introducing duality and sensitivity analysis
DESCRIPTION
Introducing Duality and Sensitivity Analysis. Merton Trucks. Optimal Product Mix: 2000 Model 101s and 1000 Model 102s Optimal Contribution: $11,000,000. How much is Engine Assembly capacity worth to Merton Trucks?. Merton Trucks (Scaled). unit = 1 hr. unit = 2 hr. unit = 2 hr. unit = 3 hr. - PowerPoint PPT PresentationTRANSCRIPT
Introducing Duality and Sensitivity Analysis
Merton Trucks
Model 101 Model 102 Availability
Contribution $3000 $5000
Eng. Assy. 1 hr 2 hr 4000 hr
Metal Stmp. 2 hr 2 hr 6000 hr
101 Assy. 2 hr 5000 hr
102 Assy 3 hr 4500 hr
Optimal Product Mix: 2000 Model 101s and 1000 Model 102sOptimal Contribution: $11,000,000
How much is Engine Assembly capacity worth to Merton Trucks?
Merton Trucks (Scaled)
Model 101 Model 102 Availability
Contribution $3000 $5000
Eng. Assy. 1/4000 unit 1/2000 unit 1 unit
Metal Stmp. 1/3000 unit 1/3000 unit 1 unit
101 Assy. 1/2500 unit 1 unit
102 Assy 1/1500 unit 1 unit
Optimal Product Mix: 2000 Model 101s and 1000 Model 102sOptimal Contribution: $11,000,000
How much is Engine Assembly capacity worth to Merton Trucks?
unit = 1 hr
unit = 2 hr
unit = 2 hr
unit = 3 hr
Worth of Engine Capacity
% Increase ContributionWorth
(Full Capacity)
Original $11,000,000
10% increase $11,800,000 $8,000,000
5% increase $11,400,000 $8,000,000
1% increase $11,080,000 $8,000,000
0.5% increase $11,040,000 $8,000,000
Suppose now, that the engine assembly capacityincreases to 4400 hours
Worth of Engine Capacity
% Increase ContributionWorth
(Full Capacity)
Original $11,800,000
Engine Assembly Capacity is now 4400 hours
Worth of Engine Capacity
% Increase ContributionWorth
(Full Capacity)
Original $11,800,000
10% increase $12,000,000 $2,000,000
Engine Assembly Capacity is now 4400 hours
Worth of Engine Capacity
% Increase ContributionWorth
(Full Capacity)
Original $11,800,000
10% increase $12,000,000 $2,000,000
5% increase $12,000,000 $4,000,000
Engine Assembly Capacity is now 4400 hours
Worth of Engine Capacity
% Increase ContributionWorth
(Full Capacity)
Original $11,800,000
10% increase $12,000,000 $2,000,000
5% increase $12,000,000 $4,000,000
1% increase $11,888,000 $8,800,000
Engine Assembly Capacity is now 4400 hours
Worth of Engine Capacity
% Increase ContributionWorth
(Full Capacity)
Original $11,800,000
10% increase $12,000,000 $2,000,000
5% increase $12,000,000 $4,000,000
1% increase $11,888,000 $8,800,000
0.5% increase $11,844,000 $8,800,000
Engine Assembly Capacity is now 4400 hours
Merton TrucksBase Engine Assembly capacity = 4000 hrs
Engine Assembly capacity = 4000 hrs
Merton Trucks
Engine Assembly capacity ↑ by 1%
Base Engine Assembly capacity = 4000 hrs
Merton Trucks
Engine Assembly capacity ↑ by 5%
Base Engine Assembly capacity = 4000 hrs
Merton Trucks
Engine Assembly capacity ↑ by 10%
Base Engine Assembly capacity = 4000 hrs
Merton Trucks (new)
Engine Assembly capacity = 4400 hrs
Base Engine Assembly capacity = 4400 hrs
Merton Trucks (new)
Engine Assembly capacity ↑ by 1%
Base Engine Assembly capacity = 4400 hrs
Merton Trucks (new)
Engine Assembly capacity ↑ by 5%
Base Engine Assembly capacity = 4400 hrs
Merton Trucks (new)
Engine Assembly capacity ↑ by 10%
Base Engine Assembly capacity = 4400 hrs
Forming the dual
Input (Primal):– Maximization objective– Non-negative decision variables– “Less than or equal to” type constraints
Output (Dual):– Minimization objective– One dual variable for each primal constraint– Non-negative dual variables– “Greater than or equal to” type constraints– One constraint for each primal variable
Some Laws
• The objective function values of optimal solutions to primal and dual problems are equal.
