Question 1. Modify and solve the capital budgeting model to account for the following additional restrictions:
(a) Project 5 must be selected if either project 1 or project 3 is selected.
(b) Projects 2 and 3 are mutually exclusive.
Question 2. Five items are to be loaded in a vessel. The weight 24, volume and value ri for item are tabulated below.
| Item i |
Unit weight wi (tons) |
Unit volume, wi |
Unit worth ri($100)
|
| 1 |
5 |
1 |
4 |
| 2 |
8 |
8 |
7 |
| 3 |
3 |
6 |
6 |
| 4 |
2 |
5 |
5 |
| 5 |
7 |
4 |
4 |
The maximum allowable cargo weight and volume are 112 tons and 109 yd3, respectively. Formulate the ILP model, and find the most valuable cargo.
Question 3. Suppose that you have 7 full wine bottles, 7 half-full, and 7 empty. You would like to divide the 21 bottles among three individuals so that each will receive exactly 7. Additionally, each individual must receive the same quantity of wine. Express the problem as ILP constraints, and find a solution. (Hint: Use a dummy objective function in which all the objective coefficients are zeros.)
Question 4. An eccentric sheikh left a will to distribute a herd of camels among his three children: Tarek receives at least one-half of the herd, Sharif gets at least one third, and Maisa gets at least one-ninth. The remainder goes to charity. The will does not specify the size of the herd except to say that it is an odd number of camels and that the named charity receives exactly one camel. Use ILP to determine how many camels the sheikh left in the estate and how many each child got.
Question 5. The three children of a farm couple are sent to the market to sell 90 apples. Karen, the oldest, carries 50 apples; Bill, the middle one, carries 30; and John, the youngest, carries only 10.The parents have stipulated five rules:
(a) The selling price is either $1 for 7 apples or $3 for 1 apple, or a combination of the two prices.
(b) Each child may exercise one or both options of the selling price.
(c) Each of the three children must return with exactly the same amount of money.
(d) Each child's income must be in whole dollars (no cents allowed).