Introducing Duality by HC120911194242

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									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 102s
              Optimal 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       unit = 1 hr
Metal Stmp.     1/3000 unit     1/3000 unit       1 unit       unit = 2 hr
101 Assy.       1/2500 unit                       1 unit       unit = 2 hr
102 Assy                        1/1500 unit       1 unit       unit = 3 hr

  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?
Worth of Engine Capacity
                                    Worth
% Increase      Contribution
                               (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 capacity
          increases to 4400 hours
Worth of Engine Capacity
      Engine Assembly Capacity is now 4400 hours

                                           Worth
% Increase          Contribution
                                      (Full Capacity)
Original            $11,800,000
Worth of Engine Capacity
      Engine Assembly Capacity is now 4400 hours

                                           Worth
% Increase          Contribution
                                      (Full Capacity)
Original            $11,800,000
10% increase        $12,000,000        $2,000,000
Worth of Engine Capacity
      Engine Assembly Capacity is now 4400 hours

                                           Worth
% Increase          Contribution
                                      (Full Capacity)
Original            $11,800,000
10% increase        $12,000,000        $2,000,000
 5% increase        $12,000,000        $4,000,000
Worth of Engine Capacity
      Engine Assembly Capacity is now 4400 hours

                                           Worth
% Increase          Contribution
                                      (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
Worth of Engine Capacity
      Engine Assembly Capacity is now 4400 hours

                                           Worth
% Increase          Contribution
                                      (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
   Merton Trucks
Base Engine Assembly capacity = 4000 hrs




  Engine Assembly capacity = 4000 hrs
   Merton Trucks
Base Engine Assembly capacity = 4000 hrs




   Engine Assembly capacity ↑ by 1%
   Merton Trucks
Base Engine Assembly capacity = 4000 hrs




   Engine Assembly capacity ↑ by 5%
   Merton Trucks
Base Engine Assembly capacity = 4000 hrs




   Engine Assembly capacity ↑ by 10%
Merton Trucks (new)
 Base Engine Assembly capacity = 4400 hrs




   Engine Assembly capacity = 4400 hrs
Merton Trucks (new)
 Base Engine Assembly capacity = 4400 hrs




    Engine Assembly capacity ↑ by 1%
Merton Trucks (new)
 Base Engine Assembly capacity = 4400 hrs




    Engine Assembly capacity ↑ by 5%
Merton Trucks (new)
 Base Engine Assembly capacity = 4400 hrs




    Engine Assembly capacity ↑ by 10%
                  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
                       Optimum
                       Value = 11 Million
                       Model_101 = 2000
                       Model_102 = 1000




      Original Model
                Sensitivity Analysis
                                                        Optimum
                                                        Value = 13 Million
                                                        Model_101 = 2000
                                                        Model_102 = 1000




                    Objective function coefficient increases
Z = 3000 Model_101 + 5000 Model_102 to Z = 4000 Model_101 + 5000 Model_102
                Sensitivity Analysis
                                                        Optimum
                                                        Value = ???
                                                        Model_101 = 2000
                                                        Model_102 = 1000




                    Objective function coefficient increases
Z = 3000 Model_101 + 5000 Model_102 to Z = 6000 Model_101 + 5000 Model_102
                Sensitivity Analysis
                                                        Optimum
                                                        Value = 17.5 Million
                                                        Model_101 = 2000
                                                        Model_102 = 1000




                    Objective function coefficient increases
Z = 3000 Model_101 + 5000 Model_102 to Z = 6000 Model_101 + 5000 Model_102
Sensitivity Analysis
                       Optimum
                       Value = 11 Million
                       Model_101 = 2000
                       Model_102 = 1000




      Original Model
                Sensitivity Analysis
                                                        Optimum
                                                        Value = 10.5 Million
                                                        Model_101 = 2000
                                                        Model_102 = 1000




                    Objective function coefficient decreases
Z = 3000 Model_101 + 5000 Model_102 to Z = 2750 Model_101 + 5000 Model_102
                Sensitivity Analysis
                                                        Optimum
                                                        Value = ???
                                                        Model_101 = 2000
                                                        Model_102 = 1000




                    Objective function coefficient decreases
Z = 3000 Model_101 + 5000 Model_102 to Z = 2000 Model_101 + 5000 Model_102
                Sensitivity Analysis
                                                        Optimum
                                                        Value = 9.5 Million
                                                        Model_101 = 2000
                                                        Model_102 = 1000




                    Objective function coefficient decreases
Z = 3000 Model_101 + 5000 Model_102 to Z = 2000 Model_101 + 5000 Model_102
Sensitivity Analysis




  Objective function coefficient change
Sensitivity Analysis
                       Optimum
                       Value = 11 Million
                       Model_101 = 2000
                       Model_102 = 1000




      Original Model
           Sensitivity Analysis
                                                   Optimum
                                                   Value = 11.4 Million
                                                   Model_101 = 1800
                                                   Model_102 = 1200




                   Engine Assy. RHS increases
Model_101 + 2 Model_102 ≤ 4000 to Model_101 + 2 Model_102 ≤ 4200
           Sensitivity Analysis
                                                   Optimum
                                                   Value = 12 Million
                                                   Model_101 = 1500
                                                   Model_102 = 1500




                   Engine Assy. RHS increases
Model_101 + 2 Model_102 ≤ 4000 to Model_101 + 2 Model_102 ≤ 4600
Sensitivity Analysis
                       Optimum
                       Value = 11 Million
                       Model_101 = 2000
                       Model_102 = 1000




      Original Model
           Sensitivity Analysis
                                                   Optimum
                                                   Value = 10.6 Million
                                                   Model_101 = 2200
                                                   Model_102 = 800




                   Engine Assy. RHS decreases
Model_101 + 2 Model_102 ≤ 4000 to Model_101 + 2 Model_102 ≤ 3800
           Sensitivity Analysis
                                                   Optimum
                                                   Value = 9 Million
                                                   Model_101 = 2500
                                                   Model_102 = 300




                   Engine Assy. RHS decreases
Model_101 + 2 Model_102 ≤ 4000 to Model_101 + 2 Model_102 ≤ 3100
Sensitivity Analysis




    Engine Assy. RHS change

								
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