# Sensitivity Analysis with Excel

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1 Sensitivity Analysis with Excel 1 Lecture Outline Sensitivity Analysis Effects on the Objective Function Value (OFV): Changing the Values of Decision Variables Looking at the Variation in OFV: Excel One- and Two-Way Data Tables, and Scenario Manager Finding the Optimum OFV: Excel Solver Effects on the Optimum: Making One Change to the Model Parameters: Changing a Decision Variable Coefficient in a Constraint (Graphical intuitions) Changing the Right Hand Side (RHS) of a Constraint: Slack & Shadow Price Changing a Decision Variable Coefficient in the Objective Function: Reduced Cost Making Multiple Changes to the Model Parameters The 100% Rule SolverTable 2 1

2 Steps in Modeling STEP 1: Formulate STEP 2: Solve (Find the Optimum) STEP 3: Do a Sensitivity Analysis 3 Problem Statement Boxers and Briefs Example - Revisited Champion Sports manufactures two types of custom men's underwear: boxers and briefs. How many boxers and how many briefs should be produced per week, to maximize profits, given the following constraints The (profit) contribution per boxer is \$3.00, compared to \$4.50 per brief. Briefs use 0.5 yards of material; boxers use 0.4 yards. 300 yards of material are available. It requires 1 hour to manufacture one pair of boxers and 2 hours for one pair of briefs. 900 labors hours are available. There is unlimited demand for boxers but total demand for briefs is 375 units per week. Each boxer uses 1 insignia logo and 600 insignia logos are in stock. 4 2

3 Algebraic Formulation 5 The Algebraic LP Formulation Revisited Variables: number of Boxers, number of Briefs. Objective function: Objective Function Coefficients maximize ( \$3.00 x Boxers ) + ( \$4.50 x Briefs ) Constraint Coefficients Constraints: Material: (.4 x Boxers ) + (.5 x Briefs ) <= 300 yards Labor: ( 1 x Boxers ) + ( 2 x Briefs ) <= 900 hrs Demand: ( 0 x Boxers ) + ( 1 x Briefs ) <= 375 units Logos: ( 1 x Boxers ) + ( 0 x Briefs ) <= 600 Non-Negativity: Boxers >= 0 Briefs >= 0 Constraints Right Hand Sides (RHS) 6 3

4 Graphical Formulation 7 The Graphical LP Formulation - Revisited Optimal Solution Boxers F Hours E D A C Logos Demand Optimum Material B Briefs \$3.00 \$4.50 Total Boxers Briefs Profit A \$1, B \$1, C \$2, D \$2, E \$2, F \$1,

5 Excel Formulation 9 Excel Formulation - Revisited Boxers and Briefs Example: Profit Maximization See boxers_and_briefs_example.xls in the downloads for today s class. 10 5

6 Excel Tools for Sensitivity Analysis One-way Data Table, Two-Way Table, Scenario Manager Solver Answer Report Slack Pertains to changing RHS of a constraint Solver Sensitivity Report Shadow Price, Allowable Increase / Decrease Reduced Cost, Allowable Increase / Decrease SolverTable Pertains to changing Objective Function Coefficients 11 Changing the Value of Decision Variables With Two- Way Data Tables The two-way data table below shows the effect, on profit, of changing the two decision variables (i.e. the number of boxers produced, and the number of briefs produced). This initial analysis assumes there are no constraints. Clearly, if constraints are taken into account, the maximum profit in the data table shown below (\$4350, for 700 boxers and 500 briefs) would change since that is not a feasible solution. 12 6

7 Steps in Modeling STEP 1: Formulate STEP 2: Solve (Find the Optimum) STEP 3: Do a Sensitivity Analysis 13 Changing the Value of Decision Variables With Two- Way Data Tables The two-way data table below shows the effect on profit of changing the number of boxers and briefs produced. Conditional formatting has been used to highlight infeasible solutions (in red) and the optimal feasible solution (in green). Don t panic! You aren t expected to know how to construct the table / formula shown below its only shown to illustrate how complex it is to create a table that shows the feasible and optimal solutions! Notice that we were lucky in this instance that the optimum solution falls at a point that was in our table: had we chosen broader intervals for number of boxers and briefs we would have found a good, but not optimal, solution. 14 7

