Supply chain management
IMSE2008 Operational Research Techniques
Problem Set: Linear Programming
Due: Monday, October 20, 2014
Please submit to the general office of the IMSE department (Ms Kate Lee HW8-17) on Oct 20
Late submission will be discounted by 20% on a daily basis
Please e-mail me msong@hku.hk if you have any questions
Problem 1. Work through the simplex method step by step to demonstrate that the following problem is unbounded. (5 marks) max 5x1 + x2 + 3x3 + 4x4
s.t. x1 – 2x2 + 4x3 + 3x4 ≤ 20 –4x1 + 6x2 + 5x3 – 4x4 ≤ 40 2x1 – 3x2 + 3x3 + 8x4 ≤ 50 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
Problem 2. Work through the simplex method step by step to find all optimal basic feasible solutions for the following problem. (5 marks) max x1 + x2 + x3 + x4
s.t. x1 + x2 ≤ 3 x3 + x4 ≤ 2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
Problem 3. Use the Big M method to solve the following problem. (5 marks) min 3x1 + 8x2 + 5x3
s.t. 3x2 + 4x3 ≥ 70 3x1 + 5x2 + 2x3 ≥ 70 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
Problem 4. The diet problem (15 marks)
Three sellers offer three foods, which contain three different nutrients. We are given the following table with the nutritional content of a unit of each food, the minimum requirement of each nutrient, and the selling price for a unit of each food.
Burger
Fried Rice
Pizza
Minimum Requirement
Carbohydrates
4
3
3
12
Fat
3
2
3
8
Protein
4
4
2
20
Unit Price
5
4
3
Design a diet so as to satisfy the minimum requirement of the nutrients and minimize the total price to purchase the foods in the diet.
(a) Formulate the diet problem as a linear programming model.
(b) Solve the LP by Excel and obtain the sensitivity analysis report.
(c) The seller of fried rice would like to increase his revenue by increasing the price of fried rice quoted to us. Is it possible for him to do so? If it is possible, how much more revenue can he obtain? If not, what should the seller do to
Problem Set: Linear Programming
Due: Monday, October 20, 2014
Please submit to the general office of the IMSE department (Ms Kate Lee HW8-17) on Oct 20
Late submission will be discounted by 20% on a daily basis
Please e-mail me msong@hku.hk if you have any questions
Problem 1. Work through the simplex method step by step to demonstrate that the following problem is unbounded. (5 marks) max 5x1 + x2 + 3x3 + 4x4
s.t. x1 – 2x2 + 4x3 + 3x4 ≤ 20 –4x1 + 6x2 + 5x3 – 4x4 ≤ 40 2x1 – 3x2 + 3x3 + 8x4 ≤ 50 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
Problem 2. Work through the simplex method step by step to find all optimal basic feasible solutions for the following problem. (5 marks) max x1 + x2 + x3 + x4
s.t. x1 + x2 ≤ 3 x3 + x4 ≤ 2 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
Problem 3. Use the Big M method to solve the following problem. (5 marks) min 3x1 + 8x2 + 5x3
s.t. 3x2 + 4x3 ≥ 70 3x1 + 5x2 + 2x3 ≥ 70 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
Problem 4. The diet problem (15 marks)
Three sellers offer three foods, which contain three different nutrients. We are given the following table with the nutritional content of a unit of each food, the minimum requirement of each nutrient, and the selling price for a unit of each food.
Burger
Fried Rice
Pizza
Minimum Requirement
Carbohydrates
4
3
3
12
Fat
3
2
3
8
Protein
4
4
2
20
Unit Price
5
4
3
Design a diet so as to satisfy the minimum requirement of the nutrients and minimize the total price to purchase the foods in the diet.
(a) Formulate the diet problem as a linear programming model.
(b) Solve the LP by Excel and obtain the sensitivity analysis report.
(c) The seller of fried rice would like to increase his revenue by increasing the price of fried rice quoted to us. Is it possible for him to do so? If it is possible, how much more revenue can he obtain? If not, what should the seller do to