Managerial Decision Modeling with Spreadsheets

Chapter 4 Linear Programming Sensitivity Analysis

Learning Objectives Understand, using graphs, impact of changes in objective function coefficients, right-hand-side values, and constraint coefficients on optimal solution of a linear programming problem. Generate answer and sensitivity reports using Excel's Solver. Interpret all parameters of reports for maximization and minimization problems. Analyze impact of simultaneous changes in input data values using 100% rule. Analyze impact of addition of new variable using pricing-out strategy.

4.1 Introduction Optimal solutions to LP problems have been examined under deterministic assumptions. Conditions in most real world situations are dynamic and changing.

After an optimal solution to problem is found, input data values are varied to assess optimal solution sensitivity. This process is also referred to as sensitivity analysis or post-optimality analysis.

4.2 Sensitivity Analysis Using GraphsHigh Note Sound Company Manufactures quality CD players and stereo receivers. Each product requires skilled craftsmanship. LP problem formulation:

Objective: maximize profit = $50C + $120Rsubject to 2C + 4R 80 3C + R 60 C, R 0 Where: C = number of CD players to make. R = number of receivers to make.(Hours of electricians' time available) (Hours of audio technicians' time available) (Non-negativity constraints)

High Note Sound Company Problem Solution

Changes in Objective Function CoefficientImpact of price change of ReceiversIf unit profit per stereo receiver (R) increased from $120 to $150, is corner point a still the optimal solution? YES ! But Profit is $3,000 = 0 ($50) + 20 ($150)

Changes in Objective Function CoefficientImpact of price change of ReceiversIf receivers profit coefficient changed from $120 to $80, slope of isoprofit line changes causing corner point (b) to become optimal. But Profit is $1,760 = 16 ($50) + 12 ($80).

Changes in Objective Function CoefficientImpact of price change of CDsIf CD players unit profit increases from $50, slope of isoprofit line changes. Optimal solution will change from corner point a to corner point b.

Changes in Right-hand-side (RHS) Value

May change feasible region size. May change or move corner points.

Increase in Electricians Available Time As available electricians time increases, corner points a and b will move closer to one other.

Further increases in available electricians timemay make this constraint redundant.

Decrease in Electricians Available Time As available electricians time decreases, corner points b and c move closer to one another from their current locations. Corner points b and c will no longer be feasible, and intersection of electricians time constraint with horizontal (C) axis will become a new feasible corner point.

Changes in Constraint Coefficients Changes in constraint coefficients (numbers in

left-hand-sides of constraint equations). Changes will have no effect on objective function but may produce significant change in shape of feasible solution region, and in optimal profit or cost.

Changes in Constraint Coefficients

4.3 Sensitivity Analysis Using Solver

Solver ReportHigh Note Sound Company

Answer ReportHigh Note Sound Company

Sensitivity Report Sensitivity report has two distinct components. (1) Table titled Adjustable Cells (2) Table titled Constraints. Tables permit one to answer several "what-if" questions regarding problem solution. Consider a change to only a single input data value.

Sensitivity information does not always apply tosimultaneous changes in several input data values.

Sensitivity ReportHigh Note Sound Company

Changes in Constraint Right-hand-side (RHS) Primary information is provided by Shadow Price Resources available: 80 hours of electricians time. 60 hours of audio technicians time.

Final Values in table reveal optimal solution requires: all 80 hours of electricians time. Only 20 hours of audio technicians time. Electricians time constraint is binding. Audio technicians time constraint is non-binding.

40 unused hours of audio technicians time are referred to as slack.

Changes in Right-hand-side (RHS)RHS of Binding Constraint If RHS of non-redundant constraint changes, size of feasible region changes. If size of region increases, optimal objective function improves. If size of region decreases, optimal objective function worsens. Relationship expressed as Shadow Price. Shadow Price is change in optimal objective function value for one unit increase in RHS.

Changes in Right-hand-side (RHS)High Note Sound Company

In case of electrician hours Shadow Price is $30.

For each additional hour of electrician time thatfirm can increase profits by $30.

Change in RHS of Non-binding Constraint

Audio technicians time has 40 unused hours. No interest in acquiring additional hours of resource. Shadow price for audio technicians time is zero. Allowable increase for RHS value is infinity (shown as 1E+30 by Solver).

Once 40 hours is lost (current unused portion, or slack) of audio technicians time, resource also becomes binding. Any additional loss of time will clearly have adverse effect on profit.

Change in Objective Function Coefficient (OFC)Adjustable Cells Reduced Cost value - shows amount one will loose if solution is forced to make an additional unit. Current value is 0. If one makes 1, firm will lose $10.

Allowable Increase - indicates if price increases by $10, one will profit by making additional CDs. Allowable Decrease infinity (1E+30) indicates if $50 is not attractive enough to make CD any price below it will not make it attractive either!

Change in Objective Function Coefficient (OFC)

4.4 Sensitivity Analysis For A Larger Maximization Example Anderson ElectronicsConsidering producing four potential products: VCRs, stereos, televisions (TVs), and DVD players: Profit per unit: VCR Stereo TV DVD $41 $32 $72 $54

Anderson Electronics LP FormulationObjective: maximize profit = $29 V + $32 S + $72 T + $54 Dsubject to3 V + 4 S + 4 T + 3 D 4700 2 V + 2 S + 4 T + 3 D 4500 1 V + 1 S + 3 T + 2 D 2500 V, S, T, D 0 (Electronic components) (Non-electronic components) (Assembly time in hours)

Where: V S T D

= = = =

number of VCRs to produce. number of Stereos to produce. number of TVs to produce. number of DVD players to produce.

