Quantitative Methods for Management Practice Exam

Course Introduction

Quantitative Methods for Management provides an introduction to the analytical tools and mathematical models used in decision-making within organizations. The course covers fundamental techniques such as linear programming, decision analysis, forecasting, inventory management, and simulation. Students learn how to apply these methods to real-world business problems, interpret quantitative data, and use statistical software for problem-solving. Emphasis is placed on practical application, critical thinking, and how quantitative analysis supports effective management decisions across various industries.

Recommended Textbook

Introduction to Operations Research 10th Edition by Frederick Hillier

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Chapter 3: Introduction to Linear Programming

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Chapter 4: Solving Linear Programming Problems: the Simplex Method

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Q1) Work through the simplex method (in algebraic form)step by step to solve the following problem.Maximize Z = x1 + 2x2 + 2x3,subject to 5x1 + 2x2 + 3x3 15 x1 + 4x2 + 2x3 12 2x1 + x3 8 and x1 0,x2 0,x3 0.

Q2) Consider the following problem.Maximize Z = 2x1 - x2 + x3,subject to x1 - x2 + 3x3 4 2x1 + x2 10 x1 - x2 - x3 7 and x1 0,x2 0,x3 0.(a)Work through the simplex method step by step in algebraic form to solve this problem.(b)Work through the simplex method step by step in tabular form to solve the problem.

Q3) Consider the following problem: Maximize Z = 2x1 + 4x2x3,subject to 3x2 - x3 30 (resource 1)2x1 - x2 + x3 10 (resource 2)4x1 +2x2 - 2x3 40 (resource 3)and x1 0,x2 0,x3 0.(a)Work through the simplex method step by step to solve the problem.(b)Identify the shadow prices for the three resources and describe their significance.

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Chapter 5: The Theory of the Simplex Method

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Q1) Consider the following problem.Maximize Z = 2x1 + 4x2 + 3x3,subject to x1 + 3x2 + 2x3 30 x1 + x2 + x3 24 3x1 + 5x2 + 3x3 60 and x1 0,x2 0,x3 0.You are given the information that x1 > 0,x2 = 0,and x3 >0 in the optimal solution.Using the given information and the theory of the simplex method,analyze the constraints of the problem in order to identify a system of three constraint boundary equations (defining equations)whose simultaneous solution must be the optimal solution (not augmente d).Then solve this system of equations to obtain this solution.

Q2) Consider the following problem.Minimize Z = x1 + 2 x2,subject to -x1 + x2 $\le$15 2 x1 + x2 $\le$ 90 x2 $\ge$ 30 and x1 $\ge$ 0,x2 $\ge$ 0. (a)Solve this problem graphically (b)Develop a table giving each of the CPF solutions and the corresponding defining equations,BF solution,and nonbasic variables.

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Chapter 6: Duality Theory

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Q1) For the following linear programming problem,use the SOB method presented in Sec.6.4 of the textbook to construct its dual problem.Minimize Z = 3 x1 + 2 x2, subject to 2 x1 + x2 $\ge$ 10 -3 x1 + 2 x2 $\le$ 6 1 x1 + 1 x2 $\ge$ 6 and x1 $\ge$ 0,x2 $\ge$ 0.

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Chapter 7: Linear Programming Under Uncertainty

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Q1) Consider the following problem.Maximize Z = 20x1 + c2x2,subject to 3x1 + a12x2 60 5x2 + a22x2 100 and x1 0,x2 0,where x1 represents the level of activity 1 and x2 represents the level of activity 2.The values of c2,a12,and a22 have not been determined yet.Only activity 1 needs to be undertaken soon whereas activity 2 will be initiated somewhat later.There are two different,but equally likely,scenarios that could unfold between now and the time activity 2 is undertaken that would lead to different values for c2,a12,and a22,as listed below.Scenario 1: c2 = 30,a12 = 4,and a22 = 7 Scenario 2: c2 = 10,a12 = 1,and a22 = 3 The goal is to use all of this information to choose a value for x1 now and to simultaneously determine a plan for choosing a value of x2 later after seeing which scenario has occurred.Use stochastic programming with recourse to formulate the appropriate model for this problem.

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Chapter 12: Integer Programming

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Q1) Decora Accessories manufactures a variety of bathroom accessories,including decorative towel rods and shower curtain rods.Each of the accessories includes a rod made out of stainless steel.However,many different lengths are needed: 12",18",24",40",and 60".Decora purchases 60" rods from an outside supplier and then cuts the rods as needed for their products.Each 60" rod can be used to make a number of smaller rods.For example,a 60" rod could be used to make a 40" and an 18" rod (with 2" of waste),or 5 12" rods (with no waste).For the next production period,Decora needs 25 12" rods,52 18" rods,45 24" rods,30 40" rods,and 12 60" rods.What is the fewest number of 60" rods that can be purchased to meet their production needs? (a)Formulate an integer programming model in algebraic form for this problem.(b)Formulate and solve an integer programming model in a spreadsheet for this problem.

