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operations research an introduction
Operations Research An Introduction 11th Edition Hamdy A. Taha - Solutions
A police car is normally on regular patrol circulating a neighborhood. During the patrol, there is a 60% chance of responding in time to the location where help is needed; else regular patrol will continue. Upon receiving a call, there is a 10% chance of cancellation (in which case regular patrol
Bank1 offers loans which are either paid when due or are delayed. If the payment on a loan is delayed by more than 4 quarters (1 year), Bank1 considers the loan a bad debt and writes it off. The following table provides a sample of Bank1’s past experience with loans.Express Bank1’s loan
A school district is in dire need to reduce expenses to meet new budgetary restrictions at its elementary schools. Two options are available: delete the physical education program (E), or delete the music program (M). The superintendent has formed a committee with equal-vote representation from the
Patients suffering from kidney failure can either get a transplant or undergo periodic dialysis. During any 1 year, 30% undergo cadaveric transplants, and 10% receive living-donor kidneys. In the year following a transplant, 30% of those who undergo the cadaveric transplants and 15% of living-donor
An engineering professor acquires a new computer every 2 years. The professor can choose from three models: M1, M2, and M3. If the present model is M1, the next computer can be M2 with probability .2, or M3 with probability .15. If the present model is M2, the probabilities of switching to M1 and
Consider the data of Problem 14-9. Suppose that cancellations intersperse new requests according to the following time-sequenced activities: Request 10 seats for fare 1, request 3 seats for fare 3, request 15 seats for fare 7, cancel 5 seats of fare 1, request 14 seats for fare 6, cancel 10 seats
A pension fund that starts with an initial capital of $250,000 has financial obligations over the next 15 years estimated at $11,000 for year 1 and growing at 8.5% per year thereafter. The fund will invest in short-, mid-, and long-term bonds per the financial data in the following table:All bonds
Acme Manufacturing produces lots of widget with .8%, 1%, 1.2%, and 1.4% defectives according to the respective probabilities .4, .3, .25, and .05. Three customers, A, B, and C, are contracted to receive batches with no more than .8%, 1.2%, and 1.4% defectives, respectively. If the defectives are
You have one fair coin and two bent ones. A flip of a fair coin has a 50-50 chance of landing heads (H). The odds for the outcome H of the bent coins are 30-70 and 90-10. A coin selected randomly is flipped. Determine the posterior probabilities head and tails after one flip. After two flips.
The network in Figure 6.41 gives the distances in miles between pairs of cities 1, 2, . . . , and 8. Use Dijkstra’s algorithm to find the shortest route between the following cities:(a) Cities 1 and 8(b) Cities 1 and 6(c) Cities 4 and 8(d) Cities 2 and 6FIGURE 6.41
Consider the Q-P curve, Q = 100/P2.(a) Compute point elasticity at any point (Q, P).(b) What is the significance of the result in (a)(c) Compute arc elasticity between current P1 = 2 and educed P2 in decrements of 10%, 20%, 30% and 40% relative to P1; meaning P2 assumes the successive values 1.8,
An old-fashioned electric toaster has two spring-loaded base-hinged doors. The two doors open outward in opposite directions away from the heating element. A slice of bread is toasted one side at a time by pushing open one of the doors with one hand and placing the slice with the other hand. After
Repeat Problem 14-10 assuming reversible fare closing. Compare the results.Problem 14-10Consider the data of Problem 14-9. Suppose that cancellations intersperse new requests according to the following time-sequenced activities: Request 10 seats for fare 1, request 3 seats for fare 3, request 15
