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operations research an introduction

Optimization In Operations Research 2nd Edition Ronald Rardin - Solutions

Consider the standard-form LP min 10x1 + 1x2 s.t. x1 - x2 + 2x3 = 3 x2 - x3 = 2 x1, x2, x3 Ú 0(a) Show that x102 = 14, 3, 12 is an appropriate point to start affine scaling Algorithm 7A.(b) Derive the associated scaled standard form corresponding to solution x102.(c) Compute the move direction x
Do Exercise 7-15 using the LP max 6x1 + 8x2 + 10x3 s.t. 9x1 - 2x2 + 4x3 = 6 2x2 - 2x3 = -1 x1, x2, x3 Ú 0 and initial solution x102 = (29, 1, 32).
Suppose that Affine Scaling Algorithm 7A has reached current solution x1112 = 13, 1, 92.Determine whether each of the following next move directions would cause the algorithm to stop and why. If not, compute x1122.(a) x = 16, 2, 22(b) x = 10, 4, -92(c) x = 16, -6, 02(d) x = 10, -2, 02
Consider solving the following standard-form primal LP by Affine Scaling Algorithm 7A starting from solution x = 12, 3, 2, 3, 1>32.(a) Verify that the given solution is an appropriate point at which to start Algorithm 7A.(b) Derive the scaled standard form in terms of variables y that comes from
Consider the standard-form LP max 13w1 - 2w2 + w3 s.t. 3w1 + 6w2 + 4w3 = 12 w1, w2, w3 Ú 0(a) Sketch the feasible space in a plot like Figure 7.6, identify an optimal extreme point, and show w112 = 11.4, 0.7, 0.92 and w2 = 10.01, 0.01, 2.97752.(b) Form the corresponding log barrier problem with
Do Exercise 7-19 for the LP min 2w1 + 5w2 - w3 s.t. w1 + 6w2 + 2w3 = 18 w1, w2, w3 Ú 0 and points w112 = 18, 1, 22, w122 = 10.01, 0.02, 8.9352.
Determine whether each of the following could be the first four barrier multiplier values m used in the outer loop of a barrier search Algorithm 7B.(a) 100,80,64,51.2(b) 100,200,100,800(c) 100,500,1000,2000(d) 600,300,150,75
Assume that barrer Algorithm 7B has computed an appropriate move direction x for its standard-form LP and a maximum feasible step size lmax. For each of the original objective functions below, determine which of the following curves best depicts how the corresponding barrier objective function will
Consider the LP min 4x1 - x2 + 2x3 s.t. 4x1 - 3x2 + 2x3 = 13 3x2 - x3 = 1 x1, x2, x3 Ú 0(a) Show that x102 = 13, 1, 22 is an appropriate point to start barrier Algorithm 7B.(b) Form the corresponding log barrier problem with multiplier m = 10.(c) Compute the move direction x that could be pursued
Do Exercise 7-23 for the LP max -x1 + 3x2 + 8x3 s.t. x1 + 2x2 + x3 = 14 2x1 + x2 = 11 x1, x2, x3 Ú 0 at x102 = 15, 1, 72
Return to the LP and x102 of Exercise 7-15.(a) Show that x102 = 14, 3, 12 is an appropriate point to start barrier Algorithm 7B.(b) Compute the move direction x that would be pursued from x102 by Algorithm 7B with m = 10 (refer to Table 7.4 for projection matrices).(c) Show that your direction is
Do Exercise 7-25 for the LP and x102 of Exercise 7-16.
Return to the LP of Exercise 7-18 and consider solving by Primal-Dual Interior Algorithm 7C.(a) Derive the corresponding dual formulation over variables v and place it in standard form by adding slack variables s as needed.(b) Show that Algorithm 7C could appropriately start with solutions x = 12,
Do Exercise 7-27 on the LPwith starting solutions x = 12, 1, 5, 3, 12 and v = 12, 52 x1 x2 X3 X4 x5 min 14 30 11 9 10 RHS s.t. 2 -1 0 3 -2 10 1 5 2 -1 1 15
Return to the standard form LP instance of Exercise 7-18.(a) Explain why this LP model is an instance of problem (LP) defined as min a nj = 1 cj xj s.t. a nj = 1 aij xj = bi i = 1,c, m xj Ú 0, j = 1, c.n(b) Following Section 7.6 and 14.1, derive a binary encoding of the instance using alphabet
Repeat Exercise 7–29 for the LP instance of(7.29).
