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operations research an introduction

Questions and Answers of Operations Research An Introduction

Consider solving (approximately) the ILP max 12x1 + 7x2 + 9x3 + 8x4 s.t. 3x1 + x2 + x3 + x4 … 3 x3 + x4 … 1 x1,c, x4 = 0 or 1 by a version of discrete improving search Algorithm 15B that always
Do Exercise 15-11 for the ILP min 20x1 + 40x2 + 20x3 + 15x4 s.t. x1 + x2 Ú 1 x1 + x4 Ú 1 x1,c, x4 = 0 or 1 Start Algorithm 15B at x = 11, 1, 1, 12, and multistart at x = 11, 0, 1, 12, and (1, 1, 1,
Return to the improving search problem of Exercise 15-11.(a) Show that x = 11, 0, 0, 02 is a local optimum.(b) Show that if a nonimproving move is allowed at x = 11, 0, 0, 02, the next iteration will
Do Exercise 15-13 for the model of Exercise 15-12.
Return to the improving search problem of Exercise 15-11, starting from x102 = 11, 0, 0, 02. Compute an approximate optimum by Tabu search Algorithm 15C, forbidding complementation of a variable for
Do Exercise 15-15 for the model of Exercise 15-12. Forbid complementation of a variable for two steps after its value changes.
Return to the improving search problem of Exercise 15-11, starting from x102 = 10, 0, 0, 12. Compute an approximate optimum by Simulated Annealing Algorithm 15D, using a temperature of q = 20,
Do Exercise 15-17 for the model of Exercise 15-12. Use random numbers 0.60, 0.87, 0.77, 0.43, 0.13, 0.19, 0.23, 0.71, 0.78, 0.83, 0.29.Start at x102 = 11, 0, 0, 02.
Return to the (TSP) and the instance of Exercise 15-5(b). Given a feasible tour, the pairwise interchange move set considers all possible swaps of city positions in the current tour. For example, one
Inform College (IC) is planning a major government issues conference with panels on topics i = 1,c, 30. Panels will be scheduled in one of t = 1,c, 6 time blocks, with 5 running simultaneously in
Silo State’s Industrial Engineering faculty is moving to new offices. Professors p = 1,c, 20 will be assigned offices among the r = 1,c, 25 rooms, with unused rooms being left for graduate
Return to the model of Exercise 15-21, and consider solving it approximately with Improving Search Algorithm 15B over each of the following move sets:M1! 5reassignments of a single professor to any
Return to the improving search problem of Exercise 15-11.(a) Show that the solutions x112 = 10, 0, 1, 02 and x122 = 10, 0, 0, 12 are eligible to belong to a genetic algorithm population for the
Do Exercise 15-23 on the model of Exercise 15-12 using x112 = 10, 1, 1, 12 and x122 = 11, 0, 1, 12.
Return again to the model of Exercise 15-11, and consider employing genetic Algorithm 15E with initial population 510, 0, 1, 02, 10, 0, 0, 12, 10, 1, 1, 02, 11, 0, 0, 026, pe = pi = 1, and pe = 2.
Do Exercise 15-25 on the model of Exercise 15-12 with initial population 510, 1, 1, 12, 11, 0, 1, 12, 10, 1, 0, 12, 11, 0, 0, 026.
Return to the model of Exercise 15-20 and consider applying Genetic Algorithm 15E.(a) First consider encoding solutions by taking sessions in i order and recording the time block t to which each is
Return to the model of Exercise 15-21 and consider applying Genetic Algorithm 15E.(a) First consider encoding solutions by taking professors in p order (adding p = 21,c, 25 for rooms assigned to
A biomedical intrumentation company sells its main product at the rate of 5 units per day. The instrument is manufactured in lots run every few days. It costs the company $2000 to setup for
As part of a study of 911 emergency calls, an analyst wishes to choose the value of parameter a in exponential probability density function d1t2!ae-at that best fits call interarrival times 80, 10,
An oil drilling company wishes to locate a supply base somewhere in the jungle area where it is presently exploring for oil. The base will service drilling sites at map coordinates (0, -30),(50,
Repeat Exercise 16-3, this time minimizing the maximum distance to any drilling site.
