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operations research an introduction
Operations Research An Introduction 8th Edition Hamdy A. Taha - Solutions
(b) The number of repairpersons needed so that the expected delay time until repair is started is less than 10 minutes.
*1. A shop uses 10 identical machines. Each machine breaks down once every 7 hours on the average. It takes half an hour on the average to repair a broken machine. Both the breakdown and repair processes follow the Poisson distribution. Determine the following:(a) The number of repairpersons needed
*4. A machine shop includes 20 machines and 3 repairpersons. A working machine breaks down randomly according to a Poisson distribution. The repair time per machine is exponential with a mean of 6 minutes. A queuing analysis of the situation shows an average of 57.8 calls for repair per 8-hour day
(b) Should the company lease a second WATS line? How much would it gain or lose over the singleWATS case by leasing an additional line?
3. A company leases a wide-area telecommunications service (WATS) telephone line for$2000 a month. The office is open 200 working hours per month. At all other times, the WATS line service is used for other purposes and is not available for company business.Access to the WATS line during business
(b) What is the schedule loss in dollars per breakdown when the number of repairpersons on duty is two? Three?
*2. Tasca Oil owns a pipeline booster unit that operates continuously. The time between breakdowns for each booster is exponential with a mean of 20 hours. The repair time is exponential with mean 3 hours. [n a particular station, two repairpersons attend 10 1 boosters. The hourly wage for each
1. Solve Example 15.9-2, assuming that C1 = $20 and C2 = $45.
4. In the solution space in Figure 7.2 (drawn to scale), express the interior point (3, 1) as a convex combination of the extreme points A, B, C, and D with each extreme point carrying a strictly positive weight. 16 S + FIGURE 7.2 Solution space for Problem 4, Set 7.1a B 2 3 4 5 6 3 2 1 D (3, 1)! 0
1. In the following sets of equations, (a) and (b) have unique (basic) solutions, (c) has an infinity of solutions, and (d) has no solution. Show how these results can be verified using graphical vector representation. From this exercise, state the general conditions for vector
2. Use vectors to determine graphically the type of solution for each of the sets of equations below: unique solution, an infinity of solutions, or no solution. For the cases of unique solutions, indicate from the vector representation (and without solving the equations algebraically)whether the
5. In the generalized simplex tableau, suppose that the X = (XI, xnf. where Xu corresponds to a typical starting basic solution (consisting of slack and/or artificial variables)with B :::: I, and let C = (C), Cn) and A = (D, I) be the corresponding partitions of C and A, respectively. Show that the
5. Using the matrix form of the simplex tableau, show that in an all-artificial starting basic solution, the procedure employed in Section 3.4.1 that calls for substituting outthe artificial variables in the objective function (using the constraint equations) actually computes the proper Zj - ci
5. Consider the matrix definition of the bounded-variables problem. Suppose that the vector X is partitioned into (Xz' Xu), where X'I represents the basic and nonbasic variables that will be substituted at upper bound during the course of the algorithm. The problem may thus be written as -C 650-0 D
4. Consider the following LP: - Maximize z = 2x + 4x2 + 4x3 3x4
*5. An LP model includes two variables Xl and X2 and three constraints of the type :5. The associated slacks are X3, X4, and xs' Suppose that the optimal basis is B = (PI> P2, P3), and its inverse is 0 -1 B-1 = 0 1 1 -1, The optimal primal and dual solutions are X = (x1, x2, x3) = (2, 6, 2) Y =
1. Modify and solve the capital budgeting model of Example 9.1-1 to account for the following additional restrictions:(a) Project 5 must be selected if either project 1 or project 3 is selected.(b) Projects 2 and 3 are mutually exclusive.
2. Five items are to be loaded in a vessel. TIle weight Wi, volume Vi, and value I"i for item i are tabulated below.Item i Unit weight, Wi (tons) Unit volume, Vi (yd3) Unit worth, ri (100 $)1 5 1 4 2 8 8 7 3 3 6 6 4 2 5 5 5 7 4 4
*3. Suppose that you have 7 full wine bottles, 7 half-full, and 7 empty. You would like to divide the 21 bottles among three individuals so that each will receive exactly 7. Additionally, each individual must receive the same quantity of wine. Express the problem as ILP constraints, and find a
*6. Once upon a time, there was a captain of a merchant ship who wanted to reward three crew members for their valiant effort in saving the ship's cargo during an unexpected storm in the high seas. The captain put aside a certain sum of money in the purser's office and instructed the first officer
7. (Weber, 1990) You have the following three-letter words: AFr, FAR,TVA, ADV, JOE, FIN, OSF, and KEN. Suppose that we assign numeric values to the alphabet starting with A = 1 and ending with Z :; 26. Each word is scored by adding numeric codes of its three letters. For example,AFT has a score of
8. Solve Problem 7 given that, in addition to the total sum being the largest, the sum of column 1 and the sum of column 2 will be the largest as well. Find the optimum solution.
