1. Ardavan Asef-Vaziri Systems and Operations Management
Introduction to
Linear
Programming
Ardavan Asef-Vaziri Systems and Operations Management

2. The Lego Production Problem
You have a set of legos
8 small bricks
6 large bricks
These are your “raw materials”.
You have to produce tables and chairs out of these legos. These are your “products”.

3. The Lego Production Problem
Weekly supply of raw materials:
8 Small Bricks
6 Large Bricks
Products:
Chair Table
Profit = 15 cents per Chair Profit = 20 cents per Table

4. Problem Formulation X1 is the number of Chairs
X2 is the number of Tables
Large brick constraint
X1+2X2  6
Small brick constraint
2X1+2X2  8
Objective function is to Maximize
15X1+20 X2
X1 ≥ 0
X2 ≥ 0

5. Linear Programming We can make Product1 and Product2.
There are 3 resources; Resource1, Resource2, Resource3.
Product1 needs one hour of Resource1, nothing of Resource2, and three hours of resource3.
Product2 needs nothing from Resource1, two hours of Resource2, and two hours of resource3.
Available hours of resources 1, 2, 3 are 4, 12, 18, respectively.
Contribution Margin of product 1 and Product2 are $300 and $500, respectively.
Formulate the Problem
Solve the problem using solver in excel

7. Problem Formulation Objective Function Z = 3 x1 +5 x2 Constraints
Resource 1
x  4
Resource 2
2x2  12
Resource 3
3 x1 + 2 x2  18
Nonnegativity
x1  0, x2  0

8. Feasible, Infeasible, and Optimal Solution
Given the following problem
Maximize Z = 3x1 + 5x2
Subject to: the following constraints
x ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≤ 18
x1, x2 ≥ 0
What combination of x1 and x2 could be the optimal solution?
A) x1 = 4, x2 = 4
B) x1 = -3, x2 = 6
C) x1 = 3, x2 = 4
D) x1 = 0, x2 = 7
E) x1 = 2, x2 = 6
Infeasible; Violates Constraint 3
Infeasible; Violates nonnegativity
Feasible; z = 3×3+ 5×4 = 29
Infeasible; Violates Constraint 2
Feasible; z = 3×2+ 5×6 = 36 and Optimal

9. Optimal Product Mix
The Omega Manufacturing Co. has discontinued the production of a certain non-profitable product line. This act created considerable excess capacity. Management is considering devoting this excess capacity to one or more of three products. The hours required from each resource for each unit of product, the available capacity (hours per week) of the the three resources, as well as the profit of each unit of product are given below.
Sales department indicates that the sales potentials for products 1 and 2 exceeds maximum production rate, but the sales potential for product 3 is 20 units per week.
Formulate the problem and solve it using excel

10. Practice (Page 304, Prob. 3)
An appliance manufacturer produces two models of microwave ovens: H and W. Both models require fabrication and assembly work: each H uses four hours fabrication and two hours of assembly, and each W uses two hours fabrication and six hours of assembly. There are 600 fabrication hours this week and 450 hours of assembly. Each H contributes $40 to profit, and each W contributes $30 to profit.
Formulate the problem as a Linear Programming problem.
Solve it using excel.
What are the final values?
What is the optimal value of the objective function?

11. Practice (Page 304, Prob. 4)
A small candy shop is preparing for the holyday season. The owner must decide how many how many bags of deluxe mix how many bags of standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3 pound raisins and 1/3 pounds peanuts, and the standard mix has 1/2 pound raisins and 1/2 pounds peanuts per bag. The shop has 90 pounds of raisins and 60 pounds of peanuts to work with. Peanuts cost $0.60 per pounds and raisins cost $1.50 per pound. The deluxe mix will sell for 2.90 per pound and the standard mix will sell for 2.55 per pound. The owner estimates that no more than 110 bags of one type can be sold.
Formulate the problem as a Linear Programming problem.
Solve it using excel.
What are the final values?
What is the optimal value of the objective function?

