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operations research an introduction

Questions and Answers of Operations Research An Introduction

Weber (1990). You have the following three-letter words: AFT, 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.
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
The three children of a farm couple are sent to the market to sell 90 apples. Karen, the oldest, carries 50 apples; Bill, the middle one, carries 30; and John, the youngest, carries only 10. The
An eccentric sheikh left a will to distribute a herd of camels among his three children:Tarek receives at least one-half of the herd, Sharif gets at least one third, and Maisa gets at least
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
Five items are to be loaded in a vessel. The weight wi, volume vi, and value ri 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
Modify and solve the capital budgeting model of Example 9.1-1 to account for the following additional restrictions:(a) Project 4 must be selected if either project 1 or project 3 is selected.(b)
Solve the Ozark University model (Problem 8-3) using the preemptive method, assuming that the goals are prioritized in the same order given in the problem.
Consider Problem 8-2, which deals with the presentation of band concerts and art shows at the NW Mall. Suppose that the goals set for teens, the young/middle-aged group, and seniors are referred to
Solve Problem 8-1 using the following priority ordering for the goals:G1 G2 G3 G4 G5.
In Example 8.2-2, suppose that the budget goal is increased to $150,000. The exposure goal remains unchanged at 45 million persons. Show how the preemptive method will reach a solution.3
Solve Problem 8-20 using the Chebyshev method proposed in Problem 8-11.
The Malco Company has compiled the following table from the files of five of its employees to study the impact on income of three factors: age, education (expressed in number of college years
In the Vista City Hospital of Problem 8-8, suppose that only the bed limits represent flexible goals and that all the goals have equal weights. Can all the goals be met?
In Problem 8-6, suppose that the market demand goal is twice as important as that of balancing the two machines, and that no overtime is allowed. Solve the problem, and determine if the goals are
In Problem 8-5, determine the solution, and specify whether or not the daily production of wheels and seats can be balanced.
In the Circle K model of Problem 8-4, is it possible to satisfy all the nutritional requirements?
In the Ozark University admission situation described in Problem 8.3, suppose that the limit on the size of the incoming freshmen class must be met, but the remaining requirements can be treated as
In Problem 8-2, suppose that the goal of attracting young/middle-aged people is twice as important as for either of the other two categories (teens and seniors). Find the associated solution, and
Consider Problem 8-1, dealing with the Fairville tax situation. Solve the problem, assuming that all five goals have the same weight. Does the solution satisfy all the goals?
Chebyshev Problem. An alternative goal for the regression model in Problem 8-10 is to minimize over bj the maximum of the absolute deviations. Formulate the problem as a GP model.
Regression analysis. In a laboratory experiment, suppose that yi is the ith observed(independent) yield associated with the dependent observational measurements xij, i = 1, 2,c, m; j = 1, 2,c, n. It
The Von Trapp family is in the process of moving to a new city where both parents have accepted new jobs. In trying to find an ideal location for their new home, the family list the following
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
Two products are manufactured on two sequential machines. The following table gives the machining times in minutes per unit for the two products:Machining time in min Machine Product 1 Product 2 1 5
Camyo Manufacturing produces four parts that require the use of a lathe and a drill press. The two machines operate 10 hours a day. The following table provides the time in minutes required by each
Mantel produces a toy carriage, whose final assembly must include four wheels and two seats. The factory producing the parts operates three shifts a day. The following table provides the amounts
Circle K Farms consumes 3 tons of special feed daily. The feed—a mixture of limestone, corn, and soybean meal—must satisfy the following nutritional requirements:Calcium. At least 0.8% but not
The Ozark University admission office is processing freshman applications for the upcoming academic year. The applications fall into three categories: in-state, out-of-state, and international. The
The NW Shopping Mall conducts special events to attract potential patrons. Among the events that seem to attract teenagers, the young/middle-aged group, and senior citizens, the two most popular are
Formulate the Fairville tax problem, assuming that the town council is specifying an additional goal, G5, that requires gasoline tax to equal at least 20% of the total tax bill.
