MASSACHUSETTS INSTITUTE OF TECHNOLOGY

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Optimization Methods in Management Science (Spring 2007) Problem Set 4 Due March 8 th, 2007 at 4:30 pm. You will need 162 points out of 189 to receive a grade of 5. Problem 1: Phase 1 & Phase 2 (30 Points) The purpose of this problem is to give you practice solving a linear program completely. Donald Trump is too busy getting ready for Wrestlemania and worrying that he may soon look like Britney. Due to this he has no time to solve a linear program to decide how much to invest in three Projects: Project A is to determine how much to sell his casino for in Las Vegas. That is why it is non positive) Project B is a machine to create his new wives each time one ages past 30. Project C is a media campaign to destroy Rosie O Donnell and The View. The variables are a, b, and c. Don t be concerned if the model doesn t make much sense. In order to determine how much he should invest in these projects he has formulated the following linear program not in standard form. min : z = a+ b+ c St : a + 2b + 3c = 3 a + 2b + 6c = 5 a 0 bc, 0 He hires you, a bright student, (Kevin Federline was busy watching The View ) as his apprentice to solve the problem. He gives you the following set of instructions. Don t mess up or else you will be fired. Part A: Page 1 of 12

2 Convert the problem from a min to a max problem Part B: Convert the problem to standard form. Be sure to show all of your steps. Part C: Give an example of a constraint that if added would make the problem infeasible. Part D: Form the auxiliary Phase 1 Problem from the linear program you formulated in part B. Part E: Write out the initial phase one tableau and indicate the initial BFS. Part F: Using the phase one method find an initial BFS of the standard form problem. For each iteration write out the tableau and indicate the pivot element. Hint: You will need three pivots Part F: What is the optimal solution to the phase 1 problem? Part G: Write out the initial tableau for the Phase 2 problem. What is the initial BFS? Part H: Solve the Phase 2 part of the problem using the simplex method. Be sure to write out each of your tableaus and indicate the pivot element on each iteration. Part I: Are there multiple optimal solutions to this problem? Problem 2: Alex and Paris in Paris Texas (28 Points) The purpose of this problem is to get you to think about the geometric and algebraic interplay of the Simplex Method. Page 2 of 12

3 Paris Hilton and Alex Trebeck were having lunch at a café in Paris, Texas. (These were two of your top celebrity choices). After talking about their respective run-ins with the law, they decide to so something more fun. They solve a maximization problem using the simplex method. However halfway through, Paris spills her Jack and Coke on the sheet of paper they were using, blurring some of the entries (these unknown entries are shown with letters in the tableau). Their tableau is shown below: z x 1 x 2 x 3 x 4 x 5 x 6 x 7 RHS C I D E F A B H G 1 3 For each statement find the ranges of the parameters to make the statements correct. If there are multiple sets of ranges that can be used indicate only one. Part A: The current basic solution is feasible. Under this condition, write the current bfs. (You can write your answer in terms of the letters A to H.) Part B: The current tableau is a unique optimal solution to the problem. Part C: x 4 is the only candidate for entering the basis and the problem is unbounded from above. Part D: The current tableau is optimal; however, their also exists an extreme ray of alternative optimal solutions Part E: Assume for this part that D=100, I=100. The current tableau is optimal and there are exactly two BFS s that are optimal. Give two sets of conditions one for each variable that enters the basis first. Part G: Page 3 of 12

4 The current tableau is not optimal, x 6 is the only candidate to enter the basis and when the iteration is performed, x 3 will leave the basis. Part H: The current BFS is degenerate. Part I: The current BFS is degenerate, x 4 enters the basis, and when the pivot is performed, the objective will not improve. (Note that degeneracy does not always lead to the objective not improving.) Problem 3: Degeneracy (33 Points) Degeneracy is one of the most confusing topics. Students struggle with it each year. This problem was designed to help you work through the theoretical aspects of degeneracy in a through manor. Consider the following tableau for a maximization problem. Part A : z x 1 x 2 x 3 x 4 RHS What is the basic feasible solution that corresponds to this tableau? What are the basic and non-basic variables? Part B: Assume x 1,x 2 are slack variables. Draw the feasible region in the x 3 x 4 plane. Part C: Perform one iteration of the simplex method. What is the solution and objective value corresponding to the tableau that you just calculated? How does this solution differ from the basic feasible solution in part A? Part D: Using your graph explained what happened geometrically in part B? Part E: Page 4 of 12

5 Fill in the Blanks: In this problem I just learned that when Blank A is present it is possible to have a Blank B reduced cost and still be at an Blank C. Part F: Now consider a similar tableau. z x 1 x 2 x 3 x 4 RHS Suppose you choose to pivot x 4 into the basis. Perform the pivot and write out the new tableau. Part G: What is the solution you found in Part E? Is it the same solution as in part A? Part H: Fill in the blanks: In this part of the problem I learned that when degeneracy is present it is possible to pivot once to obtain a Blank A solution than the current one, and a strictly Blank B objective value. In fact, the pivot can be non-degenerate. Part I: Suppose the BFS for an optimal tableau is degenerate and a nonbasic variable in row z row has a 0 coefficient. Show by example that it is possible that the LP has a single optimal solution, but possibly has more then one optimal tableau Part J: Suppose the BFS for an optimal tableau is degenerate and a nonbasic variable in row z row has a 0 coefficient. Show by example that it is possible that the LP has multiple optimal solutions but only one corner point that is optimal. Part K: Geometrically the set of optimal solutions to the LP you formulated in Part K is known as what? Page 5 of 12

