More Advanced Linear Programming Concepts and Methods pdf

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Ch 12: More Advanced Linear
Programming Concepts and
Methods
Applying Linear Programming to
Those Investments in Which The
Simplifying Assumptions of Basic
LP Analysis Do Not Hold.
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Simple Application of L. P.

In Chapter 11, Linear Programming was
applied to those investments satisfying the
following assumptions:
1. Additivity within activities: resource
consumption is constant per unit of
output; there are no economies of scale.
2. Divisibility within activities: partial
investments can be implemented. There
is no requirement to accept equipment in
discrete sizes.
3. Independence of activities: there is no
recognition of productive or financial
interdependencies.
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Extensions to the Basic
Application of L.P.

This chapter extends the basic applications of L P, to
allow investment analysis where projects take on a
more ‘real word’ flavor: ie, where some simplifying

assumptions are relaxed.These extensions include:
1. Allowing more activities and constraints
2. Recognizing indivisible investments
3. Allowing inter-year resource borrowings and
transfers
4. Recognizing interdependent projects
5. Treating mutually exclusive investments
6. Recognizing threshold investments, economies
of scale, multiple goals and investment risk.
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Explanations of the
‘Extension’ Ideas I.

More Activities and Constraints: this notion
deals with more complex resource mixes,
and more constraints, or combinations of
projects

Indivisible Investments: Most projects are
not physically divisible. For example, power
stations are not divisible, although they can
vary in size as to scale.

Inter-Year Transfers: Capital and supplies
may become available at different times, or
surplus amounts may be able to be
transferred between years.
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Explanations of the
‘Extension’ Ideas II.


Interdependent projects: projects may
provide mutual support and resources, or
infrastructure to each other.

Mutually Exclusive Investments: A casino
built on a site will preclude the construction
of an hotel or sporting facility. Only one of
these projects can appear in the LP
solution.
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Explanations of the
‘Extension’ Ideas III.

Threshold Investment, Economies of Scale,
Multiple Goals, and Risk:

Projects may have a fixed scale: eg a single
large airplane, requiring a fixed amount of
capital.

Projects may generate scale of production
economies with increased size.

Projects may have to satisfy conflicting
wealth, environmental and social concerns.

All analyses must recognize risk.
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Advanced LP Techniques Applied to

A Complex Investment Problem :
An Example, ‘Power Gen Inc’.

Power Gen Inc. has identified this set of
alternative power generating proposals:-
Constraint or Hydro- Natural Gas Natural Gas Wind - Biofuel Solar
Objective power Site A Site B Farm Panels
Capital Outlay ($M) $400 $170 $150 $100 $50 $120
Power Output (MW)
420 250 200 70 50 90
NPV ($M) => $180 $100 $80 $50 $7 $20
Alternative Generating Technologies
Power Gen Inc: Electricity Generating Investment Problem
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Advanced LP Techniques Applied to A
Complex Investment Problem :
An Example, ‘Power Gen Inc’.

In maximizing total NPV by choosing a mixture
of these generating alternatives, Power Gen Inc.
faces these constraints:-
At least 100 MW have to be
produced from renewable
resources.
At least 200 MW have to be
produced from natural gas.
Total cash and credit available is
limited to $700M.
The LP Solution For ‘Power Gen Inc’.
Constraint or objective

Hydro-
power
Natural
gas, site A
Natural
gas, site B
Windfarm Biofuel
Solar
panels
Resource
use
Sign
Resource
supply
Activity Level: Chosen 0.7 1 1 1 0 0
Capital outlay ($M) 400 170 150 100 50 120 $700
≤
$700
Renewables output (MW) 420 70 50 90 364
≥
100
Nat. gas output (MW) 250 200 450
≥
200
Max. hydro 1 0.7
≤
1
Max. nat. gas A 1 1
≤
1

Max. nat. gas B 1 1
≤
1
Max. windfarm 1 1
≤
1
Max. biofuel 1 0
≤
1
Max. solar 1 0
≤
1
NPV ($M) => 180 100 80 50 7 20 $356

Formatted Problem and LP Solution For Power Gen Inc.
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Notes On The LP Solution For
‘Power Gen Inc’.
The chosen generating methods are:
Hydro 70% of project adopted,
Natural Gas, Site A 100% of project adopted,
Natural Gas, Site B 100% of project adopted,
Windfarm 100% of project adopted,
Biofuel 0% of project adopted,
Solar Panels 0% of project adopted.
Total NPV from this selection is $M356.
Calculated as:
(0.70 x $180) + (1 x $100) + ( 1 x $80) +
(1 x $50)
Total capital outlay for this selection is $M700

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Notes On Constraints For The LP
Solution For ‘Power Gen Inc’.
Output from renewable resources at 364MW is
greater than the required minimum of 100MW.
Output from natural gas at 450MW is greater
than the required minimum of 200MW.
All projects were artificially constrained at a
maximum of 1 unit, so that more than one
project of any technology could not be chosen.
This constraint has been satisfied.
Capital outlay at $M700 is equal to the
maximum allowed of $M700.
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Note On Output For The LP
Solution Of ‘Power Gen Inc’.
The solution shows that only 70% of the Hydro scheme is to
be adopted. Such a scaled down scheme may not be
acceptable. To ensure that projects are either accepted or
rejected in their entirety, Mixed Integer Linear Programming
can be used.
‘Integer’ settings such as 0,1,2,3… allow discrete zero or
multiple selection of projects.
‘Binary’ settings with levels of 0 or 1 allow discrete zero or
unitary selection of projects.
MILP is invoked by selecting either ‘bin or ‘int’
constraints within the ‘Constraints’ selection in the ‘sign’
part of the Solver dialog box.
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Setting Integer and Binary

Constraints.
‘Binary’ constraint selected via the Solver ‘sign’ dialog box.
‘Integer’ constraint selected via the Solver ‘sign’ dialog box.
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Other LP Formulations
Mixed Integer Linear Programming can be used
to solve other complex investment problems by
careful specifications of the goals, and
imaginative definitions of the constraints.
For example:
Inter-Year Capital Transfers Introduce
activities

for borrowing and capital transfers.
Contingent Projects introduce
permission
constraints
which allow one activity to proceed only
if another is adopted.
Mutually Exclusive Projects introduce constraints
in which the total number of activities is below or
equal to a maximum level.
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Other LP Applications
Threshold investment levels – the threshold level
is set up as a binary constraint.
Economies of Scale – particular scale levels are
set up as independent activities with binary
constraints.
Multiple Goals - each goal is set up as a

constraint goal, or each goal can be individually
weighted in a total goal measure.
Risk Analysis – risky alternatives could be
constrained in the product mix; or an overall risk
measure such as ‘variance could be targeted and
minimized.
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Advanced LP Applications:
Summary
Linear programming can be used to solve selection
problems from amongst competing investment
alternatives in the face of complex constraints.
These constraints mirror real world problems, and
present a more realistic picture of actual
investment behavior, than that assumed in base
level LP analysis.
This higher level of analysis requires imaginative
definitions of both goals and constraints, and an
appreciation of Linear Programming methodology.