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introduction to operations research
Introduction To Operations Research 7th Edition Frederick S. Hillier, Gerald J. Lieberman - Solutions
For each of the following linear programming models, use the SOB method to construct its dual problem.(a) Model in Prob. 4.6-3(b) Model in Prob. 4.6-8(c) Model in Prob. 4.6-18
Consider the following problem.Minimize Z x1 2x2, subject to2x1 x2 12x1 2x2 1 and x1 0, x2 0.(a) Construct the dual problem.(b) Use graphical analysis of the dual problem to determine whether the primal problem has feasible solutions and, if so, whether its objective function is
Construct the dual problem for the linear programming problem given in Prob. 4.6-4.
Consider the primal and dual problems in our standard form presented in matrix notation at the beginning of Sec. 6.1. Use only this definition of the dual problem for a primal problem in this form to prove each of the following results.(a) If the functional constraints for the primal problem Ax b
Consider the following problem.Maximize Z x1 x2, subject to x1 2x2 10 2x1 x2 2 and x2 0 (x1 unconstrained in sign).(a) Use the SOB method to construct the dual problem.(b) Use Table 6.12 to convert the primal problem to our standard form given at the beginning of Sec. 6.1, and construct
Suppose that you also want information about the dual problem when you apply the revised simplex method (see Sec. 5.2)to the primal problem in our standard form.(a) How would you identify the optimal solution for the dual problem?(b) After obtaining the BF solution at each iteration, how would you
Consider the model given in Prob. 3.1-4.(a) Construct the dual problem for this model.(b) Use the fact that (x1, x2) (13, 5) is optimal for the primal problem to identify the nonbasic variables and basic variables for the optimal BF solution for the dual problem.(c) Identify this optimal solution
Consider the model given in Prob. 5.3-13.(a) Construct the dual problem.(b) Use the given information about the basic variables in the optimal primal solution to identify the nonbasic variables and basic variables for the optimal dual solution.(c) Use the results from part (b) to identify the
Reconsider the model of Prob. 6.1-4b.(a) Construct its dual problem.(b) Solve this dual problem graphically.(c) Use the result from part (b) to identify the nonbasic variables and basic variables for the optimal BF solution for the primal problem.(d) Use the results from part (c) to obtain the
Consider the following problem.Maximize Z 2x1 7x2 4x3, subject to x1 2x2 x3 10 3x1 3x2 2x3 10 and x1 0, x2 0, x3 0.(a) Construct the dual problem for this primal problem.(b) Use the dual problem to demonstrate that the optimal value of Z for the primal problem cannot exceed 25.(c)
Consider the following problem.Maximize Z 2x1 4x2, subject to x1 x2 1 and x1 0, x2 0.(a) Construct the dual problem, and then find its optimal solution by inspection.(b) Use the complementary slackness property and the optimal solution for the dual problem to find the optimal solution for
Follow the instructions of Prob.6.3-1 for this model.
Consider the model with two functional constraints and two variables given in Prob.
Consider the following problem.Maximize Z 6x1 8x2, subject to 5x1 2x2 20 x1 2x2 10 and x1 0, x2 0.(a) Construct the dual problem for this primal problem.(b) Solve both the primal problem and the dual problem graphically. Identify the CPF solutions and corner-point infeasible solutions for
For any linear programming problem in our standard form and its dual problem, label each of the following statements as true or false and then justify your answer.(a) The sum of the number of functional constraints and the number of variables (before augmenting) is the same for both the primal and
Consider the primal and dual problems in our standard form presented in matrix notation at the beginning of Sec. 6.1. Let y* denote the optimal solution for this dual problem. Suppose that b is then replaced by b. Let x denote the optimal solution for the new primal problem. Prove that cx y*b.
Consider the primal and dual problems in our standard form presented in matrix notation at the beginning of Sec. 6.1. Use only this definition of the dual problem for a primal problem in this form to prove each of the following results.(a) The weak duality property presented in Sec. 6.1.(b) If the
Consider the following problem.Maximize Z x1 2x2, subject tox1 x2 2 4x1 x2 4 and x1 0, x2 0.(a) Demonstrate graphically that this problem has no feasible solutions.(b) Construct the dual problem.(c) Demonstrate graphically that the dual problem has an unbounded objective function.
