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Ch 12: More Advanced Linear

Programming Concepts andMethods

Applying Linear Programming to

Those Investments in Which TheSimplifying Assumptions of Basic

LP Analysis Do Not Hold.



2

Simple Application of L. P. In Chapter 11, Linear Programming was

applied to those investments satisfying thefollowing assumptions:

1. Additivity within activities: resourceconsumption is constant per unit ofoutput; there are no economies of scale.

2. Divisibility within activities: partialinvestments can be implemented. There

is no requirement to accept equipment indiscrete sizes.

3. Independence of activities: there is norecognition 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

transfers4. 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: thisnotion deals with more complex resourcemixes, and more constraints, orcombinations of projects

Indivisible Investments: Most projects arenot physically divisible. For example,power stations are not divisible, althoughthey can vary in size as to scale.

Inter-Year Transfers: Capital and suppliesmay become available at different times,or surplus amounts may be able to betransferred between years.



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Explanations of the

‘Extension’ Ideas II.

Interdependent projects: projects mayprovide mutual support andresources, or infrastructure to eachother.

Mutually Exclusive Investments: Acasino built on a site will preclude the

construction of an hotel or sportingfacility. Only one of these projects canappear 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 asingle large airplane, requiring a fixedamount of capital.

Projects may generate scale ofproduction economies with increasedsize.

Projects may have to satisfy conflictingwealth, environmental and socialconcerns.

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 PanelsCapital 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 TechnologiesPower Gen Inc: Electricity Genera ting Investment Prob lem



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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 GenInc. faces these constraints:-

At least 100 MW have to beproduced from renewableresources.

At least 200 MW have to beproduced from natural gas.

Total cash and credit available is

limited to $700M.



The LP Solution For ‘Power Gen Inc’.

Constraint or objectiveHydro-

power

Natural

gas, site A

Natural

gas, site BWindfarm Biofuel

Solar

panels

Resource

useSign

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 isgreater than the required minimum of 100MW.

Output from natural gas at 450MW is greaterthan the required minimum of 200MW.

All projects were artificially constrained at amaximum of 1 unit, so that more than oneproject of any technology could not be

chosen. This constraint has been satisfied.

Capital outlay at $M700 is equal to themaximum 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 tobe adopted. Such a scaled down scheme may not beacceptable. To ensure that projects are either accepted orrejected in their entirety, Mixed Integer Linear Programming

can be used.

‘Integer’ settings such as 0,1,2,3… allow discrete zero ormultiple selection of projects.

‘Binary’ settings with levels of 0 or 1 allow discrete zero orunitary 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 FormulationsMixed Integer Linear Programming can be usedto solve other complex investment problems by

careful specifications of the goals, andimaginative definitions of the constraints.

For example:Inter-Year Capital Transfers -- Introduce activities

for borrowing and capital transfers.

Contingent Projects -- introduce permissionconstraints which allow one activity to proceedonly if another is adopted.

Mutually Exclusive Projects -- introduce constraintsin which the total number of activities is below orequal to a maximum level.



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Other LP ApplicationsThreshold investment levels – the threshold

level is set up as a binary constraint.

Economies of Scale – particular scale levels areset up as independent activities with binaryconstraints.

Multiple Goals - each goal is set up as aconstraint goal, or each goal can be individuallyweighted in a total goal measure.

Risk Analysis – risky alternatives could beconstrained in the product mix; or an overall riskmeasure such as ‘variance could be targeted andminimized.



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Advanced LP Applications:

Summary

Linear programming can be used to solve selectionproblems from amongst competing investmentalternatives in the face of complex constraints.

These constraints mirror real world problems, andpresent a more realistic picture of actualinvestment behavior, than that assumed in baselevel LP analysis.

This higher level of analysis requires imaginativedefinitions of both goals and constraints, and anappreciation of Linear Programming methodology.

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