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introduction to operations research
Introduction To Operations Research 7th Edition Frederick S. Hillier, Gerald J. Lieberman - Solutions
A department has one word-processing operator. Documents produced in the department are delivered for word processing according to a Poisson process with an expected interarrival time of 20 minutes. When the operator has just one document to process, the expected processing time is 15 minutes. When
A certain small grocery store has a single checkout stand with a full-time cashier. Customers arrive at the stand “randomly”(i.e., a Poisson input process) at a mean rate of 30 per hour. When there is only one customer at the stand, she is processed by the cashier alone, with an expected
Consider a single-server queueing system where some potential customers balk (refuse to enter the system) and some customers who enter the system later get impatient and renege (leave without being served). Potential customers arrive according to a Poisson process with a mean rate of 4 per hour. An
A maintenance person has the job of keeping two machines in working order. The amount of time that a machine works before breaking down has an exponential distribution with a mean of 10 hours. The time then spent by the maintenance person to repair the machine has an exponential distribution with a
A service station has one gasoline pump. Cars wanting gasoline arrive according to a Poisson process at a mean rate of 15 per hour. However, if the pump already is being used, these potential customers may balk (drive on to another service station). In particular, if there are n cars already at the
As for Property 3 of the exponential distribution, let T1, T2, . . . , Tn be independent exponential random variables with parameters 1, 2, . . . , n, respectively, and let U min{T1, T2, . . . , Tn}. Show that the probability that a particular random variable Tj will turn out to be smallest of the
For each of the following statements regarding service times modeled by the exponential distribution, label the statement as true or false and then justify your answer by referring to specific statements (with page citations) in the chapter.(a) The expected value and variance of the service times
A queueing system has two servers whose service times are independent random variables with an exponential distribution with a mean of 15 minutes. Customer X arrives when both servers are idle. Five minutes later, customer Y arrives and customer X still is being served. Another 10 minutes later,
Consider a two-server queueing system where all service times are independent and identically distributed according to an exponential distribution with a mean of 10 minutes. When a particular customer arrives, he finds that both servers are busy and no one is waiting in the queue.(a) What is the
Show that Ls1 n0 nPn Lq s1 s1 n0 Pnby using the statistical definitions of L and Lq in terms of the Pn.
You are given two queueing systems, Q1 and Q2. The mean arrival rate, the mean service rate per busy server, and the steadystate expected number of customers for Q2 are twice the corresponding values for Q1. Let Wi the steady-state expected waiting time in the system for Qi, for i 1, 2. Determine
Midtown Bank always has two tellers on duty. Customers arrive to receive service from a teller at a mean rate of 40 per hour.A teller requires an average of 2 minutes to serve a customer. When both tellers are busy, an arriving customer joins a single line to wait for service. Experience has shown
Consider the linear programming model for player 1 given near the end of Sec. 14.5 for variation 3 of the political campaign problem (see Table 14.6). Verify the optimal mixed strategies for both players given in Sec. 14.5 by applying an automatic routine for the simplex method to this model to
Consider variation 3 of the political campaign problem (see Table 14.6). Refer to the resulting linear programming model for player 1 given near the end of Sec. 14.5. Ignoring the objective function variable x3, plot the feasible region for x1 and x2 graphically (as described in Sec. 3.1). (Hint:
Consider the linear programming models for players 1 and 2 given near the end of Sec. 14.5 for variation 3 of the political campaign problem (see Table 14.6). Follow the instructions of Prob.14.5-5 for these two models.
Section 14.5 presents a general linear programming formulation for finding an optimal mixed strategy for player 1 and for player 2. Using Table 6.14, show that the linear programming problem given for player 2 is the dual of the problem given for player 1. (Hint: Remember that a dual variable with
Follow the instructions of Prob. 14.5-2 for the game having the following payoff table.
Consider the game having the following payoff table.(a) Use the approach described in Sec. 14.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem.C (b) Use the simplex method to find these optimal mixed strategies
Refer to the last paragraph of Sec. 14.5. Suppose that 3 were added to all the entries of Table 14.6 to ensure that the corresponding linear programming models for both players have feasible solutions with x3 0 and y4 0. Write out these two models. Based on the information given in Sec. 14.5, what
The A. J. Swim Team soon will have an important swim meet with the G. N. Swim Team. Each team has a star swimmer(John and Mark, respectively) who can swim very well in the 100-yard butterfly, backstroke, and breaststroke events. However, the rules prevent them from being used in more than two of
For each of the following payoff tables, use the graphical procedure described in Sec. 14.4 to determine the value of the game and the optimal mixed strategy for each player according to the minimax criterion.
