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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. 2 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. 3 4.2 Sensitivity Analysis Using Graphs High Note Sound Company • Manufactures quality CD players and stereo receivers. • Each product requires skilled craftsmanship. • LP problem formulation: Objective: maximize profit = $50C + $120R subject to 2C + 4R 80 (Hours of electricians' time available) 3C + R 60 (Hours of audio technicians' time available) C, R 0 (Non-negativity constraints) Where: C = number of CD players to make. R = number of receivers to make. 4 High Note Sound Company Problem Solution 5 Changes in Objective Function Coefficient Impact of price change of Receivers If 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) 6 Changes in Objective Function Coefficient Impact of price change of Receivers If receiver’s 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). 7 4.3 Sensitivity Analysis Using Solver • In Answer Report • Final Values: objective function, decision variables. • Binding and nonbinding constraints • Slack: Unused resource. • In Sensitivity Report • Adjustable Cells: – Objective Function Coefficients – Reduced Cost – Allowable Changes • Constraints: – Shadow Price – Allowable Changes 8 High Note Sound Company Answer Report 9 High Note Sound Company Answer Report • 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. • Binding and Non-binding Constraints: – Electricians’ time constraint is binding. – Audio technicians’ time constraint is non-binding. • 40 unused hours of audio technicians’ time are referred to as slack. 10 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 to simultaneous changes in several input data values. 11 High Note Sound Company Sensitivity Report 12 Changes in Constraint Right-hand-side (RHS) • Primary information is provided by Shadow Price • Shadow Price is change in optimal objective function value for one unit increase in RHS. • The shadow price is positive for binding constraints and is zero for nonbinding constraints. 13 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. 14 Changes in Right-hand-side (RHS) High Note Sound Company Constraints Cell Name Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease $D$8 Electricians' Time 80.00 30.00 80.00 160.00 80.00 $D$9 Audio Technicians' Time 20.00 0.00 60.00 1E+30 40.00 • In case of electrician hours, shadow price is $30. • For each additional hour of electrician time that firm can increase profits by $30. • The range of RHS for electrician time with a shadow price of $30 is (0, 240). • How to calculate shadow price and range? Excel. 15 Change in RHS of Nonbinding Constraint Constraints Cell • Name Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease $D$8 Electricians' Time 80.00 30.00 80.00 160.00 80.00 $D$9 Audio Technicians' Time 20.00 0.00 60.00 1E+30 40.00 In case of audio technicians’ time, shadow price is zero. – Audio technicians’ time has 40 unused hours. – No interest in acquiring additional hours of resource. – Allowable increase for RHS value is infinity. • Allowable decrease for RHS value is 40. – 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. • The range of RHS for audio technicians’ time with a shadow price of $0 is (20, infinite). 16 Change in Objective Function Coefficient (OFC) Adjustable Cells • Reduced Cost value - shows the difference between the marginal contribution of a decision variable and the marginal worth of the resources it uses. – Objective Function Coefficients – Shadow Prices and Resources Used • Allowable Increase and Allowable Decrease – the limits to which the objective function coefficient of a decision variable can be changed without affecting the optimality of the current solution. 