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operations research an introduction

Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions

A large copper company16 has 23 plants, each of which can burn 4 different kinds of fuels to produce the energy needed in smelting. Energy requirements at each plant p are known quantities rp. We also know the energy output ef of each ton of fuel f burned and the quantity of sulfur pollution sf
Alabama Cabinet17 runs a sawmill producing wood panels called “blanks” for cabinetmaking.Some of the wood comes from logs sawed into boards at the company’s mill, and the remainder derives from boards purchased green (undried).Lumber from both sources must be dried in the company’s kilns
The Bottles Film Festival draws thousands of people each year to view some of the latest motion pictures and award medals for the best. Planners are now selecting one of time slots t = 1,c, n for each of the j = 1,c, m films to be shown. From past experience, the festival can estimate numbers aj,
To improve tax compliance18 the Texas Comptroller’s staff regularly audits at corporate home offices the records of out-of-state corporations doing business in Texas. Texas is considering the opening of a series of small offices near these corporate locations to reduce the travel costs now
Channel 999 TV has staff to provide on-scene coverage of up to 4 high school football games this Friday. The following table indicates 3 possibilities are in the town where Channel 999 is located.At least 2 of them must be covered, as well as at least 1 out of town. The table below shows 4 of the
The Speculators Fund is a stock mutual fund investing in categories j = 1,c, n of common stocks. At least a fraction /j and at most fraction uj of the fund’s capital is invested in any category j. The fund maintains estimates, vj, of the expected annual return in capital gain and dividends for
Engineers19 are designing the location for modules i = 1,c, m from among the j = 1,c, n available sites on a computer board.They already know ai,i=! •1 0if a wire is required from module i to module i=otherwise dj,j=! distance between sites j and j=Using this information, they wish to choose a
In each of the following plots, determine whether the specified points are feasible, infeasible, local optimal, and/or global optimal in the depicted mathematical program over two continuous variables. Dashed lines indicate contours of the objective function, and solid lines show constraints.(a)
Each of the following shows the sequence of directions and steps employed by an improving search that began at y102 = 12, 0, 52. Compute the sequence of points visited by the search.(a) y112 = 13, -1, 02, l1 = 2, y122 =1 -1, 2, 12, l2 = 5, y132 = 10, 6, 02, l3 = 12(b) y112 = 11, 3, -22, l1 = 2,
Each of the following shows the sequence of points visited by an improving search. Compute the corresponding sequence of move directions assuming that all step sizes l = 1.(a) w102 = 10, 1, 12, w112 = 14, -1, 72, w122 =14, -3, 192, w132 = 13, -3, 222(b) w102 = 14, 0, 72, w112 = 14, 2, 102, w122 =1
Refer to the plots of Exercise 3-1 and determine graphically whether the following directions appear to be improving at the points indicated.(a) x = 1 -3, 32 at x112 of 3-1(a)(b) x = 10, 12 at x122 of 3-1(a)(c) x = 1 -10, 12 at x132 of 3-1(a)(d) x = 10, -32 at x122 in 3-2(b)(e) x = 12, 22 at
Refer to the plots of Exercise 3-2 and determine graphically whether the following directions appear to be feasible at the points indicated.(a) x = 1 -5, 52 at x112 of 3-2(a)(b) x = 10, 12 at x132 of 3-2(a)(c) x = 1 -10, 02 at x132 of 3-2(a)(d) x = 1 -3, -22 at x122 in 3-2(b)(e) x = 12, 52 at
Consider a mathematical program with constraints x1 - 2x2 + 3x3 … 25 x1, x2, x3 Ú 0 Determine the maximum step (possibly + ) that preserves feasibility in the direction indicated from the point specified. Also indicate whether that step indicates that the model is unbounded, assuming that
