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operations research an introduction

Questions and Answers of Operations Research An Introduction

Solve Problem 3-38, by the two-phase method.
For each case in Problem 3-37, write the corresponding Phase I objective function.
In Phase I, if the LP is of the maximization type, explain why we do not maximize the sum of the artificial variables in Phase I.
Show that the M-method will conclude that the following problem has no feasible solution.Maximize z = 2x1 + 5x2 subject to 3x1 + 2x2 Ú 6 2x1 + x2 … 2 x1, x2 Ú 0
Solve the following problem using x3 and x4 as starting basic feasible variables. As in Problem 3-39, do not use any artificial variables.Minimize z = 3x1 + 2x2 + 3x3 subject to x1 + 4x2 + x3 Ú 14
Consider the problem Maximize z = 2x1 + 4x2 + 4x3 - 3x4 subject to x1 + x2 + x3 = 4 x1 + 4x2 + x4 = 8 x1, x2, x3, x4 Ú 0 Solve the problem with x3 and x4 as the starting basic variables and without
Consider the following set of constraints:x1 + x2 + x3 = 7 2x1 - 5x2 + x3 Ú 10 x1, x2, x3 Ú 0 Solve the problem for each of the following objective functions:(a) Maximize z = 2x1 + 3x2 - 5x3.(b)
Consider the following set of constraints:-2x1 + 3x2 = 3 112 4x1 + 5x2 Ú 10 122 x1 + 2x2 … 5 132 6x1 + 7x2 … 3 142 4x1 + 8x2 Ú 5 152 x1, x2 Ú 0 For each of the following problems, develop the
In Example 3.4-1, identify the starting tableau for each of the following (independent)cases, and develop the associated z-row after substituting out all the artificial variables:*(a) The third
TORA experiment. Generate the simplex iterations of Example 3.4-1 using TORA’s Iterations 1 M@method module (file toraEx3.4-1.txt). Compare the effect of using M = 1, M = 10, and M = 1000 on the
Use hand computations to complete the simplex iteration of Example 3.4-1 and obtain the optimum solution.
TORA experiment. In Problem 3-32, use TORA to find the next-best optimal solution.
TORA experiment. Consider the following LP:Maximize z = x1 + x2 + 3x3 + 2x4 subject to x1 + 2x2 - 3x3 + 5x4 … 4 5x1 - 2x2 + 6x4 … 8 2x1 + 3x2 - 2x3 + 3x4 … 3 -x1 + x3 + 2x4 … 0 x1, x2, x3, x4
The Gutchi Company manufactures purses, shaving bags, and backpacks. The construction includes leather and synthetics, leather being the scarce raw material. The production process requires two types
Can you extend the procedure in Problem 3-9 to determine the third-best optimal value of z?
In Example 3.3-1, show how the second-best optimal value of z can be determined from the optimal tableau.
Consider the following LP:Maximize z = 16x1 + 15x2 subject to 40x1 + 31x2 … 124-x1 + x2 … 1 x1 … 3 x1, x2 Ú 0(a) Solve the problem by the simplex method, where the entering variable is the
Consider the two-dimensional solution space in Figure 3.17.(a) Suppose that the objective function is given as Maximize z = 6x1 + 3x2 If the simplex iterations start at point A, identify the path to
The following tableau represents a specific simplex iteration. All variables are nonnegative.The tableau is not optimal for either maximization or minimization. Thus, when a nonbasic variable enters
Solve the following problem by inspection, and justify the method of solution in terms of the basic solutions of the simplex method.Maximize z = 5x1 - 6x2 + 3x3 - 5x4 + 12x5 subject to x1 + 3x2 + 5x3
Consider the following LP:Maximize z = x1 subject to 5x1 + x2 = 4 6x1 + x3 = 8 3x1 + x4 = 3 x1, x2, x3, x4 Ú 0(a) Solve the problem by inspection (do not use the Gauss-Jordan row operations), and
Consider the following system of equations:x1 + 2x2 - 3x3 + 5x4 + x5 = 8 5x1 - 2x2 + 6x4 + x6 = 16 2x1 + 3x2 - 2x3 + 3x4 + x7 = 6-x1 + x3 - 2x4 + x8 = 0 x1, x2,c, x8 Ú 0 Let x5, x6, . . . , and x8
Consider the following set of constraints:x1 + 2x2 + 2x3 + 4x4 … 40 2x1 - x2 + x3 + 2x4 … 8 4x1 - 2x2 + x3 - x4 … 10 x1, x2, x3, x4 Ú 0 Solve the problem for each of the following objective
This problem is designed to reinforce your understanding of the simplex feasibility condition. In the first tableau in Example 3.3-1, we used the minimum (nonnegative)ratio test to determine the
For each of the given objective functions and the solution space in Figure 3.16, select the nonbasic variable that leads to the next simplex corner point, and determine the associated improvement in
For the solution space in Figure 3.16, all the constraints are of the type … and all the variables x1, x2, and x3 are nonnegative. Suppose that s1, s2, s3, and s4 1Ú 02 are the slacks associated
Consider the three-dimensional LP solution space in Figure 3.16, whose feasible extreme points are A, B, . . . , and J.(a) Which of the following pairs of corner points cannot represent successive
Consider the graphical solution of the Reddy Mikks model given in Figure 2.2. Identify the path of the simplex method and the basic and nonbasic variables that define this path.