6 Hey Ollie, Do you think degenerates like degeneracy? O.J Simpson always enjoyed the subject. Problem 4: The Cap n (21 Points) This problem is meant to give you practice on a fairly easy problem on sensitivity analysis. It is meant to cover the basic topics and give you some review. Cap n Crunch is trying to reduce costs for his tasty Crunchberries cereal. Currently he buys crunchy red berries and blue berries on the open market for $3 and $1 per pound respectively. To help reduce costs, he has purchased a 100-acre farm to grow his own tasty, crunchy berries. Each acre planted with berries requires a yearly allotment of water as well as a specified number of hours of labor to harvest. Additionally, each type of berry provides a different yearly yield from the harvest. All of this information is provided in the table below: Red Berries Blue Berries Water (tons per acre per year) Labor (hours per acre per year) Market Price (dollars per pound) $3.00 $1.00 Yearly Yield (pounds per acre) The Cap n s farm is fed by a well which can only pump 6,000 tons of water per year for irrigation. Additionally, the Cap n has found two Operations Research graduate students who will work for peanuts over the summer harvest season to satisfy their practicum requirements. The students provide 225 hours of labor at a fixed cost. All of the resource limitations are provided in the table below: Limit Farm Land (acres) 100 Water (tons per year) 6,000 Labor (hours per year) 225 If the Cap n is capable of using all of the berries he produces, how many acres of each type should he plant to maximize his yearly cost savings? Page 6 of 12

7 For this problem, we will use the following decision variables: x r = Number of acres planted with red berries x b = Number of acres planted with blue berries Thus, we setup our Linear Program as follows: Maximize 270 x r x b (yearly savings) subject to x r + x b 100 (limit on farm land) 50 x r + 75 x b 6000 (limit on water) 2 x r x b 225 (limit on labor) x r, x b 0 After inputting this model into Excel, we receive the following Sensitivity Report: Microsoft Excel 11.0 Sensitivity Report Worksheet: [Sensitivity Analysis.xls]Cap'n Crunch Report Created: 03/05/ :44:59 PM Adjustable Cells Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease $C$16 Acres Red $D$16 Acres Blue Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $C$24 Land Function $C$25 Water Function $C$26 Labor Function E+30 5 Please answer the following questions based on the Sensitivity Report provided above. Part A: What is the optimal solution? What is the optimal objective value (i.e. annual savings)? Part B: How much would the Cap n be willing to pay for an increase of one hour of labor from its current level of 225? Part C: How far would the price per pound of blue berries have to drop before the optimal solution would change? (Hint: Note how the objective function depends on the price of blueberries.) Page 7 of 12

8 Part D: Suppose a neighboring farmer wins 10 acres of the Cap n farmland in a poker match. How much would this reduce the objective value for the LP? Part E: The Cap n is considering planting purple berries at his farm. Currently the Cap n purchases purple berries for $4 per pound. One acre of purple berry plants is expected to yield 80 pounds of purple berries, requiring 3 hours of labor to harvest and 100 tons of water per year. Rewrite the LP including the new variable. Rewrite the LP. List any new decision variables first. Part F: Based on information from the sensitivity report, Should the Cap n plant any acres of purple berries at his farm? Please price out to demonstrate your reasoning. Part G: If the Cap n s team of crack scientists comes up with a method to increase the yearly yield of red berries to 100 pounds per acre will the current solution remain optimal? Problem 5: Nooz and Olli s New Firm (32 Points) This problem will give you practice using excel to help with solving sensitivity analysis problems. [See spreadsheet Problem Set 4.xls.] After failing in the crabbing business Nooz and Ollie have decided to move south(they fly Delta of course!). The pair has decided to open up a commercial printing firm outside of Macon Georgia. Hey Ollie, I Think were gonna like Macon Macon is great, we can take delta connection there. Did you know that was ASA s first destination Page 8 of 12

