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operations research an introduction

Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions

For the problems of Section 10.1, use the product form of the inverse to perform the revised simplex method.
Use the product form of the inverse to compute B1-1 and B2-1 for the Dakota problem that was solved by the revised simplex in Section 10.1
4 The university library has one copying machine for the students to use. Students arrive at the machine with the distribution of interarrival times shown in Table 18. The time to make a copy is uniformly distributed over the range[16, 25] seconds. Analysis of past data has shown that the number of
find the maximum flow from source to sink. Also find a cut whose capacity equals the maximum flow in the network. 50 20 30 5 27 10 3 15 2 14 19
5 A salesperson in a large bicycle shop is paid a bonus ifhe sells more than 4 bicycles a day. The probability of selling more than 4 bicycles a day is only .40. If the number of bicycles sold is greater than 4, the distribution of sales is as shown in Table 20. The shop has four different models
find the maximum flow from source to sink. Also find a cut whose capacity equals the maximum flow in the network. 3 6 1 3, 2 2 8 4 8 2
A local accounting firm in Smalltown orders boxes of floppy disks (10 disks to a box)from a store in Megalopolis. The per-box price charged by the store depends on the number of boxes purchased (see Table 2). The accounting firm uses 10,000 disks per year. The cost of placing an order is assumed to
4 The number of crimes in each of a city’s three police precincts depends on the number of patrol cars assigned to each precinct (see Table 11). Five patrol cars are available.Use dynamic programming to determine how many patrol cars should be assigned to each precinct. TABLE 11 No. of Patrol
6 A heart specialist schedules 16 patients each day, 1 every 30 minutes, starting at 9 A.M. Patients are expected to arrive for their appointments at the scheduled times.However, past experience shows that 10% of all patients arrive 15 minutes early, 25% arrive 5 minutes early, 50%arrive exactly on
7 Suppose we are considering the selection of the reorder point, R, of a (Q, R) inventory policy. With this policy, we order up to Q when the inventory level falls to R or less.The probability distribution of daily demand is given in Table 22. The lead time is also a random variable and has the
2 Use either of the approaches outlined in this section to solve the following knapsack problem: max z = 5x + 4x2 + 2x3 s.t. 4x + 3x2 + 2x3 8 x1, x2, x30; x1, x2, x3 integer
Find the maximum flow from source to sink in each network.Find a cut in the network whose capacity equals the maximum flow in the network. Also, set up an LP that could be used to determine the maximum flow in the network. 30 3 3 3 1 2 2
3 An airport hotel has 100 rooms. On any given night, it takes up to 105 reservations, because of the possibility of no-shows. Past records indicate that the number of daily reservations is uniformly distributed over the integer range[96, 105]. That is, each integer number in this range has an
1 J. R. Carrington has $4 million to invest in three oil well sites. The amount of revenue earned from site i(i 1, 2, 3)depends on the amount of money invested in site i (see Table 10). Assuming that the amount invested in a site must be an exact multiple of $1 million, use dynamic programming to
Find the maximum flow from source to sink in each network.Find a cut in the network whose capacity equals the maximum flow in the network. Also, set up an LP that could be used to determine the maximum flow in the network. 4 3 3 2 2
2 A news vendor sells newspapers and tries to maximize profits. The number of papers sold each day is a random variable. However, analysis of the past month’s data shows the distribution of daily demand in Table 16. A paper costs the vendor 20¢. The vendor sells the paper for 30¢. Any unsold
Find the maximum flow from source to sink in each network.Find a cut in the network whose capacity equals the maximum flow in the network. Also, set up an LP that could be used to determine the maximum flow in the network. FIGURE 18 Network for Problem 1 50 2 3 2 3 7 2 si
Finco has $6,000 to invest, and three investments are available. If dj dollars (in thousands)are invested in investment j, then a net present value (in thousands) of rj(dj) is obtained, where the rj(dj)’s are as follows:The amount placed in each investment must be an exact multiple of $1,000. To
5 A machine operator processes two types of jobs, A and B, during the course of the day. Analyses of past data show that 40% of all jobs are type A jobs, and 60% are type B jobs. Type A jobs have completion times that can be represented by an Erlang distribution with rate parameter 5 and shape
10 Explain how by solving a single transshipment problem you can find the shortest path from node 1 in a network to each other node in the network. TABLE 7 Box 1 2 3 4 5 6 7 Size 33 30 26 Demand 400 300 500 700 200 24 19 18 17 200 400 200
3 Given the continuous triangular distribution in Figure 13, develop a process generator using the inverse transformation method. FIGURE 13 fx) 0.2 2 8 12