17 Change in Objective Function Coefficient (OFC) High Note Sound Company 18 Change in Objective Function Coefficient (OFC) Reduced Cost for each CD player • The marginal contribution is the objective coefficient $50. • The marginal worth of the resources used: – Resources Used: 2 hours of electrician time and 3 hours of audio technician’s time. – Shadow Prices: $30 for per hour of electrician time and $0 for per hour of audio technician time. – Marginal Cost: 2 x $30 + 3 x $0 = $60. • Reduced Cost: $60 - $50 = $10 • Current value is 0. If one makes 1, firm will lose $10. 19 Change in Objective Function Coefficient (OFC) Allowable Increase and Decrease for the coefficient of CD players • Allowable Increase - indicates if the price of CD players 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! 20 Change in Objective Function Coefficient (OFC) Reduced Cost for each Stereo Receiver • The marginal contribution is the objective coefficient $120. • The marginal worth of the resources used: – Resources Used: 4 hours of electrician time and 1 hours of audio technician’s time. – Shadow Prices: $30 for per hour of electrician time and $0 for per hour of audio technician time. – Marginal Cost: 4 x $30 + 0 x $0 = $120. • Reduced Cost: $120 - $120 = $0. 21 Change in Objective Function Coefficient (OFC) Allowable Increase and Decrease for each Stereo Receiver • Allowable Increase - infinity (1E+30) indicates if $120 is profitable enough to make receiver – any price above it will also be profitable. • Allowable Decrease – $20 indicates if the price of receivers drops below than $100, it is not optimal to produce 20 receivers and no CDs. 22 4.4 Sensitivity Analysis For A Larger Maximization Example • Anderson Electronics Considering producing four potential products: VCRs, stereos, televisions (TVs), and DVD players: Profit per unit: VCR Stereo TV DVD $29 $32 $72 $54 Electronic Components Non-electronic Components Assembly time (hours) Selling price (per unit) VCR Stereo TV DVD Supply Cost 3 2 4 2 4 4 3 3 4,700 4,500 $7 $5 1 $70 1 $80 2,500 $10 3 2 $150 $110 23 Anderson Electronics LP Formulation Objective: maximize profit = $29 V + $32 S + $72 T + $54 D subject to 3 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 Where: V S T D = = = = (Electronic components) (Non-electronic components) (Assembly time in hours) number of VCRs to produce. number of Stereos to produce. number of TVs to produce. number of DVD players to produce. 24 Excel Solver Answer Report Target Cell (Max) Cell $F$8 Name Profit Original Value $0.00 Final Value $69,400.00 Adjustable Cells Cell Name Original Value Final Value $B$5 Solution value VCR 0.00 0.00 $C$5 Solution value Stereo 0.00 380.00 $D$5 Solution value TV 0.00 0.00 $E$5 Solution value DVD 0.00 1060.00 Constraints Cell Name Cell Value Formula Status $F$10 Electronic comp 4700.00 $F$10<=$H$10 Binding $F$11 Non-electronic comp 3940.00 $F$11<=$H$11 Not Binding $F$12 Assembly time 2500.00 $F$12<=$H$12 Binding Slack 0.00 560.00 0.00 25 Excel Solver Sensitivity Report Adjustable Cells Cell Name $B$5 Solution value VCR $C$5 Solution value Stereo $D$5 Solution value TV $E$5 Solution value DVD Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 0.00 -1.00 29.00 1.00 1E+30 380.00 0.00 32.00 40.00 1.67 0.00 -8.00 72.00 8.00 1E+30 1060.00 0.00 54.00 10.00 5.00 Constrains Cell Name Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease $F$10 Electronic comp 4700.00 2.00 4700.00 2800.00 950.00 $F$11 Non-electronic comp 3940.00 0.00 4500.00 1E+30 560.00 $F$12 Assembly time 2500.00 24.00 2500.00 466.67 1325.00 26 Excel Solver Sensitivity Report Adjustable Cells Non 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) 27 Excel Solver Sensitivity Report Zero 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 $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 $80: Objective Coefficient – Allocable Decrease ($72 - infinity) and Objective Coefficient – Allocable Increase ($72 + $8). 28 Excel Solver Sensitivity Report Constraints Nonzero Shadow Prices: • Electronic Components, Shadow price $2 – Each additional unit of electronic components will allow Anderson to increase its profit by $2. – The shadow price is $2 for RHS between (3750, 7500). RHS - Allocable Increase (4700 + 2800) and RHS - Allocable Decrease (4700 - 950). 