Follow the instructions of Prob. 6.1-6 for the following problem.Maximize Z x1 3x2 2x3, subject to 2x1 2x2 2x3 6 (resource 1)2x1 x2 2x3 4 (resource 2)and x1 0, x2 0, x3 0.
Consider the following problem.Maximize Z 2x1 6x2 9x3, subject to x1x1 x3 3 (resource 1)x1x2 2x3 5 (resource 2)and x1 0, x2 0, x3 0.(a) Construct the dual problem for this primal problem.(b) Solve the dual problem graphically. Use this solution to identify the shadow prices for the
Consider the following problem.Maximize Z x1 2x2 x3, subject to x1 x2 2x3 12 x1 x2 x3 1 and x1 0, x2 0, x3 0.(a) Construct the dual problem.(b) Use duality theory to show that the optimal solution for the primal problem has Z 0.
For each of the following linear programming models, give your recommendation on which is the more efficient way (probably) to obtain an optimal solution: by applying the simplex method directly to this primal problem or by applying the simplex method directly to the dual problem instead.
Construct the dual problem for each of the following linear programming models fitting our standard form.(a) Model in Prob. 3.1-5(b) Model in Prob. 4.7-6
Construct the primal-dual table and the dual problem for each of the following linear programming models fitting our standard form.(a) Model in Prob. 4.1-6(b) Model in Prob. 4.7-8
Reconsider Prob. 22.8-4.For the single-server queueing system under consideration, suppose now that service times definitely have an exponential distribution. However, it now is possible to reduce the variability of interarrival times, so we want to explore the impact of doing so.Assume now that
Follow the instructions of part (a) of Prob. 22.8-4 for an M/G/2 queueing system (two servers), with 1.6 and 1[so /(2 ) 0.8] and with 2 still being the variance of service times.
One of the main lessons of queueing theory (Chaps. 17 and 18) is that the amount of variability in the service times and interarrival times has a substantial impact on the measures of performance of the queueing system. Significantly decreasing variability helps considerably.This phenomenon is well
Reconsider Prob. 22.6-3.You now wish to begin the analysis by performing a short simulation by hand and then applying the regenerative method of statistical analysis when possible.R (a) Starting with four new tubes, simulate the operation of the two alternative policies for 5,000 hours of simulated
Consider the queueing system example presented in Sec.22.8 for the regenerative method. Explain why the point where a service completion occurs with no other customers left is not a regeneration point.
A certain single-server system has been simulated, with the following sequence of waiting times before service for the respective customers. Use the regenerative method to obtain a point estimate and 90 percent confidence interval for the steady-state expected waiting time before service.(a) 0, 5,
Reconsider Prob. 22.6-4.Suppose now that more careful statistical analysis has provided new estimates of the probability distributions of the radii of the shafts and bushings. In actuality, the probability distribution of the radius of a shaft (in inches)has the probability density function fs(x)
The probability distribution of the number of heads in 3 flips of a fair coin is the binomial distribution with n 3 and p1 2, so that P{X k}1 2k 12 3k k!(3 3!k)!1 23 for k 0, 1, 2, 3.The mean is 1.5.R (a) Obtaining uniform random numbers as instructed at the beginning of the Problems section, use
Consider the probability distribution whose probability density function is f(x)Use the method of complementary random numbers with two uniform random numbers, 0.096 and 0.569, to estimate the mean of this distribution.
Consider the probability distribution whose probability density function is f(x) Use the method of complementary random numbers with two uniform random numbers, 0.096 and 0.569, to estimate the mean of this distribution.
[For part ( f ), the true mean of the overall probability distribution of the size of an employee’s health insurance claim is $2,600.]