Consider the 2-variable LP instance in Figure 7.9 and equation (7.29).(a) State the instance in the (LP) standard form format above, fixing a = 1>4.(b) Construct the basic solution corresponding to x112 in the figure.(c) Show that movement to figure solution x122 can be understood as the simplex
The sketch that follows shows the District 88 river system in the southwestern United States.All water arises at mountain reservoir R1. The estimated flow is 294 million acre-feet. At least 24 million acre-feet of the water is contracted to go directly from R1 to nearby communities, but the
A semiconductor manufacturer has three different types of silicon wafers in stock to manufacture its three varieties of computer chips. Some wafer types cannot be used for some chips, but there are two alternatives for each chip. The table that follows shows the cost and on-hand supply of each
A national commission is deciding which military bases to close in order to save at least $85 million per year. The table that follows shows projected annual savings (millions of dollars) for each of five possible closings, together with the implied percent loss of military readiness and the number
A garden superstore sells 4 kinds of fertilizer from 20 pallet spaces aligned along a fence.Weekly demands for the 4 fertilizers are 20, 14, 9, and 5 pallets, respectively, and each pallet holds 10 bags. The sales counter is at the beginning of the fence, and every bag sold must be carried from its
Consider the multiobjective LP max x1 + 5x2 max x1 s.t. x1 - 2x2 … 2 x1 + 2x2 … 12 2x1 + x2 … 9 x1, x2 Ú 0(a) Show graphically that the optimal solutions taken separately for each objective do not coincide.(b) Determine graphically whether each of the following is an efficient point:
Do Exercise 8-5 for multiobjective LP min 5x1 - x2 min x1 + 4x2 s.t. -5x1 + 2x2 … 10 x1 + x2 Ú 3 x1 + 2x2 Ú 4 x1, x2 Ú 0 and points (4,0), (2,1), (4,4), (1,2), (5,0), (0,2).
Consider the multiobjective LP max 6x1 + 4x2 max x2 s.t. 3x1 + 2x2 … 12 x1 + 2x2 … 10 x1 … 3 x1, x2 Ú 0(a) State and solve graphically a sequence of linear programs to compute a preemptive solution giving priority to the first objective.Also verify that the result is an efficient point.(b)
Do Exercise 8-7 for the multiobjective LP min x1 + x2 min x1 s.t. 2x1 + x2 Ú 4 2x1 + 2x2 Ú 6 x1 … 4 x1, x2 Ú 0
Combine each of the following sets of objective functions into a single weighted-sum objective of the specified maximize or minimize form, weighting individual objectives in the proportions indicated.(a) min 3x1 + 5x2 - 2x3 + 19x4 max 17x2 - 28x4 min 34x2 + 34x3 Results form min, proportions
Return to the multiobjective linear program of Exercise 8-7.(a) Solve graphically with a weighted-sum objective that weights the first objective twice as heavily as the second, and verify that the result is an efficient point.(b) Solve graphically with a weighted-sum objective that weights the
Do Exercise 8-10 on the multiobjective linear program of Exercise 8-8, weighting the second objective twice the first.
Convert each of the following multiobjective optimization models to a goal program seeking the specified target levels, and minimizing the unweighted sum of all goal violations.(a) min 3x1 + 5x2 - x3 max 11x2 + 23x3 s.t. 8x1 + 5x2 + 3x3 … 40 x2 - x3 … 0 x1, x2, x3 Ú 0 Targets 20,100.(b) min
Consider the multiobjective LP max x1 max 2x1 + 2x2 s.t. 2x1 + x2 … 9 x1 … 4 x2 … 7 x1, x2 Ú 0(a) Sketch the feasible space and contours where the first objective equals 3 and the second equals 14.(b) Formulate a corresponding goal program seeking target levels 3 and 14 on the two objectives
Do Exercise
on the multiobjective LP min x2 max 5x1 + 3x2 s.t. 2x1 + 3x2 Ú 6 x1 … 5-x1 + x2 … 2 x1, x2 Ú 0 using target levels 1 and 30 and solution x = 15, 5>32.
