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operations research an introduction
Operations Research: An Introduction 10th Global Edition Hamdy A Taha - Solutions
Solve Problem 9-26 if, additionally, each receiver can handle at most 4 meters and receiver 8 can handle meters 1, 4, 8, and 10.
Gavernini and Associates (2004). Modern electric networks use automated electric utility meter reading in place of the more costly manual meter reading. In the automated system, meters from several customers are linked wirelessly to a single receiver. The meter sends monthly signals to a designated
Guéret and Associates (2002), Section 12.6. MobileCo is budgeting $15 million 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 given
Walmark Stores is in the process of expansion in the western United States. During 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 customers. In
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
The great treasures of King Tut are on display in the Giza Museum in Cairo. The layout of the museum is shown in Figure 9.7, with the different rooms joined by open doors. A guard standing at a door can watch two adjoining rooms. The museum’s security policy requires guard presence in every room.
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 18 minutes of driving time from the towns it serves. The table below
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
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 Customers served on the route 1 3, 2 2 5, 3, 4 3 2, 5, 1, 3 4 2, 3, 5 5 1, 4, 2 6 1, 3, 5 The segments of each
The world-renowned logic puzzle, Sudoku, deals with a 9 * 9 grid subdivided into 9 nonoverlapping 3 * 3 subgrids. The puzzle calls for assigning the numerical digits 1 through 9 to the cells of the grid such that each row, each column, and each subgrid contain distinct digits. Some of the cells may
A widely circulated puzzle requires assigning a single distinct digit (0 through 9) to each letter in the equation SEND + MORE = MONEY. Formulate the problem as an integer program, and find the solution. (Hint: This is an assignment model with side conditions.)
Given i = 1, 2,c, n, formulate a general ILP model (for any n) to determine the smallest number y that, when divided by the integer amount 2 + i, will always produce a remainder equal to i; that is, y mod 12 + i2 = i.
A street vendor selling electronic gadgets was robbed of all his possessions. When reporting the matter to the police, the vendor did not know the number of gadgets he had but stated that when dividing the total in lots of size 2, 3, 4, 5, or 6, there was always one gadget left over. On the other
You have a 4 * 4 grid and a total of 10 tokens. Use ILP to place the tokens on the grid such that each row and each column will have an even number of tokens.
You have three currency denominations with 11 coins each. The total worth (of all 11 coins) is 12 bits for denomination 1, 14 bits for denomination 2, and 20 bits for denomination 3. You need to buy one 30-bit item. Use ILP to determine the smallest number of coins of the three denominations needed
Graves and Associates (1993). Ulern 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 two
In Problem 9-10, suppose that the nature of the melodies dictates that songs 3 and 4 cannot be recorded on the same CD. Formulate the problem as an ILP. Would it be possible to use a 30 MB CDs to record the eight songs? If not, use ILP to determine the minimum CD capacity needed to make the
The Record-a-Song Company has contracted with a rising star to record eight songs. The sizes in MB of the different songs are 8, 10, 8, 7, 9, 6, 7, and 12, respectively. Record-a-Song uses two CDs for the recording. Each CD has a capacity of 40 MB. The company would like to distribute the songs
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, O, P, R, S, and T. Develop a model to assign a unique numeric value from 1 through 9
Solve Problem 9-7 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.
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. Each word is scored by adding numeric codes of its three letters. For example, AFT has a score of 1
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
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 parents have stipulated five rules: (a) The selling price is either $1 for 7 apples or $3 for 1 apple,
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 one-seventh. The remainder goes to charity. The will does not specify the size of the herd except to say that
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
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 3 3 6 6 4 2 5 5 5 7 4 4 The maximum allowable cargo weight and volume are 210 tons and 198 yd3,
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) Projects 2 and 4 are mutually exclusive.
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 as G1, G2, and G3, respectively. Solve the problem for each of the following priority orders:(a) G1
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 completed), and experience (expressed in number of years in the
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 met.*8-18. In Problem 8-7, suppose that production strives to meet the quota for the two products,
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 flexible goals. Further, assume that the ACT score goal is twice as important as any of the remaining
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 check if all the goals have been met.
