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operations research an introduction

Operations Research: An Introduction 10th Global Edition Hamdy A Taha - Solutions

Solve Problem 11-7, by B&B.
Solve Problem 11-2, by B&B.
Solve Example 11.3-1 using subtour 2-5-4-2 to start the branching process at node 1, using the following sequences for exploring the nodes:(a) Explore all the subproblems horizontally from left to right in each tier before proceeding to the next tier.(b) Follow each path vertically from node 1,
Warehouse order picking, Ratliff and Rosenthal (1983). In a rectangular warehouse, a stacker overhead crane is used to pick and deliver orders between specified locations in the warehouse. The tasks of the crane involve the following: (1) picking a load at a location, (2) delivering a load to a
Wallpaper cutting, Garfinkel (1977). Covering the walls of a room usually requires cutting sheets of different lengths to account for doors and windows, and the like. The sheets are cut from a single roll, and their start points must be aligned to match the repeating pattern of the roll. The amount
Automatic guided vehicle. An AGV makes a round-trip (starting and ending at the mailroom)to deliver mail to 5 departments on a factory floor. Using the mailroom as the origin (0, 0), the (x, y) locations of the delivery spots are (10, 30), (10, 50), (30, 10), (40, 40), and (50, 60) for the five
The U.S. space agency, NASA, uses satellites for imaging celestial objects. The amount of fuel needed to reposition the satellites is a function of the sequence in which the objects are imaged. The matrix ‘cij ‘ below provides units of fuel consumption used to realign the satellites with the
DNA sequencing. In genetic engineering, a collection of DNA strings, each of length 10 ft, is concatenated to form one universal string. The genes of individual DNA strings may overlap, thus producing a universal string with length less than the sum of the individual lengths. The matrix ‘Oij ‘
(Integrated circuit boards) Circuit boards (such as those used in PCs) are drilled with holes for mounting different electronic components. The boards are fed one at a time under a moving drill. The matrix ‘dij ‘ below provides the distances (in millimeters)between pairs of 6 holes of a
Meals-on-Wheels is a charity service that prepares meals in its central facility for delivery to people who qualify for the service. Ideally, all meals should be delivered within 20 min from the time they leave the kitchen. This means that the return time from the last-meal location to the kitchen
A manager has a total of 10 employees working on six projects. Projects are reviewed weekly with each employee. A project may employ more than one employee resulting in assignment overlaps, as the following table shows:Project 1 2 3 4 5 6 1 x x x 2 x x x 3 x x x x 4 x x x Employee 5 x x x 6 x x x x
A tourist in New York City uses local transportation to visit 8 sites. The start and end and the order in which the sites are visited are unimportant. What is important is to spend the least amount of money on transportation. Matrix ‘cij ‘ below provides the fares in dollars between the
Proteins clustering. Proteins are clustered using an overall measure of similarity based on protein–protein interaction information. Clustering information is used to predict unknown protein functions. By definition, the best cluster maximizes the sum of the measures of similarity between
A baseball fan wishes to visit eight major league parks in (1) Seattle, (2) San Francisco,(3) Los Angeles, (4) Phoenix, (5) Denver, (6) Dallas, (7) Chicago, and (8)Tampa before returning home to Seattle. The fan will use air transportation between the different cities.The matrix ‘pij ‘ below
Seers Service Center schedules its daily repair visits to customers. The matrix ‘Tij ‘ below gives the travel time (in minutes) between the service center (row 1 and column 1)and seven jobs. The jobs are assigned to one of the repairpersons during an 8-hr shift.At the end of the day, the
A book salesperson who lives in Basin must call once a month on four customers located in Wald, Bon, Mena, and Kiln before returning home to Basin. The following table gives the distances in miles among the different cities.Miles between cities Basin Wald Bon Mena Kiln Basin 0 125 225 155 215 Wald
In each of the following instances, describe the data elements (cities and distances)needed to model the problem as a TSP.(a) Seers Service Center schedules its daily repair visits to customers. The jobs are categorized and grouped and each group assigned to a repairperson. At the end of the
Repeat Problem 10-41 using the variable y.
