Linear Programming: Sensitivity Analysis and Interpretation of Solution

Transcription

1 8 Linear Programming: Sensitivity Analysis and Interpretation of Solution MULTIPLE CHOICE. To solve a linear programming problem with thousands of variables and constraints a personal computer can be use a mainframe computer is require the problem must be partitioned into subparts. unique software would need to be develope a Computer solution. A negative dual price for a constraint in a minimization problem means as the right-hand side increases, the objective function value will increas as the right-hand side decreases, the objective function value will increas as the right-hand side increases, the objective function value will decreas as the right-hand side decreases, the objective function value will decreas a Dual price. If a decision variable is not positive in the optimal solution, its reduced cost is what its objective function value would need to be before it could become positiv the amount its objective function value would need to improve before it could become positiv zero. its dual pric b Reduced cost. A constraint with a positive slack value will have a positive dual pric will have a negative dual pric will have a dual price of zero. has no restrictions for its dual pric c Slack and dual price

2 5. The amount by which an objective function coefficient can change before a different set of values for the decision variables becomes optimal is the optimal solution. dual solution. range of optimality. range of feasibility. c Range of optimality 6. The range of feasibility measures the right-hand-side values for which the objective function value will not chang the right-hand-side values for which the values of the decision variables will not chang the right-hand-side values for which the dual prices will not chang each of the above is tru c Range of feasibility 7. The % Rule compares proposed changes to allowed changes. new values to original values. objective function changes to right-hand side changes. dual prices to reduced costs. a Simultaneous changes 8. An objective function reflects the relevant cost of labor hours used in production rather than treating them as a sunk cost. The correct interpretation of the dual price associated with the labor hours constraint is the maximum premium (say for overtime) over the normal price that the company would be willing to pay. the upper limit on the total hourly wage the company would pay. the reduction in hours that could be sustained before the solution would chang the number of hours by which the right-hand side can change before there is a change in the solution point. a Dual price 9. A section of output from The Management Scientist is shown her 6 What will happen to the solution if the objective function coefficient for variable decreases by? Nothing. The values of the decision variables, the dual prices, and the objective function will all remain the sam The value of the objective function will change, but the values of the decision variables and the dual prices will remain the sam The same decision variables will be positive, but their values, the objective function value, and the dual prices will chang The problem will need to be resolved to find the new optimal solution and dual pric b Range of optimality

3 . A section of output from The Management Scientist is shown her Constrain t What will happen if the right-hand-side for constraint increases by? Nothing. The values of the decision variables, the dual prices, and the objective function will all remain the sam The value of the objective function will change, but the values of the decision variables and the dual prices will remain the sam The same decision variables will be positive, but their values, the objective function value, and the dual prices will chang The problem will need to be resolved to find the new optimal solution and dual pric d Range of feasibility. The amount that the objective function coefficient of a decision variable would have to improve before that variable would have a positive value in the solution is the dual pric surplus variabl reduced cost. upper limit. c Interpretation of computer output. The dual price measures, per unit increase in the right hand side, the increase in the value of the optimal solution. the decrease in the value of the optimal solution. the improvement in the value of the optimal solution. the change in the value of the optimal solution. c Interpretation of computer output. Sensitivity analysis information in computer output is based on the assumption of no coefficient chang one coefficient chang two coefficient chang all coefficients chang b Simultaneous changes. When the cost of a resource is sunk, then the dual price can be interpreted as the minimum amount the firm should be willing to pay for one additional unit of the resourc maximum amount the firm should be willing to pay for one additional unit of the resourc minimum amount the firm should be willing to pay for multiple additional units of the resourc maximum amount the firm should be willing to pay for multiple additional units of the resourc b Dual price