9. (Weber, 1990) Consider the following two groups of words:Group 1 Group 2 AREA ERST FORT FOOT HOPE HEAT SPAR PAST THAT PROF TREE STOP All the words in groups 1 and 2 can be formed from the nine letters A, E, F, H, 0, P, R, S, and T. Develop a model to assign a unique numeric value from 1 through
11. In Problem 10, suppose that the nature of the melodies dictates that songs 3 and 4 cannot be recorded on the same side. Formulate the problem as an ILP. Would it be possible to use a 25-minute tape (each side) to record the eight songs? If not, use ILP to determine the minimum tape capacity
*12. (Graves and Associates, 1993) Ulem University uses a mathematical model that optimizes student preferences taking into account the limitation of classroom and faculty resources.To demonstrate the application of the model, consider the simplified case of 10 students who are required to select
*1. ABC is an LTL (less-than-truckload) trucking company that delivers loads on a daily basis to five customers. The following list provides the customers associated with each route:Route 12 34 56 Customers served on the route 1,2,3,4 4,3,5 1,2,5 2,3,5 1,4,2 1,3,5 The segments of each route are
*2. The U of A is in the process of forming a committee to handle students' grievances. The administration wants the committee to include at least one female, one male, one student, one administrator, and one faculty member. Ten individuals (identified, for simplicity, by the letters a to j) have
3. Washington County includes six towns that need emergency ambulance service. Because of the proximity of some of the towns, a single station may serve more than one community.The stipulation is that the station must be within 15 minutes of driving time from the towns it serves. The table below
4. The treasures of King Tut are on display in a museum in New Orleans. The layout of the museum is shown in Figure 9.3, with the different rooms joined by open doors. A guard standing at a door can watch two adjoining rooms. The museum wants to ensure guard presence in every room, using the
S. Bill has just completed his exams for the academic year and wants to celebrate by seeing every movie showing in theaters in his town and in six other neighboring cities. If he travels to another town, he will stay there until he has seen all the movies he wants. The following table provides the
6. Walmark Stores is in the process of expansion in the western United States. During the next year, Walmark is planning to construct new stores that will serve 10 geographically dispersed communities. Past experience indicates that a community must be within 25 miles of a store to attract
*7. (Gueret and Associates, 2002, Section 12.6) MobileCo is budgeting 15 million dollars to construct as many as 7 transmitters to cover as much population as possible in 15 contiguous geographical communities. The communities covered by each transmitter and the budgeted construction costs are
1. Leatherco is contracted to manufacture batches of pants, vests. and jackets. Each product requires a special setup of the machines needed in the manufacturing processes. The following table provides the pertinent data regarding the use of raw material (leather) and labor time together with cost
*2. lobco is planning to produce at least 2000 widgets on three machines. The minimum lot size on any machine is 500 widgets. The following table gives the pertinent data of the situation.Machine 12 3Cost 300 100 200 Production cosUunit 210 5Capacity (units)600 800 1200 Formulate the problem as an
*3. OiIco is considering two potential drilling sites for reaching four targets (possible oil wells). The following table provides the preparation costs at each of the two sites and the cost of drilling from site i to target j (i = 1,2; j = 1,2,3,4).Drilling cost (million $) to target Site 12 12 42
4. Three industrial sites are considered for locating manufacturing plants. The plants send their supplies to three customers. The supply at the plants, the demand at the customers, and the unit transportation cost from the plants to the customers are given in the following table.Unit
5. Repeat Problem 4 assuming that the demands at each of customers 2 and 3 are changed to 800.
6. (Liberatore and Miller, 1985) A manufacturing facility uses two production lines to produce three products over the next 6 months. Backlogged demand is not allowed.However, a product may be overstocked to meet demand in later months. The following table provides the data associated with the
8. (Brown and Associates, 1987) A company uses four special tank trucks to deliver four different gasoline products to customers. Each tank has five compartments with different capacities:500,750,1200,1500, and 1750 gallons. The daily demands for the four products are estimated at 10,15,12, and 8
*10. (Barnett, 1987) Professor Yataha needs to schedule six round-trips between Boston and Washington, D.C.The route is served by three airlines: Eastern, US Air, and Continental and there is no penalty for the purchase of one-way tickets. Each airline offers bonus miles for frequent fliers.
