MATH2003J, OPTIMIZATION IN ECONOMICS,
BDIC 2022/2023, SPRING
Problem Sheet 2
Question 1:
For the following LP problems sketch the feasible sets, find all corner points and use the graphical method to solve them:
(a) max. 3x1 + 4x2 subject to 3x1 + 2x2 ≤ 6, x1 + 4x2 ≤ 4, x1 ≥ 0, x2 ≥ 0.
(b) max. 2x1 + 5x2 subject to −2x1 + 3x2 ≤ 6, 7x1 − 2x2 ≤ 14, x1 + x2 ≤ 5, x1 ≥ 0, x2 ≥ 0.
Θ (c) max. 8x1 + 9x2 subject to x1 + 2x2 ≤ 8, 2x1 + 3x2 ≤ 13, x1 + x2 ≤ 6, x1 ≥ 0, x2 ≥ 0.
Θ Question 2:
Let λ > 0 be a constant. Sketch the feasible set F determined by the inequalities: x1 + x2 ≤ λ , x1 ≥ 0, x2 ≥ 0. Also maximize and minimize the function f(x1 , x2 ) = 2x1 − 2x2 on the feasible set F.
Question 3:
The region B consists of all (x1 , x2 ) satisfying x1 − 2x2 ≤ 2, 2x1 + x2 ≤ 8, x1 ≥ 0, x2 ≥ 0. Solve the following problems with B as the feasible region:
Θ (a) max. 3x1
(b) max. x2
Θ (c) max. 2x1 + 3x2
(d) min. −2x1 + 2x2
Θ (e) min. −2x1 − 3x2
Θ Question 4:
A doctor wants to design a breakfast menu for his patients. The menu is to include two items A and B which provide a number of units of vitamin C and iron as given in the table below
|
item
|
units of vitamin C (per gram)
|
units of zinc (per gram)
|
Cost per gram
|
|
A B
|
2
1
|
2
2
|
$4
$3
|
The breakfast menu must provide at least 8 units of vitamin C and 10 units of zinc.
(a) How many grams of each item should be provided in order to meet the zinc and vitamin C requirements for the minimum cost? What will this breakfast cost?
(b) Now suppose the cost of Item B rises to $4 per gram. What is the new minimum cost of the breakfast? Describe all the possible set of breakfasts which achieve this minimum cost.