12. Assignment 6a:1 Due at the beginning of next class
The following table summarizes the key facts about two products, A and B, and the resources, Q, R, and S, required to produce them.
Resource Usage per Unit Produced
Resource
Product A
Product B
Amount of resource available Q 2 1 R S 3 4
Profit/Unit
$3000
$2000
Formulate the problem as a Linear Programming problem.
Solve it using excel.
What are the final values?
What is the optimal value of the objective function?

13. Assignment 6a:2 Due at the beginning of next class
The Apex Television Company has to decide on the number of 27” and 20” sets to be produced at one of its factories. Market research indicates that at most 40 of the 27” sets and 10 of the 20” sets can be sold per month. The maximum number of work-hours available is 500 per month. A 27” set requires 20 work-hours and a 20” set requires 10 work-hours. Each 27” set sold produces a profit of $120 and each 20” set produces a profit of $80. A wholesaler has agreed to purchase all the television sets produced if the numbers do not exceed the maximum indicated by the market research.
Formulate the problem as a Linear Programming problem.
Solve it using excel.
What are the final values?
What is the optimal value of the objective function?

14. Assignment 6a:3 Due at the beginning of next class
Ralph Edmund loves steaks and potatoes. Therefore, he has decided to go on a steady diet of only these two foods (plus some liquids and vitamins supplements) for all his meals. Ralph realizes that this isn’t the healthiest diet, so he wants to make sure that he eats the right quantities of the two foods to satisfy some key nutritional requirements. He has obtained the following nutritional and cost information:
Grams of Ingredient per Serving
Ingredient
Steak
Potatoes
Daily Requirements (grams)
Carbohydrates 5 15
≥ 50
Protein 20
≥ 40
Fat 2
≤ 60
Cost per serving $4 $2
Ralph wishes to determine the number of daily servings (may be fractional of steak and potatoes that will meet these requirements at a minimum cost.
Formulate the problem as a Linear Programming problem. Solve it using excel. What are the final values? What is the optimal value of the objective function?

15. Assignment 6a:4 Due at the beginning of next class
You are given the following linear programming model in algebraic form, where, X1 and X2 are the decision variables and Z is the value of the overall measure of performance.
Maximize Z = X1 +2 X2
Subject to
Constraints on resource 1: X1 + X2 ≤ 5 (amount available)
Constraints on resource 2: X1 + 3X2 ≤ 9 (amount available)
And
X1 , X2 ≥ 0

16. Assignment 6a:4 Due at the beginning of next class
Identify the objective function, the functional constraints, and the non-negativity constraints in this model.
Incorporate this model into a spreadsheet.
Is (X1 ,X2) = (3,1) a feasible solution?
Is (X1 ,X2) = (1,3) a feasible solution?
Use the Excel Solver to solve this model.

17. Assignment 6a:5 Due at the beginning of next class
You are given the following linear programming model in algebraic form, where, X1 and X2 are the decision variables and Z is the value of the overall measure of performance.
Maximize Z = 3X1 +2 X2
Subject to
Constraints on resource 1: 3X1 + X2 ≤ 9 (amount available)
Constraints on resource 2: X1 + 2X2 ≤ 8 (amount available)
And
X1 , X2 ≥ 0

18. Assignment 6a:5 Due at the beginning of next class
Identify the objective function, the functional constraints, and the non-negativity constraints in this model.
Incorporate this model into a spreadsheet.
Is (X1 ,X2) = (2,1) a feasible solution?
Is (X1 ,X2) = (2,3) a feasible solution?
Is (X1 ,X2) = (0,5) a feasible solution?
Use the Excel Solver to solve this model.

19. Example A Production System Manufacturing Two Products, P and Q
$90 / unit
$100 / unit P: Q:
110 units / week
60 units / week D D
Purchased Part
10 min.
5 min.
$5 / unit C B C
10 min.
5 min.
25 min. B A A
15 min.
10 min.
10 min.
RM1
RM2
RM3
$20 per
$20 per
$25 per
unit
unit
unit
Time available at each work center: 2,400 minutes per week
Operating expenses per week: $6,000