9 The firm is trying to determine the best mix of printing jobs it should seek, given its current capacity constraints in its four capital-intensive departments: typesetting, camera, pressroom, and bindery. It has classified its commercial work into three classes: A, B, C, each requiring different amounts of time in the four major departments. The production requirements in hours per unit of product are as follows: Department Class of Work A B C Typesetting Camera Pressroom Bindery Assuming these units of work are produced using regular time, the contribution to overhead and profit is $200 for each unit of Class A work, $300 for each unit of Class B, and $100 for each unit of Class C work. The firm currently has the following regular time capacity available in each department for the next time period: typesetting, 50 hours; camera, 90 hours; pressroom, 190 hours; bindery, 170 hours. In addition to this regular time, the firm could utilize an overtime shift in typesetting, which would make available an additional 40 hours in that department. The premium for this overtime (i.e., incremental costs in addition to regular time) would be $4/hour. The firm wants to find the optimal job mix for its equipment, and management assumes it can sell all it produces. However, to satisfy long-established customers, management guarantees that it will produce at least 10 units of each class of work in each time period. Assume that the firm wants to maximize its contribution to profit and overhead. Part A: Formulate as a Linear Programming problem. Parts B through E are based on the Sensitivity Report in Excel that has been provided on the associated spreadsheet. You may also solve the Linear Program multiple times in order to verify your answers, but all justifications should be based on the Sensitivity Report. For Part F, please create a separate worksheet. Part B: What is the optimal production mix? Part C: Is there any unused production capacity? Page 9 of 12

10 Part D: Is this a unique optimum? Why? What other optimal solutions exist? Hint: Take a close look at the Sensitivity Report for the row BO. There are two different clues in that row that there are multiple optimal solutions. If there are alternate solutions, Excel will provide only one of them. You can trick Excel into providing another optimal solution by changing cost coefficients just a little (say by.01%) and resolving. Try this trick to get a 2 nd optimal solution. You do not need to submit a revised worksheet. Part E: Why is the shadow price of regular typesetting different from the shadow price of overtime typesetting? Hint: Notice that the shadow price for overtime typesetting is less than the shadow price for regular typesetting. In order to see why the prices are what they are first recall the (mathematical) definition of the shadow price. Then solve the problem with an additional unit of regular typesetting and see how the optimal solution changes. Then do the same for overtime typesetting. You do not need to submit a revised worksheet. Part F: If the printing firm has a chance to sell a new type of work that requires 0 hours of typesetting, 2 hours of camera, 2 hours of pressroom, and 2 hour of bindery, what contribution is required to make it attractive? Do this in two ways: (i) Price out (ii) Create a new worksheet and resolve (this worksheet should be submitted) Part G: Suppose that both the regular and overtime typesetting capacity are reduced by 4 hours. How does the optimal solution change? Part H: Let r denote the RHS value for regular typesetting. Let z(r) denote the optimal objective value when the RHS for regular typesetting is set to be r, and all other data is the same as given in problem 2. For example, z(50) = Find z(r) for all r from 0 to 100. Solve as few linear programs as you can. How many linear programs did you solve and what values of r did you test? Provide a sketch of z(r). How does z(r) behave between the break points provided in the sensitivity report? Page 10 of 12

11 Problem 6: A night at the Oscars (35 Points) This problem will help you explore the theoretical aspects of sensitivity analysis. Part A: Eddy Murphy, feeling upset over loosing best supporting actor, asks why in every sensitivity report the reduced costs are non-positive. Please explain. Part B: Jennifer Holiday after complaining about Jennifer Hudson, asks if it is always true that the negative reduced cost of a variable equals the allowable increase. Please explain. Part C: (3 points) After winning best Actress Helen Mirren asks the producers backstage if it is possible for a linear program constraint to have multiple shadow prices. Please explain and give an example. Part D: (5 points) The cast of babble is having trouble translating the lecture notes. Can you help them come up with a Linear Program with all equality constraints such that at least one constraint has a negative shadow price and one has a positive shadow price? Part E: Joan Rivers asks Clint Eastwood on the red carpet if we plot the cost coefficient vs. optimal z value for an objective for a max problem what type of property will the plot have. Part G: (Bonus 4 Points) Ellen counters a question by asking Barbra Walters to given an example of a linear program where a nonbasic variable objective function coefficient has to be improved by more than its REDUCED COST in order for it to enter the basis. Please come up with an example to help Barbra and explain why this sometimes occurs. Problem 7: Formulation (10 Points) The question challenges you to formulate an abstract LP from scratch. It is good practice for the exam. There are m refineries with the j-th refinery having the capacity to supply a i gallons of fuel. There are n cities with a demand for fuel. The demand for city i is b i gallons of fuel. When shipping fuel via train from refinery i to city j a fraction of the fuel shipped, f ij, is consumed by the track during transportation, Formulate a linear program to meet all fuel demands while minimizing the total fuel consumed. (HINT: if you are having trouble, first try making up date for a problem with two refineries and two cities. Your formulation for this made up Page 11 of 12

12 problem may help guide you to solving the abstract problem. However, there is no partial credit for handing in the formulation for the made up problem.) Challenge Problem D: (10 Points) Consider the following LP: Maximize z = 5x + y - 12v S.T 3x + 2y + v = 10 5x + 3y + w = 16 v, w, x, y, 0 An optimal solution to this problem is the vector (x, y, v, w) = (2, 2, 0, 0) and the corresponding simplex tableau is given by: z x y v w RHS Suppose in the original problem that we change the 3 coefficient of x to 3+ε. Determine the optimal profit as a function of ε, assuming the optimal basis matrix is not changed. To determine the optimum solution, you can assume that v = w = 0, and that x and y and z are functions of ε. Note that the functions of ε are not necessarily linear when we change a coefficient of the constraint matrix. Page 12 of 12