Consider the game having the following payoff table.Use the graphical procedure described in Sec. 14.4 to determine the value of the game and the optimal mixed strategy for each player according to the minimax criterion. Check your answer for player 2 by constructing his payoff table and applying
Reconsider Prob. 14.3-1.Use the graphical procedure described in Sec. 14.4 to determine the optimal mixed strategy for each player according to the minimax criterion. Also give the corresponding value of the game.
Consider the following parlor game between two players.It begins when a referee flips a coin, notes whether it comes up heads or tails, and then shows this result to player 1 only. Player 1 may then (1) pass and thereby pay $5 to player 2 or (2) bet. If player 1 passes, the game is terminated.
For each of the following payoff tables, determine the optimal strategy for each player by successively eliminating dominated strategies. (Indicate the order in which you eliminated strategies.)
Reconsider Prob. 14.1-1.(a) Use the concept of dominated strategies to determine the best strategy for each side.(b) Without eliminating dominated strategies, use the minimax criterion to determine the best strategy for each side.
The state of a particular continuous time Markov chain is defined as the number of jobs currently at a certain work center, where a maximum of three jobs are allowed. Jobs arrive individually. Whenever fewer than three jobs are present, the time until the next arrival has an exponential
Reconsider the example presented at the end of Sec. 16.8.Suppose now that a third machine, identical to the first two, has been added to the shop. The one maintenance person still must maintain all the machines.(a) Develop the rate diagram for this Markov chain.(b) Construct the steady-state
A video cassette recorder manufacturer is so certain of its quality control that it is offering a complete replacement warranty if a recorder fails within 2 years. Based upon compiled data, the company has noted that only 1 percent of its recorders fail during the first year, whereas 5 percent of
Consider the following gambler’s ruin problem. A gambler bets $1 on each play of a game. Each time, he has a probability p of winning and probability q 1 p of losing the dollar bet. He will continue to play until he goes broke or nets a fortune of T dollars. Let Xn denote the number of dollars
A production process contains a machine that deteriorates rapidly in both quality and output under heavy usage, so that it is inspected at the end of each day. Immediately after inspection, the condition of the machine is noted and classified into one of four possible states:The process can be
Consider the inventory example presented in Sec. 16.1 except that demand now has the following probability distribution:P{D 0} 1 4, P{D 2} 1 4, P{D 1} 1 2, P{D 3) 0.The ordering policy now is changed to ordering just 2 cameras at the end of the week if none are in stock. As before, no order
Reconsider Prob. 16.6-2.Now suppose that the manufacturer keeps a spare machine that only is used when the primary machine is being repaired. During a repair day, the spare machine has a probability of 0.1 of breaking down, in which case it is repaired the next day. Denote the state of the system
A manufacturer has a machine that, when operational at the beginning of a day, has a probability of 0.1 of breaking down sometime during the day. When this happens, the repair is done the next day and completed at the end of that day.(a) Formulate the evolution of the status of the machine as a
A computer is inspected at the end of every hour. It is found to be either working (up) or failed (down). If the computer is found to be up, the probability of its remaining up for the next hour is 0.90. If it is down, the computer is repaired, which may require more than 1 hour. Whenever the
An important unit consists of two components placed in parallel. The unit performs satisfactorily if one of the two components is operating. Therefore, only one component is operated at a time, but both components are kept operational (capable of being operated) as often as possible by repairing
Consider the following inventory policy for a certain product. If the demand during a period exceeds the number of items available, this unsatisfied demand is backlogged; i.e., it is filled when the next order is received. Let Zn (n 0, 1, . . . ) denote the amount of inventory on hand minus the
Consider the inventory example introduced in Sec. 16.1, but with the following change in the ordering policy. If the number of cameras on hand at the end of each week is 0 or 1, two additional cameras will be ordered. Otherwise, no ordering will take place. Assume that the storage costs are the
In the last subsection of Sec. 16.5, the (long-run) expected average cost per week (based on just ordering costs and un-satisfied demand costs) is calculated for the inventory example of Sec. 16.1. Suppose now that the ordering policy is changed to the following. Whenever the number of cameras on
A soap company specializes in a luxury type of bath soap.The sales of this soap fluctuate between two levels—“Low” and“High”—depending upon two factors: (1) whether they advertise, and (2) the advertising and marketing of new products being done by competitors. The second factor is out
Consider the following blood inventory problem facing a hospital. There is need for a rare blood type, namely, type AB, Rh negative blood. The demand D (in pints) over any 3-day period is given by P{D 0} 0.4, P{D 1} 0.3, P{D 2} 0.2, and P{D 3} 0.1.Note that the expected demand is 1 pint, since E(D)
The leading brewery on the West Coast (labeled A) has hired an OR analyst to analyze its market position. It is particularly concerned about its major competitor (labeled B). The analyst believes that brand switching can be modeled as a Markov chain using three states, with states A and B
Reconsider Prob. 16.3-3.Use the results given in Prob.16.5-2 to find the steady-state probabilities for this Markov chain.Then find what happens to these steady-state probabilities if, at each step, the probability of moving one point clockwise changes to 0.9 and the probability of moving one point
A transition matrix P is said to be doubly stochastic if the sum over each column equals 1; that is, M i0 pij 1, for all j.If such a chain is irreducible, aperiodic, and consists of M 1 states, show that j M 11, for j 0, 1, . . . , M.