4 5. The amount by which an objective function coefficient would have to improve before it would be possible for the corresponding variable to assume a positive value in the optimal solution is called the reduced cost. relevant cost. sunk cost. dual pric a Reduced cost 6. Which of the following is not a question answered by sensitivity analysis? If the right-hand side value of a constraint changes, will the objective function value change? Over what range can a constraint s right-hand side value without the constraint s dual price possibly changing? By how much will the objective function value change if the right-hand side value of a constraint changes beyond the range of feasibility? By how much will the objective function value change if a decision variable s coefficient in the objective function changes within the range of optimality? c Interpretation of computer output TRUE/FALSE. Output from a computer package is precise and answers should never be rounde False Computer solution. The reduced cost for a positive decision variable is. True Reduced cost. When the right-hand sides of two constraints are each increased by one unit, the objective function value will be adjusted by the sum of the constraints dual prices. False Simultaneous changes. If the range of feasibility indicates that the original amount of a resource, which was, can increase by 5, then the amount of the resource can increase to 5. True Range of feasibility 5. The % Rule does not imply that the optimal solution will necessarily change if the percentage exceeds %. True Simultaneous changes 6. For any constraint, either its slack/surplus value must be zero or its dual price must be zero. True Dual price

5 5 7. A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objectiv True Dual price 8. Decision variables must be clearly defined before constraints can be written. True Model formulation 9. Decreasing the objective function coefficient of a variable to its lower limit will create a revised problem that is unbounde False Range of optimality. The dual price for a percentage constraint provides a direct answer to questions about the effect of increases or decreases in that percentag False Dual price. The dual price associated with a constraint is the improvement in the value of the solution per unit decrease in the right-hand side of the constraint. False Interpretation of computer output. For a minimization problem, a positive dual price indicates the value of the objective function will increas False Interpretation of computer output--a second example. There is a dual price for every decision variable in a model. False Interpretation of computer output. The amount of a sunk cost will vary depending on the values of the decision variables. False Cautionary note on the interpretation of dual prices 5. If the optimal value of a decision variable is zero and its reduced cost is zero, this indicates that alternative optimal solutions exist. True Interpretation of computer output 6. Any change to the objective function coefficient of a variable that is positive in the optimal solution will change the optimal solution. False Range of optimality 7. Relevant costs should be reflected in the objective function, but sunk costs should not. True Cautionary note on the interpretation of dual prices 8. If the range of feasibility for b is between 6 and 7, then if b = the optimal solution will not change from the original optimal solution. False Right-hand sides 9. The percent rule can be applied to changes in both objective function coefficients and right-hand sides at the same tim

6 6 False Simultaneous changes. If the dual price for the right-hand side of a < constraint is zero, there is no upper limit on its range of feasibility. True Right-hand sides SHORT ANSWER. Describe each of the sections of output that come from The Management Scientist and how you would use each. Interpretation of computer output. Explain the connection between reduced costs and the range of optimality, and between dual prices and the range of feasibility. Interpretation of computer output. Explain the two interpretations of dual prices based on the accounting assumptions made in calculating the objective function coefficients. Dual price. How can the interpretation of dual prices help provide an economic justification for new technology? Dual price 5. How is sensitivity analysis used in linear programming? Given an example of what type of questions that can be answere Sensitivity analysis 6. How would sensitivity analysis of a linear program be undertaken if one wishes to consider simultaneous changes for both the right-hand-side values and objective function. Simultaneous sensitivity analysis PROBLEMS. In a linear programming problem, the binding constraints for the optimal solution are 5X + Y < X + 5Y < Fill in the blanks in the following sentence: As long as the slope of the objective function stays between and, the current optimal solution point will remain optimal. Which of these objective functions will lead to the same optimal solution? ) X + Y ) 7X + 8Y ) 8X + 6Y ) 5X + 5Y Graphical sensitivity analysis

7 . The optimal solution of the linear programming problem is at the intersection of constraints and. Max s.t.. x + x x + x x + x x + x x, x < < > 6 Over what range can the coefficient of x vary before the current solution is no longer optimal? Over what range can the coefficient of x vary before the current solution is no longer optimal? Compute the dual prices for the three constraints. Graphical sensitivity analysis The binding constraints for this problem are the first and secon Min s.t. 7 x + x x + x x + x x + 5x x, x < > 75 Keeping c fixed at, over what range can c vary before there is a change in the optimal solution point? Keeping c fixed at, over what range can c vary before there is a change in the optimal solution point? If the objective function becomes Min.5x + x, what will be the optimal values of x, x, and the objective function? If the objective function becomes Min 7x + 6x, what constraints will be binding? Find the dual price for each constraint in the original problem. Graphical sensitivity analysis