8 Excel Answer Report - Revisited Boxers and Briefs Example: Profit Maximization Fortunately, as we ve seen Excel provides us with an easy mean of finding the optimal feasible solution: simply use Excel Solver. Make sure Solver is available: If not, Tools Add-Ins Solver Add-in Use it: Tools Solver Read the Answer Report to find the optimal value, and the decision variable settings at this optimum. 15 Excel Answer Report Optimum Objective Function Value Optimum Product Mix Status of the Constraints 16 8

9 Steps in Modeling STEP 1: Formulate STEP 2: Solve (Find the Optimum) STEP 3: Do a Sensitivity Analysis 17 Answer Report The Answer Report also shows us which constraints are binding and nonbinding at the optimum. Slack indicates the spare capacity on a non-binding constraint at the optimum. Slack = 0 implies constraint is binding: resources are exhausted Slack 0 implies constraint is non-binding: there are left-over resources We can see below, for instance, that: getting additional logos wouldn t help us improve profit since we already have 100 unused logos at the optimum advertising our briefs to stimulate demand (at the current price) would be money wasted, since demand for briefs is already greater than the number of briefs we should produce at the optimum. material and labor constraints are cramping our profit. 18 9

10 Slack and Binding Slack measures unused available resources Binding constraints Optimal solution lies on binding constraints Multiple binding constraints means solution is at intersection of constraints: a vertex Slack and binding constraints Slack = 0 : the resource is exhausted, the constraint is binding, the optimal vertex includes this constraint. Slack 0 : spare capacity is available, constraint is not binding 19 Excel Sensitivity Analysis Report Choose Sensitivity to see a more detailed Sensitivity Report... Pertain to Objective Function Coefficient Ranging Pertain to Right Hand Side (RHS) Ranging on Constraints 20 10

11 Excel Sensitivity Analysis Report The Excel Sensitivity Analysis Report allows you to see: over what range and under what conditions the components of a solution remain unchanged how sensitive a solution is to changes in the data, and to get an insight into how technological improvements may affect optimum values. 21 Boxers Sensitivity Analysis Changing a Decision Variable Co-Efficient in a Constraint Graphical Intuitions The Sensitivity Report doesn t tell us anything about what happens when the coefficients of a decision variable in a constraint change, but here are some graphical intuitions Hours Logos Demand Assume we changed the amount of material required per brief from 0.5 yards to 1 yard. Notice how the shape of the feasible region changes, and the optimal solution changes. Material Boxers Hours Logos Demand Material Briefs Briefs 11

12 Boxers Sensitivity Analysis Report Changing the Right Hand Side (RHS) of a Constraint Graphical Intuitions Changing the RHS of a constraint causes the constraint line to shift left or right, but does not alter the slope of the line! Hours Demand Logos Material Assume an extra 100 yards of material became available (increasing our total available to 400 yards). Notice how the material constraint shifts right. The change is greater than the allowable increase (see later) and the optimum solution vertex changes. Boxers Hours Demand Logos Material Briefs Briefs Sensitivity Analysis Changing the Right Hand Side (RHS) of a Constraint The Shadow Price What is meant by the optimal vertex? The optimal vertex is : what intersection of constraints is the optimal solution to be found at?. From the previous slide, we saw that the original optimal vertex was: at the intersection of the materials and hours constraints. However, when we increased the RHS of the materials constraint beyond the maximum allowable increase (i.e. by 100 yards), the optimal vertex shifts to: at the intersection of the logos and hours constraints. Had the change in materials been within the allowable increase, then the optimal vertex would have stayed the same (i.e. it would still have been at the intersection of the materials and hours constraints) but the optimal product mix and the optimal solution value would have changed

13 Sensitivity Analysis Changing the Right Hand Side (RHS) of a Constraint Graphical Intuitions Changing the RHS of a constraint causes the constraint line to shift to a parallel position to the left or right, but does not alter the slope of the line! Assume a extra 100 hours of labor became available (increasing our total available to 1000 hours). Notice how the labor constraint shifts right, and the optimal solution value and optimal product mix change. However, the change is less than the allowable increase for this constraint and the optimum solution vertex stays the same: the optimal solution is still at the intersection of the materials and labor hours constraints. Boxers Hours Demand Logos Material Briefs Sensitivity Analysis Changing the Right Hand Side (RHS) of a Constraint Tighten constraint: make feasible region smaller. Optimal value can only get worse (fewer choices) Loosen (relax) constraint: make feasible region larger. Optimal value can only do better (more choices) Assume b is a positive number and the constraint line has the form y = ax + b (i.e. y ax = b) then increasing the RHS (i.e. b) will cause the line to shift vertically up and will: Tighten the constraint if it s a lower-bound constraint Loosen the constraint if its an upper-bound constraint So notice that increasing the RHS may have positive or negative effects on the optimum, depending on the type of constraint! In contrast, if b is a positive number and the constraint line has the form y = ax b (i.e. ax y = b) then increasing the RHS (i.e. b) will cause the line to shift vertically down! 26 13