For each of the following combinations of objective function, point, and direction, determine whether conditions 3.21 and 3.22 show that the direction improves at the point, does not improve at the point, or that further information is required.(a) max 4y1 - 2y3 + y5, y = 11, 0, 19, 4, 62,y = 12,
Construct an improving direction from the gradient of each objective function at the point indicated.(a) max 3w1 - 2w2 + w4 at w = 12, 0, 5, 12(b) min -4w2 + 5w3 - w4, at w = 12, 2, 1, 02(c) min 1w1 + 222 - w1w2 at w = 13, 22(d) max -4w1 + 9w2 + 21w222 at w = 111, 22
Determine which of the constraints 1z1 - 222 + 1z2 - 122 … 25 [i]2z1 - z2 = 8 [ii]z1 Ú 0 [iii]z2 Ú 0 [iv]are active at each of the following solutions.(a) z = 14, 02(b) z = 16, 42
Determine whether each of the directions specified is feasible at the solution indicated to the linear constraints 3y1 - 2y2 + 8y3 = 14 6y1 - 4y2 - 1y3 … 11 y1, y2, y3 Ú 0(a) y = 10, 4, 12 at y = 12, 0, 12(b) y = 10, -4, 12 at y = 12, 0, 12(c) y = 12, 0, 12 at y = 10, 1, 22(d) y = 1 -2, 1,
State all conditions that must be satisfied by a feasible direction w at the solution indicated to each of the following systems of linear constraints.(a) 2w1 + 3w3 = 18 1w1 + 1w2 + 2w3 = 14 w1, w2, w3 Ú 0 at w = 10, 2, 62(b) Same constraints as part (a) at w = 16, 4, 22.(c) 1w1 + 1w2 = 10 2w1 -
Consider the linear program min -y1 + 5y2 s.t. -y1 + y2 … 3 y2 Ú 2 y2 Ú y1 y1, y2 Ú 0 at current solution y112 = 10, 32.(a) List the condition for a direction y to be improving at y112.(b) Show that direction y = 11, -12 satisfies your condition of part (a).(c) Determine which constraints
Do Exercise 3-12(a)–(e), on LP min 3x1 - 13x3 s.t. 11x1 + 3x2 + 4x3 = 69 x1 + x2 + x3 … 16 x1, x2, x3 Ú 0 using x112 = 13, 0, 92 and x = 1-1, 1, 22.
Consider the mathematical program max 4z1 + 7z2 s.t. 2z1 + z2 … 9 0 … z1 … 4 0 … z2 … 3(a) Show that directions z112 = 12, 02 and z122 = 1 -2, 42 are improving directions for this model at every z.(b) Beginning at z102 = 10, 02, execute Improving Search Algorithm 3A on the model. Limit
Do Exercise 3-14 for mathematical program min z1 + z2 s.t. z1 + 2z2 Ú 4 0 … z1 … 6 0 … z2 … 4 directions z112 = 10, -22, z122 = 1 -4, 22, and initial point z102 = 16, 42.
Consider the line segment between each of the following solution pairs z112 and z122. Write an algebraic expression representing all points on the line segment, show that the given z132 is on the line segment, and show that the z142 specified is not.(a) z112 = 13, 1, 02, z122 = 10, 4, 92, z132 =12,
Determine whether the feasible set for each of the following systems of constraints is convex, and if not, indicate points x112 and x122 that violate definition 3.27 .(a) 1x122 + 1x222 Ú 9 x1 + x2 … 10 x2, x2 Ú 0(b) 1x122>4 + 1x222 … 25 x1 … 9 x1 + x2 Ú 3 x1, x2 Ú 0(c) x1 - 2x2 + x3 = 2
Construct a Phase I model corresponding to each of the following, and indicate appropriate starting values for the artificial variables. Assume that all original decision variables start at wj = 0.(a) max 22w1 - w2 + 15w3 s.t. 40w1 + 30w2 + 10w3 = 150 w1 - w2 … 0 4w2 + w3 Ú 0 w1, w2, w3 Ú 0(b)
Consider the linear program min 3w1 + 7w2 s.t. w1 + w2 Ú 5 0 … w1 … 2 0 … w2 Ú 2(a) Justify by inspection that this model must be infeasible.(b) Add artifical variables to construct a Phase I version for which improving search could start with w1 = w2 = 0.(c) Explain why your Phase I model
Describe how a two-phase improving search of a model with original variables y1, y2, and y3 would proceed if Phase I search terminated as follows:(a) Global optimum y = 140, 7, 0, 9, 02(b) Global optimum y = 16, 3, 1, 0, 02(c) Local optimum y = 11, 3, 1, 0, 02(d) Local optimum y = 10, 5, 1, 2, 02,
Construct a Big-M starting model for each case in Exercise 3-18 and indicate appropriate starting values for the artificial variables.Assume that all original decision variables start at wj = 0.
