The Theory of the Simplex Method Chapter 6: Hillier and Lieberman Dr. Hurley’s AGB 328 Course

Terms to Know Dual, Primal, Primal-Dual Table, Weak Duality Property, Strong Duality Property, Complementary Solutions Property, Complementary Solutions, Complementary Optimal Solutions Property, Complementary Optimal Solution, Symmetry Property, Duality Theorem, Complementary Basic Solutions Property, Complementary Basic Solution,

Terms to Know Cont. Complementary Slackness Property, Complementary Slackness, Complementary Optimal Basic Solutions Property, Complementary Optimal Basic Solution, Primal Feasible, Dual Feasible, Sensible-Odd-Bizarre Method, Sensitivity Analysis, Sensitive Parameters, Reduced Cost, Allowable Range

Duality Theory The point of duality theory is that a maximization problem has a corresponding minimization problem and vice versa The coefficients in the maximization problem can be mapped to a set of coefficients in the minimization problem

Duality Theory—The Two Problems Primal Dual 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍= 𝑗=1 𝑛 𝑐 𝑗 𝑥 𝑗 Subject to: 𝑗=1 𝑛 𝑎 𝑖𝑗 𝑥 𝑗 ≤ 𝑏 𝑖 𝑓𝑜𝑟 𝑖=1,2, …, 𝑚 𝑥 𝑗 ≥0 𝑓𝑜𝑟 𝑗=1,2, …,𝑛 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑊= 𝑖=1 𝑚 𝑏 𝑖 𝑦 𝑖 Subject to: 𝑖=1 𝑚 𝑎 𝑖𝑗 𝑦 𝑖 ≥ 𝑐 𝑗 𝑓𝑜𝑟 𝑗=1,2, …, 𝑛 𝑦 𝑖 ≥0 𝑓𝑜𝑟 𝑖=1,2, …,𝑚 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍=𝒄𝒙 Subject to: 𝑨𝒙≤𝒃 𝒙≥𝟎 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑊=𝒃𝒚 Subject to: 𝒚𝑨≥𝒄 𝒚≥𝟎

Duality Theory-The Correspondence The coefficients in the objective function of the primal are the coefficients of the RHS of the constraints of the dual The RHS coefficients of the constraints of the primal are the coefficients in the objective function of the dual The coefficients in the functional constraints are the same in the primal as the dual

Duality Theorem Relationship between the primal and the dual: If one problem is bounded and has feasible solutions, then so does the other If one problem is unbounded with feasible solutions, then the other problem has no feasible solution If one of the problems has no feasible solution, then the other problem is either unbounded or has no feasible solutions

Why Care About the Dual Sometimes the dual is easier and quicker to solve than the primal problem The optimal value of the dual can be seen in the shadow prices of the primal

Economic Interpretation of Duality 𝑖=1 𝑚 𝑎 𝑖𝑗 𝑦 𝑖 represents the current contribution to the objective function of the mix of resources that would be consumed if 1 unit of activity j were utilized To make the best possible use of the resources, we want to have 𝑖=1 𝑚 𝑎 𝑖𝑗 𝑦 𝑖 ≥ 𝑐 𝑗

Economic Interpretation of Duality Cont. The yi’s represents the values of the resources bi’s Hence in the resource constraint of the dual, the value of the xi’s usage of the resources should be greater than or equal to their corresponding value in the primal, i.e., the ci’s 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍= 𝑐 1 𝑥 1 + 𝑐 2 𝑥 2 Subject to: 𝑎 11 𝑥 1 + 𝑎 12 𝑥 2 ≤ 𝑏 1 𝑎 21 𝑥 1 + 𝑎 22 𝑥 2 ≤ 𝑏 2 𝑥 1 ≥0, 𝑥 2 ≥0 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑍= 𝑏 1 𝑦 1 + 𝑏 2 𝑦 2 Subject to: 𝑎 11 𝑦 1 + 𝑎 21 𝑦 2 ≥ 𝑐 1 𝑎 12 𝑦 1 + 𝑎 22 𝑦 2 ≥ 𝑐 2 𝑦 1 ≥0, 𝑦 2 ≥0

Economic Interpretation of Duality Cont. We should expect that when xj > 0 (j = 1, 2, …, n), then 𝑖=1 𝑚 𝑎 𝑖𝑗 𝑦 𝑖 = cj Why? We should also expect that when xn+i > 0 (i = 1, 2, …, m), then yi = 0

Role of Duality Theory in Sensitivity Analysis Duality theory can help when you are conducting a sensitivity analysis on your model Suppose you have a change in the coefficient of a non-basic variable in your model You can check the complementary basic solution in the dual to see if the constraints in the dual are still satisfied If so, then the change in the coefficient will not change the optimal decision

Role of Duality Theory in Sensitivity Analysis Cont. Suppose you introduce a new variable to the model This is equivalent to introducing a new constraint in the dual Why? If the complementary basic solution in the dual satisfies the new constraint, the new variable does not meaningful change the dual and will end up being zero

Essence of Sensitivity Analysis Sensitivity analysis examines how robust your model and its optimal solution are to changes in the assumptions of your model These assumptions may include what variables are in the model, the value of the coefficient for each variable, the functional form of the objective function and the constraints, etc.