Solve Problem 7-55 assuming that the right-hand side is changed to b1t2 = 13 + 3t2, 6 + 2t2, 4 - t22T Further assume that t can be positive, zero, or negative.
The analysis in this section assumes that the optimal LP solution at t = 0 is obtained by the (primal) simplex method. In some problems, it may be more convenient to obtain the optimal solution by
Study the variation in the optimal solution of the following parameterized LP, given t Ú 0:Minimize z = 4x1 + x2 + 2x3 subject to 3x1 + x2 + 2x3 = 6 + 6t 4x1 + 3x2 + 2x3 Ú 12 + 4t x1 + 2x2 + 5x3
In Example 7.5-2, find the first critical value, t1, and define the vectors of B1 in each of the following cases:(a) b1t2 = 140 + 2t, 60 - 3t, 30 + 6t2T(b) b1t2 = 140 - t, 60 + 2t, 30 - 5t2T
In Example 7.5-1, suppose that the objective function is nonlinear in t 1t Ú 02 and is defined as Maximize z = 13 + 2t22x1 + 12 - 2t22x2 + 15 - t2x3 Determine the first critical value t1.
The analysis in this section assumes that the optimal solution of the LP at t = 0 is obtained by the (primal) simplex method. In some problems, it may be more convenient to obtain the optimal
Study the variation in the optimal solution of the following parameterized LP, given t Ú 0.Minimize z = 14 - t2x1 + 11 - 3t2x2 + 12 - 2t2x3 subject to 3x1 + x2 + 2x3 = 6 4x1 + 3x2 + 2x3 Ú 12 x1 +
Solve Example 7.5-1, assuming that the objective function is given as(a) Maximize z = 13 + 3t2x1 + 2x2 + 15 - 6t2x3(b) Maximize z = 13 - 2t2x1 + 12 + t2x2 + 15 + 2t2x3(c) Maximize z = 13 + t2x1 + 12
In Example 7.5-1, suppose that t is unrestricted in sign. Determine the range of t for which XB0 remains optimal.
Show that the dual of max z = 5CXaX …b, 0 6 L … X … U6 always possesses a feasible solution.
Write the dual of max z = 5CXaX =b, X unrestricted6.
An LP model includes two variables x1 and x2 and three constraints of the type …. The associated slacks are x3, x4, and x5. Suppose that the optimal basis is B = 1p1, p2, p32, and its inverse is
Consider the following LP:Maximize z = 2x1 + 4x2 + 4x3 - 3x4 subject to x1 + x2 + x3 = 4 x1 + 4x2 + + x4 = 8 x1, x2, x3, x4 Ú 0(a) Write the dual problem.(b) Verify that B = 1p2, p32 is optimal by
Consider the following LP:Maximize z = 5x1 + 12x2 + 4x3 subject to 2x1 - x2 + 3x3 = 2 x1 + 2x2 + x3 + x4 = 5 x1, x2, x3, x4 Ú 0(a) Write the dual.(b) In each of the following cases, first verify
Consider the following LP:Maximize z = 50x1 + 30x2 + 10x3 subject to 2x1 + x2 = 1 2x2 = -5 4x1 + x3 = 6 x1, x2, x3 Ú 0(a) Write the dual.(b) Show by inspection that the primal is infeasible.(c) Show
Verify that the dual problem of the numeric example given at the end of Theorem 7.4-1 is correct. Then verify graphically that both the primal and dual problems have no feasible solution.
Define the dual problem given the primal is min z = 5CX aX Úb, X Ú 06.
Prove that the dual of the dual is the primal.
Bounded Dual Simplex Algorithm. The dual simplex algorithm (Section 4.4.1) can be modified to accommodate the bounded variables as follows. Given the upper-bound constraint xj … uj for all j (if uj
Solve part (a) of Problem 7-34 using the revised simplex (matrix) version for upperbounded variables.