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
Reconsider Eddie’s Bicycle Shop described in Prob. 22.4-7.Forty percent of the bicycles require only a minor repair.The repair time for these bicycles has a uniform distribution between 0 and 1 hour. Sixty percent of the bicycles require a major repair. The repair time for these bicycles has a
A random variable X has P{X 0} 0.9. Given X 0, it has a uniform distribution between 5 and 15. Thus, E(X) 1.Obtaining uniform random numbers as instructed at the beginning of the Problems section, use simulation to estimate E(X).(a) Estimate E(X) by generating five random observations from the
Simulation is being used to study a system whose measure of performance X will be partially determined by the outcome of a certain external factor. This factor has three possible outcomes(unfavorable, neutral, and favorable) that will occur with equal probability (1 3). Because the favorable
Consider the probability distribution whose probability density function is f(x)The problem is to perform a simulated experiment, with the help of variance-reducing techniques, for estimating the mean of this distribution. To provide a standard of comparison, also derive the mean analytically.For
Reconsider Prob. 22.4-8 involving the game of craps.Now the objective is to estimate the probability of winning a play of this game. If the probability is greater than 0.5, you will want to go to Las Vegas to play the game numerous times until you eventually win a considerable amount of money.
Refer to the financial risk analysis example presented at the end of Sec. 22.6, including its results shown in Fig. 22.14.Think-Big management is quite concerned about the risk profile for the proposal. Two statistics are causing particular concern. One is that there is nearly a 20 percent chance
For one new product to be produced by the Aplus Company, bushings will need to be drilled into a metal block and cylindrical shafts inserted into the bushings. The shafts are required to have a radius of at least 1.0000 inch, but the radius should be as little larger than this as possible. With the
The Avery Co. factory has been having a maintenance problem with the control panel for one of its production processes.This control panel contains four identical electromechanical relays that have been the cause of the trouble. The problem is that the relays fail fairly frequently, thereby forcing
Look ahead at the scenario described in Prob. 22.7-5.Obtain a close estimate of the expected cost of insurance coverage for the corporation’s employees by performing 500 iterations of a simulation of an employee’s health insurance experience on a spreadsheet. Also generate the frequency
Reconsider Prob. 10.4-3, which involves trying to find the probability that a project will be completed by the deadline.Assume now that the duration of each activity has a triangular distribution that is based on the three estimates in the manner depicted in Fig. 22.10. Obtain a close estimate of
An insurance company insures four large risks. The number of losses for each risk is independent and identically distributed on the points {0, 1, 2} with probabilities 0.7, 0.2, and 0.1, respectively. The size of an individual loss has the following cumulative distribution function:F(x)Obtaining
Obtaining uniform random numbers as instructed at the beginning of the Problems section, use the acceptance-rejection method to generate three random observations from the probability density function f(x)R
Obtaining uniform random numbers as instructed at the beginning of the Problems section, use the acceptance-rejection method to generate three random observations from the triangular distribution used to illustrate this method in Sec. 22.4.
Consider the discrete random variable X that is uniformly distributed (equal probabilities) on the set {1, 2, . . . , 9}. You wish to generate a series of random observations xi (i 1, 2, . . .) of X.The following three proposals have been made for doing this. For each one, analyze whether it is a
Let r1, r2, . . . , rn be uniform random numbers. Define xi ln ri and yi ln (1 ri), for i 1, 2, . . . , n, and z n i1 xi. Label each of the following statements as true or false, and then justify your answer.(a) The numbers x1, x2, . . . , xn and y1, y2, . . . , yn are random observations from
You need to generate 10 random observations from the probability distribution P{X n}(a) Prepare to do this by generating 16 random integer numbers from the mixed congruential generator, xn1 ≡ (5xn 3) (modulo 16) and x0 1.(b) Use the single-digit random integer numbers from part (a) to generate