Describe how Big-M search of a model with original variables y1, y2, and y3 would proceed if improving search of the Big-M version produced each of the outcomes in Exercise 3-20.
Bisco’s new sugar-free, fat-free chocolate squares are so popular that the company cannot keep up with demand. Regional demands shown in the following table total 2000 cases per week, but Bisco can produce only 60% of that number.NE SE MW W Demand 620 490 510 380 Profit 1.60 1.40 1.90 1.20 The
A small engineering consulting firm has 3 senior designers available to work on the firm’s 4 current projects over the next 2 weeks. Each designer has 80 hours to split among the projects, and the following table shows the manager’s scoring(0 = nil to 100 = perfect) of the capability of each
Cattle feed can be mixed from oats, corn, alfalfa, and peanut hulls. The following table shows the current cost per ton (in dollars) of each of these ingredients, together with the percentage of recommended daily allowances for protein, fat, and fiber that a serving of it fulfills.Oats Corn Alfalfa
Several forms of gasoline are produced during the petroleum refining process, and a last step combines them to obtain market products with specified quality measures. Suppose 4 different gasolines are available, with values for the 2 indexes of quality being 99 and 210, 70 and 335, 78 and 280, and
Ronnie Runner distilleries blends i = 1,c, m scotch whiskeys to create its j = 1,c, n products with properties k = 1,c, p. Unblended whiskey i measures ai, k on scale k. Express each of the following as linear constraint(s) in these parameters and the nonnegative decision variables xi, j ! barrels
Problems are often modeled as linear programs even though some decision variables represent quantities such as the number of units processed or the number of times an alternative is used that must be integer in a physical implementation.Briefly justify this practice.
A metalworking shop needs to cut at least 37 large disks and 211 small ones from sheet metal rectangles of a standard size. Three cutting patterns are available. One yields 2 large disks with 34% waste, the second gives 5 small disks with 22% waste, and the last produces 1 large and 3 small disks
Classic Candles handmakes three models of elegant Christmas candles. Santa models require 0.10 day of molding, 0.35 day of decorating, and 0.08 day of packaging and produce $16 of profit per unit sold. Corresponding values for Christmas trees are 0.10, 0.15, 0.03, and $9, while those of gingerbread
Wobbly Office Equipment (WOE) makes two models of tables for libraries and other university facilities. Both models use the same tabletops, but model A has 4 short (18-inch) legs and model B has 4 longer ones (30-inch). It takes 0.10 labor hour to cut and shape a short leg from stock, 0.15 labor
Perfect Stack builds standard and extralong wooden palettes for a variety of manufacturers.Each model consists of 3 heavy separators of length equal to the palette. The standard model has 5 cross pieces above and 5 below the separators and requires 0.25 hour to assemble. The extralong version has 9
The figure below shows the Bill of Materials buildup to 2 finished products of bicycle rack manufacturer Hang Up (HU). For example, Product 2 uses one Assembly 3 and four Part 6’s.Each Assembly 3, in turn, is made up of 2 Part 5’s and 1 Part 7. All Products, Assemblies and Parts are produced at
Goings Engine produces diesel engines and assemblies i = 1,c, m at its plants p = 1,c, n.There is some end demand di, p for the various engines and assemblies, with the rest used in Goings production. The number of subassemblies i required to produce each assembly k is ai, k.(a) Write a system of
