MGSC 1205 Quantitative Methods I Class Seven Sensitivity Analysis Ammar Sarhan 1
How can we handle changes? We have solved LP problems under deterministic assumptions. find an optimum solution given certain constant parameters (costs, price, time, etc) How well do we know these parameters? Usually not very accurately rough estimates Conditions in most world situations are dynamic & changing prices of raw materials change product supply changes new machinery is bought to replace old employee turnover occurs 2
Sensitivity Analysis Post-optimality analysis: examining changes after the optimal solution has been reached. input data are varied to assess optimal solution sensitivity. Basic Question: How does our solution change as the input parameters change? How much does the objective function change? How much do the optimal values of the decision variables change? Do our results remain valid (If the parameters change...)? 3
Example: High Note Sound Company The company Manufactures quality CD players and stereo receivers. Each CD player sold results in $50 profit, while each receiver yields $120 profit. Each product requires skilled craftsmanship. Each CD player requires: 2 hours electrician s time and 3 hours technician s time Each receiver requires: 4 hours electrician s time and 1 hour technician s time Hours available: 80 for electrician s time, 60 for technician s time Objective: maximize profit 4
B5:C5 D6 D8:D9 5
Answer Analysis Target Cell (Max) Cell Name Original Value Final Value $D$6 Profit $0.00 $2,400.00 Cell Name Original Value Final Value $B$5 Solution value CD players 0.00 0.00 $C$5 Solution value Stereo receivers 0.00 20.00 This column indicates whether a constraint is exactly satisfied (LHS=RHS) Cell Name Cell Value Formula Status Slack $D$8 Electricians' Time 80.00 $D$8<=$F$8 Binding 0.00 $D$9 Audio Technicians' Time 20.00 $D$9<=$F$9 Not Binding 40.00 6
Answer Report Target Cell (Max) Cell Name Original Value Final Value $D$6 Profit $0.00 $2,400.00 Cell Name Original Value Final ValueThis column indicates $B$5 Solution value CD players 0.00 0.00 the amount of unused $C$5 Solution value Stereo receivers 0.00 20.00 resource Cell Name Cell Value Formula Status Slack $D$8 Electricians' Time 80.00 $D$8<=$F$8 Binding 0.00 $D$9 Audio Technicians' Time 20.00 $D$9<=$F$9 Not Binding 40.00 7
Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Final Shadow Constraint Allowable Allowable Cell Name 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 This table presents information regarding the impact of changes to the OFCs (i.e., the unit profits of $50 & $120) Allowable Increases & Allowable Decreases: they are the range of values for which we can change the OFCs, and still have current Corner Point remain as Optimal Solution This is the whole point of doing the analysis! 8
Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Final Shadow Constraint Allowable Allowable Cell Name 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 Allowable Increases & Allowable Decreases: they are the range of values for which we can change the OFCs, and still have current Corner Point remain as Optimal Solution 9
Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Final Shadow Constraint Allowable Allowable Cell Name 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 This table presents information on impact of changes in RHS. Final Value: the values of how much of each resource (constraint) is used up in reaching the optimal solution - Electricians time is binding - Technicians time is non-binding with Slack = 40. Constraint RHS: the value input for the RHS of each constraint equation 10
Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Final Shadow Constraint Allowable Allowable Cell Name 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 Changes in RHS usually affect the size of the feasible region. - If the size of feasible region increases, optimal objective function improves - If the size of feasible region decreases, optimal objective function worsens Relationship expressed as Shadow Price. Impact of changes in RHS values is measured by the Shadow Price. 11
Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Final Shadow Constraint Allowable Allowable Cell Name 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 Shadow Price is the change in optimal objective function value for one unit increase in RHS. Electrician s time is binding constraint; If Electrician s time increases, more products can be made, and profit will go up The amount of profit will change by for each additional unit of the binding resource is equal to the Shadow Price. For each additional hour of Electrician s time that firm can increase will increase the profit by $30. 12
Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Final Shadow Constraint Allowable Allowable Cell Name 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 Allowable Increases & Allowable Decreases: Validity range for the shadow price. - For what level of increase of RHS value of the electricians time constraint is the shadow price of $30 valid? - The shadow price is valid only as long as the change in the RHS is within the Allowable Increase & Allowable Decrease values. 13
Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Final Shadow Constraint Allowable Allowable Cell Name 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 Question: Assume we have an opportunity to get 50 additional hours of electricians time. However, this time will cost us an extra $20 per hour. Should we take it? Question: Can you solve the problem If we have 100 Electricians time If we have 60 Electricians time If we have 240 Electricians time If we have 0 Electricians time 14
Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Final Shadow Constraint Allowable Allowable Cell Name 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 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. Once 40 hours is lost (current unused portion, or slack) of technicians time, resource also becomes binding. Any additional loss of time will clearly have adverse effect on profit. 15
Changes in a Right-Hand Side Receivers 60 50 40 30 20 10 Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Final Shadow Constraint Allowable Allowable Cell Name 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 Technician s time = 60 (3C+ S= 60) (2C+4S = 80) Electricians time increases from 80 to 100 (2C+4S = 100) 0 20 40 60 80 CD players 16
Changes in a Right-Hand Side 60 Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease 50 $B$5 Solution value CD players 0.00-10.00 50.00 10.00 1E+30 $C$5 Solution value Stereo receivers 20.00 0.00 120.00 1E+30 20.00 Receivers 40 30 20 Final Shadow Constraint Allowable Allowable Cell Name 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 10 Electricians time decreases from 80 to 60 0 20 40 60 80 CD players 17
Changes in a Right-Hand Side Receivers 60 50 40 30 20 Summary in changes in RHS Changes in RHS usually affect the size of the feasible region. The location of the optimal corner point changes There is a range of values for each RHS in which the shadow price remains unchanged. 10 Electricians time = 80 0 20 40 60 80 CD players 18
Manufacturing Application Anderson Electronics is considering producing four potential products: VCR Stereo TV DVD Supply Elct. Components 3 4 4 3 4,700 Non-Elct. Components 2 2 4 3 4,500 Assembly time 1 1 3 2 2,500 Selling price $70 $80 $150 $110 Cost $41 $48 $78 $56 Profit $29 $32 $72 $54 Objective: Maximize profit 19
Questions Answer Report Anderson Electronics Answer Report Target Cell (Max) Cell Name Original Value Final Value $F$8 Profit $0.00 $69,400.00 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 Cell Name Cell Value Formula Status Slack $F$10 Electronic comp 4700.00 $F$10<=$H$10 Binding 0.00 $F$11 Non-electronic comp 3940.00 $F$11<=$H$11 Not Binding 560.00 $F$12 Assembly time 2500.00 $F$12<=$H$12 Binding 0.00 Question 1: What s the optimal production strategy? Question 2: What s maximum profit for A.E.? Question 3: Which resource is fully used up? Which constrain is exactly satisfied? 20
Questions Sensitivity Report Anderson Electronics Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value VCR 0.00-1.00 29.00 1.00 1E+30 $C$5 Solution value Stereo 380.00 0.00 32.00 40.00 1.67 $D$5 Solution value TV 0.00-8.00 72.00 8.00 1E+30 $E$5 Solution value DVD 1060.00 0.00 54.00 10.00 5.00 Final Shadow Constraint Allowable Allowable Cell Name 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 Question 1: What s the Shadow Price of electronic components? Question 2:What s the Allowable Range of RHS value of electronic components? Question 3: What s the impact on profit if we could increase the supply of non-electronic components by 400 units (to a total 4,900 units)? 21
Questions Sensitivity Report Anderson Electronics Sensitivity Report Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$5 Solution value VCR 0.00-1.00 29.00 1.00 1E+30 $C$5 Solution value Stereo 380.00 0.00 32.00 40.00 1.67 $D$5 Solution value TV 0.00-8.00 72.00 8.00 1E+30 $E$5 Solution value DVD 1060.00 0.00 54.00 10.00 5.00 Final Shadow Constraint Allowable Allowable Cell Name 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 Question 4: What s the impact on profit if we could increase the supply of electronic components by 400 units (to a total 5,100 units)? Question 5: What would happen if we could increase the supply of electronic components by 4000 units (to a total 8,700 units)? 22
Questions Sensitivity Report Question 6: For the question about getting an additional 400 units of electronic components, what would happen if the supplier of these 400 units wants $8 per unit (rather than the current cost of $7 per unit)? Question 7: Assume we have an opportunity to get 250 additional hours of assembly time. However, this time will cost us $15 per hour (rather than the current cost of $10 per hour). Should we take it? Question 8: If we force the production of VCR, what would be the impact on profit? Alternatively, how much profit must VCRs become before the firm should consider producing them? Question 9: Assume that there is uncertainty in the price of DVD. For what range of prices will the current production be optimal? Question 10: If DVD actually sold for less and profit per unit drops to $50, what would be the firm s new total profit? 23
Simultaneous Changes In Parameter Values Previous assumption: analyzing only one change at a time Possible to analyze impact of simultaneous changes on optimal solution only under specific condition 100% Rule: Σ (Change / Allowable change) 1 If sum does not exceed 1, information provided in sensitivity report is valid to analyze impact of changes. 24
Sensitivity Analysis Sensitivity analysis is used by management to answer a series of what if questions about inputs to LP model. Over what ranges can prices change without affecting the optimality of the present solution? Will the present solution remain the optimum solution if the amount of raw materials, production time, or storage space is suddenly changed? The amount of each type of resources needed to produce one unit of each type of product may vary slightly. Will such changes affect the optimal solution? 25
Sensitivity Analysis Sensitivity analysis is used to determine effects on the optimal solution within specified ranges for the objective function coefficients (OFCs), and right hand side (RHS) values. Basic Question: How does our solution change as the input parameters change? Allowable Range for OFCs Shadow prices Allowable Range for RHS values 26