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Unformatted text preview: IE 398 Final Exam
December 11, 2002. 8:00AM11:00AM There are 180 points total on this exam. Please put your name on all the pages of the exam. If you need more space, feel free to use the backs of the sheets. The more clearly you write your answer, the better the chance that I can grade it accurately and give it full credit. Thank youI had a lot of fun teaching this class. Good luckOn the final and beyond. Name: Potentially Useful Jibberish =1
jN aj xj b jN aj xj + M M + b aj xj b = 1 jN jN aj xj  (m  ) b + =1
jN aj xj b jN aj xj + m m + b aj xj b = 1 jN jN aj xj  (M + ) b  x > 0 = 1 x M = 1 x m x m More Potentially Useful Jibberish
Call the following problem (for x maximize f (x) subject to g1 (x) b1 g2 (x) b2 . . . gm (x) bm x1 0 x2 0 . . . xn 0. Then, if x is an optimal solution to NLP, then there exists ^ multipliers 1 , 2 , . . . , m , 1 , 2 , . . . , n that satisfy the following (KKT) conditions: gi (^) bi x ^i 0 x
m gi (^) x f (^) x  i + j xj xj i=1 i j i (bi  gi (^)) x j xj ^ n ) (P): i = 1, 2, . . . , m j = 1, 2, . . . , n j = 1, 2, . . . n i = 1, 2, . . . , m j = 1, 2, . . . , n i = 1, 2, . . . , m j = 1, 2, . . . n = 0 = = 0 0 0 0 IE398 Final Exam Name:
(35) 1 Mmmmmmmmmmmmmmm. Cake. Lee Sara bakes cheesecakes and Black Forest cakes. During any month, he can bake at most 65 cakes. The cost/cake of making each type of cake and the demand for each type of cake for the next three months is shown in Table 1. It costs $0.50 to hold a cheesecake, and $0.40 to hold a Black Forest cake in inventory for a month. Lee Sara has no cakes of any type in inventory right now. Table 1: Monthly Demand and Costs for Cakes Month 1 Demand Cost Cheese 40 $3.00 Black Forest 20 $2.50 Month 2 Month 3 Demand Cost Demand Cost 30 $3.40 20 $3.80 30 $2.80 10 $3.40 For you numbershaters out there, you can use the definitions in Table 2 instead of the numbers in the previous discussion. Please don't mix and match numbers and symbolic parameters. Table 2: Symbolic Parameters for Lee Sara Cake Problem Symbol C T j djt cjt Domain {1, 2, 3} jC j C, t T j C, t T Definition Set of cakes Set of time periods Inventory cost for cake type j Demand for cakes of type j in time period t Cost of producing one cake of type j in time period t Problem 1 Page 4 IE398 Final Exam Name: 1.1 Problem (10 points) Formulate a LP to minimize the total cost of meeting the next three months' demands for each type of cake. Answer. Problem 1 Page 5 IE398 Final Exam Name: 1.2 Problem (15 points) Suppose that in any month Lee Sara has the option of not meeting demand. However, he must pay a penalty cost of $3.60 for each cheesecake by which he does not meet his demand, and $3.00 for each Black Forest cake by which he does not meet demand. (Parameter people: Let j , j C, be the percake penalty that Lee Sara must pay for not meeting demand for cake type j.) Formulate a LP to minimize the total cost of meeting the next three months' demands for each type of cake plus the penalty costs for not meeting this demand. Answer. Problem 1 Page 6 IE398 Final Exam Name: 1.3 Problem (10 points) Suppose that Lee Sara can only let his demand slip by 5 at the penalty costs described in Problem 1.2. If his demand slips by more than 5, he must pay the larger penalty costs of $4.60 for each cheesecake and $3.50 for each Black Forest cake. (Parameter people: Let Let tj , t C be the maximum number of cakes under his demand for which Lee Sara can pay the low penalty j , and let j , (j > j ) be the higher penalty that Lee Sara must pay for not meeting his demand for cake j within tj ), Modify your answer to Problem 1.2 to accommodate this fact. Hint: You can (and should) do this as a linear program. Answer. Problem 1 Page 7 IE398 Final Exam Name: 2 You Say ToMAYTo, I Say ToMAHTo (25) Tom's Terrific Tomatoes (TTT) is in the business of selling a set P of tomatorelated products (e.g salsa, ketchup, tomato paste). In order to create these products, resources from a set R are required (e.g. tomatoes, sugar, labor, spices). To be specific, the parameter arp is the amount of resource r R that is required to produce one unit of product p P . There is a limit br on the amount of each resource r R that can be purchased in a production period. Each unit of resource r R costs cr . TTT management is expecting a demand of dp for each of its products p P . Any surplus production of a product p P must be stored at a cost of p /unit. TTT management also considers unmet demand important, so it imposes a penalty cost of p for each unit of unmet demand for product p P . 