3 I must drive from Bloomington to Cleveland. Several paths are available (see Figure 4). The number on each arc is the length of time it takes to drive between the two cities.For example, it takes 3 hours to drive from Bloomington toCincinnati. By working backward, determine the shortest path (in
8 A library must build shelving to shelve 200 4-inch high books, 100 8-inch high books, and 80 12-inch high booksEach book is 0.5 inch thick. The library has several ways to store the books. For example, an 8-inch high shelf may be built to store all books of height less than or equal to 8 inches,
2 A job shop manager wants to develop a simulation model to help schedule jobs through the shop. He has evaluated the completion times for all the different types of jobs. For one particular job, the times to completion can be represented by the following triangular distribution:Develop a process
2 A sales representative lives in Bloomington and must bein Indianapolis next Thursday. On each of the days Monday, Tuesday, and Wednesday, he can sell his wares in Indianapolis, Bloomington, or Chicago. From past experience, he believes that he can earn $12 from spending a day in Indianapolis, $16
1 Consider a continuous random variable with the following pdf:Develop a process generator for these breakdown times using the inverse transformation method and the acceptance–rejection method. f(x) = 2 1 0 x 1 1 x 3
1 Find the shortest path from node 1 to node 10 in the network shown in Figure 3. Also, find the shortest path from node 3 to node 10. FIGURE 3 7 1 /m 3 4 160 + 5 a 3 2 3 10 5 7 3
5 Suppose it costs $10,000 to purchase a new car. The annual operating cost and resale value of a used car are shown in Table 4. Assuming that one now has a new car, determine a replacement policy that minimizes the net costs of owning and operating a car for the next six years. TABLE 4 Age of Car
Use the acceptance–rejection method to generate random variates from a triangular distribution whose pdf is given by f(x) = *2 9 12- 21 + 2/1 - 1 = 85x59 2 x 6
Consider a random variable X whose pdf is given byUse the ITM to generate observations from the distribution. This distribution, called a triangular distribution, is represented graphically in Figure 9. It has the endpoints [2, 6], and its mode is at 3. We can see that 25% of the area under the
4 Use Dijkstra’s algorithm to find the shortest path from node 1 to node 4 in Figure 5. Why does Dijkstra’s algorithm fail to obtain the correct answer? Network for Problem 2 1 00 8 2 5. 2 4 12 5 10 6 3 4
Consider a random variable X that is uniformly distributed on the interval [a, b]. The pdf of this distribution is given by the functionUse the ITM to generate observations from this random variable. f(x)=b- a a x b otherwise
As mentioned in Chapter 20, the exponential distribution has important applications in the mathematical representation of queuing systems. The pdf of the exponential distribution is given byUse the inverse transformation method to generate observations from an exponential distribution. f(x) = {ex
I have a 9-oz cup and a 4-oz cup. My mother has ordered me to bring home exactly 6 oz of milk. How can I accomplish this goal? TABLE 1 Moves in the Cup-and-Milk Problem No. of Ounces in 9-oz Cup No. of Ounces in 4-oz Cup 6 0 6 4 9 1 0 1 0 4 5 0 5 4 9 0 0 0
Let us consider the Powerco example (Figure 2). Suppose that when power is sent from plant 1 (node 1) to city 1 (node 6), it must pass through relay substations (nodes 2–5).For any pair of nodes between which power can be transported, Figure 2 gives the distance(in miles) between the nodes. Thus,
1 A consulting firm is trying to determine how to minimize the annual costs associated with purchasing computer paper.Each time an order is placed an ordering cost of $20 is incurred. The price per box of computer paper depends on q, the number of boxes ordered (see Table 3). The annual holding
6 Consider a knapsack problem for whichShow that if the knapsack can hold w pounds, and w >= w*, wherethen the optimal solution to the knapsack problem must use at least one Type 1 item. L WI > C2 W2
8 A large car dealership in Bloomington, Indiana, employsfive salespeople. All salespeople work on commission; they are paid a percentage of the profits from the cars they sell.The dealership has three types of cars: luxury, midsize, and subcompact. Data from the past few years show that the car
2 Each year, Shopalot Stores sells 10,000 cases of soda.The company is trying to determine how many cases should be ordered each time. It costs $5 to process each order, and the cost of carrying a case of soda in inventory for one year is 20% of the purchase price. The soda supplier offers Shopalot
An auto repair shop always needs to have an engine analyzer available. A new engine an lyzer costs $1,000. The cost mi of maintaining an engine analyzer during its ith year of operation is as follows: m1 = $60, m2 = $80, m3 = $120. An analyzer may be kept for1, 2, or 3 years; after i years of use
9 Consider a bank with 4 tellers. Customers arrive at anexponential rate of 60 per hour. A customer goes directly into service if a teller is idle. Otherwise, the arrival joins a waiting line. There is only one waiting line for all the tellers.If an arrival finds the line too long, he or she may
find the maximum flow from source to sink. Also find a cut whose capacity equals the maximum flow in the network. 50 6 2 6 3 si 2 1 4 4