Figure 6.40 provides the communication network between two stations, 1 and 7. The probability that a link in the network will operate without failure is shown on each arc. Messages are sent from station 1 to station 7, and the objective is to determine the route that maximizes the probability of a
In the unbalanced transportation problem in Table 5.36, if a unit from a source is not shipped out (to any of the destinations), a storage cost is incurred at the rate of $5, $4, and $3 per unit for sources 1, 2, and 3, respectively. Additionally, all the supply at source 2 must be shipped out
Vista City Hospital plans the short-stay assignment of surplus beds (those that are not already occupied) 4 days in advance. During the 4-day planning period, about 30, 25, and 20 patients will require 1-, 2-, or 3-day stays, respectively. Surplus beds during the same period are estimated at 20,
Knapsack Problem. A hiker has a 5-ft3 backpack and needs to decide on the most valuable items to take on the hiking trip. There are three items from which to choose. Their volumes are 2, 3, and 4 ft3, and the hiker estimates their associated values on a scale from 0 to 100 as 30, 50, and 70,
Three inmates escorted by three guards to be transported by boat from the mainland to a penitentiary island to serve their sentences. The boat transfers no more than two persons in either direction. The inmates are certain to overpower the guards if they outnumber them anywhere at any time. Assume
Consider eight equal squares arranged in three rows, with two squares in the first row, four in the second, and two in the third. The squares of each row are arranged symmetrically about the vertical axis. Fill the squares with distinct numbers in the range 1 to 8 so that no two adjacent vertical,
Draw the network defined byN = {1, 2, 3, 4, 5, 6}A = {(1, 2), (1, 5), (2, 3), (2, 4), (3, 4), (3, 5), (4, 3),(4, 6), (5, 2), (5, 6)}
The National Parks Service is receiving four bids for logging at three pine forests in Arkansas. The three locations include 10,000, 20,000, and 30,000 acres. A single bidder can bid for at most 50% of the total acreage available. The bids per acre at the three locations are given in Table 5.32.
In the transportation problem in Table 5.35, the total demand exceeds the total supply. Suppose that the penalty costs per unit of unsatisfied demand are $5, $3, and $2 for destinations 1, 2, and 3, respectively. Use the LC starting solution, and compute the iterations leading to the optimum
Military Planning, Shepard and Associates (1988). The Red Army (R) is trying to invade the territory defended by the Blue Army (B). Blue has three defense lines and 200 regular combat units and can draw also on a reserve pool of 200 units. Red plans to attack on two fronts, north and south, and
Suppose that for budgetary reasons the TOYCO model of Example 3.6-3 must reduce the operation times for all three operations by an equal percentage.(a) Use the 100% rule to determine the percent reduction.(b) Show that any reduction ratio in the range [0, 1] will work just as well to satisfy Reddy
Ozark Electronics manufactures four models of smartphones; M1, M2, M3, and M4. The electronic assembly must now be encased using two specially treated plastic components, C1 and C2. The following table provides the data of the situation:(a) Letting x1, x2, x3, and x4 be the amounts produced of M1,
Periodic preventive maintenance is carried out on aircraft engines, where an important component must be replaced. The numbers of aircraft scheduled for such maintenance over the next six months are estimated at 200, 180, 300, 198, 230, and 290, respectively. All maintenance work is done during the
Consider the solution space in Figure 3.16 with the solution starting at the origin A = (0, 0, 0). For each of the objective functions listed below: Select the nonbasic variable that leads to the next simplex corner-point solution, identify the next solution point as A, B, . . ., or J, and
Voting on Issues. In a particular county in the State of Arkansas, four election issues are on the ballot: Build new highways, increase gun control, increase farm subsidies, and increase gasoline tax. The county includes 100,000 urban voters, 250,000 suburban, and 50,000 rural, all with varying
