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business
operations research an introduction
Optimization In Operations Research 2nd Edition Ronald Rardin - Solutions
Return to the Facilities Location Problem(FLP) of Chapter 11, definition 11.29 , and assume all parameters are integer.(a) Consider the following instance:Define a finite alphabet of symbols, and then show a binary encoding of the instance in terms of your alphabet.(b) Establish that your encoding
Now return to the Fixed Charge Network Flow Problem (FCNP) of Chapter 11, definition 11.31 , and assume all data are integer.(a) Develop and justify an expression for the length of a binary encoding for an instance in terms of the dimensions and parameters of the model.(b) State the threshold
(c) to your threshold problem 1FCNP…2 of (b).(e) How do (b), (c), and NP-Completeness of 1FLP…2 (Exercise 14-3(e)) lead to the conclusion that the threshold version 1FCNP…2 also belongs to NP-Complete.(f) Explain why (e) implies the full optimization model (FCNP) is NP-Hard.(g) What would be
The Capital Budgeting Problem (CapBud)of Section 11.2 over a set of n proposed projects j using decision variables xj = 1 if project j is chosen and = 0 otherwise can be formulated as follows:max a nj = 1rjxj ( maximize total return)s.t. a nj = 1at, j xj … bt for all t ( budget limits in times
The Multi-dimensional Knapsack Problem(MKP) over n decision variables xj = 1 is object j is chosen and = 0 otherwise, can be formulated:max a nj = 1rjxj (maximize total return)s.t. a nj = 1aijxj … bi i = 1,c, m (capacities i)xj binary j = 1,c, n(a) Develop and justify an expression for the length
Given a digraph G(V, E), the Vertex Cover problem (VCover) seeks a minimum cardinality subset of vertices that together touch every edge of E.(a) Show that the problem can be modeled in terms of decision variables xi = 1 if vertex i is in the solution and = 0 otherwise as ILP min ai Vxi s.t. ai Ie
Return to the cardinality Vertex Cover problem (VCover) of Exercise 14-7(a). One algorithm to construct an approximately optimum set VQ ! 5i V: xi = 16 for the problem can be stated as follows: (i) start with VQ d 0; then(ii) as long as there exists any edge e E with Ie ¨ VQ = 0, update VQ d VQ
Return to the Traveling Salesman Problem(TSP) and the Twice-Around heuristic of Application 14.5. Then consider an instance on the 5 points in the following plot:Arcs can be assumed to exist between all pairs of points, and distances are Euclidean.(a) Compute a minimum spanning tree of the points
Return to the lists of problems in Table 14.1. For each of the following pairs of problems in the table, establish that the first, polynomially solvable one is a special case of the NP-Hard second, and identify the special properties of the first that prevent it from yielding worst-case instances
Consider solving (approximately) the following knapsack problem by constructive search Algorithm 15A.max 11x1 + 1x2 + 9x3 + 17x4 s.t. 9x1 + 2x2 + 7x3 + 13x4 … 17 x1,c, x4 = 0 or 1(a) Determine a global optimum by inspection.(b) Explain why it is reasonable to fix variables in decreasing order of
Do Exercise 15-1 for the knapsack model(this time with increasing ratio order)min 55x1 + 150x2 + 54x3 + 180x4 s.t. 25x1 + 30x2 + 18x3 + 45x4 Ú 40 x1,c, x4 = 0 or 1
Now consider knapsack instance max 30x1 + 20x2 + 20x3 s.t. 21x1 + 20x2 + 20x3 … 40 x1, x2, x3 = 0 or 1(a) Determine a global optimum by inspection(b) Now apply Constructive Search Algorithm l5A with the greedy ratio of Exercise l5-l(b) to compute an approximate optimum.(c) Compare the results of
Do Exercise 15-3 for the knapsack instance min 10x1 + 10x2 + 3x3 s.t. 10x1 + 10x2 + 6x3 Ú 20 x1, x2, x3 = 0 or 1
Recall the Traveling Salesman Problem(TSP) of Section 11.5, which seeks a minimum total length cycle (or tour) of a given graph G(V, E) that includes all nodes of V. One constructive algorithm for the problem is the Nearest Neighbor method, which begins at one of the given points, then successively
Return to the (TSP) instance of Exercise 15-5(b). This time construct an approximately optimal tour by the Greedy Insertion strategy used on the KI Truck Routing application in Figure 15.1. Specifically begin with a 2-point tour 1–5–1.Then at each iteration insert to unvisited point closest to
Consider a simplified Vehicle Routing Problem (VRP) over the customer sites in the following plot (refer to Section 11.5).Two 5-customer routes are to be designed, both originating and returning to terminal site x1 = x2 = 0. All point-to-point distances are assumed Euclidean, and the goal is to
Classic Bin Packing (BP) considers the task of packing a collection of items j = 1,c, n of varying sizes aj into a minimum number N of bins of capacityb. The first-fit algorithm to solve instances approximately begins with no bins open. Then it takes items in arbitrary sequence, considering
Consider solving (approximately) the ILP max 5x1 + 7x2 - 2x3 s.t. x2 + x3 … 1 x1, x2, x3 = 0 or 1 by a version of discrete improving search Algorithm 15B that employs move set M = 511, 0, 02, 10, 1, 02, 10, 0, 126 and always advances to the feasible neighbor with best objective value.(a) Identify
Repeat Exercise 15-9 for the ILP min 2x1 - 11x2 + 14x3 s.t. x1 + x2 + x3 Ú 1 x1, x2, x3 = 0 or 1
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 advances to the feasible neighbor with best objective value and uses the single-complement
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, 0).