Construct the search tree in Figure 10.3 using the variable x to initiate the search.Compare the resulting amount of computations with that in Figure 10.3.
Excel experiment. Apply excelIPHeuristicGA.xls to Problem 10-37.
Carry out two iterations of Problem 10-36.
Carry out the next iteration that follows the one given in Table 10.19.
Excel experiment. Use file excels-IP-Heuristic.xls to find a solution for the following ILP:Minimize z = x1 + x2 + x3 + x4 + x5 + x6 + x7 subject to x1 + x4 + x5 + x6 + x7 Ú 20 x1 + x2 + x5 + x6 + x7 Ú 12 x1 + x2 + x3 + x6 + x7 Ú 14 x1 + x2 + x3 + x4 + x7 Ú 17 x1 + x2 + x3 + x4 + x5 Ú 18 x2 +
Carry out 5 SA iterations for the following problem:Maximize z = 99x1 + 90x2 + 58x3 + 40x4 + 79x5+ 92x6 + 102x7 + 74x8 + 67x9 + 80x10 subject to 30x1 + 8x2 + 6x3 + 5x4 + 20x5 + 12x6 + 25x7 + 24x8 + 32x9 + 29x10 … 100 All variables are binary
Carry out 5 iterations of Example 10.4-2 assuming cj = 1 for all j.
Excel experiment. Use excelTabu-IP-heuristc.xls to find a solution for the following problems:(a) Project selection problem of Example 9.1-1.(b) Set covering problem of Example 9.1-2.Compare the heuristic and exact solutions.
Carry out 10 TS iterations for each of the following problems:(a) Maximize z = 4x1 + 6x2 + 2x3 subject to 4x1 - 4x2 … 5-x1 + 6x2 … 5-x1 + x2 + x3 … 5 x1, x2, x3 Ú 0 and integer(b) Maximize z = 3x1 + x2 + 3x3 subject to-x1 + 2x2 + x3 … 4 4x2 - 3x3 … 2 x1 - 3x2 + 2x3 … 3 x1, x2, x3 Ú 0
Verify the entries in iterations 1, 2, and 3 in Table 10.15.
In the game of chess, queens move horizontally, vertically, or along a (45°) diagonal path. We need to position N queens in the 1N * N2 grid so that no queen can “take”any other queen. Design a GA for the problem starting with a random population of 4 parents and using a 1-point crossover. A
Consider the following problem:Maximize f1x, y2 = xsin14x2 + 1.1sin12y2, x = 0, 1, 2,c, 10, y = 0, 1, 2,c, 10 Carry out five GA iterations to estimate the optimum solution.
Repeat Problem 10-28 assuming that the wire is used to form a box with the maximum volume.
You have a piece of wire whose length is L = 107.1 inches and you would like to shape it into a rectangular frame. Use the genetic algorithm to determine the width and height that will yield the maximum area of the rectangle.
You have a deck of ten cards numbered 1 to10. You need to divide the ten cards into two piles such that the sum of pile 1 cards is 36 and the product of the pile 2 cards is also 36. Develop a GA for the problem using an initial population of 4 parents, 1-point crossover, and 1% mutation rate. Carry
Carry out an additional iteration of Example 10.3-6.
Carry out two additional iterations of Example 10.3-5.