True or False?(a) To balance a transportation model, it may be necessary to add both a dummy source and a dummy destination.(b) The amounts shipped to a dummy destination represent surplus at the shipping source.(c) The amounts shipped from a dummy source represent shortages at the receiving
Pollution Control. Three types of coal, C1, C2, and C3, are pulverized and mixed together to produce 50 tons per hour needed to power a plant for generating electricity. The burning of coal emits sulfur oxide (in parts per million) which must meet the EPA specifications of at most 2000 parts per
True or False?(a) The dual of the dual problem yields the original primal.(b) If the primal constraint is originally in equation form, the corresponding dual variable is necessarily unrestricted.(c) If the primal constraint is of the type ≤, the corresponding dual variable will be nonnegative
Example 1.6-1 revises the 50-50 guesstimate (prior) probability based on the parking lot observation, yielding a posterior probability which may then be used as prior probability for a succeeding observation. Suppose that the succeeding observation is represented by the statistics given in Problem
Assembly-Line Balancing. A product is assembled from three different parts. The parts are manufactured by two departments at different production rates as given in the following table:Determine the maximum number of final assembly units that can be produced weekly.Assembly units = min{units of part
Financial institutions use predictive models to decide whether to grant loans. The goal is to reduce the risk of issuing bad loans. The overall objective is to compute a score representing the credit risk of the loan seeker. With this goal in mind, what types of data are needed to develop the
Show that, in general, the Newton–Raphson method (Section 20.1.2) when applied to a strictly concave quadratic function will converge in exactly one step. Apply the method to the maximization of f(x) = 4x₁ + 6x₂ 4x₁ + 6x₂ - 2x² - 2x₁x₂ - 2x²
Approximate the following problem as a mixed integer program.subject to Maximize z = e-x₁ + x₁ + (x₂ + 1)²
Show how the following problem can be made separable.subject to Maximize z = X₁ X₂ + x3 + X₁ X3
Show how the following problem can be made separable.subject to Minimize z = ²x₁ + x² + (x3 - 2 2)²
Show how the following problem can be made separable.subject to Maximize z = ex₁x² + x²x3 + x4
Show that in separable convex programming, it is never optimal to have xki > 0 when xk – 1, i is not at its upper bound.
Solve as a separable convex programming problem.subject to Minimize z x₁ + x₂ + x²
Solve the following as a separable convex programming problem.subject to Minimize z (x₁ - 2)² + 4(x₂ - 6)²
Show that the following vectors are linearly dependent.(a)(b) 1 -2 3 -2 4 42 -2 1 -2 -1
Consider the partitioned matricesFind AB using partitioned matrix manipulation. A 1 5 7 2-69 3 4 7 2 9 1 B 2 3 12 3 1 4 5 6 7 09
Givenfind A−1 using the following methods:(a) Adjoint matrix method.(b) Row operations method.(c) Product form of the inverse.(d) Partitioned matrix method with p = 2 and q = 1. A || 1 1 12 2 1 2 2 1
Consider the nonsingular matrixSuppose that P1 is replaced with the vector V1 = P2 + P3, rendering the singular matrix B = (V1, P2 , P3 ). Show how the product form inverse method discovers the singularity of B. A = (P₁, P₂, P3) = 2 1 2 02 1 404
Show that the following quadratic form is negative definite. Q(x₁, x₂) = 6x₁ + 3x₂ - 4x₁x₂ - 2x² - 3x² - 27 4
Show that the following quadratic form is positive definite. Q(X₁, X2, X3) = 2x² + 2x² + 3x² + 2x1x₂ + 2x2x3
Show that the quadratic functionis strictly convex. f(x₁, x₂, x3) = 5x² + 5x² + 4x² + 4x₁x₂ + 2x₂x3
Show that the function f(x) = ex is strictly convex over all real values of x.
In Problem D-12, show that −f( x1, x2 , x3) is strictly concave.Problem D-12:Show that the quadratic functionis strictly convex. f(x₁, x₂, x3) = 5x² + 5x² + 4x² + 4x₁x₂ + 2x₂x3
In the tickets example, identify and evaluate a fourth feasible alternative.