14 Sensitivity Analysis Report Changing the Right Hand Side (RHS) of a Constraint The Shadow Price The Shadow Price for a constraint is the change in the optimal objective function value per unit increase in the Right Hand Side (RHS) of a given constraint. The Shadow Price only remains valid within the Allowable Increase and Decrease shown for that Shadow Price. 27 Shadow Prices Example A stain is found on 15 yards of material, reducing material from 300 to 285 yards. How does this affect optimal profit? New optimal profit = Old optimal profit - (shadow price x yards) = \$2,400 (\$5/yard x 15 yards) = \$2,325 Notice the Allowable range for the Materials Usage constraint. What could you say about a stain on 60 yards? 28 14

15 Shadow Prices Example Labor is willing to negotiate 100 additional hours of production work. How much of a premium should management pay for overtime hours? Addition to optimal profit = shadow price x hours = \$1/hour x 100 hours = \$100 If \$9.28 for 900th hour, then \$10.28 for 901st hour. But remember, the productivity rate will change when people work longer hours 29 Increase Labor by 100 hours Boxers F Hours Logos E D G A Demand C B Material Briefs \$3.00 \$4.50 Total Boxers Briefs Contribution A \$1, B \$1, C \$2, D \$2, E \$2, F \$1, G \$2,

16 Shadow Prices Example How much would you pay for one additional insignia logo? Nothing: logos are not constraining the solution! Notice the range for logos: what is 1E+30? 31 Shadow Price / RHS Ranging Shadow price Shadow price = marginal change to objective function value of increasing constraint RHS by 1 unit Scaling issues: changing one model unit (e.g. model may be in millions of units ) Slack and shadow price Shadow price = 0 implies constraint is not binding Shadow price 0 implies Slack = 0 Slack and allowable increase/decrease How much you must tighten a constraint to make it bind Notice that, for a non-binding constraint, either the allowable increase or the allowable decrease will be equal to the slack, since either adding or subtracting the slack to / from the RHS will make the constraint bind

17 Shadow Price / RHS Ranging Shadow price on a constraint: Change in optimal objective function value per unit change in righthand-side of the constraint zero if constraint is non-binding Shadow price and RHS ranging: Allowable increase / decrease = Range of RHS coefficients for which shadow prices remain valid. Optimal value (production mix) will change: If binding constraints are moved, the optimal mix changes The objective function value changes The shadow price allows us to predict new optimal value. Need to resolve the model to get the new mix (decision variables). 33 Allowable Increase / Decrease for the RHS of a Constraint Allowable increase/decrease in constraints: How much you can tighten or relax a constraint RHS and remain binding (or non-binding) (see also: Slack) Range in RHS coefficients for which the shadow price remains valid (see also: Shadow Price) Feasible Region: How much you can reshape the feasible region without changing the optimal vertex. (The optimal mix will always change, but the optimal vertex stays the same within the allowable increase/decrease.) 34 17

18 Allowable Increase / Decrease for the RHS of a Constraint Change within allowable increase/decrease Optimal vertex (intersection of the binding constraints) is unchanged. Optimal value of objective function is updated by Shadow Price. Optimal product mix (value of decision variables) is calculated by re-testing the model. Change outside allowable increase/decrease Start over. 35 Shadow Price, Slack, and Binding 36 18

19 Shadow Price in Maximization vs Minimization Problems For maximization problems: increase in the objective function value is good (+ve shadow price is good, and helps the optimum solution) decrease in the objective function value is bad (-ve shadow price is bad, and hurts the optimum solution) For minimization problems: increase in the objective function value is bad (+ve shadow price is bad, and hurts the optimum solution) decrease in the objective function value is good (-ve shadow price is good, and helps the optimum solution) 37 Shadow Prices in a Minimization Problem Our objective in the Big Mac Attack Problem is to achieve least-cost in meeting our Recommended Daily Allowance (RDA) requirements: including both upper and lower limits