In Example 7.3-1, do the following:(a) In Iteration 1, verify that XB = 1x4, x12T = 152, 32 2T by using matrix manipulation.(b) In Iteration 2, show how B-1 can be computed from the original data of
Consider the matrix definition of the bounded-variables problem. Suppose that the vector X is partitioned into (Xz, Xu), where Xu represents the basic and nonbasic variables that will be substituted
In the following problems, some of the variables have positive lower bounds. Use the bounded algorithm to solve these problems.(a) Maximize z = 3x1 + 2x2 - 2x3 subject to 2x1 + x2 + x3 … 8 x1 + 2x2
Solve the following problems by the bounded algorithm:(a) Minimize z = 6x1 - 2x2 - 3x3 subject to 2x1 + 4x2 + 2x3 … 8 x1 - 2x2 + 3x3 … 7 0 … x1 … 2, 0 … x2 … 2, 0 … x3 … 1(b) Maximize
Consider the following linear program:Maximize z = 2x1 + x2 subject to x1 + x2 … 3 0 … x1 … 2, 0 … x2 … 2 Problems 335(a) Solve the problem graphically, and trace the sequence of extreme
Revised Dual Simplex Method. The steps of the revised dual simplex method (using matrix manipulations) can be summarized as follows:Step 0. Let B0 = I be the starting basis for which at least one of
Solve the following using the two-phase revised simplex method:(a) Problem 7-28(c).(b) Problem 7-28(d).(c) Problem 7-29 (ignore the given starting XB0).
Solve the following LP by the revised simplex method given the starting basic feasible vector XB0 = 1x2, x4, x52T.Minimize z = 7x2 + 11x3 - 10x4 + 26x6 subject to x2 - x3 + x5 + x6 = 3 x2 - x3 + x4 +
Solve the following LPs by the revised simplex method:(a) Maximize z = 6x1 - 2x2 + 3x3 subject to 2x1 - x2 + 2x3 … 2 x1 + 4x3 … 4 x1, x2, x3 Ú 0(b) Maximize z = 2x1 + x2 + 2x3 subject to 4x1 +
In Example 7.2-1, summarize the data of iteration 1 in the tableau format of Section 3.3.
Consider the LP Maximize z = CX subject to 1a, I2 X =b, X Ú 0 Define XB as the current basic vector with B as its associated basis and CB as its vector of objective coefficients. Show that if CB is
Consider the LP Maximize z = CX subject to aX …b, X Ú 0, where b Ú 0 After obtaining the optimum solution, it is suggested that a nonbasic variable xj can be made basic (profitable) by reducing
Consider the LP, maximize z = CX subject to aX …b, X Ú 0, where b Ú 0. Suppose that the entering vector pj is such that at least one element of B -1pj is positive.(a) If pj is replaced with apj,
What are the relationships between extreme points and basic solutions under degeneracy and nondegeneracy? What is the maximum number of iterations that can be performed at a given extreme point
In applying the feasibility condition of the simplex method, suppose that xk = 0 is a basic variable and that xj is the entering variable with 1B -1pj2k 0. Prove that the resulting basic solution
Consider the general LP in equation form with m equations and n unknowns. Determine the maximum number of adjacent extreme points that can be reached from a nondegenerate extreme point (all basic
Consider the implementation of the feasibility condition of the simplex method. Specify the mathematical conditions for encountering a degenerate solution (at least one basic variable = 0) for the
Consider an LP in which the variable xk is unrestricted in sign. Prove that by substituting xk = xk- - xk+, where xk- and xk+ are nonnegative, it is impossible that the two variables replace one
Using the matrix form of the simplex tableau, show that in an all-artificial starting basic solution, the procedure in Section 3.4.1 that substitutes out the artificial variables in the objective
In an all-slack starting basic solution, show using the matrix form of the tableau that the mechanical procedure used in Section 3.3 in which the objective equation is set as z - a nj = 1 cj xj = 0
Prove that if zj - cj 7 0 16 02 for all the nonbasic variables xj of a maximization(minimization) LP problem, then the optimum is unique. Else, if zj - cj equals zero for a nonbasic xj, then the
Prove that, in any simplex iteration, zj - cj = 0 for all the associated basic variables.