A foundry smelts steel, aluminum, and cast iron scraps to produce two types of metal ingots, I and II, with specific limits on the aluminum, graphite, and silicon contents. Aluminum and silicon briquettes may be used in the smelting process to meet the desired specifications. The following tables
An oil company distills two types of crude oil, A and B, to produce regular and premium gasoline and jet fuel. There are limits on the daily availability of crude oil and the minimum demand for the final products. If the production is not sufficient to cover demand, the shortage must be made up
All-Natural Coop makes three breakfast cereals, A, B, and C, from four ingredients: rolled oats, raisins, shredded coconuts, and slivered almonds. The daily availabilities of the ingredients are 5 tons, 2 tons, 1 ton, and 1 ton, respectively. The corresponding costs per ton are $100, $120, $110,
A hardware store packages DIY (do-it-yourself) bags of screws, bolts, nuts, and washers. Screws come in 100-lb boxes and cost $110 each, bolts come in 100-lb boxes and cost $150 each, nuts come in 80-lb boxes and cost $70 each, and washers come in 30-lb boxes and cost $20 each. The DIY package
The city of Fayetteville is embarking on an urban renewal project that will include lowerand middle-income row housing, upper-income luxury apartments, and public housing. The project also includes a public elementary school and retail facilities. The size of the elementary school (number of
Lewis (1996). Monthly bills in a household are received monthly (e.g., utilities and home mortgage), quarterly (e.g., estimated tax payments), semiannually (e.g., insurance), or annually (e.g., subscription renewals and dues). The following table provides the monthly bills for next year.To account
A realtor is developing a rental housing and retail area. The housing area consists of efficiency apartments, duplexes, and single-family homes. Maximum demand by potential renters is estimated to be 500 efficiency apartments, 300 duplexes, and 250 single-family homes, but the number of duplexes
The city council of Fayetteville is in the process of approving the construction of a new 200,000-ft2 convention center. Two sites have been proposed, and both require exercising the “eminent domain” law to acquire the property. The following table provides data about proposed (contiguous)
A refinery manufactures two grades of jet fuel, F1 and F2, by blending four types of gasoline, A, B, C, and D. Fuel F1 uses gasolines A, B, C, and D in the ratio 1:1:2:4, and fuel F2 uses the ratio 2:2:1:3. The supply limits for A, B, C, and D are 1000, 1200, 900, and 1500 bbl/day, respectively.
Two alloys, A and B, are made from four metals, I, II, III, and IV, according to the following specifications:The four metals are extracted from three ores according to the following data:Develop an LP model to determine how much of each type of alloy should be produced, and find the solution using
In the bus scheduling example suppose that buses can run either 8- or 12-hour shifts. If a bus runs for 12 hours, the driver is paid for the extra hours at 150% of the regular hourly pay. Do you recommend the use of 12-hour shift buses (in conjunction with the 8-hour shift buses)? Solve the new
A company has contracted to produce two products, A and B, over the months of June, July, and August. The total production capacity (expressed in hours) varies monthly. The following table provides the basic data of the situation:The production rates in units per hour are .75 and 1 for products A
The demand for an item over the next four quarters is 300, 400, 450, and 250 units, respectively. The price per unit starts at $20 in the first quarter and increases by $2 each quarter thereafter. The supplier can provide no more than 400 units in any one quarter. Although one can take advantage of
HiRise Construction can bid on two 1-year projects. The following table provides the quarterly cash flow (in millions of dollars) for the two projects.HiRise has cash funds of $1 million at the beginning of each quarter and may borrow at most $1 million at a 10% nominal annual interest rate. Any
TORA Experiment. Consider the Diet Model, and let the objective function be given as Minimize z = .8x1 + .8x2 Use TORA to show that the optimum solution is associated with two distinct corner points and that both points yield the same objective value. In this case, the problem is said to have
TORA Experiment. In the Reddy Mikks model, use TORA to show that the removal of the raw material constraints (constraints 1 and 2) would esult in an unbounded solution space. What can be said in this case about the optimal solution of the model?