The River City Police Department uses work shifts in which officers work 5 of the 7 days of the week with 2 successive days off. For example, a shift might work Sunday through Thursday and then have Friday and Saturday off. A total of 6 officers must be on duty Monday, Tuesday, Wednesday, and
Mama’s Kitchen serves from 5:30 a.m. each morning until 1:30 p.m. Tables are set and cleared by busers working 4-hour shifts beginning on the hour from 5 a.m. through 10 a.m. Most are college students who hate to get up in the morning, so Mama’s pays $7 per hour for the 5, 6, and 7 a.m. shifts,
The MacKensie’s daughter will begin college 4 years from today. Her parents want to invest $10,000 at the beginning of each of the 4 years to accumulate a fund that can help pay the cost. Each year they expect to have available both certificates of deposit returning 5% after 1 year and ones
The Big Gear (BG) transmission company buys and distributes replacement transmissions for large, 18-wheeler trucks. For the next 4 months, the company anticipates demands of 100, 130, 95, and 300 units, respectively. During the first month, units can be purchased from BG’s supplier at a cost of
Seasons Greetings (SG) manufacturers artificial holiday trees decorated with embedded multi-colored lights suitable for homes or stores, tree sales vary seasonally, with 1 thousand demanded in the first quarter of the year, 5 thousand in the second, 10 thousand in the third, and 7 thousand in the
Down Hill Ski (DHS) manufactures high speed racing skis for the most adventurous of skiers. Ski sales vary seasonally, with 7 thousand pairs demanded in the first quarter of the year, 2 thousand in the second, 1 thousand in the third, and 10 thousand in the fourth. Corresponding profits to the
Ace Green Windows (AGW) manufactures environmentally efficient windows as replacements of those in existing homes. It has just received a contract for the next 6 months, requiring 100, 250, 190, 140, 220, and 110 units in months 1 through 6, respectively. Production costs for windows vary with time
Global Minimum manufactures bikini swimming suits. Their business is highly seasonal, with expected demands being 2800, 500, 100, and 850 dozen suits over the four quarters of next year. The company can produce 1200 dozen suits per quarter, but inventories must be built up to meet larger demands at
A company manufactures parts i = 1,c, m in weeks t = 1,c, n, where each unit of part i requires ai, k units of production resource k = 1,c, q and has value vi. Production resource capacities bk cannot be exceeded in any period and part demands di,t must be met. Express each of the following as
The following table shows observed electrical power consumption at several different levels of a factory’s operation.Level 2 3 5 7 Power 1 3 3 5 Engineers want to fit the estimating relationship power = b0 + b1level to these data in a way that minimizes the sum of the absolute deviations between
The following figure shows the ceiling locations of 3 sensors in a new factory relative to a coordinate system (in feet) with origin at the lower left.A control box will be located along the long (lower in the figure) wall with fiber-optic cables running rectilinearly to each sensor. Designers want
Repeat Exercise 4-22, this time choosing a fit that minimizes the maximum deviation between observed and predicted power requirements.
Repeat Exercise 4-23, this time minimizing the length of the longest cable.