2. A machine is used to produce two interchangeable products. The daily capacity of the machine can produce at most 20 units of product 1 and 10 units of product 2. Alternatively, the machine can be adjusted to produce at most 12 units of product 1 and 25 units of product 2 daily. Market analysis
*3. Gapco manufactures three products, whose daily labor and raw material requirements are given in the following table.Product 12 3Required daily labor(hr/unit)3 45 Required daily raw material(lb/unit)4 36
4. lobco Shop has 10 outstanding jobs to be processed on a single machine. The following table provides processing times and due dates. All times are in days and due time is measured from time 0:Job Processing time Due time 1 10 20 2 3 98 3 13 100 4 15 34 5 9 50 6 22 44 7 17 32 8 30 60 9 12 80 10
5. In Problem 4, suppose that job 4 cannot be processed until job 3 has been completed.Also, machine settings for jobs 7 and 8 necessitate processing them one right after the other (Le., job 7 immediately succeeds or immediately precedes 8). lobco's objective is to process all ten jobs with the
6. laco owns a plant in which three products are manufactured. The labor and raw material, requirements for the three products are given in the following table. Product Required daily labor (hr/unit) Required daily raw material (Ib/unit) 1 3 4 2 4 3 3 S 6 Daily availability 100 100 The profits per
7. UPak is a subsidiary of an LTL (Iess-than-truck-load) transportation company. Customers bring their shipments to the UPak terminal to be loaded on the trailer and can rent space up to 36 ft. The customer pays for the exact linear space (in foot increments)the shipment occupies. No partial
8. Show how the nonconvex shaded solution spaces in Figure 9.5 can be represented by a set of simultaneous constraints. Find the optimum solution that maximizes z = 2x\ + 3X2 subject to the solution space given in (a).
9. Suppose that it is required that any k out of the following m constraints must be active:Show how this condition may be represented.
10. In the following constraint, the right-hand side may assume one of values, hI> b2, ..• , and bm'Show how this condition is represented.
1. Solve the ILP of Example 9.2-1 by the B&B algorithm starting with X2 as the branching variable. Start the procedure by solving the subproblem associated with X2 :s [x;].
2. Develop the B&B tree for each of the following problems. For convenience, always select Xl as the branching variable at node O.*(a) Maximize z = 3Xl + 2X2 subject to 2x( + 5X2 :5 9 4xl + 2X2 S 9 Xl, X;
(b) Maximize z = 2xI + 3X2 subject to 5XI + 7x2 ::5 35 4Xl + 9Xl ::5 36 Xl> X2 ~ 0 and integer
(c) Maximize z = Xl + Xl subject to 2XI + 5X2 ::5 16 6Xl + 5X2 ::5 27 Xl> Xl ~ 0 and integer
*(d) Minimize z = 5x1 + 4Xl subject to 3Xl + 2Xl ~ 5 2x( + 3X2 ~ 7 xl> X2 2:: 0 and integer
(e) Maximize z = 5xI + 7X2 subject to 2Xl + X2 ::5 13 5XI + 9X2 ::5 41 xl> X2 ~ 0 and integer
*3. Repeat Problem 2, assuming that Xl is continuous.
4. Show graphically that the following ILP has no feasible solution, and then verify the result using B&B.Maximize z = 2Xl + Xl subject to lOXI + lOx2 ::5 9 lOx! + 5Xl ~ 1 Xl, X2 ~ 0 and integer
5. Solve the following problems by B&B.Maximize z = 18xl + 14xl + 8x3 + 4X4 subject to 15Xj + 12xl + 7X3 + 4X4 + Xs ::5 37 Xt> Xl,X3,X4,XS = (0,1)
6. Convert the following problem into a mixed ILP and find the optimum solution.Maximize z = Xl + 2X2 + 5x)subject to
7. TORA/Solver/AMPL Experiment. The following problem is designed to demonstrate the bizarre behavior of the B&B algorithm even for small problems. In particular, note how many subproblems are examined before the optimum is found and how many are needed to verify optimality.Minimize y subject to
8. TORA Experiment. Consider the following ILP:Maximize z = 18xI + 14x2 + 8x3 subject to Xl, X2, X3 nonnegative integers Use TORA's B&B user-guided option to generate the search tree with and without activating the objective-value bound. What is the impact of activating the objective-value bound on
*9. TORA Experiment. Reconsider Problem 8 above. Convert the problem into an equivalent 0-1 ILP, then solve it with TORA's automated option. Compare the size of the search trees in the two problems.