2.1 Problem (10 points) Formulate a linear programming model that will tell TTT management how to minimize the combination of resource, surplus, and unmet demand costs, subject to their resource constraints. Answer. Problem 2 Page 8 IE398 Final Exam Name: Suppose now that the demand is not known for certain until after the products are made. Instead, TTT forecasts a number of demand scenarios S = {1, 2, . . . S} where dps is the demand for product p P under scenario s S. The probability of a scenario s S occurring is s . 2.2 Problem (3 points) Referring to your answer to problem 2.1, which of the variables are firststage variables, and which of the variables are secondstage variables. If it helps, you can assume that all of the resources that TTT purchases are used in the production of products. Answer. 2.3 Problem (12 points) Formulate a linear programming "supermodel" that will tell TTT management how to minimize their expected costs under the varying demand scenarios. Answer. Problem 2 Page 9 IE398 Final Exam Name:
(50) 3 Problem Categorization and KKT Categories of problems that we have discussed in class are the following: Table 3: Optimization Problem Categories A B C D E Linear Programming Mixed Integer Linear Programming Unconstrained Optimization Nonlinear Constrained Optimization Mixed Integer Nonlinear Constrained Optimization 3.1 Problem (2 points) Consider the following problem: maximize 3x1 + x2 + 7x3 subject to x1  x2 x2  x3 2x1 + x2 + 5x3 x1 x2 x3 3 7 8 0 0 0 What problem category in Table 3 best describes this problem? Answer. 3.2 Problem (2 points) Is the optimization instance in Problem 3.1 "easy" of "hard"? Answer. Problem 3 Page 10 IE398 Final Exam Name: 3.3 Problem (7 points) Write the KKT conditions for the optimization instance in Problem 3.1. Answer. Problem 3 Page 11 IE398 Final Exam Name: 3.4 Problem (6 points) Use your KKT conditions from Problem 3.3 to deduce whether or not x1 = 1/2, x2 = 7, x3 = 0 is an optimal solution to the optimization instance in Problem 3.1. Answer. Problem 3 Page 12 IE398 Final Exam 3.5 Problem (2 points) maximize Name: 3x2 + 2x2 1 2 subject to x1 x2 x1 + x2 x1 , x 2 , x 3 , 3 7 6 0 What problem category in Table 3 best describes this problem? Answer. 3.6 Problem (2 points) Is the optimization instance in Problem 3.5 "easy" of "hard"? Why? Answer. 3.7 Problem (2 points) minimize f (x1 , x2 ) = 8x3 + 16x2  2x1 x2 1 2 subject to x1 + 3x2 6 x1 , x 2 0 What problem category in Table 3 best describes this problem? Answer. Problem 3 Page 13 IE398 Final Exam Name: 3.8 Problem (2 points) Is the function f (x1 , x2 ) in Problem 3.7 convex for all x? Answer. 3.9 Problem (2 points) Is the function f (x1 , x2 ) in Problem 3.7 convex for all feasible x? Answer. 3.10 Problem (100 points) Is the optimization instance in Problem 3.7 "easy" of "hard". Why? Answer. 3.11 Problem (7 points) Write the KKT Conditions for the optimization instance in Problem 3.7. Answer. Problem 3 Page 14 IE398 Final Exam Name: 3.12 Problem (6 points) Use your KKT conditions from Problem 3.11 to verify whether or not x1 = 0, x2 = 2 is an optimal solution to the problem. Answer. Problem 3 Page 15 IE398 Final Exam 3.13 Problem (2 points) maximize Name: 3x1 + 2x2 + 7x3  2x4 + 5x5 subject to 2x1  5x2 x2  5x4 x3 x3 + x4  x5 x1 , x 2 , x 3 , x 4 , x 5 x1 , x 2 , x 3 , x 4 , x 5 = = = = 3 7 6 9 0 integer What problem category in Table 3 best describes this problem? Answer. 3.14 Problem (2 points) Is the optimization instance in Problem 3.13 "easy" of "hard". Answer. Problem 3 Page 16 IE398 Final Exam 3.15 Problem (2 points) minimize Name: 3x2 + 2x2 + 7x3  2x4 + 5x5 1 2 subject to 2x1  x4 = 3 2 2 x1 + x2 = 8 2 x3 , x 4 , x 5 integer What problem category in Table 3 best describes this problem? Answer. 3.16 Problem (2 points) Is the optimization instance in Problem 3.15 "easy" of "hard". Answer. Problem 3 Page 17 IE398 Final Exam Name:
(30) 4