8 8. Excel s Solver tool has been used in the spreadsheet below to solve a linear programming problem with a maximization objective function and all < constraints. Input Section Objective Function Coefficients X Y 6 s # # # 5 Avail Usage Slack.789E- -.69E Output Section s Profit # # # Give the original linear programming problem. Give the complete optimal solution. Spreadsheet solution of LPs

9 5. 9 Excel s Solver tool has been used in the spreadsheet below to solve a linear programming problem with a minimization objective function and all > constraints. Input Section Objective Function Coefficients X 5 Y s # # # Req' Usage Slack.5E E- Output Section s Profit # # # 6. Give the original linear programming problem. Give the complete optimal solution. Spreadsheet solution of LPs Use the spreadsheet and Solver sensitivity report to answer these questions. What is the cell formula for B? What is the cell formula for C? What is the cell formula for D? What is the cell formula for B5? What is the cell formula for B6? What is the cell formula for B7? g. What is the optimal value for x? h. What is the optimal value for x? i. Would you pay $.5 each for up to 6 more units of resource? j. Is it possible to figure the new objective function value if the profit on product increases by a dollar, or do you have to rerun Solver?

10 Input Information Var. Var. 5 Profit 5 (type) < < > Output Information s Profit Resources Avail. = Total Used Slk/Surp Microsoft Excel 7. Sensitivity Report Worksheet: [xs7basxls]sheet Changing Cells Cell $B$ $C$ Name s s Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease Shadow Price R.H. Side Allowable Increase s Cell $B$5 $B$6 $B$7 7. Name constraint Used constraint Used constraint Used Final Value.7698 Spreadsheet solution of LPs Use the following Management Scientist output to answer the questions. LINEAR PROGRAMMING PROBLEM MAX X+5X+X S.T. ) X+5X+X>9 ) 6X+7X+8X<5 ) 5X+X+X< OPTIMAL SOLUTION Allowable Decrease E+

11 Objective Function Value = 76. Value Reduced Cost X X X..889 Slack/Surplu s. Dual Price OBJECTIVE COEFFICIENT RANGES X X X No No. 5.. No RIGHT HAND SIDE RANGES No Give the solution to the problem. Which constraints are binding? What would happen if the coefficient of x increased by? What would happen if the right-hand side of constraint increased by? 8. Interpretation of Management Scientist output Use the following Management Scientist output to answer the questions. MIN X+5X+6X S.T. ) X+X+X<85 ) X+X+X>8 ) X+X+X> Objective Function Value = Value Reduced Cost X X X 8.5. Slack/Surplu s Dual Price

12 OBJECTIVE COEFFICIENT RANGES X X X No 6. No RIGHT HAND SIDE RANGES 8 No No What is the optimal solution, and what is the value of the profit contribution? Which constraints are binding? What are the dual prices for each resource? Interpret. Compute and interpret the ranges of optimality. Compute and interpret the ranges of feasibility. 9. Interpretation of Management Scientist output The following linear programming problem has been solved by The Management Scientist. Use the output to answer the questions. LINEAR PROGRAMMING PROBLEM MAX 5X+X+5X S.T. ) X+5X+8X< ) 9X+5X+X<5 OPTIMAL SOLUTION Objective Function Value = 7 Value Reduced Cost X X X 8 Slack/Surplu s Dual Price..