10. AMPL Experiment. In the following 0-1 ILP use interactive AMPL to generate the associated search tree. In each case, show how the z-bound is used to fathom subproblems.Maximize z = 3Xl + 2X2 - 5x3 - 2X4 + 3xs subject to Xl + Xl + X3 + 2X4 + X5 5 4 7xI + 3X3 - 4X4 + 3X5 5 8 11Xl - 6X2 + 3X4 -
1. In Example 9.2-2, show graphically whether or not each of the foHowing constraints can form a legitimate cut:*(a) Xl + 2xz :5 10(b) 2xl + X2 :5 10(c) 3X2:5 10(d) 3xl + X2 :5 15
2. In Example 9.2-2, show graphically how the following two (legitimate) cuts can lead to the optimum integer solution:XI + 2X2 ~ 10 3Xl + x2 ::S 15(cut I)(cut II)
3. Express cuts I and II of Example 9.2-2 in terms of Xl and X2 and show that they are the same ones used graphically in Figure 9.10.
4. In Example 9.2-2, derive cut II from the x3-row. Use the new cut to complete the solution of the example.
5. Show that, even though the following problem has a feasible integer solution in Xl and X2, the fractional cut would not yield a feasible solution unless all the fractions in the constraint were eliminated.Maximize z = Xl + 2X2 subject to Xl + ~X2 ::s ¥X[,X2 ~ 0 and integer
6. Solve the following problems by the fractional cut, and compare the true optimum integer solution with the solution obtained by rounding the continuous optimum.*(a) Maximize z = 4XI + 6X2 + 2X3 subject to 4xI - 4X2 s: 5-Xl + 6X2 ::s 5- Xl + X2 + X3 ::s 5 Xl> X2, X3 ~ 0 and integer
(b) Maximize z = 3Xl + X2 + 3X3 subject to- Xl + 2X2 + X3 ::s 4 4X2 - 3X3 ::s 2 Xl - 3X2 + 2x3 s: 3 Xl, X2, X3 ~ 0 and integer
1. An automatic guided vehicle (AGV) is used to deliver mail to 5 departments located on a factory floor. The trip starts at the mail sorting room and makes the delivery round to the different departments before returning to themailroom.Using the mailroom as the origin (0,0), the (x, y) locations
*1. Solve Example 10.1-1, assuming the following routes are used:d(1,2) = 5,d(1,3) = 9,d(I,4) = 8 d(2,S) = 10,d(2,6) = 17 d(3,5) = 4, d(3, 6) = 10 d(4,5) = 9,d(4,6) = 9 d(5,7) = 8 d(6,7) = 9
2. I am an avid hiker. Last summer, I went with my friend G. Don on a 5-day hike-and-camp trip in the beautiful White Mountains in New Hampshire.We decided to limit our hiking to an area comprising three well-known peaks: Mounts Washington, Jefferson, and Adams.Mount Washington has a 6-mile
1. For Problem 1, Set 1O.1a, develop the backward recursive equation, and use it to fmd the optimum solution. S 14 9 12 3 12 FIGURE 10.3 Network for Problem 3, Set 10.2a 2. For Problem 2, Set 10.1a, develop the backward recursive equation, and use it to find the optimum solution. *3. For the
1. What relationships bind the stages together?
2. What information is needed to make feasible decisions at the current stage without reexamining the decisions made at previous stages?
1. In Example 10.3-1, determine the optimum solution, assuming that the maximum weight capacity of the vessel is 2 tons then 5 tons.