Obtaining uniform random numbers as instructed at the beginning of the Problems section, generate two random observations from each of the following probability distributions.(a) The exponential distribution with mean 4(b) The Erlang distribution with mean 4 and shape parameter k 2 (that is,
Obtaining uniform random numbers as instructed at the beginning of the Problems section, generate three random observations (approximately) from a normal distribution with main 0 and standard deviation 1.(a) Do this by applying the central limit theorem, using three uniform random numbers to
The random variable X has the cumulative distribution function F(x) whose value or derivative F(x) is shown below for various values of x.F(0) 0.F(x)1 8, for 0 x 2.P{X 2}1 2, so F(2)3 4.F(x)1 4, for 2 x 3.F(3) 1.Generate four random observations from this probability distribution by
The game of craps requires the player to throw two dice one or more times until a decision has been reached as to whether he (or she) wins or loses. He wins if the first throw results in a um of 7 or 11 or, alternatively, if the first sum is 4, 5, 6, 8, 9, or 10 and the same sum reappears before a
Eddie’s Bicycle Shop has a thriving business repairing bicycles. Trisha runs the reception area where customers check in their bicycles to be repaired and then later pick up their bicycles and pay their bills. She estimates that the time required to serve a customer on each visit has a uniform
Each time an unbiased coin is flipped three times, the probability of getting 0, 1, 2, and 3 heads is 18, 3 8, 3 8, and 18, respectively.Therefore, with eight groups of three flips each, on the average, one group will yield 0 heads, three groups will yield 1 head, three groups will yield 2 heads,
Suppose that random observations are needed from the triangular distribution whose probability density function is f(x) (a) Derive an expression for each random observation as a function of the uniform random number r.(b) Generate five random observations for this distribution by using the
Obtaining uniform random numbers as instructed at the beginning of the Problems section, generate three random observations from each of the following probability distributions.(a) The random variable X has P{X 0}1 2. Given X 0, it has a uniform distribution between 5 and 15.(b) The distribution
Obtaining uniform random numbers as instructed at the beginning of the Problems section, generate three random observations from each of the following probability distributions.(a) The uniform distribution from 25 to 75.(b) The distribution whose probability density function is f(x)(c) The
Apply the inverse transformation method as indicated below to generate three random observations from the uniform distribution between 10 and 40 by using the following uniform random numbers: 0.0965, 0.5692, 0.6658.(a) Apply this method graphically.(b) Apply this method algebraically.(c) Write the
Reconsider the coin flipping game introduced in Sec. 22.1 and analyzed with simulation in Figs. 22.1, 22.2, and 22.3.(a) Simulate one play of this game by repeatedly flipping your own coin until the game ends. Record your results in the format shown in columns B, D, E, F, and G of Fig. 22.1. How
Reconsider Prob. 22.3-1.Suppose now that you want to convert these random integer numbers to (approximate) uniform random numbers. For each of the three parts, give a formula for this conversion that makes the approximation as close as possible.
Read the articles about all four applications of simulation mentioned in Prob. 22.2-1.For each one, write a one-page summary of the application and the benefits it provided.
Section 22.2 introduced four actual applications of simulation that are described in articles in Interfaces. (The citations for the two that also use queueing models are given in Sec. 18.6.) Select one of these applications and read the corresponding article Write a two-page summary of the
Vistaprint produces monitors and printers for computers.In the past, only some of them were inspected on a sampling basis. However, the new plan is that they all will be inspected before they are released. Under this plan, the monitors and printers will be brought to the inspection station one at a
Hugh’s Repair Shop specializes in repairing German and Japanese cars. The shop has two mechanics. One mechanic works on only German cars and the other mechanic works on only Japanese cars. In either case, the time required to repair a car has an exponential distribution with a mean of 0.2 day.
View the second demonstration example (Simulating a Queueing System with Priorities) in the simulation area of your OR Tutor. Then enter this same problem into the interactive routine for simulation in your OR Courseware. Interactively execute a simulation run for 20 minutes of simulated time.