13 OBJECTIVE COEFFICIENT RANGES X X X 9.86 No RIGHT HAND SIDE RANGES Give the complete optimal solution. Which constraints are binding? What is the dual price for the second constraint? What interpretation does this have? Over what range can the objective function coefficient of x vary before a new solution point becomes optimal? By how much can the amount of resource decrease before the dual price will change? What would happen if the first constraint's right-hand side increased by 7 and the second's decreased by 5?. Interpretation of Management Scientist output LINDO output is given for the following linear programming problem. MIN X + X + 9 X SUBJECT TO ) 5 X + 8 X + 5 X >= 6 ) 8 X + X + 5 X >= 8 END LP OPTIMUM FOUND AT STEP OBJECTIVE FUNCTION VALUE ) 8 VARIABLE VALUE REDUCED COST X X X 8 ROW SLACK OR SURPLUS DUAL PRICE ) ) - NO. ITERATIONS= RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ. COEFFICIENT RANGES

14 VARIABLE X X X CURRENT COEFFICIENT 9 INCREASE INFINITY 5 INFINITY DECREASE RIGHTHAND SIDE RANGES ROW INCREASE INFINITY DECREASE INFINITY 5 What is the solution to the problem? Which constraints are binding? Interpret the reduced cost for x. Interpret the dual price for constraint. What would happen if the cost of x dropped to and the cost of x increased to?. CURRENT RHS 6 8 Interpretation of LINDO output The LP problem whose output follows determines how many necklaces, bracelets, rings, and earrings a jewelry store should stock. The objective function measures profit; it is assumed that every piece stocked will be sol measures display space in units, constraint measures time to set up the display in minutes. s and are marketing restrictions. LINEAR PROGRAMMING PROBLEM MAX X+X+5X+5X S.T. ) ) ) ) X+X+X+X<8 X+5X+X< X+X<5 X+X+X>5 OPTIMAL SOLUTION Objective Function Value = 775. Value Reduced Cost X X X X Slack/Surplu s 6. Dual Price OBJECTIVE COEFFICIENT RANGES

15 5 X X X X 87.5 No No RIGHT HAND SIDE RANGES No Use the output to answer the questions. g. h. i. j. k. l.. How many necklaces should be stocked? Now many bracelets should be stocked? How many rings should be stocked? How many earrings should be stocked? How much space will be left unused? How much time will be used? By how much will the second marketing restriction be exceeded? What is the profit? To what value can the profit on necklaces drop before the solution would change? By how much can the profit on rings increase before the solution would change? By how much can the amount of space decrease before there is a change in the profit? You are offered the chance to obtain more spac The offer is for 5 units and the total price is 5. What should you do? Interpretation of Management Scientist output The decision variables represent the amounts of ingredients,, and to put into a blen The objective function represents profit. The first three constraints measure the usage and availability of resources A, B, and C. The fourth constraint is a minimum requirement for ingredient. Use the output to answer these questions. g. h. i. j. How much of ingredient will be put into the blend? How much of ingredient will be put into the blend? How much of ingredient will be put into the blend? How much resource A is used? How much resource B will be left unused? What will the profit be? What will happen to the solution if the profit from ingredient drops to? What will happen to the solution if the profit from ingredient increases by? What will happen to the solution if the amount of resource C increases by? What will happen to the solution if the minimum requirement for ingredient increases to 5? LINEAR PROGRAMMING PROBLEM MAX X+6X+7X

16 6 S.T. ) ) ) ) X+X+5X< X+X+X<8 5X+5X+8X<6 +X> OPTIMAL SOLUTION Objective Function Value = 66. Value Reduced Cost X X X 6.. Slack/Surplu s 8.. Dual Price. -.6 OBJECTIVE COEFFICIENT RANGES X X X No.75 No No 9.6 RIGHT HAND SIDE RANGES No No 6. Interpretation of Management Scientist output The LP model and LINDO output represent a problem whose solution will tell a specialty retailer how many of four different styles of umbrellas to stock in order to maximize profit. It is assumed that every one stocked will be sol The variables measure the number of women's, golf, men's, and folding umbrellas, respectively. The constraints measure storage space in units, special display racks, demand, and a marketing restriction, respectively. MAX X + 6 X + 5 X +.5 X SUBJECT TO ) X + X + X + X <= ).5 X + X <= 5 ) X + X + X <= 7 5) X + X >= END