Return to the multiobjective LP of Exercise 8-13 with target levels 3 and 14 for the two objectives.(a) State and solve graphically [in 1x1, x22 space] a sequence of linear programs to compute a preemptive goal programming solution, prioritizing objectives in the sequence given and using the given
Return to the multiobjective LP of Exercise 8-14. Do Exercise 8-15 on the model using target levels of 1 and 30 and a revised objective 1 target of 3.
Return to the multiobjective District 88 water allocation problem of Exercise 8-1.(a) State and solve with class optimization software a sequence of linear programs to compute a preemptive solution for the model of part (a) taking objectives in the order given.(b) State and solve with class
Do Exercise 8-17 on the silicon wafer problem of Exercise 8-2 using target levels of 30,000 for cost and 13,000 for match.
Do Exercise 8-17 on the base closing problem of Exercise 8-3 using target levels of 3% for readiness loss and 12,000 for displaced workers.
Do Exercise 8-17 on the pallet allocation problem of Exercise 8-4 using target levels of 10 for refills and 2500 for walking.
Professor Proof is reconsidering where to invest the $50,000 left in his retirement account.One available bond fund with estimated return of 5% per year has a published risk rating of 20.The other option is a hedge fund with estimated return of 15% per year but a risk rating of 80 (it may be a
A major auto manufacturer is planning the marketing campaign to introduce its new Bambi model of hybrid sport utility vehicle. Bambis will be introduced with a television advertising campaign totaling $200 million. The table below shows the 3 kinds of programs being considered for placement of the
The State Highway Patrol (SHP) has only 60 officers to patrol j = 1,c, 10 major highway segments in rural and suburban areas of the state.Officers work i = 1,c, 7 different shift patterns across a 24 hour, 7 day work week. Exceptions can be made on late-night shifts i = 1 and 2, but all other
An architect is trying to decide what mix of single, double, and luxury rooms to include in a new hotel. Only $10 million is available for the project, and single rooms cost $40,000 to build, double rooms$60,000, and luxury rooms $120,000 each. Business travelers (in groups of one or more) average
Trustees of a major state university are trying to decide how much tuition4 should be charged next year from four categories of fulltime students: in-state undergraduates, in-state graduates, out-of-state undergraduates, and outof-state graduates. The following table shows present tuition levels
A university library5 must cut annual subscription expenses sj to some or all scientific journals j = 1,c, 40 to absorb a $5000 per year budget cut. One consideration will be the sum of published counts cj of the number of times other journals cite papers in journal j, which is a measure of how
The household sector is the largest consumer of energy in India,6 requiring at least 108 kilowatthours per day in a large city such as Madras.This energy can be obtained from sources j = 1 kerosene, j = 2 biogas, j = 3 photovoltaic, j = 4 fuelwood-generated electricity, j = 5 biogas-generated
The table that follows lists tasks like those required of the U.S. Food and Drug Administration (FDA) in carrying out its responsibilities to regulate drug laboratories, along with an importance weighting of each tasks and the estimated minimum number of staff hours required to accomplish the task
The Lake Lucky dam7 retains water in the reservoir lake of the same name to prevent flooding downstream and to assure a steady year-round supply of water for wildlife and nearby cities. The best available projections call for inflows it (cubic meters of water) to arrive at the reservoir on upcoming
A retail store has hired a marketing consultant to help it decide how to evaluate the quality8 of its service. The store would like to evaluate customer perceptions about i = 1, employee attitude;i = 2, employee competence; i = 3, product quality; and i = 4, sensory (look, sound, smell)appeal of