3 A firm buys a product using the price schedule given in Table 5. The company estimates holding costs at 10% of purchase price per year and ordering costs at $40 per order.The firm’s annual demand is 460 units.a Determine how often the firm should order.b Determine the size of each order.c At
10 Jobs arrive at a workshop, which has two work centers(A and B) in series, at an exponential rate of 5 per hour.Each job requires processing at both these work centers, first on A and then on B. Jobs waiting to be processed at each center can wait in line; the line in front of work center A has
find the maximum flow from source to sink. Also find a cut whose capacity equals the maximum flow in the network. 6 6 3 9 7 8 50 2 6 7 6 4 6 si
4 A hospital orders its thermometers from a hospital supply firm. The cost per thermometer depends on the order size q, as shown in Table 6. The annual holding cost is 25%of the purchasing cost. Let EOQ80 be the EOQ if the cost per thermometer is 80¢, and let EOQ79 be the EOQ if the cost per
2 Suppose that a new car costs $10,000 and that the annual operating cost and resale value of the car are as shown in Table 13. If I have a new car now, determine a replacement policy that minimizes the net cost of owning and operating a car for the next six years. TABLE 13 Age of Car Resale
3 It costs $40 to buy a telephone from a department store.The estimated maintenance cost for each year of operation is shown in Table 14. (I can keep a telephone for at most five years.) I have just purchased a new telephone, and my old telephone has no salvage value. Determine how to minimize the
15 Fly-by-Night Airlines is considering flying three flights.The revenue from each flight and the airports used by each flight are shown in Table 11. When Fly-by-Night uses an airport, the company must pay the following landing fees(independent of the number of flights using the airport):airport 1,
Widgetco is about to introduce a new product (product 3). One unit of product 3 is produced by assembling 1 unit of product 1 and 1 unit of product 2. Before production begins on either product 1 or 2, raw materials must be purchased and workers must be trained. Before products 1 and 2 can be
1 What problem would arise if the network in Figure 39 were a portion of a project network? FIGURE 39 Network for Problem 1 1 2 3
5 Consider the (simplified) list of activities and predecessors that are involved in building a house (Table 20).a Draw a project network, determine the critical path, find the total float for each activity, and find the free float for each activity.b Suppose that by hiring additional workers, the
Ohm City Appliances has three types of TVs delivered from Springfield TV. Figure 14 gives the annual demand, unit purchasing cost, annual holding cost (as a percentage of purchase cost), the fixed cost of ordering a product, and the fixed cost of placing an order.Determine an ordering policy that
5 To graduate from State University, Angie Warner needs to pass at least one of the three subjects she is taking this semester. She is now enrolled in French, German, and statistics. Angie’s busy schedule of extracurricular activities allows her to spend only 4 hours per week on
1 Square City Appliance orders four types of washing machines. Table 7 gives the annual demand, purchasing cost, and annual holding cost (as a percentage of purchase cost), and the fixed cost of ordering a product. Determine an ordering policy that minimizes the sum of fixed and holding costs. Each
8 The LINDO output in Figure 41 can be used to determine the critical path for Problem 5. Use this output to do the following:a Draw the project diagram.b Determine the length of the critical path and the critical activities for this project. TABLE 24 Activity Cost per Day of Reducing Duration of
3 A camera store sells an average of 100 cameras per month.The cost of holding a camera in inventory for a year is 30%of the price the camera shop pays for the camera. It costs$120 each time the camera store places an order with its supplier. The price charged per camera depends on the number of
13 The government is going to build a high-speed computer in Austin, Texas. Once the computer is designed(D), we can select the exact site (S), the building contractor(C), and the operating personnel (P). Once the site isselected, we can begin erecting the building (B). We can start manufacturing
4 A company inventories two items. The relevant data for each item are shown in Table 9. Determine the optimal inventory policy if no shortages are allowed and if the average investment in inventory is not allowed to exceed $700. If this constraint could be relaxed by $1, by how much would the
Find the critical path, total float, and free float for each activity in the project networks FIGURE 44 B3 F4 G5 6 15. A3 C4 J4 I 2 5 E3 D3 4 K2 H6 L6 8
4 Each year, Data Corporal produces as many as 400 computers in Boston and 300 computers in Raleigh. Los Angeles customers must receive 400 computers, and 300 computers must be supplied to Austin customers. Producing a computer costs $800 in Boston and $900 in Raleigh.Computers are transported by
12 A job shop has four jobs that must be processed on a single machine. The due date and processing time for each job are given in Table 19. Use dynamic programming to determine the order in which the jobs should be done so as to minimize the total lateness of the jobs. (The lateness of a job is