17 7 OBJECTIVE FUNCTION VALUE ) 8. VARIABLE VALUE REDUCED COST X X X X 6.5 ROW SLACK OR SURPLUS DUAL PRICE ) ) ) 5) RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ. COEFFICIENT RANGES VARIABLE X X X X CURRENT COEFFICIENT INCREASE.5.5 INFINITY DECREASE.5 INFINITY.5.5 RIGHTHAND SIDE RANGES ROW 5 CURRENT RHS 5 7 INCREASE 8 INFINITY DECREASE 6 8 Use the output to answer the questions. g. h. i. j. k. l. How many women's umbrellas should be stocked? How many golf umbrellas should be stocked? How many men's umbrellas should be stocked? How many folding umbrellas should be stocked? How much space is left unused? How many racks are used? By how much is the marketing restriction exceeded? What is the total profit? By how much can the profit on women's umbrellas increase before the solution would change? To what value can the profit on golf umbrellas increase before the solution would change? By how much can the amount of space increase before there is a change in the dual price? You are offered an advertisement that should increase the demand constraint from 7 to 86 for a total cost of $. Would you say yes or no? Interpretation of LINDO output

18 8. Eight of the entries have been deleted from the LINDO output that follows. Use what you know about linear programming to find values for the blanks. MIN 6 X X + X SUBJECT TO ) 5 X + 5 X + X >= ) X + X + 8 X >= END LP OPTIMUM FOUND AT STEP OBJECTIVE FUNCTION VALUE ) 6.5 VARIABLE VALUE REDUCED COST X X X ROW SLACK OR SURPLUS DUAL PRICE ) ) NO. ITERATIONS= RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ. COEFFICIENT RANGES VARIABLE X X X CURRENT COEFFICIENT INCREASE.5 5 DECREASE RIGHTHAND SIDE RANGES ROW 5. CURRENT RHS INCREASE DECREASE Interpretation of LINDO output Portions of a Management Scientist output are shown below. Use what you know about the solution of linear programs to fill in the ten blanks. LINEAR PROGRAMMING PROBLEM MAX X+9X+7X S.T. ) X+5X+X<5

19 9 ) X+X+X<6 ) X+X+X<8 ) X+X+X>6 OPTIMAL SOLUTION Objective Function Value = 6. Value Reduced Cost X X X..5 Slack/Surplu s Dual Price 5. OBJECTIVE COEFFICIENT RANGES X X X 5.. No No.5 RIGHT HAND SIDE RANGES Interpretation of Management Scientist output Note to Instructor: The following problem is suitable for a take-home or lab exam. The student must formulate the model, solve the problem with a computer package, and then interpret the solution to answer the questions. 6. A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape, the girl's Sea Sprite, and the boy's Trail Blazer. It is assumed that every bike ordered will be sold, and their profits, respectively, are, 5,, and. The LP model should maximize profit. There are several conditions that the store needs to worry about. One of these is space to hold the inventory. An adult s bike needs two feet, but a child's bike needs only one foot. The store has 5 feet of spac There are hours of assembly time availabl The child's bike need hours of assembly time; the Open Trail needs 5 hours and the Cityscape needs 6 hours. The store would like to place an order for at least 75 bikes. Formulate a model for this problem. Solve your model with any computer package available to you. How many of each kind of bike should be ordered and what will the profit be? What would the profit be if the store had more feet of storage space?

20 g. h. 7. If the profit on the Cityscape increases to $5, will any of the Cityscape bikes be ordered? Over what range of assembly hours is the dual price applicable? If we require 5 more bikes in inventory, what will happen to the value of the optimal solution? Which resource should the company work to increase, inventory space or assembly time? Formulation and computer solution A company produces two products made from aluminum and copper. The table below gives the unit requirements, the unit production man-hours required, the unit profit and the availability of the resources (in tons). Product Product Available Aluminum Copper 6 Man-hours Unit Profit 5 6 The Management Scientist provided the following solution output: OBJECTIVE FUNCTION VALUE = 5 VARIABLE VALUE REDUCED COST X X 6.. CONSTRAINT SLACK/SURPLUS DUAL PRICE.. RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ. COEFFICIENT RANGES VARIABLE X X CURRENT COEFFICIENT 5 6 INCREASE 5. DECREASE RIGHTHAND SIDE RANGES CONSTRAINT CURRENT RHS 6.. INCREASE. INFINITY. DECREASE... What is the optimal production schedule? Within what range for the profit on product will the solution in (a) remain optimal? What is the optimal profit when c = 7? Suppose that simultaneously the unit profits on x and x changed from 5 to 55 and 6 to 65 respectively. Would the optimal solution change? Explain the meaning of the "DUAL PRICES" column. Given the optimal solution, why should the dual price for copper be? What is the increase in the value of the objective function for an extra unit of aluminum?