A steel workpiece is to be cut9 in a lathe to a target depth of 0.04 inch. Empirical studies with this type of steel express the properties of the resulting product as finish !0.41n3.97f 3.46d0.91 power !nf 0.75d0.90 time !nf where v is the cutting speed in revolutions per minute, f is the feed
Trees in available wild forest10 stands j =1,c, 300 have been measured in terms of desirable traits i = 1,c, 12 such as rate of growth, disease resistance, and wood density. Computations of averages and standard deviations in traits over the stands then produced numbers zi, j of standard deviations
Soar, a retailer catering to upscale suburbanites, 11 wishes to be represented in all the major malls j = 1,c, 5 in the Atlanta region.However, Soar wants to divide its investment budget b in proportion to three major measures of mall attractiveness: weekly patronage, pj; average patron annual
Shipley Company12 is designing a new photoresist, which is a chemical coating used in the photoengraving of silicon chips. Important characteristics of photoresists are i = 1, high flow(liquefication) temperature; i = 2, low minimum line-space resolution; i = 3, retention of unexposed areas when
South African wildlife management officials13 must decide annual quotas (in thousands of tons)for harvesting of pilchards, anchovies, and other fish of the pelagic family in the rich waters to the west of the Cape of Good Hope. One consideration is the maximization of fishing industry income, which
The state of Calizona will soon be opening centers14 in 12 of its county seats i = 1,c, 100 to promote solid waste recycling programs. Each selected site will service a district of surrounding counties.The state wants to minimize the sum of distances di, j populations pi in counties i must travel
Every year the U.S. Navy must plan the reassignment of thousands of sailors finishing one tour of duty in specialties i = 1,c, 300 and preparing for their next one, k. The reassignment may also require a move of the sailor and his or her dependents from present base location j = 1,c, 25 to a new
A segment of automatically controlled highway is being equipped with sensors spaced equally along its length. The maximum error in estimating traffic volume, which occurs halfway between any two sensors, can be expressed as (d/s)2, where d is the length of the segment and s is the number of
Return to the problem of Exercise 1-1 and assume that d = 10, p = 3.5, and b = 14. Determine(if necessary by trying all the possibilties) whether each of the following is a feasible and/or optimal solution.(a) s = 4(b) s = 6(c) s = 2
A factory has two production lines available to make a product. The first can produce one lot of the product in t1 hours at cost c1, and the second requires t2 hours and cost c2. The plant manager wishes to find the least costly way to produce b lots in a total of at most T hours. An integer number
Return to the problem of Exercise 1-3 and assume that t1 = 10, t2 = 20, c1 = 500, c2 = 100, b = 3, and T = 40. Determine (if necessary by trying all the possibilties) whether each of the following is a feasible and/or optimal solution.(a) x1 = 0, x2 = 3(b) x1 = 2, x2 = 1(c) x1 = 3, x2 = 0
A university wishes to purchase the maximum number of computer workstations that can be accommodated within the available laboratory floor space and budget. Determine whether each of the following outcomes is most likely the result of closed-form optimization, exact numerical optimization,
Explain why the first alternative of each pair constitutes more complete analysis of an optimization model than the second.(a) Closed—form versus numerical optimization(b) Exact optimization versus heuristic optimiza tion(c) Heuristic optimization versus simulation of some specific solutions
Explain why a model that admitted one of the preferred choices in Exercise
would not necessarily be more appropriate than one allowing only the alternative.