Find the critical path, total float, and free float for each activity in the project networks FIGURE 43 A2 2 B6 3 C4 4 E2 Fl D4 G2 6 HI
15 Consider the project diagram in Figure 42. This project must be completed in 90 days. The time required to complete each activity can be reduced by up to five days at the costs given in Table 27.Formulate an LP whose solution will enable us to minimize the cost of completing the project in 90
6 Workco must have the following number of workers available during the next three months: month 1, 20; month 2, 16; month 3, 25. At the beginning of month 1, Workco has no workers. It costs Workco $100 to hire a worker and$50 to fire a worker. Each worker is paid a salary of$140/month. We will
8 Daisymay Van Line moves people between New York, Philadelphia, and Washington, D.C. It takes a van one day to travel between any two of these cities. The company incurs costs of $1,000 per day for a van that is fully loaded and traveling, $800 per day for an empty van that travels, $700 per day
4 Indianapolis Airlines has been told that it may schedule six flights per day departing from Indianapolis. The destination of each flight may be New York, Los Angeles,or Miami. Table 20 shows the contribution to the company’s profit from any given number of daily flights from Indianapolis to
6 A company needs to have a working machine during each of the next six years. Currently, it has a new machine.At the beginning of each year, the company may keep the machine or sell it and buy a new one. A machine cannot be kept for more than three years. A new machine costs $5,000.The revenues
3 In this problem, we explain why the MST algorithm works. Define S = minimum spanning tree Ct = nodes connected after iteration t of MST algorithm has been completed Ct= nodes not connected after iteration t of MST algorithm has been completed At = set of arcs in minimum spanning tree after t
9 For a computer to work properly, three subsystems of the computer must all function properly. To increase the reliability of the computer, spare units may be added to each system. It costs $100 to add a spare unit to system 1, $300 to system 2, and $200 to system 3. As a function of the number of
10 During any year, I can consume any amount that does not exceed my current wealth. If I consume c dollars during a year, I earn ca units of happiness. By the beginning of the next year, the previous year’s ending wealth grows by a factor k.a Formulate a recursion that can be used to maximize
2 For the MCNFP in Figure 62, find a bfs. FIGURE 62 00 $11 2 25 $10 00 $7 $8 (40) 1 25 $12 3 25 $7 00 $5 6 (-40) $5 25 $6 10 5 00 $3
3 Find the optimal solution to the MCNFP in Figure 63 using the bfs in Figure 64 as a starting basis FIGURE 63 30 $5 2 4(-22) (32) 1 25 $15 00 $5 20 $10 * $14 12 $4 12 $11 3 5 (-10) 00 $5 FIGURE 64 20 2 12 3 10 12 10 5
4 Find a bfs for the network in Figure 65. FIGURE 65 3 (10) (10) (1 15 (10) 2 13 24 12 6 9 7 4 6 (-20) 24 15 5
5 Find the optimal solution to the MCNFP in Figure 66 using the bfs in Figure 67 as a starting basis. FIGURE 66 30 $10 75 $15 50 $10 60 $30 100 $45 40 $10 (-25) (100) 40 $5 5 100 $45 50 $10 40 $25 40 $10 50 $30 3 8 (-75) 30 $5 60 $8 80 $15 6 FIGURE 67 50 2 25 25 30 25 50 75 6 25
1 A truck must travel from New York to Los Angeles. As shown in Figure 68, a variety of routes are available. The number associated with each arc is the number of gallons of fuel required by the truck to traverse the arc.a Use Dijkstra’s algorithm to find the route from New York to Los Angeles
2 Telephone calls from New York to Los Angeles are transported as follows: The call is sent first to either Chicago or Memphis, then routed through either Denver or Dallas, and finally sent to Los Angeles. The number of phone lines joining each pair of cities is shown in Table 39.a Formulate an LP
3 Before a new product can be introduced, the activities in Table 40 must be completed (all times are in weeks).a Draw the project diagram.b Determine all critical paths and critical activities.c Determine the total float and free float for each activity.d Set up an LP that can be used to determine
6 A company produces a product at two plants, 1 and 2.The unit production cost and production capacity during each period are given in Table 41. The product is instantaneously shipped to the company’s only customer according to the unit shipping costs given in Table 42. If a unit is produced and
7 A project is considered completed when activities A–F have all been completed. The duration and predecessors of each activity are given in Table 43. The LINDO output in Figure 69 can be used to determine the critical path for this project.a Use the LINDO output to draw the project
8 State University has three professors who each teachfour courses per year. Each year, four sections of marketing, finance, and production must be offered. At least one section of each class must be offered during each semester (fall and spring). Each professor’s time preference and preference
9 During the next two months, Machineco must meet (on time) the demands for three types of products shown in Table 45. Two machines are available to produce theseproducts. Machine 1 can only produce products 1 and 2, and machine 2 can only produce products 2 and 3. Each machine can be used for up
What Is LINGO?
