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# 75 Study Guide for IE 33500 at Purdue

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Date Created: 02/06/15

IE 335 Operations Research Optimization Fall 2008 Review Problems for Final Exam Problem 1 While running branch and bound on an integer linear program with only binary variables we eventually reach the following node which still needs to be processed After simplifying our problem based on the branching that has already occurred we are left with the candi date problem of node 27 with some coef cients left unspeci ed the coef cients Cl Cz a1 and a2 minimize 15 61x5 62X7 subject to alxs azm 2 15 X5 6 0 lX7 E 0 l The current incumbent has an objective value of 34 For each of the following parts provide example values for the parameters 61 Cz a1 and a2 so that the condition stated is satis ed Assume that the parameters only take integer values a Node 27 is terminated and the LP relaxation of the node s candidate problem is infeasible b Node 27 is terminated the solution to the LP relaxation of the node s candidate problem is integer and the incumbent does not change c Node 27 is terminated the solution to the LP relaxation of the node s candidate problem is integer and the incumbent is updated d Node 27 is terminated and the solution to the LP relaxation of the node s candidate problem is not integer e Node 27 is not terminated Problem 2 An integer linear program with only binary variables has been partially solved using branch and bound Below is the current enumeration tree infeasible by solving by bound 137 denotes the optimal value of the LP relaxation of node j s candidate problem Note that a and 3 are the active nodes in the current enumeration tree a Does this enumeration tree result from a maximization or a minimization problem b What is the incumbent c Provide an interval for the optimal cost d If all the cost coef cients are integer what can you say about the incumbent Problem 3 To graduate from Basketweavers University with a major in operations research a student must complete at least two math courses at least two OR courses and at least two computer courses Some courses can be used to ful ll more than one requirement Calculus can ful ll the math requirement Oper ations Research math and OR requirements Data Structures computer and math requirements Business Statistics math and OR requirements Computer Simulation OR and computer requirements Introduction to Computer Programming computer requirement and Forecasting OR and math requirements Some courses are prerequisites for others Calculus is a prerequisite for Business Statistics Introduction to Computer Programming is a prerequisite for Computer Simulation and Data Structures and Business Statistics is a prerequisite for Forecasting Formulate an integer linear program that minimizes the number of courses needed to satisfy the major requirements Problem 4 Challenging Governor Blue of the state of Berry is attempting to get the state legislature to gerrymander Berry s congressional districts The state consists of ten cities and the numbers of registered Republicans and Democrats in thousands in each city are shown in the table below Republicans Democrats City 1 80 34 City 2 60 44 City 3 40 44 City 4 20 24 City 5 40 114 City 6 40 64 City 7 70 14 City 8 50 44 City 9 70 54 City 10 70 64 Berry has ve congressional representatives and therefore 5 districts To form congressional districts cities must be grouped according to the following restrictions 3 All voters in a city must be in the same district 3 Each district must contain between 150000 and 250000 voters there are no independent voters Governor Blue is a Democrat Assume that each voter always votes a straight party ticket Formulate an integer linear program to help Governor Blue maximize the number of Democrats who will congressional seats 9amp1 a Laquot 2 39 0 1J as l M Na Wen Ed u LP 6 MAL a WOQA M tau amp X2 6 O resl C 0 2954 cud M A M 4w weuk39aws L d dyax X X4 7 I 0 Xb Knit 16 c Icro c1 Ioa M 01 M 16 NI LPA ld VkM I me Add pmm iMLyAMLWEhwmwahwisIm a 3339 gum FLlt WW 2 8m wt ML MAMquot 6 Pk fw 39 l6 wwwh a 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