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Chapter 14: Metaheuristics

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Q1) Suppose you are applying a simulated annealing algorithm to a certain problem,where T is the parameter that measures the tendency to accept the current candidate to be the next trial solution.You now have come to an iteration where the current value of T is T = 4,the value of the objective function for the current trial solution is 40,and the value of the objective function for the current candidate to be the next trial solution is 36.(a)Using the standard move selection rule,determine the probability of accepting this candidate to be the next trial solution when the objective is maximization of the objective function.(b)What is this probability when the objective is instead minimization of the objective function?

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Chapter 16: Decision Analysis

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Q1) You are given the opportunity to guess whether a coin is fair or two-headed,where the prior probabilities are 0.5 for each of these possibilities.If you are correct,you win $5; otherwise,you lose $5.You are also given the option of seeing a demonstration flip of the coin before making your guess.You wish to use Bayes' decision rule to maximize expected profit.(a)Develop a decision analysis formulation of this problem by identifying the decision alternatives,states of nature,and payoff table.(b)What is the optimal decision alternative,given that you decline the option of seeing a demonstration flip? (c)Find EVPI.d)Calculate the posterior distribution if the demonstration flip is a tail.Do the same if the flip is a head.

(e)Determine your optimal policy.(f)Now suppose that you must pay to see the demonstration flip.What is the most that you should be willing to pay? (g)Draw the complete decision tree for this problem and label all the branches.Write the cash flow incurred at each branch or terminal node underneath that branch or node.Write the probability associated with each branch emanating from an event node above that branch.At each decision node,insert a 1 or 2 inside the node to indicate whether the optimal decision there is to choose the upper branch (indicated by a 1)or to choose the lower branch (indicated by a 2).Next to each decision node and event node,write the expected payoff at that node if the optimal policy is being followed.

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Chapter 17: Queueing Theory

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Q1) The Copy Shop is open 5 days per week for copying materials that are brought to the shop.It has three identical copying machines that are run by employees of the shop.Only two operators are kept on duty to run the machines,so the third machine is a spare that is used only when one of the other machines breaks down.When a machine is being used,the time until it breaks down has an exponential distribution with a mean of 2 weeks.If one machine breaks down while the other two are operational,a service representative is called in to repair it,in which case the total time from the breakdown until the repair is completed has an exponential distribution with a mean of 0.2 week.However,if a second machine breaks down before the first one has been repaired,the third machine is shut off while the two operators work together to repair this second machine quickly,in which case its repair time has an exponential distribution with a mean of only 1/15 week.If the service representative finishes repairing the first machine before the two operators complete the repair of the second,the operators go back to running the two operational machines while the representative finishes the second repair,in which case the remaining repair time has an exponential distribution with a mean of 0.2 week.(a)Letting the state of the system be the number of machines not working,construct the rate diagram for this queueing system.(b)Use the balance equations to find the steady-state distribution of the number of machines not working.(c)What is the expected number of operators available for copying?

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Chapter 18: Inventory Theory

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Q1) Cindy Stewart and Misty Whitworth graduated from business school together.They now are inventory managers for competing wholesale distributors,making use of the scientific inventory management techniques they learned in school.Both of them are purchasing 85-horsepower speedboat engines for their inventories from the same manufacturer.Cindy has found that the setup cost for initiating each order is $200 and the unit holding cost is $400.Cindy has learned that Misty is ordering 10 engines each time.Cindy assumes that Misty is using the basic EOQ model and has the same setup cost and unit holding cost as Cindy.Show how Cindy can use this information to deduce what the annual demand rate must be for Misty's company for these engines.

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Chapter 20: Simulation

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Q1) The employees of General Manufacturing Corp.receive health insurance through a group plan issued by Wellnet.During the past year,40 percent of the employees did not file any health insurance claims,40 percent filed only a small claim,and 20 percent filed a large claim.The small claims were spread uniformly between 0 and $2,000,whereas the large claims were spread uniformly between $2,000 and $20,000.Based on this experience,Wellnet now is negotiating the corporation's premium payment per employee for the upcoming year.You are an operations research analyst for the insurance carrier,and you have been assigned the task of estimating the average cost of insurance coverage for the corporation's employees.(a)Use the random numbers 0.4071,0.5228,0.8185,0.5802,and 0.0193 to simulate whether each of five employees files no claim,a small claim,or a large claim.Then use the random numbers 0.9823,0.0188,0.8771,0.9872,and 0.4129 to simulate the size of the claim (including zero if no claim was file d).Calculate the average of these claims to estimate the mean of the overall distribution of the size of employee's health insurance claims.(b)Formulate and apply a spreadsheet model to simulate the cost for 300 employees' health insurance claims.Calculate the average of these random observations.(c)The true mean of the overall probability distribution of the size of an employee's health insurance claim is $2,600.Compare the estimates of this mean obtained in parts a and b with the true mean of the distribution.

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