Suppose that GA is used to find the maximum of F(x), x = 0, 1,c, 275. Let x = 107 and x = 254 represent parents P1 and P2.(a) Represent P1 and P2 as binary codes.(b) Use uniform crossover to create C1 and C2.(c) Create C1 and C2 using a 1-point crossover.(d) Create C1 and C2 using a 2-point
Consider the well-known six-hump camelback function:f1x, y2 = 4x2 - 2.1x4 + x6/3 + xy - 4y2 + 4y4, -3 … x … 3, -2 … y … 2 The exact global minima are 1 -.08984, .712662 and 1.08984, -.712662 with f* = -1.0316. Apply five SA iterations to estimate the minima of f(x, y). Start with 1x0, y02 =
Scheduling conflicting classroom courses. A simplified version of college course scheduling calls for assigning eight courses (1, 2, . . . , 8) in the least possible number of time periods. Table 10.22 assigns “x” to conflicting courses (those that cannot be scheduled in the same time
Map-coloring problem. The coloring problem deals with determining the least number of colors for painting the regions of a map such that no two adjacent regions will have the same color. Figure 10.7(a) provides an example of a 6-region map. The problem can be modeled as a network in which the nodes
Timetable scheduling. Consider a case of developing a timetable of teaching 5 classes(C) by 5 teachers (T). The teachers provide the following preferences for teaching classes (first of the list is most desired):T1: C2-C3-C1-C5 T2: C2-C1-C4-C5 T3: C1-C4-C5-C3 T4: C4-C2-C5-C3 T5: C2-C5-C3-C1 The
Carry out four additional iterations of the job sequencing problem in Example 10.3-4.
Solve Example 10.3-3 to estimate the maximum solution point. Use x0 = 2 and t = 3.
Carry out five additional iterations in Example 10.3-3.
Cartographic label placement, Yamamoto and Associates (2002). Unambiguous placement of the names of cities, streets, lakes, and rivers on printed maps has long been a time-consuming manual process. With the advent of online map generation (as in Google and MapQuest), the manual process is not a
Constrained Minimal Spanning Tree, Glover (1990). Section 6.2 presents an optimum algorithm for finding the minimal spanning tree that links all the nodes of a network(by definition, a tree contains no cycles). In a practical setting, it may be necessary to impose interdependence restrictions on
Warehouse allocation. Consider the case of 4 warehouses and 5 stores. The fixed cost of opening a warehouse is 20 ($ thousand). The transportation cost, cij, of shipments between the warehouses and the stores is summarized in Table 10.21.(a) Formulate the problem as an ILP, and find the optimum
Repeat Problem 10-12 for the following Boolean expressions:(B1 and B5) or (B3 and B9) and (B2 or B10)B3 or B6 and (B7 or B9 and B10)B4 and B7 and B8 B2 or B3 and B4 and B5 or B8 and (B1 or B6)(B3 and B4 and B10) or (B5 and B7) or (B9 and B10)B1 or (B4 and B7) or B8(B3 and B5 or B6) or (B1 or B8 and
Consider 10 Boolean variables, Bi, i = 1, 2,c, 10. Each variable assumes the value T (true) or F (false). Next, consider the following six expressions (the notation Bi defines not Bi):(B1 and B3 and B8) or (B4 and B10) and B6 B2 and B7(B2 or B5) and (B1 or B4 or B6)(B1 and B3 or B4) or (B5)(B4 and
Apply TS with t = 3 iterations to solve the 5-job sequencing problem using the data in Table 10.20. (Hint: You may find it convenient to use file excelJobSequencing.xls to compute the cost functions.)
Consider the following function:f1x2 = .01172x6 - .3185x5 + 3.2044x4 - 14.6906x3 + 29.75625x2 - 19.10625x The function has multiple maxima and minima in the range x = 1, 2,c, 10. Apply 10 TS iterations to estimate the maximum and minimum. Use x0 = 5 and tabu tenure period t = 2 iterations.
Solve Example 10.2-1 to estimate the maximum solution point. Use x0 = 8 and t = 2.
The height of a cylindrical water tank must be at least twice as much as its base diameter. Neither the diameter nor the height can exceed 10 ft. The volume of the tank must be at least 300 ft3. The cost of the elevated structure on which the tank is installed is proportional to the area of the
Apply the uniform sampling heuristic to estimate the minimum solution of the following two-variable function: f1x2 = 3x2 + 2y2 - 4xy - 2x - 3y, 0 … x … 5, 0 … y … 5.