An electronics assembly firm is planning its production staff needs to make a new modem. It has measured one test worker assembling the unit and observed the following data:Through unit 2 6 20 25 40
The following shows a series of measurements of the height (in inches) of a new genetically engineered tomato plant versus the number of weeks after the plant was replanted outdoors.Week 1 2 4 6 8 10
The university motor pool3 provides a large number of cars n for faculty and staff traveling on university business. Motor pool cars have an average annual cost of f dollars per car for fixed
Once a site for a new service facility has been chosen, the limits of its market area must be determined,4 along with the corresponding facility size. Assume (i) that the facility is to be located at
Renewing highway pavement markings5 costs c dollars per mile but reduces social costs from delays, accidents, and other effects of declining marking performance over time. Suppose that new markings
The number of potential patrons pi of a new movie theater complex has been estimated from census data for each of the surrounding counties i = 1,c, 15. However, the fraction of potential patrons from
Denoting by nt the number of universities using a textbook through semester t of its availability 1n0 = 02, the number of new adoptions in any single semester t can be estimated 1a + bnt - 121m - nt
Major aircraft parts undergo inspection and overhaul6 every t1 flying hours, and replacement every t2. Experience shows the cost of overhauling a particular model of jet engine can be expressed as
Determine whether each of the following functions is smooth on the specified domain.(a) f1x2! x4 + 3x - 19 for all x(b) f1x2!min 52x - 1, 2 - x6 for all x(c) f1x2! x - 5 for x 7 0(d) f1x2!3x + ln
Each of the following plots shows a function f1x2. Determine graphically whether each indicated point is an unconstrained local maximum, an unconstrained global maximum, an unconstrained local
Each of the following plots shows contours of a smooth function f1x1, x22. Determine graphically whether each indicated point is an unconstrained local maximum, an unconstrained global maximum, an
Use golden section Algorithm 16A to find an optimum of the NLP min 10x +70 xs.t. 1 … x … 10 to within an error of{1.
Use golden section Algorithm 16A to find an optimum of the NLP max 500 - x1x - 2023 s.t. 0 … x … 12 to within an error of {1.
Suppose that we were given only the lower limit of 1 in the NLP of Exercise 16-16. Apply 3-point pattern Algorithm 16B to compute a corresponding upper limit with which golden section search could
Do Exercise 16-18 for the NLP of Exercise 16-17 using d = 2 and d = 5.
Use quadratic fit Algorithm 16C to compute an optimum for the NLP of Exercise 16-16 within an error tolerance of 2. Start with the 3-point pattern 51, 2, 106.
Use quadratic fit Algorithm 16C to compute an optimum for the NLP of Exercise 16-17 within an error tolerance of 4. Start with 3-point pattern 50, 3, 126.
Consider the 1-variable function f1x2! x3 -3x2 + 11x at current point x = 3.(a) Derive the first-order Taylor approximation to f1x + l2.(b) Derive the second-order Taylor approximation to f1x +
Do Exercise 16-22 for function f1x2!18x - 20 ln1x2 at x = 16.
Consider the 2-variable function f1x1, x22!1x123 - 5x1x2 + 61x222 with current point x = 10, 22 and move direction x = 11, -12.(a) Derive the first-order Taylor approximation to f1x + lx2.(b)
Do Exercise 16-24 for function f1x1, x22!13x1 - 6x1x2 + 8>x2, x = 12.12 andx = 13.12.
For each of the following unconstrained NLPs, either verify that the given x is a stationary point of the objective function or give a directionΔx that improves at x.(a) min 1x122 + x1x2 - 6x1 -
For each of the following functionsf, use conditions 16.19 to 16.22 to classify the specified x as definitely local maximum, possibly local maximum, definitely local minimum, possibly local minimum,
Determine whether each of the following functions is convex, concave, both, or neither over the domain specified.(a) f1x1, x22! ln 1x12 + 20 ln 1x22 over x1, x2 7 0(b) f1x2!x sin 1x2 over x [0,
Use convexity/concavity to establish that each of the following solutions x is either an unconstrained global maximum or an unconstrained global minimum of the f indicated, and explain which.(a)
Consider the unconstrained NLP max x1x2 - 51x1 - 224 - 31x2 - 524(a) Use graphing software to produce a contour map of the objective function for x1 [1, 4], x2 [2, 8].(b) Compute the move
Do Exercise 16-30 for the unconstrained NLP min 1000 x1 + x2+ 1x1 - 422 + 1x2 - 1022 starting from x102 = 13, 12, and plotting x1 [2, 11], x2 [0, 15].
Return to the unconstrained optimization of Exercise 16-30 starting from x102 = 13, 72.(a) Write the second-order Taylor approximation to the objective function at x102 for unknown x and l = 1.(b)
Do Exercise 16-32 on the NLP of Exercise 16-31 starting from x102 = 113, 12.
