Chapter 6 Supplement

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 lt 12 (constraint 1)
• 4X1 3X2 lt 24 (constraint 2)
• X1 gt 0
• X2 gt 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
X2
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 lt 500 (Milling)
• 5X1 4X2 lt 350 (Lathe)
• 3X1 2X3 lt 150 (Drill)
• X3 lt 20 (Sales Potential)
• X1 gt 0, X2 gt 0, X3 gt 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 1742.857
• Variable Value Reduced Cost
• X1 26.19048 0.0000000
• X2 54.76190 0.0000000
• X3 20.00000 0.0000000
• Row Slack or Surplus Dual Price
• PROFIT 1742.857 1.000000
• MILLING 0.0000000 2.857143
• LATHE 0.0000000 0.8571429
• DRILL 31.42857 0.0000000
• SALESPOT 0.0000000 0.7142857

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
• X1 30.00000 0.7500000
  15.00000
• X2 12.00000 12.00000
  0.6000000
• X3 15.000 INFINITY
  0.7142857
• Righthand Side Ranges
• Row Current Allowable
  Allowable
• RHS Increase
  Decrease
• MILLING 500.0000 55.00000 137.5000
• LATHE 350.0000 183.3333
  73.33334
• DRILL 150.0000 INFINITY
  31.42857
• SALESPOT 20.00000 27.50000 20.00000

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 gt 200 (Carbo)
• 30X1 80X2 60X3 gt 180 (Protein)
• 10X1 20X2 60X3 gt 150 (Vitamin)
• X1 gt 0, X2 gt 0, X3 gt 0

26
Linear Programming Problem 2 Solution Using
LINGO Software

• Objective value 60.42857
• Variable Value Reduced Cost
• X1 1.142857 0.0000000
• X2 0.0000000 4.428571
• X3 2.428571 0.0000000
• Row Slack or Surplus Dual Price
• COST 60.42857 1.000000
• CARBOHY 0.0000000 -0.1928571
• PROTEIN 0.0000000 -0.1214286
• VITAMIN 7.142857 0.0000000

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
• X1 21.00000 12.75000
  9.299998
• X2 18.00000 INFINITY
  4.428571
• X3 15.00000 2.818181
  5.666667
• Righthand Side Ranges
• Row Current Allowable
  Allowable
• RHS Increase
  Decrease
• CARBOHY 200.0000 25.00000 80.00000
• PROTEIN 180.0000 120.0000
  6.000000
• VITAMIN 150.0000 7.142857
  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)