20 Shadow Prices in a Minimization Problem Sodium was an upper bound constraint. It has a negative shadow price: increasing the RHS of the sodium constraint would loosen the constraint and allow us to achieve a better optimum: i.e. a lower meal cost. Vitamin C was an lower bound constraint. It has a positive shadow price: increasing the RHS of the Vitamin C constraint would tighten the constraint and force us to a worse optimum: i.e. a higher meal cost. Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease \$D\$21 TOTAL Protein E E+30 \$E\$21 TOTAL Fat E E \$F\$21 TOTAL Sodium E \$G\$21 TOTAL VitaminA E \$H\$21 TOTAL VitaminC E E \$I\$21 TOTAL VitaminB E E+30 \$J\$21 TOTAL VitaminB E E+30 \$K\$21 TOTAL Niacin E E+30 \$L\$21 TOTAL Calcium E \$M\$21 TOTAL Iron E Sensitivity Analysis Report Changing a Decision Variable Co-Efficient in the Objective Function The allowable increase and decrease for the Adjustable Cells (i.e. for the objective function coefficients) tells us how much the objective function coefficients can change before the optimal solution vertex changes. Note that the optimal solution value changes as the objective function coefficients change, but the optimal product mix and vertex stays the same within the objective function coefficient s allowable increase/decrease range

21 Sensitivity Analysis Report Changing an Objective Function Coefficient Graphical Intuitions Changing an objective function coefficient changes the slope of the objective function. Optimal solution value always changes but, within the allowable increase / decrease, the optimal product mix will not: within the allowable increase decrease the isoprofit line just swivels around a single point! If the slope of the objective function changes beyond the allowable increase / decrease then the optimal vertex will change, and the optimal product mix will change, as illustrated to the right. At right we see the relative contribution of briefs (i.e. the coefficient of briefs in the objective function) increasing, causing the objective function to become steeper, until eventually (beyond the allowable increase) the optimal product mix and optimal vertex shift. The new optimal solution favors more briefs in the optimal product mix. Boxers Hours Demand Logos Material Briefs 41 Sensitivity Analysis Report Changing an Objective Function Coefficient Graphical Intuitions Multiple Optima Notice that, for a certain combination of objective function coefficients, the objective function can becomes tangent to a segment of a constraint line (in this case, to a segment of the Hours constraint). In this case, every point along the tangential line segment is an optimum, so multiple optimal product mixes are available, all with the same optimal solution value. Notice that for an isoprofit line with multiple optima, the allowable increase and decrease are zero, since it is impossible for the line to swivel around a single point and any change in the objective function coefficients will cause the optimal vertex to shift! Boxers Hours Demand Logos Material Briefs 42 21

22 Summary Changing a Decision Variable Co-Efficient in the Objective Function A coefficient is associated with each decision variable The allowable increase / decrease for each coefficient is the range over which the coefficient can vary without changing the product mix (i.e. without changing the vertex at which the optimal solution is found) The following do change: Objective Function Value Shadow Prices Reduced Costs Need to resolve the model to find this information Can use to understand flexibility in relative pricing of a product. 43 Summary Changing a Decision Variable Co-Efficient in the Objective Function Changes slope of isoprofit / isocost curve (in 2D) Changes decision variable contribution to objective function Does NOT change shape of feasible region In 2-dimensional case, the slope of the objective function changes, but, within the allowable increase / decrease range, the optimal solution still resides at the same extreme vertex of the feasible region 44 22

23 Changing a Decision Variable Co-Efficient in the Objective Function Example A management consultant offers to improve efficiency in the production of boxers. This would increase the contribution by \$0.50 to \$3.50. What is the new mix? What is the increase in weekly profit? No change in the product mix. Change in weekly profit = \$0.50 x 500 boxers/week = \$250 (\$2,650 total) Note the range: What could you say about \$1 increase? 45 Changing a Decision Variable Co-Efficient in the Objective Function Example The contribution of briefs decreases by \$0.75 to \$3.75. What is the new mix? What is the decrease in weekly profit? No change in the product mix. Change in weekly profit = \$0.75 x 200 briefs/week = \$150 (\$2,250 total) 46 23

