (X1 = 1.0; X3 = 1.0; X4 = 1.0; X6 =
.969697, and so on).
2. The rest of the variables (X2, X5, X8, S1, S2,
S3, S5, S6, and S11) are non-basic variables,
which means that they take a zero value. A value
of zero for X1, X3,and X8 means that these three
projects are completely rejected in the optimal
solution. A value of zero for S1 and S2 implies
that the budgets of 50 in year 1 and 20 in year 2
are fully exhausted on the six accepted projects.