21 Man-hours were not figured into the unit profit as it must pay three workers for eight hours of work regardless of the number of man-hours use What is the dual price for man-hours? Interpret. On the other hand, aluminum and copper are resources that are ordered as neede The unit profit coefficients were determined by: (selling price per unit) - (cost of the resources per unit). The units of aluminum cost the company $. What is the most the company should be willing to pay for extra aluminum? g. 8. Interpretation of solution Given the following linear program: MAX s.t. 5x + 7x x x + x x + x x, x < < < > The graphical solution to the problem is shown below. From the graph we see that the optimal solution occurs at x = 5, x =, and z = 6. X 8 X + X < 8 7 MAX 5X + 7X 6 X < 6 5 Optimal X = 5, X = Z = 6 X + X < Calculate the range of optimality for each objective function coefficient. Calculate the dual price for each resourc Introduction to sensitivity analysis Consider the following linear program: MAX x + x ($ Profit) X

22 s.t. x + x x + x x x, x < < 8 < > The Management Scientist provided the following solution output: OPTIMAL SOLUTION Objective Function Value = Value Reduced Cost X X.. Slack/Surplu s.6 Dual Price.. OBJECTIVE COEFFICIENT RANGES X X RIGHT HAND SIDE RANGES No What is the optimal solution including the optimal value of the objective function? Suppose the profit on x is increased to $7. Is the above solution still optimal? What is the value of the objective function when this unit profit is increased to $7? If the unit profit on x was $ instead of $, would the optimal solution change? If simultaneously the profit on x was raised to $5.5 and the profit on x was reduced to $, would the current solution still remain optimal? Interpretation of solution Consider the following linear program: MIN s.t. 6x + 9x ($ cost) x + x < 8 x + 7.5x > x > x, x >

23 The Management Scientist provided the following solution output: OPTIMAL SOLUTION Objective Function Value = 7. Value Reduced Cost X X.5. Slack/Surplu s.5 Dual Price OBJECTIVE COEFFICIENT RANGES X X No RIGHT HAND SIDE RANGES No 55.. What is the optimal solution including the optimal value of the objective function? Suppose the unit cost of x is decreased to $. Is the above solution still optimal? What is the value of the objective function when this unit cost is decreased to $? How much can the unit cost of x be decreased without concern for the optimal solution changing? If simultaneously the cost of x was raised to $7.5 and the cost of x was reduced to $6, would the current solution still remain optimal? If the right-hand side of constraint is increased by, what will be the effect on the optimal solution? Interpretation of solution SOLUTIONS TO PROBLEMS. -5/ and -/5 Objective functions ), ), and ).. < c <. 5 < c <.5 Dual prices are.5,.5,

24 ..8 < c < < c <.5 x = 5, x = 5, z = 75 s and will be binding. Dual prices are.,,. (The first and third values are negativ). Max s.t. 5. X + 6Y X + 5Y X + Y X + Y X, Y < < < > 6 8 The complete optimal solution is X =., Y =, Z = 7., S =, S =, S =.667 Min s.t. 5X + Y X + Y X + 5Y 9X + 8Y X, Y > > > > 6 5 The complete optimal solution is X = 9.6, Y = 7., Z = 76.8, S =, S = 5., S = 6. g. h. i. j. =B8*B =C8*C =B+C =B*B+C*C =B5*B+C5*C =B6*B+C6*C yes no 7. x =., x =, x =, s =, s =, s =., z = 76. s and are binding. The value of the objective function would increase by. The value of the objective function would decrease by x =, x = 8, x =, s = 5, s =, s =, Z = is binding. Dual prices are,, and -.5. They measure the improvement in Z per unit increase in each right-hand sid.5 < c < < c < 6 5 < c < As long as the objective function coefficient stays within its range, the current optimal solution point will not change, although Z coul 8 < b < - < b < 8 < b < As long as the right-hand side value stays within its range, the currently binding constraints will remain so, although the values of the decision variables could chang The dual variable values will remain the sam 9. x =, x =, x = 8, s =, s =, z = 7 s and are binding. Dual price =.. A unit increase in the right-hand side of constraint will increase the value of the objective function by..

