1. Chapter 6 Supplement
Linear Programming

2. Linear Programming
Linear Programming (LP) deals with the problems of allocating limited resources among competing activities in the best possible way (optimal)
A linear program consist of a linear objective function and a set of linear constraints

3. Linear Programming Model
Objective: the goal of an LP model is maximization or minimization
Decision variables: amounts of either inputs or outputs
Constraints: limitations that restrict the available alternatives
Parameters: numerical values

4. Linear Programming Assumptions
Linearity: the impact of decision variables is linear in constraints and objective function
Divisibility: noninteger values of decision variables are acceptable
Certainty: values of parameters are known and constant
Nonnegativity: negative values of decision variables are unacceptable

5. Linear Programming Application Procedure
Parameter Estimation
Problem Formulation
Optimal Solution
Graphical Method
Simplex Method
Computer Solution
Other Methods
Sensitivity Analysis

6. Linear Programming Application Areas
Production
Inventory
Financial
Marketing
Distribution
Sports
Agriculture

7. Linear Programming: Some Definitions
Solution: A solution is a set of values of the decision variables
Feasible Solution: A feasible solution is a solution for which all the constraints are satisfied
Optimal Solution: An optimal solution is a feasible solution which optimizes the objective function

8. Linear Programming: Types of Solutions
Single Optimal Solution
Multiple Optimal Solutions
No Optimal Solution

9. Graphical Linear Programming
Set up objective function and constraints in mathematical format
Plot the constraints
Identify the feasible solution space
Plot the objective function
Determine the optimum solution

10. Graphical Linear Programming
Maximize Z = 4X1 + 5X2
Subject to
X1 + 3X2 < 12 (constraint 1)
4X1 + 3X2 < 24 (constraint 2)
X1 > 0
X2 > 0

11. Linear Programming Example
Plot
Constraint 1
X1 + 3X2 = 12

12. Linear Programming Example
Add
Constraint 2
4X1 + 3X2 = 24
Constraint 1
X1 + 3X2 = 12
Solution space

13. Linear Programming Example
Z = 60
Z = 40
Z = 20 X1

14. LP Formulation and Computer Solution: Problem 1

15. Linear Programming Problem 1: Formulation
Let Xi be the number of units of product type i to be produced per week, i = 1, 2, 3
Maximize Z = 30X1 + 12X2 + 15X3
Subject to
9X1 + 3X2+ 5X3 < 500 (Milling)
5X1 + 4X < 350 (Lathe)
3X X3 < 150 (Drill)
X3 < 20 (Sales Potential)
X1 > 0, X2 > 0, X3 > 0

16. Slack and Surplus
Binding constraint: a constraint that forms the optimal corner point of the feasible solution space
Slack: when the optimal values of decision variables are substituted into a less than or equal to constraint and the resulting value is less than the right side value
Surplus: when the optimal values of decision variables are substituted into a greater than or equal to constraint and the resulting value exceeds the right side value

17. Linear Programming Problem 1: Solution Using LINGO Software
Objective value:
Variable Value Reduced Cost X X X
Row Slack or Surplus Dual Price
PROFIT
MILLING
LATHE
DRILL
SALESPOT

18. Sensitivity Analysis
Range of optimality: the range of values for which the solution quantities of the decision variables remains the same
Range of feasibility: the range of values for the fight-hand side of a constraint over which the shadow price (dual price) remains the same
Shadow prices: negative values indicating how much a one-unit decrease in the original amount of a constraint would decrease the final value of the objective function

19. Linear Programming Problem 1: Solution Using LINGO Software
Ranges in which the basis is unchanged:
Objective Coefficient Ranges
Current Allowable Allowable
Variable Coefficient Increase Decrease X X
X INFINITY
Righthand Side Ranges
Row Current Allowable Allowable
RHS Increase Decrease
MILLING
LATHE
DRILL INFINITY
SALESPOT

20. Linear Programming Problem 1: Solution Using EXCEL (a)

21. Linear Programming Problem 1: Solution Using EXCEL Software (b)

22. Linear Programming Problem 1: Solution Using EXCEL Software (c)

23. Linear Programming Problem 1: Solution Using EXCEL Software (d)

24. LP Formulation And Computer Solution: Problem 2

25. Linear Programming Problem 2: Formulation
Let X1 X2 X3 be the kilograms of corn, tankage, and alfalfa, respectively.
Minimize Z = 21X1 + 18X2 + 15X3
Subject to
90X1 + 20X2+ 40X3 > 200 (Carbo)
30X1 + 80X2 + 60X3 > 180 (Protein)
10X1 + 20X2 + 60X3 > 150 (Vitamin)
X1 > 0, X2 > 0, X3 > 0

26. Linear Programming Problem 2: Solution Using LINGO Software
Objective value:
Variable Value Reduced Cost X X X
Row Slack or Surplus Dual Price
COST
CARBOHY
PROTEIN
VITAMIN

27. Linear Programming Problem 2: Solution Using LINGO Software
Ranges in which the basis is unchanged:
Objective Coefficient Ranges
Current Allowable Allowable
Variable Coefficient Increase Decrease X
X INFINITY X
Righthand Side Ranges
Row Current Allowable Allowable
RHS Increase Decrease
CARBOHY
PROTEIN
VITAMIN INFINITY

28. Linear Programming Problem 2: Solution Using EXCEL Software (a)

29. Linear Programming Problem 2: Solution Using EXCEL Software (b)

30. Linear Programming Problem 2: Solution Using EXCEL Software (c)

31. Linear Programming Problem 2: Solution Using EXCEL Software (d)