View the first demonstration example (Simulating a Basic Queueing System) in the simulation area of your OR Tutor.D,I (a) Enter this same problem into the interactive routine for simulation in your OR Courseware. Interactively execute a simulation run for 20 minutes of simulated time.Q (b) Use the
The Rustbelt Manufacturing Company employs a maintenance crew to repair its machines as needed. Management now wants a simulation study done to analyze what the size of the crew should be, where the crew sizes under consideration are 2, 3, and 4. The time required by the crew to repair a machine
Consider the M/M/1 queueing theory model that was discussed in Sec. 17.6 and Example 2, Sec. 22.1. Suppose that the mean arrival rate is 5 per hour, the mean service rate is 10 per hour, and you are required to estimate the expected waiting time before service begins by using simulation.R (a)
Jessica Williams, manager of Kitchen Appliances for the Midtown Department Store, feels that her inventory levels of stoves have been running higher than necessary. Before revising the inventory policy for stoves, she records the number sold each day over a period of 25 days, as summarized
The weather can be considered a stochastic system, because it evolves in a probabilistic manner from one day to the next.Suppose for a certain location that this probabilistic evolution satisfies the following description:The probability of rain tomorrow is 0.6 if it is raining today.The
Use the uniform random numbers in cells C10:C15 of Fig. 22.1 to generate six random observations for each of the following situations.(a) Throwing an unbiased coin.(b) A baseball pitcher who throws a strike 60 percent of the time and a ball 40 percent of the time.(c) The color of a traffic light
Reconsider the prototype example of Sec. 21.1. Suppose now that the production process using the machine under consideration will be used for only 4 more weeks. Using the discounted cost criterion with a discount factor of 0.9, find the optimal policy for this four-period problem.
Reconsider Prob. 21.5-7.Suppose now that the company will be producing either of these chemicals for only 4 more months, so a decision on which pollution control process to use 1 month hence only needs to be made three more times. Find an optimal policy for this three-period problem.
For Prob. 21.5-7, use two iterations of the method of successive approximations to approximate an optimal policy.
Reconsider Prob. 21.5-7.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
A chemical company produces two chemicals, denoted by 0 and 1, and only one can be produced at a time. Each month a decision is made as to which chemical to produce that month. Because the demand for each chemical is predictable, it is known that if 1 is produced this month, there is a 70 percent
For Prob. 21.5-4, use three iterations of the method of successive approximations to approximate an optimal policy.
Reconsider Prob. 21.5-4.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
The price of a certain stock is fluctuating between $10,$20, and $30 from month to month. Market analysts have predicted that if the stock is at $10 during any month, it will be at $10 or$20 the next month, with probabilities 45 and 15, respectively; if the stock is at $20, it will be at $10, $20,
For Prob. 21.5-1, use three iterations of the method of successive approximations to approximate an optimal policy.
Reconsider Prob. 21.5-1.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
Joe wants to sell his car. He receives one offer each month and must decide immediately whether to accept the offer.Once rejected, the offer is lost. The possible offers are $600, $800, and $1,000, made with probabilities 58, 1 4, and 18, respectively (where successive offers are independent of
Suppose now that the number of pints of blood delivered (on a regular delivery) can be specified at the time of delivery (instead of using the old policy of receiving 1 pint at each delivery). Thus, the number of pints delivered can be 0, 1, 2, or 3(more than 3 pints can never be used). The cost of
Consider the blood-inventory problem presented in Prob.
Use the policy improvement algorithm to find an optimal policy for Prob. 21.2-9.
Use the policy improvement algorithm to find an optimal policy for Prob. 21.2-8.
Use the policy improvement algorithm to find an optimal policy for Prob. 21.2-7.
Use the policy improvement algorithm to find an optimal policy for Prob. 21.2-6.
Use the policy improvement algorithm to find an optimal policy for Prob. 21.2-5.
Use the policy improvement algorithm to find an optimal policy for Prob. 21.2-4.
Use the policy improvement algorithm to find an optimal policy for Prob. 21.2-3.
Use the policy improvement algorithm to find an optimal policy for Prob. 21.2-2.
Use the policy improvement algorithm to find an optimal policy for Prob. 21.2-1.
Reconsider Prob. 21.2-9.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
Reconsider Prob. 21.2-8.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
Reconsider Prob. 21.2-7.(a) Formulate a linear programming model for finding an optimal policy.C (b) Use the simplex method to solve this model. Use the resulting optimal solution to identify an optimal policy.
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