Consider the following LP:Maximize z = c1 x1 + c2 x2 + c3 x3 + c4 x4 subject to p1 x1 + p2 x2 + p3 x3 + p4 x4 = b x1, x2, x3, x4 Ú 0 The vectors p1, p2, p3, and p4 are shown in Figure 7.4. Assume
In the matrix simplex tableau, suppose that X = 1XI, XII2T, where XII corresponds to a typical starting basic solution (consisting of slack and/or artificial variables) with B = I, and let C = 1CI,
The following is an optimal LP tableau:Basic x1 x2 x3 x4 x5 Solution z 0 0 0 3 2 ?x3 0 0 1 1 -1 2 x2 0 1 0 1 0 6 x1 1 0 0 -1 1 2 The variables x3, x4, and x5 are slacks in the original problem. Use
In the following LP, compute the entire simplex tableau associated with XB = 1x1, x2, x52T.Minimize z = 2x1 + x2 subject to 3x1 + x2 - x3 = 2 4x1 + 3x2 - x4 = 4 x1 + 2x2 + x5 = 2 x1, x2, x3, x4, x5
Consider the following LP:Maximize z = 5x1 + 12x2 + 4x3 subject to x1 + 2x2 + x3 + x4 = 10 2x1 - 2x2 - x3 = 2 x1, x2, x3, x4 Ú 0 Check if each of the following matrices forms a (feasible or
In Example 7.1-3, consider B = 1p3, p42. Show that the corresponding basic solution is feasible, and then generate the corresponding simplex tableau.
True or False?(a) The system BX = b has a unique solution if B is singular.(b) The system BX = b has no solution if B is singular and b is independent of B.(c) The system BX = b has an infinity of
Consider the following system of equations:£1 23 x1 + £0 21 x2 + £1 42 x3 + £2 00 x4 = £3 42Determine if any of the following combinations forms a basis:(a) (p1, p2, p3)(b) (p1, p3, p4)(c)
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,
In the following sets of equations, (a) and (b) have unique (basic) solutions, (c) has an infinite number of solutions, and (d) has no solution. Show how these results can be verified using graphical
In the solution space in Figure 7.3 (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 the
Determine graphically the extreme points of the following convex set:Q = 5x1, x2 x1 + x2 … 3, x1 Ú 0, x2 Ú 06 Show that the entire feasible solution space can be determined as a convex
Show that the set Q = 5x1, x2 x1 Ú 1 or x2 Ú 26 is not convex.
Show that the set Q = 5x1, x2 x1 + x2 … 3, x1 Ú 0, x2 Ú 06 is convex. Is the nonnegativity condition essential for the proof?
The estimates (a, m,b) are listed in the following table:Project (a) Project (b)Activity (a, m,b) Activity (a, m,b) Activity (a, m,b) Activity (a, m, b)1-2 (5, 6, 8) 3-6 (3, 4, 5) 1-2 (1, 3, 4) 3-7
Consider Problem
Use LP to determine the critical path for the project networks in Figure 6.45.
Use LP to determine the critical path for the project network in Figure 6.44.
(Job shop scheduling) Three jobs, J1, J2, and J3, are processed on 3 machines, M1, M2, and M3, according to the following sequences (processing times are shown in parentheses):J1: M3132 - M1142 -
Compute the floats and identify the red-flagged activities for the projects (a) and (b) in Figure 6.30, then develop the time schedules under the following conditions:Project (a)(i) Activity (1, 5)
In the project of Example 6.5-2 (Figure 6.28), assume that the durations of activities B and F are changed from 6 and 11 days to 20 and 25 days, respectively.(a) Determine the critical path.(b)
In Example 6.5-4, use the floats to answer the following:(a) If activity B is started at time 1, and activity C is started at time 5, determine the earliest start times for E and F.(b) If activity B
For each of the following activities, determine the maximum delay in the starting time relative to its earliest start time that will allow all the immediately succeeding activities to be scheduled
What are the total and free floats of a critical activity? Explain.
Given an activity (i, j) with duration Dij and its earliest start time i and its latest completion time j, determine the earliest completion and the latest start times of (i, j).
Determine the critical path for the project in Problem 6-51.
Determine the critical path for the project in Problem 6-50.

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