SPRING TRIMESTER EXAMINATION - 2019/2020
MATH2003J Optimisation in Economics
1. (a) Determine whether each of the following statements is True or False. No explanation is needed when answering 1(a)(i) to 1(a)(v).
(i) Let p be a critical point of a twice diferentiable function f : Rn Y R. If det(H(p)) = 0, where H(p) denotes the Hessian matrix of f, then f has a saddle point at p. [1]
(ii) The following is a linear programming problem:
Maximize z = 2x1 - x2 subject to
x1 +2x1x2 ≥ -10
3x1(2) +5x2 ≤ 18
x1, x2 ≥ 0 [1]
(iii) The set S = [2, 5] n {8} is convex. [1]
(iv) If we solve the non-linear programming problem of maximizing f(x, y, z) subject to two diferent constraints g1 (x, y, z) ≤ c1 and g2 (x, y, z) ≤ c2 using the Kuhn-Tucker method, we have to consider 4 diferent cases. [1]
(v) Let S ⊆ Rn be a convex set. It is possible for a function f : S Y R to be neither concave nor convex on S. [1]
(b) Determine whether each of the following statements is True or False.
Briefly justify your answers to questions 1(b)(i) to 1(b)(v).
(i) There is no solution to the optimization problem:
Maximize z = f(x, y) subject to
x +2y ≤ 1
2x +3y ≥ 6
x,y ≥ 0. [3]
(ii) The set S = {(x, y) e R2 : 4x2 +9y2 ≤ 36, x ≤ 0} is bounded. [3]
(iii) Let S = {(x, y) e R2 : g1 (x, y) ≤ c1 , g2 (x, y) ≤ c2 , g3 (x, y) ≤ c3 } be a set in R2 where g1 (x, y) ≤ c1 , g2 (x, y) ≤ c2 and g3 (x, y) ≤ c3 are linear inequalities. Then S is always bounded. [3]
(iv) Let f : S Y R be defined by
f(x, y) = 2y - 5x2 - 4xy - y2
where S = {(x, y) e R2 : x2 + y2 ≤ 100} is a convex set. Then f is convex on S. [3]
(v) This question refers to 1(b)(iv). The function f has a maximum at (-2, 5) on S. [3]
2. In this question, please substitute T by your UCD Student ID. For example, if your UCD Student ID is 12345678, then T = 12345678.
(a) Let f : R3 → R be the function
f(x, y, z) = x2 − 4x + xy2 + yz2 + z2 .
(i) Determine whether (T, 0, , − 1, √3) are critical point(s) off. [4]
(ii) Classify the nature of the critical point(s) obtained in 2(a)(i). [9]
O(b)Use the graphical method to maximize f(x, y) = 2x +3y subject to
x + y ≤ 10
2x + y ≤ 18
x +2y ≤ 16
x, y ≥ 0. [7]
3. Solve the following linear programming problem by the simplex method:
Minimize z = −5x1 − 2x2 subject to
2x1 + x2 ≤ 21
x1 +2x2 ≥ 18
x1 , x2 ≥ 0. [20]
4. (a) Consider the following linear programming problem: Maximize z = 8x1 +7x2 subject to
x1 + x2 ≤ 10
2x1 + x2 ≤ 18
x1 , x2 ≥ 0.
(i) Solve the above problem with the simplex method.
(ii) Formulate the dual problem.
(iii) Determine the optimal solution to the dual problem from tableau of the original problem.
(b) A shampoo company manufactures shampoo A and shampoo B. Each litre of shampoo A sells for €8 and each litre of shampoo B sells for €7. The production and sales of both items involve labour and herbs. The constraints of each of these resources is illustrated in the table below.
|
Resource
|
Shampoo A
|
Shampoo B
|
Total availability
|
|
Labour
Herbs
|
1 time unit
2 units
|
1 time unit
1 unit
|
10 time units
18 units
|
The owner of the company wish to determine the combination of production of shampoo A and shampoo B that will maximize the company’s revenue.
(i) Let t1 and t2 represents the working time of labour and amount of herbs respectively. Determine the values of t1 and t2 required to maximize the company’s revenue. [2]
(ii) By using the results in 4(a), or otherwise, find the shadow price of the labour per time unit and the shadow price of the herbs per unit. [2]
5. Let f : R2 → R be defined by f(x, y) = 4x2 − y2 + 5 and consider the constraints x2 + y2 ≤ 4 and x ≤ 1.
(a) Sketch the feasible set in the plane and explain why f attains extrema
(maximum and minimum) subject to the above constraints. [4]
(b) Use the Kuhn-Tucker method to find the maximum and the minimum of f subject to the above constraints. [16]