Linear Programming Exam Review
Course Introduction
Linear Programming is a mathematical optimization technique used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. This course covers foundational concepts including the formulation of linear programming problems, graphical and simplex methods for finding solutions, duality theory, sensitivity analysis, and real-world applications in fields such as operations research, economics, and engineering. Students will learn to model diverse optimization problems, interpret solution outputs, and understand the practical significance of constraints and objective functions in decision-making scenarios.
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) Consider the following problem: Maximize Z = 2x<sub>1</sub> + 4x<sub>2</sub>x<sub>3</sub>,subject to 3x<sub>2</sub> - x<sub>3</sub> 30 (resource 1)2x<sub>1</sub> - x<sub>2</sub> + x<sub>3</sub> 10 (resource 2)4x<sub>1</sub> +2x<sub>2</sub> - 2x<sub>3</sub> 40 (resource 3)and x<sub>1</sub> 0,x<sub>2</sub> 0,x<sub>3</sub> 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.
Q2) Consider the following problem.Maximize Z = 2x<sub>1</sub> - x<sub>2</sub> + x<sub>3</sub>,subject to x<sub>1</sub> - x<sub>2</sub> + 3x<sub>3</sub> 4 2x<sub>1</sub> + x<sub>2</sub> 10 x<sub>1</sub> - x<sub>2</sub> - x<sub>3</sub> 7 and x<sub>1</sub> 0,x<sub>2</sub> 0,x<sub>3</sub> 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) Work through the simplex method (in algebraic form)step by step to solve the following problem.Maximize Z = x<sub>1</sub> + 2x<sub>2</sub> + 2x<sub>3</sub>,subject to 5x<sub>1</sub> + 2x<sub>2</sub> + 3x<sub>3</sub> 15 x<sub>1</sub> + 4x<sub>2</sub> + 2x<sub>3</sub> 12 2x<sub>1</sub> + x<sub>3</sub> 8 and x<sub>1</sub> 0,x<sub>2</sub> 0,x<sub>3</sub> 0.
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Chapter 5: The Theory of the Simplex Method
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Q1) Consider the following problem.Minimize Z = x<sub>1</sub> + 2 x<sub>2</sub>,subject to -x<sub>1</sub> + x<sub>2</sub> \(\le\)15 2 x<sub>1</sub><sub> </sub>+ x<sub>2</sub> \(\le\) 90 x<sub>2</sub><sub> </sub>\(\ge\) 30 and x<sub>1</sub><sub> </sub>\(\ge\) 0,x<sub>2</sub> \(\ge\) 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.
Q2) Consider the following problem.Maximize Z = 2x<sub>1</sub> + 4x<sub>2</sub> + 3x<sub>3</sub>,subject to x<sub>1</sub> + 3x<sub>2</sub> + 2x<sub>3</sub> 30 x<sub>1</sub> + x<sub>2</sub> + x<sub>3</sub> 24 3x<sub>1</sub> + 5x<sub>2</sub> + 3x<sub>3</sub> 60 and x<sub>1</sub> 0,x<sub>2</sub> 0,x<sub>3</sub> 0.You are given the information that x<sub>1</sub> > 0,x<sub>2</sub> = 0,and x<sub>3</sub> >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.
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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 x<sub>1</sub> + 2 x<sub>2</sub>,<sub> </sub> subject to 2 x<sub>1</sub> + x<sub>2</sub> \(\ge\) 10 -3 x<sub>1</sub> + 2 x<sub>2</sub> \(\le\) 6 1 x<sub>1</sub> + 1 x<sub>2</sub> \(\ge\) 6 and x<sub>1</sub><sub> </sub>\(\ge\) 0,x<sub>2</sub> \(\ge\) 0.
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Chapter 7: Linear Programming Under Uncertainty
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Q1) A certain linear programming problem has four functional constraints in inequality form such that their right-hand sides (the b<sub>i</sub>)are uncertain parameters.Therefore,chance constraints with \(\alpha\) = 0.95 have been introduced in place of these constraints.After next substituting the deterministic equivalents of these chance constraints and solving the resulting new linear programming model,its optimal solution is found to satisfy two of these deterministic equivalents with equality whereas there is some slack in the other two deterministic equivalents.Determine the lower bound and the upper bound on the probability that all of these four original constraints will turn out to be satisfied by the optimal solution for the new linear programming model so this solution actually will be feasible for the original 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) Customers arrive at a fast food restaurant with one server according to a Poisson process at a mean rate of 30 per hour.The server has just resigned,and the two candidates for the replacement are X (fast but expensive)and Y (slow but inexpensive).Both candidates would have an exponential distribution for service times with X having a mean of 1.2 minutes and Y having a mean of 1.5 minutes.Restaurant revenue per month is given by $6,000/W where W is the expected waiting time (in minutes)of a customer in the system.Determine the upper bound on the difference in their monthly compensations that would justify hiring X rather than Y.
Q2) Consider a self-service model in which the customer is also the server.Note that this corresponds to having an infinite number of servers available.Customers arrive according to a Poisson process with parameter \(\lambda\),and service times have an exponential distribution with parameter \(\mu\).
(a)Find L<sub>q</sub> and W<sub>q</sub>.
(b)Construct the rate diagram for this queueing system.
(c)Use the balance equations to find the expression for P<sub>n</sub> in terms of P<sub>0</sub>.
(d)Find P<sub>0</sub>.
(e)Find L and W.
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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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