1. As the leader of an oil-exploration drilling venture, you must determine the leastcost selection of 5 out of 10 possible sites. Label the sites S 1
,S 2
,…,S 10
, and the exploration costs associated with each as C 1
,C 2
,…,C 10
. Regional development restrictions are such that: (a) Exploring sites S 1
and S 7
will prevent you from exploring site S 8
; (b) Evaluating site S 3
or S 4
prevents you from exploring site S 5
; and (c) Of the group S 5
,S 6
,S 7
,S 8
, only two sites may be explored. Formulate this problem as an integer programming problem.
Let's define binary decision variables for each site, such that:
X1 = 1 if site S1 is selected, 0 otherwise
X2 = 1 if site S2 is selected, 0 otherwise
...
X10 = 1 if site S10 is selected, 0 otherwise
The objective is to minimize the total exploration cost, which can be formulated as:
Minimize: C1*X1 + C2*X2 + ... + C10*X10
Subject to the following constraints:
1. Exploring sites S1 and S7 prevents exploring site S8:
X1 + X7 <= 1
2. Evaluating site S3 or S4 prevents exploring site S5:
X3 + X4 <= 1
3. Only two sites from the group S5, S6, S7, S8 can be explored:
X5 + X6 + X7 + X8 <= 2
4. The number of selected sites must be 5:
X1 + X2 + ... + X10 = 5
Also, it is important to note that X1, X2, ..., X10 should be binary variables.
To formulate this problem as an integer programming problem, we can use binary decision variables for each site. Let's denote the decision variable for each site i as x_i, where x_i takes a value of 1 if site i is selected and 0 otherwise.
We want to select 5 out of the 10 possible sites. So, the objective function for this problem is to minimize the total cost of the selected sites:
Minimize: C_1*x_1 + C_2*x_2 + ... + C_10*x_10
Subject to the following constraints:
1. Exploring sites S1 and S7 prevents exploring site S8:
x_1 + x_7 <= 1
2. Evaluating site S3 or S4 prevents exploring site S5:
x_3 + x_4 <= 1
3. Only two out of the group S5, S6, S7, S8 may be explored:
x_5 + x_6 + x_7 + x_8 <= 2
4. Total number of selected sites should be 5:
x_1 + x_2 + ... + x_10 = 5
5. Binary constraints on decision variables:
x_1, x_2, ..., x_10 binary (0 or 1)
This formulation ensures that the given conditions are satisfied while minimizing the cost of selected sites.