In Figure 3.3, suppose that the objective function is changed to Maximize z = 4x1 + 7x2 Identify the path of the simplex method and the basic and nonbasic variables that define this path.
Consider the following LP:Maximize z = x1 + 3x2 subject to x1 + x2 … 2-x1 + x2 … 4 x1 unrestricted x2 Ú 0(a) Determine all the basic feasible solutions of the problem.(b) Use direct substitution
Consider the following LP:Maximize z = 2x1 + 3x2 + 5x3 subject to-6x1 + 7x2 - 9x3 Ú 4 x1 + x2 + 4x3 = 10 x1, x3 Ú 0 x2 unrestricted Conversion to the equation form involves using the substitution
Show algebraically that all the basic solutions of the following LP are infeasible.Maximize z = x1 + x2 subject to x1 + 2x2 … 3 2x1 + x2 Ú 8 x1, x2 Ú 0
Determine the optimum solution for each of the following LPs by enumerating all the basic solutions.(a) Maximize z = 2x1 - 4x2 + 5x3 - 6x4 subject to x1 + 4x2 - 2x3 + 8x4 … 2-x1 + 2x2 + 3x3 + 4x4
Consider the following LP:Maximize z = 2x1 + 3x2 subject to x1 + 3x2 … 12 3x1 + 2x2 … 12 x1, x2 Ú 0(a) Express the problem in equation form.(b) Determine all the basic solutions of the problem,
In an LP in which there are several unrestricted variables, a transformation of the type xj = xj- - xj+, xj-, xj+ Ú 0 will double the corresponding number of nonnegative variables.We can, instead,
JoShop manufactures three products whose unit profits are $2, $5, and $3, respectively.The company has budgeted 80 hrs of labor time and 65 hrs of machine time for the production of the three
Two products are manufactured in a machining center. The production times per unit of products 1 and 2 are 10 and 12 minutes, respectively. The total regular machine time is 2400 minutes per day. The
McBurger fast-food restaurant sells quarter-pounders and cheeseburgers. A quarterpounder uses a quarter of a pound of meat, and a cheeseburger uses only .2 lb. The restaurant starts the day with 250
Show that the m equations an j=1 aijxj = bi, i = 1, 2,c, m are equivalent to the following m + 1 inequalities:a nj=1 aijxj … bi, i = 1, 2,c, m an j=1 aa mi=1 aijbxj Ú a mi=1 bi
Show how the following objective function can be presented in equation form:Minimize z = max 5|x1 - x2 + 3x3|, | -x1 + 3x2 - x3|6 x1, x2, x3 Ú 0(Hint: 0 a 0 … b is equivalent to a … b and a Ú
Two different products, P1 and P2, can be manufactured by one or both of two different machines, M1 and M2. The unit processing time of either product on either machine is the same. The daily
Consider the following inequality 22x1 - 4x2 Ú -7 Show that multiplying both sides of the inequality by -1 and then converting the resulting inequality into an equation is the same as converting it
In the diet model (Example 2.2-2), determine the surplus amount of feed consisting of 525 lb of corn and 425 lb of soybean meal.
In the Reddy Mikks model (Example 2.2-1), consider the feasible solution x1 = 2 tons and x2 = 2 tons. Determine the value of the associated slacks for raw materials M1 and M2.