Reconsider Prob. 16.2-1.Suppose now that the given probabilities, 0.5 and 0.9, are replaced by arbitrary values, and , respectively. Solve for the steady-state probabilities of the state of the weather in terms of and .
Consider the Markov chain that has the following (onestep) transition matrix.P(a) Determine the classes of this Markov chain and, for each class, determine whether it is recurrent or transient.(b) For each of the classes identified in part (b), determine the period of the states in that class.
Determine the period of each of the states in the Markov chain that has the following (one-step) transition matrix.P
Given the following (one-step) transition matrix of a Markov chain, determine the classes of the Markov chain and whether they are recurrent.P
Given each of the following (one-step) transition matrices of a Markov chain, determine the classes of the Markov chain and whether they are recurrent.(a) P(b) P
Given the following (one-step) transition matrices of a Markov chain, determine the classes of the Markov chain and whether they are recurrent.
A particle moves on a circle through points that have been marked 0, 1, 2, 3, 4 (in a clockwise order). The particle starts at point 0. At each step it has probability 0.5 of moving one point clockwise (0 follows 4) and 0.5 of moving one point counterclockwise. Let Xn (n 0) denote its location on
Suppose that a communications network transmits binary digits, 0 or 1, where each digit is transmitted 10 times in succession. During each transmission, the probability is 0.99 that the digit entered will be transmitted accurately. In other words, the probability is 0.01 that the digit being
Reconsider Prob. 16.2-1.C (a) Use the routine Chapman-Kolmogorov Equations in your OR Courseware to find the n-step transition matrix P(n) for n 2, 5, 10, 20.(b) The probability that it will rain today is 0.5. Use the results from part (a) to determine the probability that it will rain n days from
Reconsider Prob. 16.2-2.Suppose now that whether or not the stock goes up tomorrow depends upon whether it increased today, yesterday, and the day before yesterday. Can this problem be formulated as a Markov chain? If so, what are the possible states?Explain why these states give the process the
Consider the second version of the stock market model presented as an example in Sec. 16.2. Whether the stock goes up tomorrow depends upon whether it increased today and yesterday.If the stock increased today and yesterday, it will increase tomorrow with probability 1. If the stock increased
Assume that the probability of rain tomorrow is 0.5 if it is raining today, and assume that the probability of its being clear (no rain) tomorrow is 0.9 if it is clear today. Also assume that these probabilities do not change if information is also provided about the weather before today.(a)
At a small but growing airport, the local airline company is purchasing a new tractor for a tractor-trailer train to bring luggage to and from the airplanes. A new mechanized luggage system will be installed in 3 years, so the tractor will not be needed after that.However, because it will receive
Use the algorithm described in Sec. 9.3 to find the shortest path through each of the following networks, where the numbers represent actual distances between the corresponding nodes
Reconsider the networks shown in Prob. 9.3-3.Use the algorithm described in Sec. 9.4 to find the minimum spanning tree for each of these networks.
For networks (a) and (b), use the augmenting path algorithm described in Sec. 9.5 to find the flow pattern giving the maximum flow from the source to the sink, given that the arc capacity from node i to node j is the number nearest node i along the arc between these nodes.
