Sensitivity Analysis for LPs - Webinar

Transcription

1 Sensitivity Analysis for LPs - Webinar 25/01/2017 Arthur d Herbemont

2 Agenda > I Introduction to Sensitivity Analysis > II Marginal values : Shadow prices and reduced costs > III Marginal ranges : RHS ranges and Coefficient ranges 2

3 Introduction > When do we need a sensitivity analysis? Uncertainty in input data Impact of different policy scenarios > How could we perform that on a mathematical optimization model? Or by-hand, running the model several times with small changes in input data Or by doing a post-optimality analysis, without being forced to re-solve entirely the model Discrete Continuous > How could we perform a post-optimality analysis? By extracting and studying marginal values By extracting and studying right hand side and coefficient ranges 3

4 A small producing chips model > Objective function to maximize 2x x 2 Vegetable Chips x 2 > Constraints x 2 = 165 2x 1 Slicing : Frying : 2x 1 + 4x min 4x 1 + 5x min Packing : 4x 1 + 2x min x 1 0 and x 2 0 Organic Chips x 1 > Solving x 1 = x 2 = See also : AIMMS Optimization Modeling book 4

5 A small producing chips model > Objective function to maximize 2x x 2 Vegetable Chips x 2 > Constraints Slicing : Frying : 2x 1 + 4x min 4x 1 + 5x min x 2 = 165 2x 1 Objective function ( 57. 5, 50 ) Packing : 4x 1 + 2x min x 1 0 and x 2 0 Organic Chips x 1 > Optimal Solution x 1 = 57.5 kg x 2 = 50 kg See also : AIMMS Optimization Modeling book 5

6 A small producing chips model > Objective function to maximize 2x x 2 Vegetable Chips x 2 > Constraints Slicing : Frying : 2x 1 + 4x min 4x 1 + 5x min x 2 = 165 2x 1 Objective function ( 57. 5, 50 ) Packing : 4x 1 + 2x min x 1 0 and x 2 0 Organic Chips x 1 > Optimal Solution x 1 = 57.5 kg x 2 = 50 kg See also : AIMMS Optimization Modeling book 6

7 Shadow price of a constraint > What if our packing capacity wasn t 330 min, but 331 min? Packing : 4x 1 + 2x min The shadow price of a constraint is the rate of change of the objective function from a one unit increase in its right hand side. Constraint Shadow price Right Hand Side Slicing 0.00 $/min 345 min Frying 0.17 $/min 480 min Packing 0.33 $/min 330 min 7

8 Shadow price of a constraint > What if our packing capacity wasn t 330 min, but 331 min? Our objective function will increase by = 0.33$! 190 $ => $ > Could we increase (or decrease) the RHS indefinitely? No! Shadow prices are local quantities, only mathematically valid right on the edge of the optimal solution. => We need to study the Right Hand Side range. 8

9 Reduced cost of a decision variable > Could we have the same approach with variables bounds? The reduced cost of a decision variable is the rate of change of the objective function for a one unit increase in its bound. Constraint Reduced cost Right Hand Side x $/kg 0 kg x $/kg 0 kg x 1 0 kg x 2 0 kg Objective function : 2x x 2 > Optimal Solution x 1 = 57.5 kg x 2 = 50 kg 9

10 Reduced costs > When can we use reduced costs? Constraint Reduced cost Right Hand Side Objective coefficient x 1 0 kg 0.00 $/kg 0 kg 2 $/kg x 2 0 kg 0.00 $/kg 0 kg 1.5 $/kg x 2 0 kg 0.5 $/kg 0 kg 0.5 $/kg Objective function : 2x x 2 Objective function : 2x x 2 > If x 2 was 0.5 $/kg more expensive, we would sell it. > If we were bound to sell x 2, we would decrease our profit by 0.5 $/kg. 10

11 Shadow price and reduced cost interpretation examples > Cost of a constraint for the system (what is the most binding constraint?) > The marginal price of the electricity (spot price) > The value of a raw material for our system > The value of relaxing or tightened a constraint related to the quality of a product > The effect of small changes in capacity > 11

12 Shadow prices : When are we not able to use them? 1. Degeneracy x 2 Another constraint Objective function 2. Right Hand Side ranges x 1 The Shadow price is not valid outside of its Right Hand Side range 12

13 Shadow prices and Right Hand Side ranges > Packing : 4x 1 + 2x min min Constraint Shadow price Right Hand Side range Packing 0.33 $/min min 13

14 Shadow prices and Right Hand Side ranges > Packing : 4x 1 + 2x min Constraint Shadow price Right Hand Side range Packing 0.33 $/min min > Examples of a sensitivity question I want my objective to be accurate within 1 % You have to know your packing capacity within a min range, at least. Calculation : = 1.9 $ and How much an increase of +100 min on packing will improve my profit? According to the RHS range, 33 $. How much does your machine cost?... = 5.75 min 14

15 Coefficient ranges > What if x 1 selling price was not 2 $/kg, but between 1. 9 and 2. 1? Variable Objective Coefficient Coefficient ranges x 1 2 $/kg $/kg x $/kg $/kg The coefficient range of a decision variable is the range of possible values its objective function coefficient can take without modifying the optimal solution. 15

16 Coefficient ranges > What if x 1 selling price was not 2 $/kg, but between 1. 9 and 2. 1? Variable Objective Coefficient Coefficient ranges x 1 2 $/kg $/kg x $/kg $/kg The coefficient range of a decision variable is the range of possible values its objective function coefficient can take without modifying the optimal solution. x 2 > Optimal solution unchanged x 1 = 57.5 kg x 2 = 50 kg 50 Objective Function value < 190$ Objective Function value = 190$ Objective Function value > 190$ x 1

17 Coefficient ranges > What if x 1 selling price was not 2 $/kg, but between 1. 9 and 2. 1 $/kg? The optimal solution would be unchanged, thus The profit would be accurate within ±3% > Optimal Solution unchanged x 1 = 57.5 kg x 2 = 50 kg How much should I be precise on x 1 selling price? Not a lot regarding your production plan. But maybe a lot regarding your profit, because we know that YourProfit SellingPrice = SellingPrice $ 17

18 Another larger example > Power System Balancing Generators Demand Hydro Biomass Coal Gas 500 MW 300 MW 200 MW 100 MW 20 $/MWh 30 $/MWh 35 $/MWh 50 $/MWh 18

19 To conclude Notion Math implication Economic interpretation AIMMS identifiers Shadow Prices RHS ranges Coefficient ranges Constraint is binding or not Shadow price validity domain Optimal solution unchanged The cost of a certain constraint for your model Sensitivity of your solution regarding input data change Sensitivity of your solution regarding a coefficient change in the objective function > Limit : Only available for a single change in input data 19

20 To go further > AIMMS Optimization Modeling Book online > Model Building in Mathematical Programming H.P. Williams > Shadow price ranges > Value ranges > Parametric programming > Stochastic programming, robust optimization Dealing with Uncertainty in Optimization-Based Decision Support Applications using AIMMS 20