MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Transcription

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Introduction to Optimization (Spring 2005) Problem Set 4, Due March 8 th, 2005 You will need 88 points out of 111 to receive a grade of 2.5. Note you have 1.5 weeks to complete this problem set; thus it is somewhat longer. This is also the last problem set before the first midterm exam. Also please make a xerox of your HW so you can check your answers, since the HW will not be retuned until after the midterm. The answers to this HW will be posted on Tuesday night on March 8 th. Late assignments will not be accepted. Problem 1: Simplex Extension (6 Points) Use the internet to look up what the Klee and Minty cube is and its relation to the simplex algorithm. Here are some suggested websites, for more in depth look you might want to check out google scholar ( which is an academic paper search engine. campuscgi.princeton.edu/~rvdb/java/pivot/kleeminty.html Optimization/RunningTimeSimplex.ps Write a few sentences about what you found out. What did Klee and Minty show? What were their backgrounds? What is interesting about their famous example? What year did they complete their work? Can you find another example though research that illustrates the same idea? If so please give the LP. In order to get full credit you need to find: 1. the year the example was developed; 2. The LP they formulated; 3. What they showed about the running time of simplex algorithm; 4. One other interesting fact (your choice).

2 Problem 2: Simplex Review (10 Points) This problem shows that the simplex method can indeed cycle. We consider a problem described in terms of the following initial tableau. Z x1 x2 x3 x4 x5 x6 x7 RHS 1-3/4 20-1/ / / / a) Using the following two pivoting rules (described below) continue to pivot until you arrive back at the tableau above. Please perform your first pivot by hand and not using Excel. After the first pivot you are free to use Excel (especially CyTools) to speed up the process. Turn in each intermediate tableau you arrive at. Pivoting Rules: 1. Each time we perform a pivot, we pivot in the variable with the most negative coefficient in the z-row. 2. If ties occur when performing the min ratio test, pivot out the variable with smallest subscript (i.e., if x 3 and x 5 simultaneously take on value 0 after the next pivot, pivot out x 3 since its index is 3 which is less than 5.) b) The simplex algorithm cycles on this example with the pivoting rule given above. Show how to perform a different sequence of pivots and obtain the optimal solution. (HINT: you can use Excel to determine the optimum solution, and knowing the optimal solution may help you to determine a good choice for the entering variables.) Problem 3: Degeneracy (10 Points) Suppose the BFS for an optimal tableau is degenerate and a nonbasic variable in row zero has a 0 coefficient. Show by examples that either of the following cases may occur. Case 1: The LP has multiple optimal solutions Case 2: The LP has a single optimal solution Hint for Case 2: degeneracy. Problem 4: Simplex Pivoting (12 Points) Suppose we are solving a maximization problem and the variable x 3 is about to leave the basis. a) What can you say about the z-row coefficient of x 3?

3 b) After the pivot is carried out that the z-row coefficient of x 3 can not be less then zero. Explain why. c) Is it true that a variable that has just left the basis can not reenter on the very next iteration? Briefly explain Problem 5: Sensitivity Analysis by Hand (32 Points) Radio Inc. manufactures 2 types of radios. The only resource that is required to produce radios is labor. Laborer 1 is willing to work up to 40 hours per week and is paid $5 per hour. Laborer 2 is willing to work up to 50 hours per week and is paid $6 per hour. The price as well as the resources required to produce each radio type are given in the table that follows. Radio A Sells for $25. It takes one hour of labor for laborer 1 to make one unit and two hours of labor for laborer 2 to make one unit. The raw materials needed to produce one unit cost $5. Radio B Sells for $22. It takes 2 hour of labor for laborer 1 to make one unit and two hours of labor for laborer 2 to make one unit. The raw materials needed to produce one unit cost $5. a) Letting x i be the number of type radio i produced each week, formulate an LP that Radio inc. must solve in order to determine the optimal number of each radio to produce. Solve the LP to find the optimum corner point. You should use the graphical technique where the optimum isoprofit line is shifted parallel to itself until the optimum corner point is found. Hint: the optimal solution to the LP you formulated is 80. b) For what values of the price of a type 1 radio would the current basis remain optimal, all other data remaining unchanged? c) For what values of the price of a type 2 radio would the current basis remain optimal, all other data remaining unchanged? d) If laborer 1 were only willing to work 30 hours per week would the current basis remain optimal? That is, would the same two constraints be binding at the new optimal solution? Explain your answer e) If laborer 2 were willing to work up to 60 hours per week would the current basis remain optimal? f) If laborer 1 were willing to work an additional hour, what is the most Radio Inc. should pay the laborer for the hour of work? g) If laborer 2 were willing to work only 48 hours, what would Radio Inc. s profits be? Verify your answer by determining the number of each type of radios that will be produced. h) A third type of radio (type 3) is being considered for production. The specification for a type 3 radio are as follows:

4 a. Price $30 b. 2 hours from laborer 1 c. 2 hours from laborer 2 d. cost of raw materials $3 Should Radio Inc. manufacture any type 3 radios? That is, would it be profitable to manufacture type 3 radios (possibly by altering the number of type 1 and type 2 radios made.) Show how to determine the answer without finding the new optimal solution. Problem 6: Basic Sensitivity Analysis Using Excel (21 Points) [See spreadsheet Problem4.xls.] A commercial printing 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, 200 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 35 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.

5 a) Formulate as a linear optimization problem. Answer the questions (b) to (e) using the sensitivity report from Excel given on the associated spreadsheet. You may also solve the LP multiple times in order to verify your answers, but all justifications should be based on the sensitivity report. Except for part f ii, you do not have to hand in an Excel print out. b) What is the optimal production mix? c) Is there any unused production capacity? 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 that there are multiple optima in that row. 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 have to hand in a revised spreadsheet. What are the shadow prices and the ranges for both optimal solutions? 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 have to hand in a revised spreadsheet. 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) Modify spreadsheet and resolve (hand in the revised formulation) g) Suppose that both the regular and overtime typesetting capacity are reduced by 4 hours. How does the solution change? 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. So, z(50) = Find z(r) for all r from 0 to 100. Solve as few linear programs as you can. HINT: the shadow price for regular typesetting is valid in the range from 48 to 78. By using the sensitivity analysis report for the original linear program, you can determine z(r) when r is between 48 and 78. Then solve the linear program for r = 78.1 and create another sensitivity report for that linear program. (And do the same for r = 47.9). And so on, until you

6 have determined z(r) for all r = 0 to 100. Please list the small number of values of r for which you solved a linear program to get z(r). You do not have to hand in a revised spreadsheets or sensitivity reports. Problem 7: Sensitivity Analysis (20 Points) The Concrete Corporation has the capability of producing four types of concrete blocks. Each block must be subjected to four processes: batch mixing, mold vibrating, inspection, and yard drying. The plant manager desires to maximize profits during the next month. During the upcoming thirty days, he has 900 machine hours available on the batch mixer, 1000 hours on the mold vibrator, and 350 man-hours of inspection time. Yard-drying time is unconstrained. The production director has formulated his problem as a linear program with the following initial tableau: Basic Variables x 1 x 2 x 3 x 4 x 5 x 6 x 7 RHS z x x x where x 1, x 2, x 3, x 4 represent the number of pallets of the four types of blocks. After solving by the simplex method, the final tableau is: Basic Variables x 1 x 2 x 3 x 4 x 5 x 6 x 7 RHS z x x x a) By how much must the profit on a pallet of number 3 blocks be increased before it would be profitable to manufacture them? b) What minimum profit on x 2 must be realized so that it remains in the production schedule? c) If the 900 machine-hours capacity on the batch mixer is uncertain, for what range of machine hours will it remain feasible to produce blocks 1 and 2? (You should use the tableau information given above to compute the range. However, you can verify using the spreadsheet that the basic solution is infeasible outside of the range.) d) A competitor located next door has offered the manager additional batch-mixing time at a rate of $3.50 per hour. Should he accept this offer?

7 e) The owner has approached the manager about producing a new type of concrete block, called block number 5, that would require 3 hours of batch mixing, 3 hours of molding, and 1 hour of inspection per pallet. What should be the profit per pallet if block number 5 is to be included in the optimal schedule? Challenge Problem D(6 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.

8 Addendum Problem Set 4 Problem 5: It takes one hour of labor 1 and two hours of laborer 2 to make a type A radio It takes two hours of labor 1 and one hours of laborer 2 to make a type B radio The cost of raw materials for a type B radio is $4 NOT $5 As a hint if you get stuck see review problem 22 in chapter 6 Problem 6: The number of available overtime hours is 40 NOT 35