The American Edwards Laboratories(AEL)10 manufactures artificial human heart valves from pig hearts. One of the things making planning complex is that the size of pig hearts is highly variable, depending on breed, age when slaughtered, feed mix, and so on. The following(fictitious) table shows the
Midville Manufacturing assembles heavy duty materials handling carts to meet demand of 500 units in the first quarter of each year, 1200 in the second, 1000 in the third, and 300 in the fourth. Elementary components, which consist of wheels, steering yokes, and carrying platforms, are first
A construction contractor has undertaken a job with 7 major tasks. Some of the tasks can begin at any time, but others have predecessors that must be completed first. The following table shows those predecessor task numbers, together with the minimum and maximum time (in days)allowed for each task,
Import Books, Incorporated (IBI)11 stocks several thousand titles in its main warehouse.The titles can be categorized by sales volume, with i = 1 requiring a stored inventory of 0 to 20 books, i = 2 requiring 21 to 40, i = 3 requiring 41 to 100, and i = 4 requiring 101 to 200. The number of titles
Radiation therapy planning for cancer treatment begins with computer images of several body tissues. A tumorous target is identified along with surrounding healthy tissues. The treatment goal is to get maximize the total radiation received by a target tumor t = 0 while avoiding damage to
Dairy cows12 in most countries calve on a regular annual basis. Their milk output varies over the year accordingly, with a peak reached a few months after calving followed by a decline to almost zero in the tenth month. Knowing these facts, farmers in an agricultural cooperative are trying to plan
Blue Bell13 is planning its monthly production of a particular type of men’s jeans. Demands di are know for the i = 1,c, 75 different fabric parts needed to make all the combinations of waist and inseam sizes being produced. Such parts are cut from fabric laid out on cutting tables in 60 to 70
To assess the impact on the U.S. coal market of different pollution control strategies, the Environmental Protection Agency (EPA)14 wants to determine, for assumed control regimes, how much coal from supplies si in different mining regions i = 1,c, 24 will be extracted, how much will then be
Quantas Airways Ltd.15 must schedule its hundreds of reservation salesclerks around the clock to have at least rt on duty during each 1-hour period starting at (24-hour) clock hour t = 0,c, 23. A shift beginning at time t extends for 9 hours with 1 hour out for lunch in the fourth, fifth, or sixth
An Indian reservation irrigation project16 must decide how much water to release through the gate at the top of its main canal in each of the upcoming 4-hour periods t = 1,c, 18. Ideal canal outflows, rt , are known for each time period, and the total outflow over all 18 periods should equal or
Major shopping mall developer Homart17 is selecting the tenant mix for its next facility.Stores of product types i = 1,c, 20 are being considered for arrangement into the new mall’s sectors j = 1,c, 5. Each sector will have 150 thousand square feet, and an allowance of ci per square foot will be
Once the configuration of molds is fixed, the planning of production of aluminum ingots18 reduces to allocating the time of furnaces j = 1,c, n among alloys i = 1,c, m and ingot sizes s = 1,c, p. Yields aj, s of ingots of size s producible from furnace j during the entire planning period can be
S&S operates its large supermarkets19 on a 24-hour per day basis using only part-time cashiers working shifts of 2 to 5 hours per day. All shifts start on the hour. The required number rh of cashiers on duty at a given store is known for (24-clock) hours h = 0,c, 23, and managers can also estimate
The transmitted gray-scale value gi, j of pixels i = 1,c, m, j = 1,c, n, in a digital space satellite photo20 is distorted by both the usual random noise and a known problem with the video camera that effectively multiplies the value for pixel (i, j) by a blurring factor bi, j. Engineers want to
The Hanshin expressway21 serves the Osaka– Kobe area of Japan. Due to heavy congestion, the number of vehicles entering at each ramp j = 1,c, 38 of the expressway is controlled by a system that reevaluates the situation every 5 minutes based on current queue lengths qj at each ramp and estimated
Industrial engineers are planning the layout of cells i = 1,c, 18 in a rectangular manufacturing facility of x = 1000 by y = 200 feet with a 6-foot-wide, two-way conveyor system along the y = 0 boundary22. It has already been decided that cells will be sequenced along the conveyor in the same order