29 Excel Solver Sensitivity Report Constraints Nonzero Shadow Prices: • Assembly Time, Shadow price $24 – Each additional hour of assembly time will allow Anderson to increase its profit by $24. – The shadow price is $24 for RHS between (1175, 2966.67). RHS - Allocable Increase (2500 + 466.67) and RHS - Allocable Decrease (2500 - 1325). 30 Excel Solver Sensitivity Report Constraints Zero Shadow Price: • Non-electronic components, Shadow price $0 – Nonbinding constraint, 560 units of unused resources – The shadow price is $0 for RHS between (3940, infinite). RHS - Allocable Increase (infinite) and RHS - Allocable Decrease (4500 - 560). 31 4.5 Simultaneous Changes Using the 100% Rule Possible 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 units in electronic component, 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. 32 4.5 Simultaneous Changes In Parameter Values Anderson 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). 33 4.5 Simultaneous Changes In OFC Values Anderson Electronics • 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. 34 4.6 Pricing-out New Variables • 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)? 35 Could Anderson Electronics Propose a New Product? Anderson Electronics • Anderson Electronics considers a new product, home-theater system (HTS). Could the company propose this new product? • Answer to such question involves a procedure called pricing-out. 36 Pricing-Out Procedure 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. • The actual cost is 5 x $7 + 4 x $5 + 4 x 10 = $95. • The net profit is $175 - $95 = $80. 37 Pricing-out procedure Home-Theater System (HTS) • Resources required to make this player: – No longer available to meet existing production plan (380 stereos and 1060 DVD players) for $69,400 total profit. • Checking validity of the 100% Rule: Calculate ratio of reduction in each resource’s availability to allowable decrease for that resource. Sum of ratios = 5/950 + 4/560 + 4/1325 = 0.015 < 1 • Profit loss if the resources are used for 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. 38 Pricing-out procedure Home-Theater System (HTS) • 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. • The unit profit of HTS is $80. Therefore, Anderson Electronics should not propose this new product. 39 Revised Excel Layout Anderson Electronics V S T T H VCR Stereo TV DVD HTS Solution value 0.00 380 0.00 1060 0.00 Selling price per unit $70 $80 $150 $110 $175 $147,000 <-- Revenue Cost price per unit $41 $48 $78 $56 $95 $77,600 <-- Cost Profit $29 $32 $72 $54 $80 $69,400 <-- Objective Constraints Cost Electronic comp 3 4 4 3 5 4700.00 <= 4700 $7 Non-electronic comp 2 2 4 3 4 3940.00 <= 4500 $5 Assembly time 1 1 3 2 4 2500.00 <= 2500 $10 LHS Sign RHS 40 Revised Excel Solver Sensitivity Report Adjustable Cells Cell Name Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 0.00 -1.00 29.00 1.00 1E+30 380.00 0.00 32.00 40.00 1.67 0.00 -8.00 72.00 8.00 1E+30 Solution value DVD 1060.00 0.00 54.00 10.00 5.00 Solution value HTS 0.00 -26.00 80.00 26.00 1E+30 $B$5 Solution value VCR $C$5 Solution value Stereo $D$5 Solution value TV $E$5 $F$5 Constraints Cell Name Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease $G$10 Electronic comp 4700.00 2.00 4700.00 2800.00 950.00 $G$11 Non-electronic comp 3940.00 0.00 4500.00 1E+30 560.00 $G$12 Assembly time 2500.00 24.00 2500.00 466.67 1325.00 41 4.7 Sensitivity Analysis - Minimization Example Burn-Off Diet Drink • Plans to introduce miracle drink that will magically burn fat away. Ingredient A Ingredient B Ingredient C Ingredient D Requirement Chemical X 3 4 8 10 At least 280 units Chemical Y 5 3 6 6 At least 200 units Chemical Z 10 25 20 40 At most 1,050 units 4 cents 7 cents 6 cents 3 cents Cost per ounce 42 Burn-Off Diet Drink LP Formulation Objective: minimize daily dose cost in cents. 