25 5 As long as c <, the solution will be unchange 5 The sum of percentage changes is 7/8 + (-5)/(-5) < so the solution will not chang x =, x = 8, x =, s =, s =, z = 8 is binding. c would have to decrease by or more for x to become positiv Increasing the right-hand side by will cause a negative improvement, or increase, of in this minimization objective function. The sum of the percentage changes is (-)/(-) + /5 < so the solution would not chang. g. h. i. j k. l Say no. Although 5 units can be evaluated, their value (5) is less than the cost (5).. g. h. i. j rerun Z = 76 Z = 68. Z = 5. g. h. i. j. k. l Yes. The dual price is.5 for additional units. The value of the ad ()(.5)= exceeds the cost of.. It is easiest to calculate the values in this order. x =, x = 5, reduced cost =, reduced cost =, row slack =, row slack =, c allowable decrease =.5, allowable increase = infinity 5. x = because the reduced cost is positiv x = after plugging into the objective function The second reduced cost is. s = and s = from plugging into the constraints..

26 6 The fourth dual price is -6.5 from plugging into the dual objective function, which your students might not understand fully until Chapter 6. The lower limit for constraint is and for constraint is 58, from the amount of slack in each constraint. There are no upper limits for these constraints. 6. MAX X + 5 X + X + X SUBJECT TO ) X + X + X + X <= 5 ) 5 X + 6 X + X + X <= ) X + X + X + X >= 75 OBJECTIVE FUNCTION VALUE ) 685 VARIABLE VALUE REDUCED COST X X X X 75 ROW SLACK OR SURPLUS DUAL PRICE ) ) ) NO. ITERATIONS= RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ. COEFFICIENT RANGES VARIABLE X X X X CURRENT COEFFICIENT 5 INCREASE INFINITY DECREASE.5 INFINITY INFINITY RIGHTHAND SIDE RANGES ROW g. CURRENT RHS 5 75 INCREASE INFINITY 5 5 DECREASE 5 5 Order Open Trails, Cityscapes, 75 Sea Sprites, and Trail Blazers. Profit will be No. The $ increase is below the reduced cost. to 5 It will decrease by 5.

27 7. 7 h. Assembly tim 6 product, product, Profit = $5 Between $5 and $75; at $7 the profit is $58 No; total % change is 8 /% < % Dual prices are the shadow prices for the resources; since there was unused copper (because S = ), extra copper is worth $ $ $; this is the amount extra man-hours are worth The shadow price is the "premium" for aluminum -- would be willing to pay up to $ + $ = $ for extra aluminum g. 8. Ranges of optimality: / < c < 7 and 5 < c < 5/ Summarizing, the dual price for the first resource is, for the second resource is, and for the third is 9. x =. and x =., and z = $.. Optimal solution will not chang Optimal profit will equal $9.6. Because is outside the range of.5 to 9., the optimal solution likely would chang Sum of the change percentages is 5% + % = 9%. Since this does not exceed % the optimal solution would not chang. x =.5 and x =., and the objective function value = 7.. is within this range of to, so the optimal solution will not chang Optimal total cost will be $.. x can fall to.5 without concern for the optimal solution changing. Sum of the change percentages is 9.7%. This does not exceed %, so the optimal solution would not chang The right-hand side remains within the range of feasibility, so there is no change in the optimal solution. However, the objective function value increases by $.5.