Swift Chemical Company23 mines phosphate rock, collects it in inventory piles i = 1,c, 8, and blends it to meet contracts with customers k = 1,c, 25 at profit pik per ton.The critical measure of phosphate content in rock is its BPL. Piles correspond to different average BPL contents bi per ton,
Any convex 3-dimensional object (i.e., a body such that the line segment between any two points in its volume falls entirely within the volume)with flat sides can be described as the set of points (x, y, z) satisfying a series of linear constraints24.For example, a 3-by 5-by 9-meter box with one
The principal export of Iceland25 is fish which are very perishable and subject to high day-to-day variation in the size of the catch available for processing. Each day processing begins at any packing plant with estimates bf of the kilograms of raw fish species f = 1,c, 10 that will be available
The U.S. Air Force (USAF)26 must procure aircraft types i = 1,c, 10 and associated munition types j = 1,c, 25 to meet anticipated sortie requirements against target types k = 1,c, 15 in weather conditions classes/ = 1,c, 8. Targets k are assigned a value rk, and tk, / are anticipated under weather
North American Van Lines27 maintains a fleet of several thousand truck tractors, each of which is owned by one of its contract truckers.Tractors can be anywhere in the range i = 0,c, 9 years old. Every 4-week planning period t = 1,c, 13, tractors may be purchased new at price p, sold to contractors
The College County Election Board(CCEB) is planning for election machines needed for the coming national election in the county’s 4 voting precincts. Each machine can be expected to service 100 voters on election day, but the challenge is that the number of voters at each precinct is
Although it is only August, the Big View(BV) electronics company is placing orders now for holiday shopping season sales of a new super-large, flat-screen TV. Orders will be delivered from the overseas manufacturer to the company’s sites in 3 regional shopping malls.Due to the global nature of
The Zoom automobile company28 is planning capacities and configurations of 3 plants to produce 4 new models being introduced for the coming market cycle. The first part of the following table shows the configuration options available at each plant, along with the implied capacities (in thousands of
The Regional Power Alliance (RPA) is a profit-making operator of electric power generators g = 1,c, G. Each generator g can be utilized or left idle in any of the t = 1,c, T months over which it plans its operation. If generator g is utilized, there is a fixed setup cost of fg each time, and the
Eli Daisy produces Wozac in huge batches by heating a chemical mixture in a pressurized container. Each time a batch is processed, a different amount of Wozac is produced.The amount produced is the process yield (measured in pounds). Daisy is interested in understanding the factors that influence
Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions(if any exist). 5 x1 + x4 = 5 x2 +2x=5 x3 + 0.5x4 1 2x3 + x4 = 3
Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions(if any exist). 6 2x2 + 2x3 = 4 x1 + 2x2 + x2 x3 = 4 x3 = 0 X3
Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions(if any exist). 7 x1 + x2 = 2 -x2 + 2x3 = 3 *2 + *3 = 3
Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions(if any exist). 8x1 + x2 + x3 = 1 x2 + 2x3 + x4 = 2 x4 = 3
Show that any set of vectors containing the 0 vector is a linearly dependent set.
Show that the set of vectors V = {[1 0], [0 1]} is a linearly independent set of vectors.
Show that V = {[1 2], [2 4]} is a linearly dependent set of vectors.
Show that rank A = 0 for the following matrix = [] A = 0 0
Show that rank A = 1 for the following matrix: A = = 2 2
Show that rank A = 2 for the following matrix: 1 = [ ]] A
Find [100] rank A = 0 0 2 1 [0 2 3]
Determine whether V = {[1 0 0], [0 1 0], [1 1 0]} is a linearly independent set of vectors.
Determine the following set of vector is linearly independent or linearly dependent V = {[1 0 1], [1 2 1], [2 2 2]}
Determine the following set of vector is linearly independent or linearly dependent V = {[2 1 0], [1 2 0], [3 3 1]}
Determine the following set of vector is linearly independent or linearly dependent V = {[2 1], [1 2]}
Determine the following set of vector is linearly independent or linearly dependent V = {[2 0], [3 0]}
Determine the following set of vector is linearly independent or linearly dependent 45 V = 2
Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions(if any exist). 4 2x1 x2 + x3 + x4 = 6 x1 + x2 + x3 = 4
Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions(if any exist). 3 x1 + x2 2x1 + x2 1 = 3 3x + 2x2 = 4
Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions(if any exist). 2 x1 + x2 + x3 = 4 x1 + 2x2 = 6

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