A typical large oil and gas company operates many exploration and production projects, which cost several billion dollars every year. Each year the company is faced with the problem of where best to invest their capital and which combination of projects should be selected from several possible project mixes. They have the difficult task of portfolio selection from a large number of competing projects for immediate or future operation under a limited amount of capital to invest.

Consider a small firm with 4 competing oil production projects. Table 1 presents the production, pollution cost, and the net present value for these projects. They contact you to help select the best combination of projects under a certain amount of capital investment, while fulfilling the firm's goals.

(a) Formulate an integer programme to maximise the net present value (NPV) subject to a pollution cost limit stating the firm can spend no more than 32 million (Mf) on its pollution and a production level stating the firm must produce at least 73 million barrels (Mbbl). Your integer programme should include the definition of all variables and constraints.

Table 1. Project specifications Project - NPV (M) - Pollution cost (Mf) - Production (Mbbl) A - 25 - 11 - 28

B - 20 - 9 - 20

C - 19 - 14 - 25 D - 28 - 17 - 30

Solution: We define four binary variables xA,xB,xC,xD as follows:

xA := (1 if Project A is selected, 0 otherwise.)

xB := (1 if Project B is selected, 0 otherwise.)?

xC := (1 if Project C is selected, 0 otherwise.)

xD := (1 if Project D is selected, 0 otherwise.)

max: 25xA + 20xB + 19xC + 28xD s.t. 11xA +9xB +14xC +17xD <=32,

28xA +20xB +25xC +30xD>= 73,

xA,xB,xC,xD {0,1}.

(b) Suppose that there are the following additional constraints: (i) Spending on Project A or B prevents you from spending on Project D. (maximum two constraints)

(ii)If production level is assured, the pollution cost should not be considered.

Extend your integer program to satisfy these constraints.

(c) Now suppose that the oil company would like to examine the possibility of expanding oil production. They need to find the minimum of the following fixed-cost function additionally incurred: ''The possible expansion will cost the company 20 M as a fixed cost and oil production will cost of 10 /barrel. If the company decides not to produce any oil itself, there won't be any expansion cost." Formulate a new integer programme, with new decision variables, to only determine the minimum cost of the new expansion.