4.1 Relax! Which of These Things is Less Than The Other? The questions in this section refer to the optimal objective function values of the following problems with variables x n . The functions f : n and g : n m , and the parameter b m do not change from problem to problem. The vector e n is the vector of all 1's. zA = max f (x) subject to g(x) b xj 0 xj 1 j = 1, 2, . . . n j = 1, 2, . . . n zB = max f (x) subject to g(x) b xj {0, 1} j = 1, 2, . . . n zC = max f (x) subject to g(x) eT x xj xj = b 1 0 1 j = 1, 2, . . . n j = 1, 2, . . . n zD = max f (x) subject to g(x) b xj 0 j = 1, 2, . . . n There are now a series of questions asking whether the values zA , zB , zC , zD are comparable. Answer each problem with either a relation (, =, ) if the values can be compared, or the word incomparable if we cannot be sure of the relation between the numbers. Problem 4 Page 18 IE398 Final Exam 4.1 Problem (3 points) zA zB ? Answer. Name: 4.2 Problem (3 points) zA zC ? Answer. 4.3 Problem (3 points) zB zC ? Answer. 4.4 Problem (3 points) zA zD ? Answer. 4.5 Problem (3 points) zB zD ? Answer. Problem 4 Page 19 4.6 Problem (3 points) Give one objective function (values for c1 , c2 ) for which no branching will be required by the branchandbound procedure for solving this integer programming instance Answer. subject to Problem 4 maximize Consider the following integer programming problem, whose feasible region is depicted in Figure 1: 4.2 IE398 Final Exam Good relaxations x2 2 3 1 Figure 1: Feasible Region 1 2x1  3x2 x1 + x2 x1 x2 c1 x1 + c2 x2 Name: 2 3 5 0 0 3 x1 Page 20 IE398 Final Exam Name: 4.7 Problem (3 points) Give one objective function (values for c1 , c2 ) for which branching will be required by the branchandbound procedure for solving this integer programming instance Answer. 4.8 Problem (9 points) Give the "best possible" formulation you can for this problem. A "best possible" formulation is one in which no branching will be required, irregardless of the objective function. Answer. Problem 4 Page 21 IE398 Final Exam Name:
(40) 5 Machine Scheduling Consider the following machine scheduling problem (MSP). We have a set M = {1, 2, . . . , m} of machines and a set N = {1, 2, . . . , n} of jobs that must be performed on the machines. Each machine i has a capacity of bi units of work, and each job j requires aij units of work to be completed if it is scheduled on machine i. All jobs must be assigned to exactly one machine. 5.1 Problem (7 points) Suppose that there is a cost cij for scheduling job j N on machine i M . Formulate an integer programming model to determine a least cost assignment of jobs to machines not exceeding the machine capacities. Answer. Problem 5 Page 22 IE398 Final Exam Name: 5.2 Problem (7 points) Suppose there is a fixed cost of operating a machine that is in order to schedule any jobs to run on machine i M , then you must pay a fixed cost of hi . Write an integer programming model to determine a least cost assignment of jobs to machines not exceeding the machine capacities in this case. (There is no "job assignment" cost cij as in Problem 5.1. Answer. Problem 5 Page 23 IE398 Final Exam Name: 5.3 Problem (8 points) Suppose if you operate machine 1 or 2 (or both), then you must operate at most 1 of machines 3 and 4. Modify your answer to Problem 5.2 in order to model this restriction. Answer. Problem 5 Page 24 IE398 Final Exam Name: 5.4 Problem (8 points) Suppose that if you operate k or more machines, then you must pay a penalty of . Modify your answer to Problem 5.2 in order to model this restriction. Answer. 5.5 Problem (2 points) Now suppose that each job j N has a duration (or length) dj . Using the variables you defined in Problem 5.1, write an expression for ti , the total length of all the jobs scheduled on machine i M . Answer. Problem 5 Page 25 IE398 Final Exam Name: 5.6 Problem (8 points) The makespan of a schedule is the length of time it takes to finish all the jobs. That is, it is the maximum of the lengths of the ti , i M . Formulate (from scratch don't rely on the answers to any previous problem) a model that will minimize the makespan of an assignment of jobs to machines not exceeding the machine capacities. Answer. Problem 5 Page 26 ...
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 Spring '08
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