24 Reduced Cost Associated with each decision variable. Amount by which profit contribution of variable must be improved before the variable will have a positive value in the solution. Or, rate at which the objective function value will deteriorate if a variable currently at zero is forced to increase by a small amount. Zero if the variable already appears in the optimal solution. 47 Reduced Cost (RC) Unit cost (penalty or loss) in optimal objective function value of forcibly including a Decision Variable (DV) not in the optimal solution. Necessary change in DV coefficient ( reduction in price) so the DV is part of the optimal solution. Rate at which the optimal objective function value deteriorates when a non-optimal DV is required. Reduced cost of a DV happens to be equivalent to the shadow price of the non-negativity constraint for that DV. Why? Because forcing a variable into a solution is the same as increasing the RHS of its non-negativity constraint (e.g. from 0 to 1). RC = 0 implies DV is part of the optimal solution. RC 0 implies DV does not contribute to the optimal objective function value, and that forcing that DV into the solution would worsen the optimal solution

25 Reduced Costs Example Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease \$B\$11 Production Boxers 500 \$ \$C\$11 Production Padded 0 (\$1.00) 6 1 1E+30 \$D\$11 Production Briefs 200 \$ New line: padded briefs, 1 yard of material and 2 hours of labor. Contribution is \$6.00 per padded brief. Forced to produce one unit of padded brief per week, what would be cost? Reduced cost is -\$1.00 per padded brief. 49 Amended Model To Demonstrate Reduced Cost Force the model to construct ONLY boxers: Objective function: Maximize ( \$10.00 x Boxers ) + ( \$1 x Briefs ) Constraints: Material: ( 1 x Boxers ) + ( 0.5 x Briefs ) 300 yards Logos: ( 1 x Boxers ) + ( 0 x Briefs ) 600 logos Labor: ( 1 x Boxers ) + ( 2 x Briefs ) 900 hrs Demand: ( 0 x Boxers ) + ( 1 x Briefs ) 375 units Boxers 0 Briefs

26 Reduced Cost Example The optimal solution to the amended model is: 300 Boxers, 0 Briefs for Optimal Value: \$3,000 Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease \$C\$3 Decision Variables Boxers E \$D\$3 Decision Variables Briefs E+30 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease \$E\$6 Material \$E\$7 Logos E \$E\$8 Labor E \$E\$9 Demand E Now, the President comes to visit Penn, but he has forgotten his briefs so we must manufacture at least one pair. What is the penalty? Think carefully 51 Reduced Cost Example (Amended Model) Maximize: ( \$10.00 x Boxers ) + ( \$1 x Briefs ) Material: ( 1 x Boxers ) + ( 0.5 x Briefs ) 300 yards Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease \$C\$3 Decision Variables Boxers E \$D\$3 Decision Variables Briefs E+30 Constraints Final Shadow Constraint Allowable Allowable Only Binding Constraint Cell Name Value Price R.H. Side Increase Decrease \$E\$6 Material \$E\$7 Logos E \$E\$8 Labor E \$E\$9 Demand E Every boxer contributes \$10.00 Every brief contributes \$1.00 If we make 1 brief we get \$1 more profit, but we lose 0.5 yards of material, which costs us 0.5 boxers (i.e. \$5 of boxer profits) Thus the marginal loss is \$4 (= \$1 - \$5). Notice that the labor, logo, and demand constraints weren t binding so producing an extra pair of briefs doesn t cost us anything there! 52 26

27 Multiple Optima We saw earlier that one indicator of multiple optima was a zero allowable increase and decrease, since that implied that the objective function overlapped with a line segment (rather than merely being tangent to a single point), and thus could not pivot within a range around a single point. As second indicator of multiple optima is finding a decision variable with a Final Value (at the optimum) of zero, and a Reduced Cost of zero. This means that the variable can be forced into the optimal solution at no cost, and therefore an alternative optimum is available. 53 Reduced Cost in Maximization vs Minimization Problems Reduced cost tells you the effect on the objective function value of forcing a variable into the optimal solution. Forcing a variable into the optimal solution always worsens the solution, irrespective of whether its a max or min problem. However: a worse solution in a max problem involves a lower optimal value (i.e. negative reduced cost) a worse solution in a min problem involves a higher optimal value (i.e. positive reduced cost). This is why: max problems have negative reduced costs min problems have positive reduced costs 54 27