28 Suppose the bfs for an optimal tableau is degenerate, and a nonbasic variable in row 0 has a zero coefficient.Show by example that either of the following cases may hold:Case 1 The LP has more than one optimal solution.Case 2 The LP has a unique optimal solution.
27 Suppose we have found an optimal tableau for an LP, and the bfs for that tableau is nondegenerate. Also suppose that there is a nonbasic variable in row 0 with a zero coefficient. Prove that the LP has more than one optimal solution.
26 Carco uses robots to manufacture cars. The following demands for cars must be met (not necessarily on time, but all demands must be met by end of quarter 4): quarter 1—600; quarter 2—800; quarter 3—500; quarter 4—400. At the beginning of the quarter, Carco has two robots. Robots can be
25 Ghostbusters, Inc., exorcises (gets rid of) ghosts.During each of the next three months, the company will receive the following number of calls from people who want their ghosts exorcised: January, 100 calls; February, 300 calls; March, 200 calls. Ghostbusters is paid $800 for each ghost
24 During the next four quarters, Dorian Auto must meet(on time) the following demands for cars: quarter 1—4,000;quarter 2—2,000; quarter 3—5,000; quarter 4—1,000. At the beginning of quarter 1, there are 300 autos in stock, and the company has the capacity to produce at most 3,000 cars per
22 Monroe County is trying to determine where to place the county fire station. The locations of the county’s four major towns are given in Figure 31. Town 1 is at (10, 20);town 2 is at (60, 20); town 3 is at (40, 30); town 4 is at (80, 60). Town 1 averages 20 fires per year; town 2, 30
21 Shoemakers of America forecasts the following demand for each of the next six months: month 1—5,000 pairs;month 2—6,000 pairs; month 3—5,000 pairs; month 4—9,000 pairs; month 5—6,000 pairs; month 6—5,000 pairs. It takes a shoemaker 15 minutes to produce a pair of shoes.Each shoemaker
20 A bus company believes that it will need the following number of bus drivers during each of the next five years:year 1—60 drivers; year 2—70 drivers; year 3—50 drivers;year 4—65 drivers; year 5—75 drivers. At the beginning of each year, the bus company must decide how many drivers
19 Suppose we are solving a maximization problem and the variable xr is about to leave the basis.a What is the coefficient of xr in the current row 0?b Show that after the current pivot is performed, the coefficient of xr in row 0 cannot be less than zero.c Explain why a variable that has left the
17 You are given the tableau shown in Table 74 for a maximization problem. Give conditions on the unknowns a1, a2, a3,b, and c that make the following statements true:a The current solution is optimal.b The current solution is optimal, and there are alternative optimal solutions.c The LP is
15 Consider a maximization problem with the optimal tableau in Table 73. The optimal solution to this LP is z 10, x3 3, x4 5, x1 x2 0. Determine the second-best bfs to this LP. (Hint: Show that the second-best solution must be a bfs that is one pivot away from the optimal solution.)
12 Productco produces three products. Each product requires labor, lumber, and paint. The resource requirements, unit price, and variable cost (exclusive of raw materials) for each product are given in Table 70. Currently, 900 labor hours, 1,550 gallons of paint, and 1,600 board feet of lumber are
10 Suppose that in the Dakota Furniture problem, 10 types of furniture could be manufactured. To obtain an optimal solution, how many types of furniture (at the most) would have to be manufactured?
Use Excel Solver to find the optimal solution to the following problems:Problem 5 of Section 3.12
Use Excel Solver to find the optimal solution to the following problems:Problem 3 of Section 3.12
Use Excel Solver to find the optimal solution to the following problems:Problem 5 of Section 3.11
Use Excel Solver to find the optimal solution to the following problems:Problem 4 of Section 3.11

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