Determination of the Optimum LP Solution by Enumerating All Feasible Corner Points. The remarkable observation gleaned from the graphical LP solution is that the optimum, when finite, is always associated with a corner point of the feasible solution space. Show how this idea is applied to the Reddy
In Example 1.6-1, suppose instead that the restaurant rating is 70% and the probability of having equal rating among all restaurant in town estimated at .4. Given the guesstimate initial probability of .5 that any restaurant in town will offer a good meal, how would the ratings evidence change the
Some of us may remember a police car lurking on a side of the road, engine running, and a radar gun trained on passing traffic. And if one is caught speeding, the ensuing process of issuing an infraction (or ticket) may take 20 minutes, during which time other speeding cars would zip by with no
Repeat Problem 1-18 using the same cost table and the following raw data:Problem 1-18In Example 1.4-2, consider the unit transportation costs ($) among the five stores as given in the following table:Find two different feasible solutions per the transportation model in Figure 1-3 and compare the
In Example 1.4-2, consider the unit transportation costs ($) among the five stores as given in the following table:Find two different feasible solutions per the transportation model in Figure 1-3 and compare the costs.Example 1.4-2The following data for a hypothetical 5-store situation is used to
A 2017 study estimates that New York City traffic, particularly in Lower Manhattan, traffic crawls at less than 5 miles per hour, down 28% from five years ago. Faced with this alarming traffic congestion, the city is implementing two programs in highly-congested selected areas (e.g., Lower
The scarcest resource in a police department is its cadre of officers. Smart use of this limited resource depends on identifying critical areas of police jurisdictions where the presence of law enforcement agents is most needed. Analytics based on manipulating data can improve the chances of the
Turnover is costly to employers and efforts must be made to keep it in check. For that to happen, employers must learn how to achieve this goal based on facts in the form of pertinent data that relates to the work experience of the employees during their tenure the company.(a) What specific
A queueing study in a department store reveal the following information about the percent of idleness and customer waiting time as a function of the of the number of cashiers:The store manager wants to keep the average waiting time below 5 minutes and simultaneously maintain the percentage of
You have 10 identical cartons each holding 10 water bottles. All bottles weigh 10 oz. each, except for one defective carton in which each of the 10 bottles weighs 9 oz. only. A scale is available for weighing. Of course, weighing the cartons randomly one at a time will eventually locate the
The squares of a rectangular board of 11 rows and 9 columns are numbered sequentially 1 through 99 with a hidden monetary reward between $0 and $20 assigned to each square. A game using the board requires the player to choose a square by selecting any two-digit number and then subtracting the sum
A (two-dimensional) “pyramid” is constructed in four layers: The bottom layer consists of (equally spaced) dots 1, 2, 3, and 4; the next layer includes dots 5, 6, and 7; the following layer has dots 8 and 9; and the top layer has dot 10. You want to invert the pyramid (i.e., bottom layer has
During the construction of a house, six joists of 24 feet each must be trimmed to the correct length of 23 feet. The operations for cutting a joist involve the following sequence:Three persons are involved: Two loaders must work simultaneously on operations 1, 2, and 5, and one cutter handles
You have four chains, each consisting of three solid links. You need to make a bracelet by connecting all four chains. It costs 2 cents to break a link and 3 cents to re-solder it.(a) Identify two feasible solutions and evaluate them.(b) Determine the cheapest cost for making the bracelet.
In a baseball game, Jorge is the pitcher and Jack the batter. Jorge can throw either a fast or a curve ball at random. If Jack predicts Jorge’s throw correctly, his batting score stands at .500 for a curve ball and .300 for a fast ball. Else, if he misses, then his batting average drops to .200
Four friends, Amy, Ravi, Ali, and Tao, are standing on the east bank of a river and wish to cross to the west side using a canoe. The canoe can hold at most two people at a time. Amy, being the most athletic, can row across the river in 1 minute. Ravi, Ali, and Tao would take 2, 5, and 10 minutes.