4A + 7B + 6C + 3D Subject to A + B + C + D 36 (Daily dose requirement) 3A + 4B + 8C + 10D 280 (Chemical X requirement) 5A + 3B + 6C + 6D 200 (Chemical Y requirement) 10A + 25B + 20C + 40D 1050 (Chemical Z max limit) A, B, C, D 0 43 Excel Solution A B C D Ingr A Ingr B Ingr C Ingr D 10.250 0.000 4.125 21.625 4 7 6 3 130.625 Daily dosage 1 1 1 1 36.00 >= 36 Chemical X 3 4 8 10 280.00 >= 280 Chemical Y 5 3 6 6 205.75 >= 200 Chemical Z 10 25 20 40 1050.00 <= 1050 LHS Sign RHS Number of ounces Cost (cents) <-- Objective Constraints 44 Solver Answer Report Burn-Off Diet Drink Target Cells Cell $F$6 Name Cost (cents) Original Value 0.000 Final Value 130.625 Adjustable Cells Cell Name Original Value Final Value $B$5 Number of ounces Ingr A 0.000 10.250 $C$5 Number of ounces Ingr B 0.000 0.000 $D$5 Number of ounces Ingr C 0.000 4.125 $E$5 Number of ounces Ingr D 0.000 21.625 Constraints Cell Name Cell Value Formula Status Slack 1050.000 $F$11<=$H$11 Binding 0.000 36.000 $F$8>=$H$8 Binding 0.000 Chemical X 280.000 $F$9>=$H$9 Binding 0.000 Chemical Y 205.750 $F$10>=$H$10 Not Binding 5.750 $F$11 Chemical Z $F$8 Daily dosage $F$9 $F$10 45 Solver Sensitivity Report Adjustable Cells Cell Name Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease $B$5 Number of ounces Ingr A 10.250 0.000 4.000 3.500 2.500 $C$5 Number of ounces Ingr B 0.000 5.688 7.000 1E+30 5.688 $D$5 Number of ounces Ingr C 4.125 0.000 6.000 15.000 2.333 $E$5 Number of ounces Ingr D 21.625 0.000 3.000 3.800 1E+30 Constraints Cell Name Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 1050.000 -0.238 1050.000 47.143 346.000 36.000 3.750 36.000 16.500 1.278 Chemical X 280.000 0.875 280.000 41.000 11.000 Chemical Y 205.750 0.000 200.000 5.750 1E+30 $F$11 Chemical Z $F$8 Daily dosage $F$9 $F$10 46 Change in Objective Function Coefficient (OFC) Nonzero Reduced Cost: Ingredient B • The reduced cost of ingredient B is $5.688. – Each ounce of ingredient B used to make the drink will cause the total cost per daily dosage to increase by 5.688 cents. – The current cost of ingredient B is 7 cents. If the cost of ingredient B is lower by 5.688 cents, then it becomes costeffective to use this ingredient. – When the cost of ingredient B is above 1.312 cents (=75.688), the current corner point solution remains optimal. 47 Change in Objective Function Coefficient (OFC) Zero Reduced Cost: Ingredient C • The reduced cost of ingredient C is $0. – The current cost of ingredient C is 6 cents per ounce. The range for the cost coefficient of this ingredient is between 3.667 cents (=6-2.333) and 21 cents (=6+15). – When the cost of ingredient C is between this range, the current corner point solution remains optimal. 48 Changes in Right-hand-side (RHS) Nonzero Shadow Price: Chemical X • The shadow price of chemical X is 0.875. • For each additional unit of chemical X required to be present in the drink, the total cost will increase by 0.875 cents. • The shadow price remains to be 0.875 if the requirement for chemical X is between 269 units (=280-11) and 321 units (=280+41). 49 Changes in Right-hand-side (RHS) Nonzero Shadow Price: Chemical Z • The shadow price of chemical Z is -0.238. • Each unit increase in the maximum limit allowed for chemical Z will reduce the total cost by 0.238 cents. • The shadow price remains to be -0.238 if the maximum limit is between 704 units (=1050-346) and 1097.143 units (=1050+47.143). 50 Simultaneous Changes In Parameter Values Burn-Off can decrease the minimum requirement for chemical X by 5 units provided the maximum limit allowed for chemical Z is reduced by 50 units. • The sum of each proportion of change to allowable change is 5/11 + 50/346 = 0.399 < 1 • Since sum does not exceed 1, information provided in sensitivity report is valid to analyze impact of changes. • The reduced cost from the change in chemical X is 0.875 x 5 = 4.375 cents. • The reduced cost from the change in chemical Z is 0.238 x 50 = 11.9 cents. • The net impact is an increase in total cost of 7.525 cents (=11.9-4.375). 51 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. 52