28 Changing Multiple Parameters Simultaneously: The 100% Rule We've only changed on parameter at a time. What happens if we change more than one? Use the 100% rule for simultaneous changes in constraint RHSs and Decision Variable coefficients. Calculate each change as a percentage (%) of its respective allowable increase/decrease. If the accumulated (absolute value) % changes are less than 100%, then you sum their shadow price / reduced cost impacts. Otherwise, you must recalculate the LP 55 The 100% Rule 100% also holds for objective coefficients. Can also be combined for changes in constraint functions. If a single value is outside of range, or if the sum of ratios > 1, then you need to re-compute a solution (resolve the LP) for new constraints. Remember: Change constraint functions, production mix changes Change objective function (within allowable increase / decrease), production mix does not

29 The 100% Rule Example If material decreases from 300 to 290 yards and labor increases from 900 to 1000 hours, what is the change in weekly contribution? 100 / labor hours + 10 / 52.5 material = = < So, change in Objective Function Value = (\$1/hr x 100 hrs) - (\$5/yard x 10 yards) = \$50 So, new Objective Function Value = \$2,450 (= \$2,400 + \$50) NOTE: The solution changes to boxers and briefs (need to resolve to get this information) % Rule Example: Pricing out a New Product Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease \$D\$4 Yards_Material \$D\$5 Logos E \$D\$6 Hours_Labor \$D\$7 Demand_Briefs E Product designers offer a new line of padded briefs that require 1 yard of material and 2 hours of labor. Contribution would be \$6.00 per brief. Should management introduce this line? Percentage changes: 1 / 52.5 material 2 / 60 labor hours = = < Loss from one padded brief = reducing relevant constraints (\$5.00/yard x 1 yard) + (\$1.00/hour x 2 hours) = \$7.00 Do not produce! \$7.00 cost exceeds the \$6.00 contribution 58 29

30 SolverTable SolverTable is an Excel Add-In that allows you to produce 2-way data tables which look at the sensitivity of the optimum solution to changes in any 2 parameters. Question: How does SolverTable differ from a regular 2-way data table? Answer: They re pretty much the same, except: SolverTable reruns Excel Solver for each combination of parameter values, whereas a regular 2-way data table could not do that. SolverTable does not auto-update it merely pastes values. So SolverTable would need to be rerun if other model parameters (besides the 2 you are testing) change. In contrast, a regular 2-way data table uses formulas, and the values of these formulas automatically update as model parameters change. 59 SolverTable Getting it: Download it from solver_table.html Installing It: Follow the instruction on the web-page above. Then open Excel and go to Tools Add-Ins Solver Table Add-in Using it: Lay out your 2-way table, putting the formula to evaluate under the different scenarios in the top left hand corner, like you would in a regular 2-way data table. Go to Tools SolverTable Warning: Because of the complexity of the 2-way data tables and SolverTables in the Boxers and Briefs example spreadsheet, it could take you up to 20 minutes to run the scenario analysis. Press Escape repeatedly at any time, or Ctrl+Break, if you wish to terminate the analysis

31 SolverTable Example Changing Multiple Decision Variables Coefficients in a Constraint The example below shows the results of running SolverTable to investigate the effects of changing per-unit labor requirements for boxers and briefs on the optimal solution. In other words, it shows the effect of changing the coefficients of the decision variables in the labor constraint. The top table shows the effect on the optimal solution value (i.e. optimal profit in dollars). The bottom table shows the effect on the optimal product mix. You can see that increasing the number of hours required per product type (boxers or brief) decreases the amount of that product type in the optimal mix. Effect on Optimal Solution Value Effect on Optimal Product Mix 61 Key Points Constraint Right Hand Sides: Slack and Shadow Price Changes within the allowable / never alter whether a constraint is (non) binding. Change constraint RHS: value of decision variables (product mix) changes. Objective Function Coefficients: Reduced Cost Changes within the allowable / never alter the optimal vertex. Changes within the allowable / : value of decision variables (product mix) does not change

32 Optimal product mix (Optimal decision variable values) Sensitivity Analysis Report Summary Non-zero if the variable is not in the solution. Zero if it is. Allowable objective function coefficient range: Solution vertex stays the same. Usage of resource (Left Hand Side of constraint) Allowable constraint RHS range: Shadow price is valid. Increase in optimal objective function value per unit increase in right hand side (RHS) of constraint. Z = (shadow price) ( RHS) 63 Sensitivity Analysis Why Should You Care? Uncertainty welcome to the real world. Slack what to do with unused resources? Iteration constraints are rarely fixed e.g. begin with a budget allocation and then evaluate alternatives 64 32

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