2. Solve the cargo-loading problem of Example 10.3-1 for each of the following sets of data:*(a) Wt = 4, rl = 70, ~ = 1, rz = 20, w3 = 2, r3 = 40, W = 6(b) WI = 1, rl = 30, Wz = 2, rz = 60, W3 = 3, r3 = 80, W = 4
3. In the cargo-loading model of Example 10.3-1, suppose that the revenue per item includes a constant amount that is realized only if the item is chosen, as the following table shows:Find the optimal solution using DP. (Hint: You can use the Excel file excelSetupKnapsack.xls to check your
4. A wilderness hiker must pack three items: food, first-aid kits, and clothes. The backpack has a capacity of 3 ft3. Each unit of food takes 1 ft3. A first-aid kit occupies V4 ft3 and each piece of cloth takes about 1/2fe. The hiker assigns the priority weights 3,4, and 5 to food, first aid, and
A student must select 10 electives from four different departments, with at least one course from each department. The 10 courses are allocated to the four departments in a manner that maximizes "knowledge."The student measures knowledge on a lOO-point scale and comes up with the following
*7. Habitat for Humanity is a wonderful charity organization that builds homes for needy families using volunteer labor. An eligible family can chose from three home sizes: 1000, 1100, and 1200 ftz. Each size house requires a certain number of labor volunteers. The Fayetteville chapter has received
8. Sheriff Bassam is up for reelection in Washington county. The funds available for the campaign are about $10,000. Although the reelection committee would like to launch the campaign in all five precincts of the county, limited funds dictate otherwise. The following table lists the voting
9. An electronic device consists of three components. The three components are in series so that the failure of one component causes the failure of the device. The reliability(probability of no failure) of the device can be improved by installing one or two standby units in each component. The
10. Solve the following model by DP:n Maximize z = IT)'i i=1 subject to)'1 + Y2 + ... + )'n = C Yj ;:::: 0, j = 1, 2, ... , n(Hint: This problem is similar to Problem 9, except that the variables, Yj, are continuous.)
11. Solve the following problem by DP:Minimize Z = YI + y~ + .. , + Y~subject to nITYi = C i=1 Yi > 0, i = 1, 2, ... , n
12. Solve the following problem by DP:Maximize z = (YI + 2)2 + Y2Y3 + (Y4 - 5)2 subject to YI + Y2 + )'3 + Y4 :s; 5)'i ;:::: °and integer, i = 1,2,3,4 13. Solve the following problem by DP:Minimize z = max{f(yd,f(}2),· .. ,f(YII)}subject to Yl + }2 + ... + Yn = c Yi 2 0, i = 1,2, ... , n Provide
1. Solve Example 10.3.2 for each of the following minimum labor requirements:*(a) bI = 6, bz = 5, b3 = 3, b4 = 6, bs = 8(b) bl = 8, b2 = 4, b3 = 7, b4 = 8, bs = 2
2. In Example 10.3-2, if a severance pay of $100 is incurred for each fired worker, determine the optimum solution.
*3. Luxor Travel arranges I-week tours to southern Egypt. The agency is contracted to provide tourist groups with 7,4,7, and 8 rental cars over the next 4 weeks, respectively. Luxor Travel subcontracts with a local car dealer to supply rental needs. The dealer charges a rental fee of $220 per car
4. GECO is contracted for the next 4 years to supply aircraft engines at the rate of four engines a year. Available production capacity and production costs vary from year to year. GECO can produce five engines in year 1, six in year 2, three in year 3, and five in year 4. The corresponding
*2. My son, age 13, has a lawn-mowing business with 10 customers. For each customer, he cuts the grass 3 times a year, which earns him $50 for each mowing. He has just paid $200 for a new mower.The maintenance and operating cost of the mower is $120 for the first year in service, and increases by
4. Consider the equipment replacement problem over a period of it years. A new piece of equipment costs c dollars, and its resale value after t years in operation is s(t) = n - t for n > l and zero otherwise. The annual revenue is a function of the age t and is given bv ret) = n2 - (2 for n > l and
5. Solve Problem 4, assuming that the equipment is 1 year old and that n = 4, c = $6000, ret) = 1 : t"
1. Solve Example 10.3-4, assuming that 71 = .085 and 72 = .08. Additionally, assume that PI = $5000, Pz = $4000, P3 =: $3000, and P4 = $2000.
2. An investor with an initial capital of $10,000 must decide at the end of each year how much to spend and how much to invest in a savings account. Each dollar invested returns ex = $1.09 at the end of the year. The satisfaction derived from spending $y in anyone year is quantified by the
3. A farmer owns k sheep. At the end of each year, a decision is made as to how many to sell or keep. The profit from selling a sheep in year i is p;. The sheep kept in year i will double in number in year i + 1. The farmer plans to sell out completely at the end of n years.*(a) Derive the general
(b) Solve the problem for n = 3 years, k = 2 sheep, PI = $100, P2 = $130. and P3 = $120.
1. Solve the following problems by DP.(a) Maximize z = 4x} + 14x2 subject to 2Xl + 7X2 === 21?xl + 2X2 === 21
(b) Maximize z = 8XI + 7X2 subject to 2x} + X2 S 8 5xl + 2X2 === 15 Xl> X2 ~ 0 and integer
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