Return to the unconstrained optimization of Exercise 16-31 and consider BFGS Algorithm 16F starting at x102 = 12, 32.(a) Compute the first direction that would be pursued by Algorithm 16F.(b)
Do Exercise 16-34 on the NLP of Exercise 16-31 starting from x102 = 16, 12 and using l =0.32 in part (b).
Consider the unconstrained NLP min max510 - x1 - x2, 6 + 6x1 - 3x2, 6 - 3x1 + 6x26(a) Explain why Nelder–Mead search is appropriate for solving this unconstrained optimization.(b) Do 3 iterations
Do Exercise 16-36 for the NLP max min 520 - x1 - x2, 6 + 3x1 - x2, 6 - x1 + 3x26 starting with ensemble (0, 0), (1, 2), (2, 2).
Compute the Nelder–Mead Algorithm 16G ensemble that would result from applying the shrinking step to each of the following (y112 best objective value, etc.).(a) y112 = 11, 2, 12, y122 = 15, 4, 52,
Chilled-water building cooling systems12 operate as indicated in the following sketch.Water flows at a rate of F1 gallons per minute around the lower loop, entering the chiller at temperature T1,1
The figure below shows a system of reservoirs and hydroelectric dams of the sort operated by large utilities such as California’s PG&E.Each node is a reservoir with a power plant releasing
Do Exercise 17-59 on the NLP of Exercise 17-38, again using x102 = 11, 12 and all multipliers v1 = 0 in part (d).
Return to the NLP of Exercise 17-36, and consider solving it by Sequential Quadratic Programming Algorithm 17E.(a) Using dual variables v1, roll constraints into the objective function to formulate
A water distribution system14 is a network with (positive = forward or negative = reverse)flows xi,j, in pipes between nodes i, j = 0,c, m representing storage tanks and pipe intersections.Pressures
Do Exercise 17-56 for the posynomial geometric program min 10> 1x1x2x322 s.t. 121x122x2 + 4x3 … 1 0.1x21x1 + x2x3 … 1 1x1x220.333 … 1 x1, x2, x3 7 0
Consider the standard-form posynomial geometric program min 3>1x1 + x1x2 + 10> 1x323 s.t. 0.5x1x2> 1x322 … 1 0.167x1 + 0.251x120.4x2 + 0.0833x3 … 1 x1, x2, x3 7 0(a) Change variables to convert
Consider the trivial separable program min 21x - 322 s.t. 0 … x … 6(a) Verify that the model is a convex program.(b) Verify by inspection that an optimal solution occurs at x* = 3.(c) Form a
Form linear programming approximations 17.59 to each of the following separable programs using breakpoints u1, 0 = 0, u1,1 = 1, u1,2 = 3, u2, 0 = 0, u2,1 = 2, u2,2 = 4.(a) min x1> 14 - x12 + 1x2 -
Do Exercise 17-52 for the NLP of Exercise 17-48 starting from solution x102 = 12, 12.
Return to the NLP of Exercise 17-46, and consider solving by active set Algorithm 17D starting from solution x102 = 10, 12.(a) Demonstrate that the model is a quadratic program by deriving the c0,c,
Do Exercise 17-50 for the equalityconstrained quadratic program max - 1x122 - 81x222 - 21x322 + 10x2x3+ 14x1 - 8x2 + 20x3 s.t. x1 + 4x3 = 4-x2 + 3x3 = 1
Consider the equality-constrained quadratic program min 61x122 + 21x222 - 6x1x2 + 41x322+ 5x1 + 15x2 - 16x3 s.t. x1 + 3x2 - 2x3 = 2 3x1 - x2 + x3 = 3(a) Identify the Q,c, A, and b of
Do Exercise 17-47 on the standard-form NLP of Exercise 17-48(a).
Do Exercise 17-46 for nonlinear program max 500 - 31x1 + 122 + 2x1x2 - 1x2 - 1022 s.t. x1 - x2 … 1 x2 … 5 x1, x2 Ú 0 using basis 5x1, x46 and standard-form starting solution x102 = 12, 1, 0, 42.
Return to the standard form NLP of Exercise 17-46(a).(a) Apply reduced gradient Algorithm 17C to compute an optimal solution starting from the x102 = 10, 1, 8, 72.(b) Graph your progress in a plot of
Consider the nonlinear program min 1x1 - 822 + 21x2 - 422 s.t. 2x1 + 8x2 … 16 x1 … 7 x1, x2 Ú 0(a) Introduce slack variables x3 and x4 to place the model in standard form for reduced gradient
Do Exercise 17-44 using reciprocal barrier functions.