Excel Solver Set-up and Solution

Excel Solver Answer Report

Excel Solver Sensitivity Report

Excel Solver Sensitivity ReportAdjustable CellsNon Zero value decision variables, Stereos and DVDs: Produce 380 Stereos with unit profit of $32. Decision should not change as profit is between $31.33 and $72: Objective Coefficient Allocable Decrease ($32 - $1.67) and Objective Coefficient Allocable Increase ($32+$40)

Produce 1060 DVDs with unit profit of $54. Decision should not change as profit is between $49 and $64: Objective Coefficient Allocable Decrease ($54 - $5) and Objective Coefficient Allocable Increase ($54+$10)

Excel Solver Sensitivity ReportZero value decision variables, VCRs and TVs:Produce 0 VCRs with unit cost of $1.00 (Reduced Cost).

Decision to make 0 should not change as profit is below $29 but should change over and $30:Objective Coefficient Allocable Decrease ($29 - infinity) and

Objective Coefficient Allocable Increase ($29 + $1).Produce 0 TVs with unit cost of $8.00 (Reduced Cost). Decision to make 0 should not change as profit is below $72 but should change over and $80: Objective Coefficient Allocable Decrease ($72 - infinity) and Objective Coefficient Allocable Increase ($72 + $8).

Constraints on the Sensitivity ReportAnderson Electronics

Constraints on the Sensitivity ReportAnderson Electronics

4.5 Simultaneous Changes In Parameter ValuesPossible to analyze impact of simultaneous changes on optimal solution only under specific condition: (Change / Allowable change) 1 If decrease RHS from 4,700 to 4,200, allowable decrease is 950. The ratio is: 500 / 950 = 0.5263 If increase 200 hours (from 2,500 to 2,700) in assembly time, allowable increase is 466.67. The ratio is: 200 / 466.67 = 0.4285 The sum of these ratios is: Sum of ratios = 0.5263 + 0.4285 = 0.9548 < 1 Since sum does not exceed 1, information provided in sensitivity report is valid to analyze impact of changes.

4.5 Simultaneous Changes In Parameter ValuesAnderson Electronics Decrease of 500 units in electronic component availability reduces size of feasible region and causes profit to decrease. Magnitude of decrease is $1,000 (500 units x $2 per unit).

Increase of 200 hours of assembly time results in larger feasible region and net increase in profit. Magnitude of increase is $4,800 (200 hours x $24 per hour).

Net impact of both changes simultaneously is an increase in profit by $3,800 ( $4,800 - $1,000).

4.5 Simultaneous Changes In Parameter ValuesAnderson Electronics Simultaneous Changes in OFC Values What is impact if selling price of DVDs drops by $3 per unit and at same time selling price of stereos increases by $8 per unit? For current solution to remain optimal, allowable decrease in DVD players is $5, while allowable increase in OFC for stereos is $40. Sum of ratios is: Sum of ratios = $3 / $5 + $8 / $40 = 0.80 < 1 $3 decrease in profit per DVD player causes total profit to decrease by $3,180 (i.e., $3 x 1,060). $8 increase in unit profit of each stereo results in total profit of $3,040 (i.e., $8 x 380).

Net impact is a decrease in profit of only $140 to a new value of $69,260.

4.6 Pricing-out New VariablesAnderson Electronics Information given in sensitivity report can be used to study impact of introduction of new decision variables (products). For example: If problem is re-solved with a new product in model, will it be recommend that a new product be made? Or, will it be recommend that a new product not be made, and continue making same products (that is, stereos and DVD players)?

Could Anderson Electronics Propose a New Product? Home-Theater System (HTS) Requires: 5 units of electronic components 4 units of non-electronic components 4 hours of assembly time.

Selling price: $175 per unit.Answer to such questions involves a procedure called pricing-out. Resources required to make this player: No longer available to meet existing production plan (380 stereos and 1060 DVD players) for $69,400total profit.

Checking Validity of the 100% RuleAnderson Electronics Calculate ratio of reduction in each resources availability to allowable decrease for that resource. Sum of ratios = 5/950 + 4/560 + 4/1325 = 0.015 < 1 Required Profit on Each HTS: 5 x shadow price of electronic components + 4 x shadow price of non-electronic components + 4 x shadow price of assembly time or 5 x $2 + 4 x $0 + 4 x $24 = $106 Profit contribution of each HTS has to at least make up shortfall in profit. OFC for HTS must be at least $106 in order for optimal solution to have non-zero value.

Revised Excel Layout and Solver Entries

Revised Excel Solver Sensitivity ReportAnderson Electronics

4.7 Sensitivity Analysis - Minimization ExampleBurn-Off Diet Drink Plans to introduce miracle drink that will magically burn fat away.

Burn-Off Diet Drink LP FormulationObjective: minimize daily dose cost in cents.

4A + 7B + 6C + 3DSubject toA + B + C + D 363A + 4B + 8C + 10D 280 5A + 3B + 6C + 6D 200

(Daily dose requirement)(Chemical X requirement) (Chemical Y requirement)

10A + 25B + 20C + 40D 1050 (Chemical Z max limit) A, B, C, D 0

Excel Solution

Solver Answer ReportBurn-Off Diet Drink

Solver Sensitivity ReportBurn-Off Diet Drink

Summary Sensitivity analysis used by management to answer series of what-if questions about LP model inputs. Tests sensitivity of optimal solution to changes: Profit or cost coefficients, and Constraint RHS values.

Explored sensitivity analysis graphically (with two decision variables). Discussed interpretation of information: In answer and sensitivity reports generated by Solver. In reports used to analyze simultaneous changes in model parameter values. Determine potential impact of new variable in model.