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 return the search to this same point.
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 one step after its value changes, and limiting the search to t max = 5 moves.
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, limiting the search to t max = 4 moves, and resolving probabilistic decisions with (uniform [0,1]) random
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 swap of a current tour 1–2–5–4–3–1 would be 1–2–3–5–3–1.(a) Begin with tour
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 each block. To make the conference as convenient as possible, IC has surveyed prospective attendees and
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 assistants. Walking distances dr, r have been computed between all pairs of rooms 1r, r 2.The department
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 office6 M2! 5reassignments of a single professor to a vacant office6 M3! 5swaps of the assignments of
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 problem.(b) Construct all possible crossover results(all cut points) for the x112 and x122 of part
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. Construct and evaluate each member of the next generation population, with crossover after component 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 assigned.Sketch what would happen if two solutions like this were combined in crossover. What kinds
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 graduate assistants), then recording the room r to which each is assigned. Sketch what would happen if
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 production of a lot and $40 per unit per day to hold finished instruments in inventory between runs.The
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, 14, 26, 40, and 22 minutes.(a) Formulate a 1-variable unconstrained NLP to choose a maximum
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, -10), (70, 20), and (30, 50) with helicopter supply runs. The company wishes to choose a location that
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 Average time 8.4 5.5 4.2 3.7 3.1 Experience shows that learning curves, which describe the ability
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 Height 9 15 22 33 44 52 Researchers wish to fit this experience with an S-shaped logistics curve
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 expenses such as depreciation, insurance, and licensing, plus a variable operating cost of vm cents per
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 the center of a circular market area and sized to cover uniform density d calls per unit area out
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 yield maximum performance p max and that performance t days after renewal can be expressed p1t2!p max
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 any i who will actually use the complex varies inversely with its (straight line)distance from the
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 - 12, where a and b are parameters relating to the rapidity of success, and m is the maximum number
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 increasing nonlinear function a1t12b of the overhaul cycle, and operating costs per hour are an
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 1x2 for x 7 0(e) f1x1, x22!x1ex2 for all x1, x2(f) f1x1, x22! 0 x1 + 1 + 0 x2 - 3 for x1, x2 Ú 0.
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 minimum, an unconstrained global minimum, or none of the above over the domain depicted. (9)* (c)+ () (1)x
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 unconstrained local minimum, an unconstrained global minimum, or none of the above over the domain
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 begin using each of the following initial step sizes d.(a) d = 0.5(b) d = 16
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 + l2.(c) Plot the original function and both Taylor series approximations in the vicinity of x.How accurate
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) Derive the second-order Taylor approximation to f1x + lx2.(c) Plot the original function and both Taylor
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 - 8x2, x = 18, -102(b) max 101x122 + 2 ln 1x22, x = 11, 22(c) min 16x1 - x1x2 + 21x222, x = 13, 02(d) max
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, and/or definitely neither.(a) f1x1, x22!31x122 - x1x2 + 1x222 - 11x1, x = 12, 12(b) f1x1, x22! -
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, 2p](c) f1x2!x1x - 222 over all x Ú 0(d) f1x2! 1x - 824 + 132x over all x(e) f1x1c. . x52!3x1 + 11x2 -
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) f1x1, x22! 1x1 - 522 + x1x2 + 1x2 - 722 at x = 12, 62(b) f1x1, x22! 500 - 81x1 + 122 - 21x2 - 122 +4x1x2
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 direction that would be pursued by gradient search Algorithm 16D at x102 = 11, 32.(c) State the line search
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) Compute the Newton direction x at x102 and verify that it is a stationary point of your second-order
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) Assuming that the optimal step is l =0.026 in that direction, compute the new solution x112, the next
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 (moves) of Nelder–Mead Algorithm 16G, starting from initial ensemble(5, 0), (10, 5), (5, 5).(c)
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, y132 = 13, 2, 72, y142 = 17, 2, 72(b) y112 = 110, 8, 102, y122 = 14, 6, 22, y132 = 10, 0, 02, y142 =
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 and being cooled to temperature T1,2 before passing through the building. An upper loop flowing at
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 water on the downstream side. The water then requires one month to reach the reservoir of the next dam
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 the Lagrangian.(b) Formulate the second-order the Lagrangian as a function of move direction x as in
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 at the nodes i can be measured in hydraulic“head,” which is the height to which water will rise
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 this geometric program into a convex program.(b) Use class optimization software to solve your convex
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 linear programming approximation 17.61 using u0 = 0, u1 = 2, u2 = 6.(d) Solve your LP approximation of
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 - 122 s.t. 2x1 + x2 Ú 2 41x1 + 123 - 91x222 … 25 0 … x1 … 3 0 … x2 … 4 (b) max 500 - 1x1 -
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, Q, a112,c, a142, b1,c, b4, G, L, and E of general symmetric form 17.50 .(b) State and solve as a
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 (symmetric)quadratic program standard form.(b) State Karush–Kuhn–Tucker optimality conditions for the model as a
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 the feasible 1x1, x22.
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 Algorithm 17C.(b) Show that x2 and x4 form a basic set of variables for your standard form.(c)
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 unconstrained model in part (a) for all m Ú 0 must be global minima.(c) Determine whether the barrier
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 model.
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.
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