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Chapter 24 – Multicriteria Capital
Budgeting and Linear Programming
Linear
programming is a mathematical
procedure, usually carried out by computer
software, to find an optimal combination to
maximize a goal function subject to
constraints.
Simple LP Problem
We cannot invest more than $3,000 in division 1 or more
than $2,000 in division 2. The amount invested in
division 1 must be at least twice as great as the amount
invested in division 2. x1 and x2 are the amounts
invested in divisions 1 and 2. The LP model is
Maximize
Subject to
.2x1+ .3x2
x1 ≤ $3,000
x2 ≤ $2,000
-x1 + 2x2 ≤ $0
x1 ≥ $0
x2 ≥ $0
Dual values
Each
constraint has a dual value, which is
typically computed by the same software
that finds the original solution
Dual values tells us how much the objective
function could be increased if we exceeded
a particular constraint by one unit
Sensitivity Analysis
We
often solve a linear programming
problem several times, with constraints
changed. As a result of this analysis, we
might go back and look at ways to
overcome a particular constraint
Solving LP problems with Excel
Excel
Solver can be used to solve LP
problems
You may need to install Solver because it is
not automatically installed
Solver Illustration
Maximize
X1
X2
Sum
Limit
0.2
0.3
0
1
0
0
<=
3,000
0
1
0
<=
2,000
-1
2
0
<=
0
1
0
0
=>
0
0
1
0
=>
0
0
0
Subject to
Variable
values
Multiple Goals and Constraints
Managers can list multiple goals, such as NPV,
sales growth, EPS growth, etc.
To put the goals in the objective function, each
must be given a weight
Considerations can also be converted to
constraints. We might have a constraint that EPS
must increase at least 5% a year, for example
Capital rationing
Set
the objective function as maximizing
wealth as of some specified future date,
given a fixed amount of capital available,
and specified infusions of capital each year,
if any.
Must specify reinvestment opportunity rates
in future periods, at least in terms of
estimated external opportunity rates
Other Programming Techniques
Integer
programming
Goal programming
Chance-constrained programming
Quadratic programming