Do Exercise 17-42 for the NLP of Exercise 17-38. Start at x102 = 11.8, 1.82 with multiplier m = 8, and decrease with factor b = 14.
Do Exercise 17-42 using reciprocal barrier functions.
Consider solving the NLP of Exercise 17-36 by barrier methods.(a) Use logarithmic barrier functions to reduce this problem to an unconstrained barrier model.(b) Explain why local minima of the
Do Exercise 17-40 using reciprocal barrier functions.
Determine whether barrier methods can be applied to each of the NLPs in Exercise 17-23, and if so, use log barrier functions to reduce the constrained optimization model to an unconstrained barrier
Do Exercise 17-38 using squared penalty functions. Stop the search in part (f) when total constraint violation … 0.2.
Do Exercise 17-36 for the NLP max 100 - 81x122 - 3 1x2 - 322 s.t. x2 Ú 2>x1 0 … x1 … 2 0 … x2 … 2 Start at x102 = 12, 22 with multiplier m = 0.5, and increase by the factor b = 4.
Do Exercise 17-36 using squared penalty functions. Stop the search in part (f) when the total constraint violation is … 0.2.
Consider the NLP min 21x1 - 322 - x1x2 + 1x2 - 522 s.t. 1x122 + 1x222 … 4 0 … x1 … 2, x2 Ú 0 with optimal solution x* = 11.088, 1.6782.(a) Use unsquared penalty functions to reduce this
Do Exercise 17-34 using squared penalty functions.
Use absolute value (unsquared) penalty functions to reduce each NLP of Exercise 17-30 to an unconstrained penalty model.
Do Exercise 17-32 for NLP max 2 ln1x12 + 8 ln1x22 s.t. 4x1 + x2 = 8 x1, x2 Ú 1 with nonoptimal point x = 11, 42, improving feasible direction x = 1 -1, 42, and global optimum x* = 11, 42.
Consider the NLP min 151x122 + 41x222 s.t. 3x1 + 2x2 = 8 x1, x2 Ú 0(a) State the KKT optimality conditions for this model.(b) Verify that at solution x = 10, 42 there exists an improving feasible
For each mathematical program in Exercise 17-30, determine whether principle 17.26 assures that a KKT point is a global optimum.
State the Karush–Kuhn–Tucker optimality conditions for each of the following mathematical programs.(a) min 141x1 - 922 + 31x2 - 522 + 1x3 - 1122 s.t. 2x1 + 18x2 - x3 = 19 6x1 + 8x2 + 3x3 … 20
Do Exercise 17-28 for the NLP max 300 - 51x1 - 2022 - 41x2 - 622 s.t. x1 + x2 = 8 and part (d) extra constraint x2 Ú 0.
Consider the nonlinear program min 81x1 - 222 + 21x2 - 122 s.t. 32x1 + 12x2 = 126(a) Form the Lagrangian function for this model.(b) Write stationary conditions for the Lagrangian.(c) Solve your
Demonstrate that each of the following NLPs is a posynomial geometric program by placing the model in standard form and detailing the sets Ki, and associated coefficients dk and ak,j.(a) min
Determine whether each of the following is a posynomial.(a) 23x1 - 34x2 + 60x3(b) 54x1 + 89x2 + 52x3(c) 7x1x2> 1x322.3 + 41x1(d) 44x1> ln 1x22 + e-x3
Determine whether each of the following NLPs is a quadratic program, and if so, identify the c and Q of matrix objective function form c # x + xQx.(a) min x1x2 + 134>x3 + ln1x12 s.t. x1 + 4x2 - x3
Determine which of the NLPs in Exercise 17-23 are separable programs.
Determine whether each of the following NLP’s is a convex program.(a) max l n1x12 + 3x2 s.t. x1 Ú 1 2x1 + 3x2 = 1 1x122 + 1x222 … 9(b) min x1 + x2 s.t. x1, x2 … 9-5 … x1 … 5-5 … x2 …
The commander of a battlefront11 must plan how to employ his f frontline and r reserve firepower to minimize the advance achieved over days t = 1,c, 14 by an attack of opposing forces with
Three urban neighborhoods are mutually connected by freeways admitting traffic in both directions. Net output bi,k (per hour) at each neighborhood k of vehicles originating at i can be estimated from
Each day qi tons of freight arrive by sea10 in Japan bound for in-country regions i = 1,c, 50.These goods may arrive at any of the major ports j = 1,c, 17, but the internal transportation cost per
A stirred tank reactor9 is a tank equipped with a large stirring device that is used in the chemical and biochemical industry to produce chemical reactions. A series of 5 such tanks will be used to

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