An engineer is working on a design problem with n parameters to be chosen, and each has 3 possible values. A highly valid descriptive model is available, and he is thinking of choosing the best design simply by using the model to evaluate the effect of every combination of design parameter
Determine whether each of the following could probably be validly modeled only as a random variable or if a deterministic quantity would suffice.(a) The number of inches of rainfall in a city over the next 14 days(b) The average 14-day rainfall in a city(c) The market price of a common stock 1 week
The Notip Table Company sells two models of its patented five-leg tables. The basic version uses a wood top, requires 0.6 hour to assemble, and sells for a profit of $200. The deluxe model takes 1.5 hours to assemble because of its glass top, and sells for a profit of $350. Over the next week the
Wiley Wiz is a mutual fund manager trying to decide how to divide up to $12 million between domestic and foreign stocks. Domestic stocks have been returning 11% per year and foreign 17%. Naturally, Wiley would like to maximize the annual return from his investments. Still, he wants to exercise some
The Tall Tree lumber company owns 95,000 acres of forestland in the Pacific northwest, at least 50,000 of which must be aerially sprayed for insects this year. Up to 40,000 acres could be handled by planes based at Squawking Eagle, and up to 30,000 acres could be handled from a more distant
The Fast Food Fantasy (Triple-F) hamburger chain is attempting to respond to customer demand for more healthy food by introducing a new birdburger made from a combination of beef and chicken. The new burger should weight at least 125 grams and have at most 350 calories, 15 grams of fat, and 360
Sun Agriculture (SunAg) operates a farm of 10,000 acres in the dry southwestern part of the United States. In the next season SunAg can plant acres in either vegetables, which return a profit of approximate $450 per acre, or cotton, which returns$200 per acre. As a precaution against bad weather,
The Kazak Film company needs to cut 15 long rolls and 10 short rolls of film from stock pieces. Each stock piece can be cut in one of two patterns. The first produces 5 long and 2 short rolls; the second yields 3 long and 5 short. Once any part of a piece of stock is cut, anything that remains is
A factory is building a 500-square feet open rectangular cooling pool for water exhausted from its main process. The pool will be 8 feet deep, its length should be at least twice its width, and there is room for a width of at most 15 feet. We wish to choose the feasible design that minimizes cost
An architect is designing a cylindrical hotel that will have 150,000 square feet of floor space.She wishes to make the hotel have as many 10-feet-high floors as possible, but the height of the building should not exceed 4 times its diameter or it might be unstable. (Fractional numbers of floors are
Consider a linear program over constraint set x1 + x2 Ú 2-x1 + x2 Ú 0 x2 … 2 x1, x2 Ú 0(a) Identify the feasible set in a 2-dimensional plot.(b) Devise a linear objective function for which the LP has a unique optimal solution, and illustrate graphically.(c) Devise a linear objective function
Do Exercise 2-9 for a linear program over constraints 2x1 + 3x2 Ú 6 x1 Ú 0
Write each of the following as compactly as possible using summation and “for all” indexed notation.(a) min 3y3,1 + 3y3, 2 + 4y4,1 + 4y4, 2(b) max 1y1, 3 + 2y2, 3 + 3y3, 3 + 4y4,3(c) max a1 y1, 4 + a2 y2, 4 + c + apyp, 4(d) min d1 y1 + d2 y2 + c + dt yt(e) y1, 1 + y1, 2 + y1, 3 + y1, 4 = s1 y2,
Suppose that the decision variables of a mathematical programming model are xi, j, t! amount of product i produced on manufacturing line j during week t where i = 1,c, 17; j = 1,c, 5; t = 1,c, 7.Use summation and “for all” indexed notation to write expressions for each of the following systems
Repeat Exercise 2-12, this time coding the variables and constraints in AMPL [Tables 2.4 and 2.5].
Suppose that the decision variables of a mathematical programming model are xi, j, t! acres of land plot i allocated to crop j in year t where i = 1,c, 47; j = 1,c, 9; t = 1,c, 10.Use summation and “for all” indexed notation to write expressions for each of the following systems of constraints
Repeat Exercise 2-14, this time coding the variables and constraints in AMPL as in Table 2.4.
Assuming that decision variables are y1, . . . , y3, identify the objective functionf, constraint functions gi, and right-hand sides bi of the general mathematical programming format 2.27 corresponding to each of the following optimization models.(a) max 1y122y2>y3 s.t. y1 + y2 + y3 + 7 = 20 2y1 Ú
Taking the xj as variables, and all other symbols as given constants, determine whether each of the following is a linear or a nonlinear constraint, and briefly explain why.(a) 3x1 + 2x2 - x17 = 9(b) x1 + x3 = 4x6 + 9x7(c) a>x9 + 10x13 … 100(d) x4>a + bx13 Ú 29(e) a 7 j = 1bj1xj22 … 10 (f) log
Assuming that the wj are decision variables and all other symbols are constant, determine whether each of the following is a linear program(LP) or a nonlinear program (NLP), and briefly explain why.(a) min 3w1 + 8w2 - 4w3 s.t. a 3j = 1hjwj = 9 0 … wj … 10, j = 1,c, 3(b) min 5w1 + 23>w2 s.t. 9w1
Determine whether a discrete or a continuous variable would be most appropriate to model each of the following quantities.(a) Amount of electricity consumed(b) Whether a plant should be closed.(c) Process used to manufacture(d) Number of storms in the coming hurricane season.