The corner points of a square are designated ass A, B, C, and D. A circle passes through corner points C and D and touches the midpoint of side AB. Develop a mathematical model that determines which of the two geometric figures has the larger perimeter.
Consider a square of perimeter 4L. Next, construct a rectangle with its lower left corner coinciding with the square’s. The perimeter of the rectangle is also 4L, with base L + x, 0 < x < L. Using only geometry, prove that the area of the rectangle is less than the area of the square for
In the garden fence problem, assume perimeter L = 50 ft , and let x be the width of the rectangle. Compute the garden area in x-increments of 10 ft.; i.e., x = 10, 20, 30, 40. Can this approach be used to determine the optimum solution? Explain.
In Example 6.3-5, use Floyd’s algorithm to determine the shortest routes between each of the following pairs of nodes:(a) From node 5 to node 1.(b) From node 3 to node 5.(c) From node 5 to node 3.(d) From node 5 to node 2.Example 6.3-5For the network in Figure 6.15, find the shortest routes
Apply Floyd’s algorithm to the network in Figure 6.43. Arcs (7, 6) and (6, 4) are unidirectional, and all the distances are in miles. Determine the shortest route between the following pairs of nodes:(a) From node 1 to node 7.(b) From node 7 to node 1.(c) From node 6 to node 7.FIGURE 6.43
Six kids, Joe, Kay, Jim, Bob, Rae, and Kim, play a variation of hide and seek. The hiding place of a child is known only to a select few of the other children. A child is then paired with another with the objective of finding the partner’s hiding place. This may be achieved through a chain of
Chicken feed is transported by trucks from three silos to four farms. Some of the silos cannot ship directly to some of the farms. The capacities of the other routes are limited by the number of trucks available and the number of trips made daily. The following table shows the daily amounts of
A parent has five (teenage) children and five household chores to assign to them. Past experience has shown that forcing chores on a child is counterproductive. With this in mind, the children are asked to list their preferences among the five chores, as the following table shows:The parent’s
Four factories are engaged in the production of four types of toys. The following table lists the toys that can be produced by each factory.All toys require approximately the same per-unit labor and material. The daily capacities of the four factories are 250, 180, 300, and 100 toys, respectively.
In Example 6.3-6, use LP to determine the shortest routes between the following pairs of nodes:(a) Node 1 to node 5.(b) Node 2 to node 5.Example 6.3-6In the network of Example 6.3-4, determine the shortest route from node 1 to node 2—that is, s = 1 and t = 2. Figure 6.16 shows a unit flow
The academic council at the U of A is seeking representation from among six students who are affiliated with four honor societies. The academic council representation includes three areas: mathematics, art, and engineering. At most two students in each area can be on the council. The following
Develop the linear program and the associated minimum-cost flow network for the employment scheduling model of Example 6.6-3 using the following 5-month demand data. The per-worker costs of recruiting and maintaining a worker for periods of 1 to 5 months are $50, $70, $85, $100, and $130,
Use vectors to determine graphically the type of solution for each of the sets of equations below: unique solution, an infinite number of solutions, or no solution. For the cases of unique solutions, indicate from the vector representation (and without solving the equations algebraically) whether
In the solution space in Figure 7.6 (drawn to scale), express the interior point (3, 1) as a convex combination of the extreme points A, B, C, and D by determining the weights associated with each extreme point.FIGURE 7.6 6 5 4 3 2 1 0 X2 с D (3, 1) A 1 2 3 4 B 5 6 X1
In the following LP, compute the entire simplex tableau associated with XB = (x1, x2, x3)T.Minimize z = 2x1 + x2subject to 3x1 + X2 - Х3 4x1 + 3x2 x1 + 2x2 -Х4 = 3 = 6 +X; = 3 х1, х2, Х3, X4, X5 > 0
An opinion survey involves designing and printing questionnaires, hiring and training personnel, selecting participants, mailing questionnaires, and analyzing the data. Construct the project network, stating all assumptions.
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