Allocation of Aircraft to Routes. Consider the problem of assigning aircraft to four routes according to the following data:Aircraft type Capacity(passengers)Number of aircraft Number of daily trips
Loading Structure, Stark and Nichole (1972). The overhead crane in Figure 2.14 with two lifting yokes is used to transport mixed concrete to a yard for casting concrete barriers.The concrete bucket
Water Quality Management, Stark and Nicholes (1972). Four cities discharge wastewater into the same stream. City 1 is upstream, followed downstream by city 2, then city 3, and then city 4. Measured
Military Planning, Shepard and Associates (1988). The Red Army (R) is trying to invade the territory defended by the Blue Army (B). Blue has three defense lines and 200 regular combat units and can
Leveling the Terrain for a New Highway, Stark and Nicholes (1972). The Arkansas Highway Department is planning a new 10-mile highway on uneven terrain as shown by the profile in Figure 2.13. The
Fitting a Straight Line into Empirical Data (Regression). In a 10-week typing class for beginners, the average speed per student (in words per minute) as a function of the number of weeks in class is
Traffic Light Control, Stark and Nicholes (1972). Automobile traffic from three highways, H1, H2, and H3, must stop and wait for a green light before exiting to a toll road.The tolls are $4, $5, and
Pollution Control. Three types of coal, C1, C2, and C3, are pulverized and mixed together to produce 50 tons per hour needed to power a plant for generating electricity. The burning of coal emits
Assembly-Line Balancing. A product is assembled from three different parts. The parts are manufactured by two departments at different production rates as given in the following
Voting on Issues. In a particular county in the State of Arkansas, four election issues are on the ballot: Build new highways, increase gun control, increase farm subsidies, and increase gasoline
Shelf Space Allocation. A grocery store must decide on the shelf space to be allocated to each of five types of breakfast cereals. The maximum daily demand is 110, 80, 150, 85, and 100 boxes,
Two alloys, A and B, are made from four metals, I, II, III, and IV, according to the following specifications:Alloy Specifications Selling price ($)A At most 80% of I 200 At most 30% of II At least
A foundry smelts steel, aluminum, and cast iron scraps to produce two types of metal ingots, I and II, with specific limits on the aluminum, graphite, and silicon contents.Aluminum and silicon
Shale Oil refinery blends two petroleum stocks, A and B, to produce two high-octane gasoline products, I and II. Stocks A and B are produced at the maximum rates of 450 and 700 bbl/hr, respectively.
Hawaii Sugar Company produces brown sugar, processed (white) sugar, powdered sugar, and molasses from sugarcane syrup. The company purchases 4000 tons of syrup weekly and is contracted to deliver at
In the refinery situation of Problem 2-71, suppose that the distillation unit actually produces the intermediate products naphtha and light oil. One bbl of crude A produces.35 bbl of naphtha and .6
An oil company distills two types of crude oil, A and B, to produce regular and premium gasoline and jet fuel. There are limits on the daily availability of crude oil and the minimum demand for the
A refinery manufactures two grades of jet fuel, F1 and F2, by blending four types of gasoline, A, B, C, and D. Fuel F1 uses gasolines A, B, C, and D in the ratio 1:1:2:4, and fuel F2 uses the ratio
All-Natural Coop makes three breakfast cereals, A, B, and C, from four ingredients:rolled oats, raisins, shredded coconuts, and slivered almonds. The daily availabilities of the ingredients are 5
A hardware store packages handyman bags of screws, bolts, nuts, and washers. Screws come in 100-lb boxes and cost $120 each, bolts come in 100-lb boxes and cost $175 each, nuts come in 80-lb boxes
Hi-V produces three types of canned juice drinks, A, B, and C, using fresh strawberries, grapes, and apples. The daily supply is limited to 200 tons of strawberries, 90 tons of grapes, and 150 tons
Suppose that an additional 100 acres of land can be purchased for $450,000, which will increase the total acreage to 900 acres.Is this a profitable deal for Realco?
Consider the Realco model of Problem
Realco owns 900 acres of undeveloped land on a scenic lake in the heart of the Ozark Mountains. In the past, little or no regulation was imposed upon new developments around the lake. The lake shores
The city of Fayetteville is embarking on an urban renewal project that will include lowerand middle-income row housing, upper-income luxury apartments, and public housing.The project also includes a
A city will undertake five urban renewal housing projects over the next 5 years. Each project has a different starting year and a different duration. The following table provides the basic data of
The city council of Fayetteville is in the process of approving the construction of a new 180,000-ft2 convention center. Two sites have been proposed, and both require exercising the “eminent
A realtor is developing a rental housing and retail area. The housing area consists of efficiency apartments, duplexes, and single-family homes. Maximum demand by potential renters is estimated to be
A large department store operates 7 days a week. The manager estimates that the minimum number of salespersons required to provide prompt service is 12 for Monday, 18 for Tuesday, 20 for Wednesday,
In an LTL (less-than-truckload) trucking company, terminal docks include casual workers who are hired temporarily to account for peak loads. At the Omaha, Nebraska dock, the minimum demand for casual
In Problem 2-56, suppose that no volunteers will start at 2:00 p.m. or 7:00 p.m. to allow for lunch and dinner. Develop the LP, and determine the optimal schedule using AMPL, Solver, or TORA.