The Forest Service can build a firewatch tower on any of 8 mountains. Using decision variables xj = 1 if a tower is built on mountain j and = 0 otherwise, write constraint(s) enforcing each of the following requirements.(a) In all 3 sites will be selected.(b) At least 2 of mountains 1, 2, 4, and 5
The National Science Foundation (NSF)has received 4 proposals from professors to undertake new research in OR methods. Each proposal can be accepted for funding next year at the level (in thousands of dollars) shown in the following table or rejected. A total of $1 million is available for the
The state Department of Labor is considering the establishment of area job training centers at up to 4 sites. The following table shows the land cost (in thousands of dollars) of the 4 sites and indicates with an * the sites that could provide adequate service to each of the five regions of the
Assuming that the zj are decision variables, determine whether each of the following mathematical programs is best described as a linear program (LP), a nonlinear program(NLP), an integer linear program (ILP), or an integer nonlinear program (INLP), and briefly explain why.(a) max 3z1 + 14z2 + 7z3
Return to the mathematical programs of Exercise 2-23. Determine which of each of the following pairs of those models would normally be most tractable, and briefly explain why.(a) Model (a) versus (d)(b) Model (b) versus (d)(c) Model (c) versus (d)(d) Model (e) versus (f)(e) Model (a) versus (g)
Consider the multiobjective optimization model max x1 max -3x1 + x2 s.t. -x1 + x2 … 4 x1 … 8 x1, x2 Ú 0(a) Compute graphically a solution that is optimal if only the first objective is considered.(b) Compute graphically a solution that is optimal if only the second objective is considered.(c)
Do Exercise 2-25 for the multiobjective optimization model min x1 + 10x2 min 12x1 + 5x2 s.t. x1 + x2 Ú 4 x1, x2 Ú 0
Mexican Communications11 is choosing cable for a new 16,000-meter telephone line. The following table shows the diameters available, along with the associated cost, resistance, and attenuation of each per meter.Diameter(0.1 mm)Cost($/m)Resistance(ohms/m)Attenuation(db/m)4 0.092 0.279 0.00175 5
The city of Lancaster’s water distribution system12 has 3 wells for water supply. There are 10 pumps at these 3 wells. It is estimated that a pumping rate of 10,000 gallons per minute is needed to satisfy the city’s total water demand. There are limits on how much water can be pumped from each
A small engineering consulting firm13 is establishing its plan for the next year. The director and the three partners are to meet to decide which projects to pursue.Preliminary research has been done on eight projects. The expected profit for each project is given in the following table together
A major expansion of the Brisbane airport14 will require moving substantial quantities of earth from 4 sites where it is surplus to 7 locations where it is needed. The following table shows the haul distances(hundreds of meters) between points, as well as the quantity available (m3) at each surplus
The state highway department would like to have a formula for estimating the snow removal cost of each snow event as a function of the number of inches of snow to fall. A sample of n falls fj and corresponding removal costs cj, j = 1,c, n, has been collected from history. Now the department would
The Blue Hills Home Corporation (BHHC)15 employs 22 remedial education teachers to service special needs students at 22 schools in the St. Louis area. BHHC assigns its teachers to schools for an entire year and is presently making decisions for the current year. One assignment consideration is
Professor Proof is trying to decide which of 6 needed teaching assistant tasks he will assign to each of his 2 graduate assistants. Naturally, one assistant would probably be better at some tasks and the other assistant better at others. The following table shows his scoring of their
Fast Food Fantasy (Triple-F) cooks different types of hamburgers j = 1,c, 4 in batches.A batch of burger j consists of at most uj units and requires the entire cooking grill for tj minutes.Assuming that the hourly demand for each burger is a known quantity dj, Triple-F would like to decide the best
The Kitty Railroad is in the process of planning relocations of freight cars among the 5 regions of the country to get ready for the fall harvest. The following table shows the cost of moving a car between each pair of regions, along with the number of cars in each at present and the number needed

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