Questions and Answers of Operations Research An Introduction

Suppose that you throw 2 fair dice simultaneously. If there is a match, you receive 50 cents. Otherwise, you pay 10 cents. Determine the expected payoff of the game.
Prove the formulas for the mean and variance of the binomial distribution.
Customers shopping at Walmark Store are both urban and suburban. Urban customers arrive at the rate of 5 per minute, and suburban customers arrive at the rate of 7 per minute. Arrivals are totally
Customers arrive at a service facility according to a Poisson distribution at the rate of four per minute. What is the probability that at least one customer will arrive in any given 30-second
The inside diameter of a cylinder is normally distributed with a mean of 1 cm and a standard deviation of .01 cm. A solid rod is assembled inside each cylinder. The diameter of the rod is also
An automatic device is used to count the volume of traffic at a busy intersection. The arrival time is recorded and translated into an absolute time starting from zero. The following table provides
The weights of individuals who seek a helicopter ride in an amusement park have a mean of 180 lb and a standard deviation of 15 lb. The helicopter can carry five persons but has a maximum weight
The following data represent the period (in seconds) needed to transmit a message.Use Excel to construct a suitable histogram. Test the hypothesis that these data are drawn from a uniform
Consider Problem 16-2. If the police car is currently at a call scene, determine the probability that an apprehension will take place in two patrols.Problem 16-2A police car is normally on regular
Consider Problem 16-1. Determine the probability that the professor will purchase the current model in 4 years.Problem 16-1An engineering professor acquires a new computer every 2 years. The
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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.
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
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%
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,
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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.,
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
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
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
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
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
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 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.
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 $1 million at the
TORA Experiment. Consider the Diet Model, and let the objective function be given as Minimize z = .8x1 + .8x2 Use TORA to show that the optimum solution is associated with two distinct corner points
TORA Experiment. In the Reddy Mikks model, use TORA to show that the removal of the raw material constraints (constraints 1 and 2) would esult in an unbounded solution space. What can be said in this
Determination of the Optimum LP Solution by Enumerating All Feasible Corner Points. The remarkable observation gleaned from the graphical LP solution is that the optimum, when finite, is always
In Example 1.6-1, suppose instead that the restaurant rating is 70% and the probability of having equal rating among all restaurant in town estimated at .4. Given the guesstimate initial probability
Some of us may remember a police car lurking on a side of the road, engine running, and a radar gun trained on passing traffic. And if one is caught speeding, the ensuing process of issuing an
Repeat Problem 1-18 using the same cost table and the following raw data:Problem 1-18In Example 1.4-2, consider the unit transportation costs ($) among the five stores as given in the following
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
In the garden fence problem, assume perimeter L = 50ft , 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
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
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
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

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