A hospital employs volunteers to staff the reception desk between 8:00 a.m. and 10:00 p.m.Each volunteer works three consecutive hours except for those starting at 8:00 p.m. who work for two hours
In the bus scheduling example suppose that buses can run either 8- or 12-hr shifts. If a bus runs for 12 hr, the driver must be paid for the extra hours at 150% of the regular hourly pay. Do you
Two products are manufactured sequentially on two machines. The time available on each machine is 8 hours per day and may be increased by up to 4 hours of overtime, if necessary, at an additional
The manufacturing process of a product consists of two successive operations, I and II.The following table provides the pertinent data over the months of June, July, and August:June July August
A company has contracted to produce two products, A and B, over the months of June, July, and August. The total production capacity (expressed in hours) varies monthly.The following table provides
The demand for an item over the next four quarters is 280, 400, 450, and 300 units, respectively. The price per unit starts at $20 in the first quarter and increases by $1 each quarter thereafter.
The demand for ice cream at All-Flavors Parlor during the three summer months (June, July, and August) is estimated at 500, 600, and 400 20-gallon cartons, respectively. Two wholesalers, 1 and 2,
A manufacturer produces three models, I, II, and III, of a certain product using raw materials A and B. The following table gives the data for the problem.Requirements per unit Raw material I II III
Four products are processed sequentially on three machines. The following table gives the pertinent data of the problem.Manufacturing time (hr) per unit Machine Cost per hr ($) Product 1 Product 2
Toolco has contracted with AutoMate to supply their automotive discount stores with wrenches and chisels. AutoMate’s weekly demand consists of at least 1570 wrenches and 1250 chisels. Toolco cannot
Lewis (1996). Bills in a household are received monthly (e.g., utilities and home mortgage), quarterly (e.g., estimated tax payments), semiannually (e.g., insurance), or annually(e.g., subscription
A gambler plays a game that requires dividing bet money among four choices. The game has three outcomes. The following table gives the corresponding gain or loss per dollar for the different options
A business executive has the option to invest money in two plans: Plan A guarantees that each dollar invested will earn $.70 a year later, and plan B guarantees that each dollar invested will earn $2
In anticipation of the immense college expenses, Joe and Jill started an annual investment program on their child’s eighth birthday that will last until the eighteenth birthday. They plan to invest
HiRise Construction can bid on two 1-year projects. The following table provides the quarterly cash flow (in millions of dollars) for the two projects.Cash flow (in millions of $) at Project January
Investor Doe has $10,000 to invest in four projects. The following table gives the cash flow for the four investments.Cash flow ($1000) at the start of Project Year 1 Year 2 Year 3 Year 4 Year 5 1
Fox Enterprises is considering six projects for possible construction over the next four years. Fox can undertake any of the projects partially or completely. A partial undertaking of a project will
Develop AMPL models for the following problems:(a) The diet problem of Example 2.2-2 and find the optimum solution.(b) Problem 2-22.(c) Problem 2-34.
In the Reddy Mikks model, suppose that a third type of paint, named “marine,” is produced.The requirements per ton of raw materials M1 and M2 are .7 and .95 ton, respectively. The daily demand
Develop the Excel Solver model for the following problems:(a) The diet model of Example 2.2-2.(b) Problem 2-21.(c) Problem 2-34.
Modify the Reddy Mikks Solver model of Figure 2.4 to account for a third type of paint named “marine.” Requirements per ton of raw materials 1 and 2 are .6 and .85 ton, respectively.The daily
TORA Experiment. Consider the Diet Model, and let the objective function be given as Minimize z = .8x1 + .8x2 Use TORA to show that the optimum solution is associated with two distinct corner points,
An industrial recycling center uses two scrap aluminum metals, A and B, to produce a special alloy. Scrap A contains 6% aluminum, 3% silicon, and 4% carbon. Scrap B has 3% aluminum, 6% silicon, and
Day Trader wants to invest a sum of money that would generate an annual yield of at least $10,000. Two stock groups are available: blue chips and high tech, with average annual yields of 10% and 25%,
OilCo is building a refinery to produce four products: diesel, gasoline, lubricants, and jet fuel. The minimum demand (in bbl/day) for each of these products is 14,000, 30,000, 10,000, and 8000,

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