• If there is excess of a resource at an optimal solution, then its shadow price is zero.
• If the shadow price of a resource is positive, then it has got completely used up at an optimal solution.
• It is possible for a resource to get completely used up at an optimal solution and still have a zero shadow price.
Reduced Costs
The reduced cost of a coefficient of a decision variable in the objective function is the minimum amount by which the coefficient should be reduced in order that the decision variable achieves a non-zero level in an optimal solution.
• The reduced cost for a decision variable already at non-zero value in an optimal solution is ZERO.
• For a minimization problem, reduced costs are either ZERO or POSITIVE.
• For a maximization problem, reduced costs are either ZERO or NEGATIVE.
• A decision variable at zero level can have a reduced cost of zero.
Sensitivity Analysis
Sensitivity analysis tells us what changes are possible in the coefficients of a linear programming model without changing the optimal basis.
• We are concerned with changing only one coefficient and keeping all others fixed.
• We are bothered only about the set of constraints that define the optimal solution – they should not change. Otherwise, the optimal solution can change, the objective function value can change.
Sensitivity Analysis
Original Model
OptimumValue = 11 MillionModel_101 = 2000Model_102 = 1000
Sensitivity Analysis
Objective function coefficient increasesZ = 3000 Model_101 + 5000 Model_102 to Z = 4000 Model_101 + 5000 Model_102
OptimumValue = 13 MillionModel_101 = 2000Model_102 = 1000
Sensitivity Analysis
Objective function coefficient increasesZ = 3000 Model_101 + 5000 Model_102 to Z = 6000 Model_101 + 5000 Model_102
OptimumValue = ???Model_101 = 2000Model_102 = 1000
Sensitivity Analysis
Objective function coefficient increasesZ = 3000 Model_101 + 5000 Model_102 to Z = 6000 Model_101 + 5000 Model_102
OptimumValue = 17.5 MillionModel_101 = 2000Model_102 = 1000
Sensitivity Analysis
Original Model
OptimumValue = 11 MillionModel_101 = 2000Model_102 = 1000
Sensitivity Analysis
Objective function coefficient decreasesZ = 3000 Model_101 + 5000 Model_102 to Z = 2750 Model_101 + 5000 Model_102
OptimumValue = 10.5 MillionModel_101 = 2000Model_102 = 1000
Sensitivity Analysis
Objective function coefficient decreasesZ = 3000 Model_101 + 5000 Model_102 to Z = 2000 Model_101 + 5000 Model_102
OptimumValue = ???Model_101 = 2000Model_102 = 1000
Sensitivity Analysis
Objective function coefficient decreasesZ = 3000 Model_101 + 5000 Model_102 to Z = 2000 Model_101 + 5000 Model_102
OptimumValue = 9.5 MillionModel_101 = 2000Model_102 = 1000
Sensitivity Analysis
Objective function coefficient change
Sensitivity Analysis
Original Model
OptimumValue = 11 MillionModel_101 = 2000Model_102 = 1000
Sensitivity Analysis
Engine Assy. RHS increasesModel_101 + 2 Model_102 ≤ 4000 to Model_101 + 2 Model_102 ≤ 4200
OptimumValue = 11.4 MillionModel_101 = 1800Model_102 = 1200
Sensitivity Analysis
Engine Assy. RHS increasesModel_101 + 2 Model_102 ≤ 4000 to Model_101 + 2 Model_102 ≤ 4600
OptimumValue = 12 MillionModel_101 = 1500Model_102 = 1500
Sensitivity Analysis
Original Model
OptimumValue = 11 MillionModel_101 = 2000Model_102 = 1000
Sensitivity Analysis
Engine Assy. RHS decreasesModel_101 + 2 Model_102 ≤ 4000 to Model_101 + 2 Model_102 ≤ 3800
OptimumValue = 10.6 MillionModel_101 = 2200Model_102 = 800
Sensitivity Analysis
Engine Assy. RHS decreasesModel_101 + 2 Model_102 ≤ 4000 to Model_101 + 2 Model_102 ≤ 3100
OptimumValue = 9 MillionModel_101 = 2500Model_102 = 300
Sensitivity Analysis
Engine Assy. RHS change