Consider the problem of forming a maximum-area rectangle out of a piece of wire of length 100 inches.(a) Excel experiment. Use excelContVarHeuristic.xls with uniform sampling to generate 5 iterations of the continuous variable heuristic to estimate the dimensions of the rectangle. Start with a base
Excel experiment. Consider the following function:f1x2 = .01172x6 - .3185x5 + 3.2044x4 - 14.6906x3 + 29.75625x2 - 19.10625x The function has multiple maxima and minima in the range 0 … x … 10. Use excelContVarHeuristic.xls to estimate the maximum and minimum of the function using uniform
Re-solve the problem of Example 10.2-3 to estimate the maximum value of F(x) using uniform sampling. Next, use the solution from uniform sampling as a starting solution for the application of normal sampling.
Re-solve the problem of Example 10.2-2 to estimate the maximum value of F(x).
Re-solve the problem of Example 10.2-1 to estimate the maximum value of F(x).Repeat the calculations using x = 7 as a starting solution.
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 = 4x1 + 6x2 + 2x3 subject to 4x1 - 4x2 … 5 -x1 + 6x2 … 5 -x1 + x2 + x3 … 5 x1, x2, x3 Ú 0 and integer (b) Maximize z
Show that, even though the following problem has a feasible integer solution in x1 and x2, the fractional cut would not yield a feasible solution unless all the fractions in the constraint were eliminated.Maximize z = x1 + 2x2 subject to x1 + 12 x2 … 13 4x1, x2 Ú 0 and integer
In Example 9.2-2, derive cut II from the x3-row. Use the new cut to complete the solution of the example.
Express cuts I and II of Example 9.2-2 in terms of x1 and x2, and show that they are the same ones used graphically in Figure 9.6.
In Example 9.2-2, show graphically how the following two (legitimate) cuts can lead to the optimum integer solution:x1 + 2x2 … 10 1Cut I2 3x1 + x2 … 15 1Cut II2
In Example 9.2-2, show graphically whether or not each of the following constraints can form a legitimate cut:*(a) x1 + 2x2 … 10(b) 2x1 + x2 … 10(c) 3x2 … 10(d) 3x1 + x2 … 15
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 = 3x1 + 2x2 - 5x3 - 2x4 + 3x5 subject to x1 + x2 + x3 + 2x4 + x5 … 4 7x1 + 3x3 - 4x4 + 3x5 … 8 11x1 - 6x2 + 3x4 -
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.
TORA Experiment. Reconsider Problem
TORA Experiment. Consider the following ILP:Maximize z = 18x1 + 14x2 + 8x3 subject to 15x1 + 12x2 + 7x3 … 43 x1, 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
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 21x1
Convert the following problem into a mixed ILP, and find the optimum solution:Maximize z = x1 + 2x2 + 5x3 subject to -x1 + 10x2 - 3x3 Ú 15 2x1 + x2 + x3 … 10 x1, x2, x3 Ú 0
Solve the following problems by B&B:Maximize z = 18x1 + 14x2 + 8x3 + 4x4 subject to 15x1 + 12x2 + 7x3 + 4x4 + x5 … 37 x1, x2, x3, x4, x5 = 10, 12
Show graphically that the following ILP has no feasible solution, and then verify the result using B&B.Maximize z = 2x1 + x2 subject to 10x1 + 9x2 … 8 8x1 + 6x2 Ú 1 x1, x2 Ú 0 and integer
Repeat Problem 9-56, assuming that x1 is continuous.
Develop the B&B tree for each of the following problems. For convenience, always select x1 as the branching variable at node 0.*(a) Maximize z = 3x1 + 2x2 subject to 2x1 + 5x2 … 18 4x1 + 2x2 … 18 x1, x2 Ú 0 and integer(b) Maximize z = 2x1 + 3x2 subject to 7x1 + 5x2 … 36 4x1 + 9x2 … 35 x1,
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 … [x2 * ].11
Give the binary variables y1, y2, . . . , yn, such that if xi = 1, then xi - 1 or xi + 1 must equal 1, i = 1, 2, . . . , n, where y0 and yn + 1 define the variable yn.
