More Advanced Linear Programming Concepts and Methods pdf

Recognizing threshold investments, economies of scale, multiple goals and investment risk...  More Activities and Constraints: this notion deals with more complex resource mixes, and

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.

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

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.

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.

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.

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.

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

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

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

Formatted Problem and LP Solution For Power Gen Inc.

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) +

Total capital outlay for this selection is $M700

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.

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’

Setting Integer and Binary

Constraints.

‘Binary’ constraint selected via the Solver ‘sign’ dialog box.

‘Integer’ constraint selected via the Solver ‘sign’ dialog box.

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

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.

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.