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 is desired to determine a linear regression fit into these data points. Let bj, j = 0, 1,c, n, be
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 goals:(a) It should be as close as possible to Mrs. Von Trapp’s place of work (within 14 mile).(b) It
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,
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 3 2 6 2 The daily production quotas for the two products are 80 and 60 units, respectively. Each
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 part:Production time in min Part Lathe Drill press 1 5 3 2 6 2 3 4 6 4 7 4 It is desired to balance
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 produced of each part in the three shifts:Units produced per run Shift Wheels Seats 1 500 300 2 600 280
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 more than 1.2%.Protein. At least 22%.Fiber. At most 5%.The following table gives the nutritional
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 male–female ratios for in-state and out-of-state applicants are 1:1 and 3:2, respectively. For
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 band concerts and art shows. Their costs per presentation are$1500 and $3000, respectively. The
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 the dual simplex method (Section 4.4.1). Show how the parametric analysis can be carried out in this
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 … 8 - 2t x1, x2, x3 Ú 0
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 solution by the dual simplex method (Section 4.4.1). Show how the parametric analysis can be carried out in
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 + 2x2 + 5x3 … 8 x1, x2, x3 Ú 0
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 + 2t2x2 + 15 - t2x3
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 = 5CX aX …b, 0 6 L … X … U6 always possesses a feasible solution.
Write the dual of max z = 5CX aX =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 B-1 = £0 -1 1 0 1 0 1 1 -1 The optimal primal and dual solutions are XB = 1x1, x2, x32T = 11, 3, 12T
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 computing zj - cj for all nonbasic pj.(c) Find the associated optimal dual solution.
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 that the given basis B is feasible for the primal. Next, using Y = CBB-1, compute the associated dual
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 that the dual in (a) is unbounded.(d) From Problems 7-42 and 7-43, develop a general conclusion
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 is infinite, replace it with a sufficiently large upper-bound M), the LP problem is converted to a
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 the problem.Then verify the given values of basic x4 and x=2 using matrix manipulation.
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 at upper bound during the course of the algorithm.The problem may thus be written as a1 -Cz -Cu 0 Dz
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 - x3 Ú 3 1 … x1 … 3, 0 … x2 … 3, 2 … x3(b) Maximize z = x1 + 2x2 subject to-x1 + 2x2 Ú
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 z = 3x1 + 5x2 + 2x3 subject to x1 + 2x2 + 2x3 … 10 2x1 + 4x2 + 3x3 … 15 0 … x1 … 4, 0 …
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 points leading to the optimal solution. (You may use TORA.)(b) Solve the problem by the upper-bounding
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 the elements of XB0 is negative (infeasible).Step 1. Compute XB = B-1b, the current values of the
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 + 3x6 = 4 x1 + x2 - 3x3 + x4 + x5 = 6 x1, x2, x3, x4, x5, x6 Ú 0
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 + 3x2 + 8x3 … 12 4x1 + x2 + 12x3 … 8 4x1 - x2 + 3x3 … 8 x1, x2, x3 Ú 0(c) Minimize z = 2x1 + x2
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 replaced with the new coefficients DB, the values of zj - cj for the basic vector XB will remain
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 the resource requirements per unit of xj to 1a of their original values, a 7 1. Since the
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, where a is a positive scalar, and provided xj remains the entering variable, find the relationship
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 assuming no cycling?
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 remains feasible even if 1B -1pj2k is negative.
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 variable are 7 0) of the solution space.
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 first time, for continuing to obtain a degenerate solution in the next iteration, and for removing
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 another in an alternative optimum solution.
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 function (using the constraint equations) actually computes the zj - cj for all the variables in the
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 automatically computes the proper zj - cj for all the variables in the starting tableau.
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 problem has an alternative optimum solution.
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 that the basis B of the current iteration is comprised of p1 and p2.(a) If the vector p1 enters the
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, CII2 and a = 1D, I2 be the corresponding partitions of C and a, respectively. Show that the matrix
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 matrix manipulations to reconstruct the original LP, and then compute the optimum objective value.
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 Ú 0
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 infeasible) basis: (p1, p3),(p1, p4), (p2, p3), (p3, p4).
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 solutions if B is singular and b is dependent.
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