The diagram to the right depicts a system of aqueducts that originate at three rivers (nodes R1, R2, and R3) and terminate at a major city(node T), where the other nodes are junction points in the system.Using units of thousands of acre feet, the following tables show the maximum amount of water
One track of the Eura Railroad system runs from the major industrial city of Faireparc to the major port city of Portstown. This track is heavily used by both express passenger and freight trains.The passenger trains are carefully scheduled and have priority over the slow freight trains (this is a
Consider the maximum flow problem shown next, where the source is node A, the sink is node F, and the arc capacities are the numbers shown next to these directed arcs.(a) Use the augmenting path algorithm described in Sec. 9.5 to solve this problem.C (b) Formulate and solve a spreadsheet model for
Reconsider the maximum flow problem shown in Prob.
Formulate this problem as a minimum cost flow problem, including adding the arc A F.Use F 20.
Reconsider Prob 9.3-1.Now formulate this problem as a minimum cost flow problem by showing the appropriate network representation.
The Makonsel Company is a fully integrated company that both produces goods and sells them at its retail outlets. After production, the goods are stored in the company’s two warehouses until needed by the retail outlets. Trucks are used to transport the goods from the two plants to the
The Audiofile Company produces boomboxes. However, management has decided to subcontract out the production of the speakers needed for the boomboxes. Three vendors are available to supply the speakers. Their price for each shipment of 1,000 speakers is shown on the next page.In addition, each
Consider the minimum cost flow problem shown above, where the bi values (net flows generated) are given by the nodes, the cij values (costs per unit flow) are given by the arcs, and the uij values (arc capacities) are given to the right of the network. Do the following work manually(a) Obtain an
Reconsider the minimum cost flow problem formulated in Prob. 9.6-1.(a) Obtain an initial BF solution by solving the feasible spanning tree with basic arcs A B, A C, A F, B D, and E F, where two of the nonbasic arcs (E C and F D) are reverse arcs.D,I (b) Use the network simplex method yourself
Reconsider the minimum cost flow problem formulated in Prob. 9.6-2.(a) Obtain an initial BF solution by solving the feasible spanning tree that corresponds to using just the two rail lines plus factory 1 shipping to warehouse 2 via the distribution center.D,I (b) Use the network simplex method
Reconsider the minimum cost flow problem formulated in Prob. 9.6-3.Starting with the initial BF solution that corresponds to replacing the tractor every year, use the network simplex method yourself (without an automatic computer routine) to solve this problem.
For the P & T Co. transportation problem given in Table 8.2, consider its network representation as a minimum cost flow problem presented in Fig. 8.2. Use the northwest corner rule to obtain an initial BF solution from Table 8.2. Then use the network simplex method yourself (without an automatic
Consider the Metro Water District transportation problem presented in Table 8.12.(a) Formulate the network representation of this problem as a minimum cost flow problem. (Hint: Arcs where flow is prohibited should be deleted.)D,I (b) Starting with the initial BF solution given in Table 8.19, use
Consider the transportation problem having the following parameter table:Formulate the network representation of this problem as a minimum cost flow problem. Use the northwest corner rule to obtain an initial BF solution. Then use the network simplex method yourself (without an automatic computer
Consider the minimum cost flow problem shown below, where the bi values are given by the nodes, the cij values are given by the arcs, and the finite uij values are given in parentheses by the arcs. Obtain an initial BF solution by solving the feasible spanning tree with basic arcs A C, B A, C D,
Reconsider Prob.
Christine has done more detailed planning for this project and so now has the following expanded activity list:Construct the new project network.Activity Activity Description Predecessors Duration A Select location — 2 weeks B Obtain keynote speaker — 1 weeks C Obtain other speakers B 2 weeks D
Construct the project network for a project with the following activity list.Immediate Estimated Activity Predecessors Duration A — 1 months B A 2 months C B 4 months D B 3 months E B 2 months F C 3 months G D, E 5 months H F 1 months I G, H 4 months J I 2 months K I 3 months L J 3 months M K 5
You and several friends are about to prepare a lasagna dinner. The tasks to be performed, their immediate predecessors, and their estimated durations are as follows:Tasks that Task Task Description Must Precede Time A Buy the mozzarella cheese* 30 minutes B Slice the mozzarella A 5 minutes C Beat 2
Consider Christine Phillip’s project involving planning and coordinating next spring’s sales management training program for her company as described in Prob.