Consider the following objective function Minimize z = min52x1 + x2, 4x1 - 3x2 x1 Ú 1, x2 Ú 06 Use auxiliary binary variables to convert the objective function z into an analytically manageable format that eliminates the min function.
In the following constraint, the right-hand side may assume one of values, b1, b2, . . . , and bm.g1x1, x2,c, xn2 … 1b1, b2 ,c, or bm2 Show how this condition is represented.
Suppose that it is required that any k out of the following m constraints must be active:gi1x1, x2,c , xn2 … bi, i = 1, 2,c , m Show how this condition may be represented.
Consider the binary variable yi, i = 1, 2,c, n. Express the following condition as a set of simultaneous ILP constraints: If i = k, then yk = 1, and all the remaining variables equal zero.
Given the binary variables x1, x2, x3, x4, and x5, if x1 = 1 and x2 = 0, then x3 = 1, x4 = 1, and x5 = 1. Formulate the condition as simultaneous constraints.*9-49. Suppose that product zw occurs in a constraint, where z and w are binary variables. Show how this term can be linearized.
Show how the nonconvex shaded solution spaces in Figure 9.8 can be represented by a set of simultaneous constraints. Find the optimum solution that maximizes z = 2x1 + 3x2 subject to the solution space given in (a).
A manufacturing process uses four interchangeable raw materials. The raw materials differ in properties, which leads to different output units per unit of raw material. They also differ in cost and lot sizes. The following table summarizes the data of the situation:Material 1 Material 2 Material 3
N queens problem. In the game of chess, queens attack by moving horizontally, vertically, or diagonally. It is desired to place N queens on an (N * N)-grid so that no queen can“take” any other queen. Formulate the problem as an integer program, and solve with AMPL (or any other software) for N
UPak is a subsidiary of an LTL 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 shipment is allowed, in the sense
Jaco 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(lb/unit)1 3 4 2 4 3 3 5 6 Daily availability 100 100 The profit per unit for
In Problem 9-41, 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 (i.e., job 7 immediately succeeds or precedes job 8). Jobco’s objective is to process all ten jobs with the smallest
Jobco 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 (day) Due time (day)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
Gapco manufactures three products, whose daily labor and raw material requirements are given in the following table.Product Required daily labor(hr/unit)Required daily raw material(lb/unit)1 3 4 2 4 3 3 5 6 The profits per unit of the three products are $20, $25, and $18, respectively. Gapco has
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 40 units of product 2. Alternatively, the machine can be adjusted to produce at most 45 units of product 1 and 25 units of product 2 daily. Market analysis
Barnett (1987). Professor Yataha needs to schedule eight 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. Eastern
A household uses at least 3000 minutes of long-distance telephone calls monthly and can choose to use the services of any of three companies: A, B, and C. Company A charges a fixed monthly fee of $10 and 5 cents per minute for the first 1000 minutes and 4 cents per minute for all additional
A company uses four special tank trucks to deliver four different gasoline products to customers. Each tank has five compartments with different capacities: 500, 750, 1200, 1500, and 1750 gallons. The daily demands for the four products are estimated at 10, 15, 12, and 8 thousand gallons. Any
Jarvis and Associates (1978). Seven cities are being considered as potential locations for the construction of at most four wastewater treatment plants. The following table provides the data for the situation. Missing links indicate that a pipeline cannot be constructed.Cost ($) of pipeline
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
Repeat Problem 9-31 assuming that the demands at each of customers 2 and 3 are changed to 800.
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:In addition to the
Oilco 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 1i = 1, 2; j = 1, 2, 3, 42:Drilling cost ($ million) to target Site 1 2 3 4
Jobco is planning to produce at least 2000 widgets on three machines. The minimum lot size on any machine is 600 widgets. The following table gives the pertinent data of the situation.Machine Setup cost ($) Production cost/unit ($) Capacity (units)1 300 2 650 2 100 10 850 3 200 5 1250 Formulate the
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 and

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