Essence of Sensitivity Analysis Cont. One important aspect in sensitivity analysis is to identify any sensitive parameters The way to determine if a parameter is sensitive is to ascertain how much it can change before your optimal decision changes

Different Cases to Sensitivity Analysis Case 1: Changes in bi Case 2a: Changes in coefficients of non- basic variables Case 2b: Introduction of a new variable Case 3: Changes in the coefficient of a basic variable Case 4: Introduction of a new constraint

A Single Change to a bi or a c j To see if a single change to the bi or cj causes a change in your optimal solution, you can examine the allowable range of the parameter

Allowable Range The allowable range of a functional constraint is the range of values for this right-hand side over which this constraint’s shadow price remains valid. The bottom end of the range is calculated by: Constraint RH Side – Allowable Decrease The upper end of the range is calculated by: Constraint RH Side + Allowable Increase

The 100 Percent Rule for Simultaneous changes in RHS Coefficients This is a rule that tells you how much of each constraint is allowed to change simultaneously before the shadow prices change This rule says that if the sum of the proportions of parameter change divided by allowable changes in absolute value terms of all the coefficients does not exceed 100%, then the original shadow prices will be valid If it changes by more than 100%, you cannot be sure.

The 100 Percent Rule for Simultaneous changes in Objective Function Coefficients This is a rule that tells you how much of each constraint is allowed to change simultaneously before the optimal might change. This rule says that if the sum of the proportions of parameter change divided by allowable changes in absolute value terms of all the coefficients does not exceed 100%, then the original optimal solution was still be optimal. If it changes by more than 100%, you cannot be sure.

Calculating a Percentage Change The percentage change for a value from the 100% rule can be calculated as: (New Value – Old Value) / Allowable Change For example: when 300 changes to 600 and the allowable change is 900 you get a proportional change of (600-300)/900 which equals approximately 33.33%

Sensitivity Analysis and Spreadsheet Models There are three major ways you can use to conduct a sensitivity analysis on your spreadsheet model You can solve the model multiple different times using different variables for the coefficient(s) of interest You can use Solver Table You can use the Sensitivity Report in Solver

Solving the Excel Model Multiple Times with Multiple Parameters Whenever you change a parameter in the model you must tell Excel to resolve the problem by going to Solver When doing this type of sensitivity analysis, you want to change the parameters in a way that will allow you to find the key points quickly You could use some form of divide and conquer to find the key changing points You could establish a particular interval to help find the sensitive points

Solver Table Solver Table is a tool developed by the textbook authors to solve the model multiple times using different parameters How would you go about finding an operable version?

Solver Table Cont. Solver Table can change up to two parameters at a time. In class activity: Build a sensitivity chart for changing the prices of windows. Examine prices that range from $100 to $1000. Use the Solver Table to find the price of windows that changes the optimal solution from 2,6 to 4,3.

Excel Side Note You can represent a solution set in a single cell by placing an & in front of the variable you want to add. For example:="("&C12&", "&D12&")” gives (2, 6) in the same cell.

Solver’s Sensitivity Report Solver has another way of finding the parameters that will change the optimal solution This is done by using Solver’s Sensitivity Report To get the Sensitivity Report, you need to highlight the report after you have used Solver

Analyzing the Sensitivity Report To find the range of the variable before the optimal solution will change, you can use the Solver information in the following way. The bottom end of the range on the coefficient is: Objective coefficient – Allowable Decrease The upper end of the range of the coefficient is: Objective coefficient + Allowable Increase

Solver Report for Wyndor Microsoft Excel 14.0 Sensitivity Report Worksheet: [Wyndor Glass.xls]Wyndor Report Created: 10/20/2013 2:51:26 PM Variable Cells   Final Reduced Objective Allowable Cell Name Value Cost Coefficient Increase Decrease $C$12 Batches Produced Doors 2 3000 4500 $D$12 Batches Produced Windows 6 5000 1E+30 Constraints Shadow Constraint Price R.H. Side $E$7 Plant 1 Used 4 $E$8 Plant 2 Used 12 1500 $E$9 Plant 3 Used 18 1000

Analyzing the Sensitivity Report Cont. In the Wyndor example the price of the doors could increase to $7500 or decrease to $0 before the optimal solution would change In the Wyndor example the price of the windows could increase an infinite amount or decrease to $2000 before the optimal solution would change

Sensitive Parameters A parameter is considered a sensitive parameter if small changes lead to a change in the optimal solution These parameters are the ones you will focus on to make sure you have them as close to correct as possible

In-Class Activity Given the model above: 𝑀𝑎𝑥 8 𝑥 1 +6 𝑥 2 Subject to 1 𝑥 1 +4 𝑥 2 ≤200 1 𝑥 1 +1 𝑥 2 ≤50 3 𝑥 1 +1 𝑥 2 ≤60 𝑥 1 ≥0, 𝑥 2 ≥0 Given the model above: Set-up the dual problem Solve the dual using the tabular form and show that the optimal of the dual is the shadow of the primal Conduct a sensitivity analysis of the dual problem using Excel Solver including explaining the allowable ranges, shadow prices If all the objective coefficients in the dual decreased by 4%, can you definitely say whether the optimal would change without resolving the model