After constructing the project network, she now is ready for the following steps.(a) Find all the paths and path lengths through this project network. Which of these paths is a critical path?(b) Find the earliest times, latest times, and slack for each activity. Use this information to determine
Refer to the activity list given in Prob. 10.2-2 as Christine Phillips does more detailed planning for next spring’s sales man agement training program for her company. After constructing the project network (described in the back of the book as the answer for Prob. 10.2-2), she now is ready for
Reconsider the Reliable Construction Co. project introduced in Sec. 10.1, including the complete project network obtained in Fig. 10.7 at the end of Sec. 10.3. Note that the estimated durations of the activities in this figure turn out to be the same as the mean durations given in Table 10.4 (Sec.
Follow the instructions for Prob. 10.3-6 except use the optimistic estimates in Table 10.4 instead.
Follow the instructions for Prob. 10.3-6 except use the crash times given in Table 10.7 (Sec. 10.5) instead.
Reconsider Prob. 10.4-2.For each of the 10 activities, here are the three estimates that led to the estimates of the mean and variance of the duration of the activity (rounded to the nearest integer) given in the table for Prob. 10.4-2.(Note how the great uncertainty in the duration of these
In particular, enter the three estimates for each activity, and the template immediately will display the estimates of the means and variances of the activity durations. After indicating each path of interest, the template also will display the approximate probability that the path will be
The Lockhead Aircraft Co. is ready to begin a project to develop a new fighter airplane for the U.S. Air Force. The company’s contract with the Department of Defense calls for project completion within 100 weeks, with penalties imposed for late delivery.The project involves 10 activities (labeled
The Lockhead Aircraft Co. is ready to begin a project to develop a new fighter airplane for the U.S. Air Force. The company’s contract with the Department of Defense calls for project completion within 100 weeks, with penalties imposed for late delivery.The project involves 10 activities (labeled
Reconsider the Tinker Construction Co. problem presented in Prob. 10.5-1.While in college, Sean Murphy took an OR course that devoted a month to linear programming, so Sean has decided to use linear programming to analyze this problem.(a) Consider the upper path through the project network.
Reconsider the Electronic Toys Co. problem presented in Prob. 10.4-5.Sharon Lowe is concerned that there is a significant chance that the vitally important deadline of 57 days will not be met. Therefore, to make it virtually certain that the deadline will be met, she has decided to crash the
The mean critical path gives an estimate that the project will finish in 51 days. However, Sharon knows from the earlier analysis that some of the pessimistic estimates are far larger than the means, so the project duration might be considerably longer than 51 days. Therefore, to better ensure that
Good Homes Construction Company is about to begin the construction of a large new home. The company’s President, Michael Dean, is currently planning the schedule for this project. Michael has identified the five major activities (labeled A, B, . . . , E) that will need to be performed according
Reconsider the Lockhead Aircraft Co. problem presented in Prob. 10.4-6 regarding a project to develop a new fighter airplane for the U.S. Air Force. Management is extremely concerned that current plans for this project have a substantial likelihood(roughly a probability of 0.5) of missing the
The corresponding mean critical path provides an estimate that the project will finish in 100 weeks. However, management understands well that the high variability of activity durations means that the actual duration of the project may be much longer. Therefore, the decision is made to require that
Reconsider Prob. 10.5-4 involving the Good Homes Construction Co. project to construct a large new home. Michael Dean now has generated the plan for how to crash this project (as given as an answer in the back of the book). Since this plan causes all three paths through the project network to be
Consider the following problem:Maximize Z 4x1 x1 2 10x2 x2 2, subject to x1 2 4x2 2 16 and x1 0, x2 0.(a) Is this a convex programming problem? Answer yes or no, and then justify your answer.(b) Can the modified simplex method be used to solve this problem? Answer yes or no, and then
Consider the following nonconvex programming problem.Minimize f(x) sin 3x1 cos 3x2 sin(x1 x2), subject to x1 2 10x2 1 10x1 x2 2 100 and x1 0, x2 0(a) If SUMT were applied to this problem, what would be the unconstrained function P(x; r) to be minimized at each iteration?(b) Describe how
Reconsider the convex programming model with an equality constraint given in Prob. 13.6-14.(a) If SUMT were to be applied to this model, what would be the unconstrained function P(x; r) to be minimized at each iteration?D,C (b) Starting from the initial trial solution (x1, x2, x3) (3 2, 3 2, 2),
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