Ch 12: More Advanced Linear Programming Concepts and Methods

1. 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.

2. Simple Application of L. P. • In Chapter 11, Linear Programming was applied to those investments satisfying the following assumptions: • Additivity within activities: resource consumption is constant per unit of output; there are no economies of scale. • Divisibility within activities: partial investments can be implemented. There is no requirement to accept equipment in discrete sizes. • Independence of activities: there is no recognition of productive or financial interdependencies.

3. 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: • Allowing more activities and constraints • Recognizing indivisible investments • Allowing inter-year resource borrowings and transfers • Recognizing interdependent projects • Treating mutually exclusive investments • Recognizing threshold investments, economies of scale, multiple goals and investment risk.

4. 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.

5. 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.

6. 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.

7. 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:-

8. 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.

9. The LP Solution For ‘Power Gen Inc’.

10. 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 capital outlay for this selection is $M700 Total NPV from this selection is $M356. Calculated as: (0.70 x $180) + (1 x $100) + ( 1 x $80) + (1 x $50)

11. 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. Capital outlay at $M700 is equal to the maximum allowed of $M700. All projects were artificially constrained at a maximum of 1 unit, so that more than one project of any technology could not bechosen. This constraint has been satisfied.

12. 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 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.

13. Setting Integer and Binary Constraints. ‘Integer’ constraint selected via the Solver ‘sign’ dialog box. ‘Binary’ constraint selected via the Solver ‘sign’ dialog box.

14. 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.

15. 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.

16. 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.