Project on Operational research about investment under uncertainty
Module Learning Outcome/s Covered:
LO1: be able to apply a variety of more advanced OR methods
LO3: be able to use mathematical programming software and interpret the computed output
Project Description
The purpose of this project is to develop an analytical framework for decision-making under uncertainty. More specifically, assume that a firm is contemplating investment in a project, a power plant, facing price uncertainty. The project has the following characteristics:
1. There is a variable operating cost, c in £/unit. Hence, if the project has installed capacity Q, then the operating cost per unit of time is cQ.
2. There is an investment cost, I in £ that depends on the size of the project, denoted by Q. The investment cost function can be expressed as I(Q) = aQb, where a and b are positive constants.
3. The output price, Pt, is stochastic and follows a geometric Brownian motion, which is indicated below.
dPt = µPtdt + σPtdZt, P0 ≡ P > 0
4. The firm can choose not only the time of investment but also the size of the project.
5. The project has operational flexibility, i.e., the firm can abandon the project.
The objective is to determine the optimal investment policy and analyse how it is affected by different forms of managerial discretion. More specifically, how is the decision to invest affected by the firm's option to abandon operations or choose the size of the project?
For the successful completion of this coursework you are required to:
i. apply a variety of more advanced OR methods, e.g., dynamic programming.
ii. be aware of problems and experienced in implementing OR solutions.
iii. be able to use mathematical programming software and interpret the computed output.
1 Introduction
In this section, you should provide real-world evidence on why this problem is relevant to the industry. Evidence on this should be obtained from both the popular press, e.g. NY Times, The Economics, Wall Street Journal, etc., and academic articles.
2 Literature Review
Here, you should discuss thoroughly the novelties of this project and comment on why it may reflect a meaningful contribution for the industry and the academic literature. Therefore, you should conduct a literature review with which you will identify what has already been done in the area of investment and capacity sizing under price uncertainty.
The literature review should begin with early work first before discussing more recent find- ings. Each work that is discussed must be cited at the end of the document and the list of citations should be in alphabetic order. Check the end of the document to see an example of how the style of citations should be.
3 Assumptions
In this section, you should introduce the assumptions of the model (some of which are already introduced in the first page). Additionally, any notation that is used in the analytical part of the project must also be introduced here.
4 Model
This is the most important part of the project and consists of 4 problems. Each problem should be addressed in a different subsection and each new subsection should indicate clearly how it extends the previous one.
1. Begin with the most simple scenario in which the firm has only discretion over the time of investment and not the size of the project, and faces only price uncertainty, as this is reflected in (1). Derive analytical expressions for the value of the active project, the value of the option to invest, and the optimal investment threshold.
2. Since the project has an operating cost, c, it may be optimal to terminate it at some point if the price drops below c. Hence, in a different subsection, you should introduce operational flexibility in the form of an embedded abandonment option. The objec- tive is to determine how this option impacts the firm's decision to invest.
3. Next, assume that the firm can choose not only when to invest but also how big the project should be. Hence, continue in a different subsection and allow for discretion over capacity. Following the same approach as in the previous steps, derive the expected value of the active project, the value of the option to invest, the optimal investment threshold, and the optimal capacity of the project.
4. Like in step 2, you should extend the analysis in step 3 by allowing for an embedded option to abandon the project should the output price drop below the operating cost. The objective here is to determine the impact of the embedded abandonment option on both the optimal investment threshold and the corresponding optimal capacity.
5 Numerical Results
In this section, you should provide numerical results for each of the cases that were analysed in Section 4. The results should be presented in the form of graphs. Before each graph, you should provide a thorough discussion of the intuitive interpretation of the numerical results as well as insights for investors and policy-makers.
6 Appendix
In this section, you should provide all the analytical derivations of the formulas presented in the analytical part of the paper. You should not include these derivations in Section 4.