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operations research an introduction

Questions and Answers of Operations Research An Introduction

A baker has 30 oz of flour and 5 packages of yeast. Baking a loaf of bread requires 5 oz of flour and 1 package of yeast. Each loaf of bread can be sold for 30¢. The baker may purchase additional
2 Explain why the Phase I LP will usually have alternative optimal solutions.
1 Use the two-phase simplex method to solve the Section 4.12 problems.
Bevco manufactures an orange-flavored soft drink called Oranj by combining orange soda and orange juice. Each ounce of orange soda contains 0.5 oz of sugar and 1 mg of vitamin C. Each ounce of orange
6 Consider an LP (maximization problem) in which each basic feasible solution is nondegenerate. Suppose that xi is the only variable in our current tableau having a negative coefficient in row 0.
5 Show that if Bland’s Rule to prevent cycling is applied to Problem 4, then cycling does not occur.
4 Use the simplex method to solve Problem 10 of Section 3.3.
3 Suppose that in solving an LP, we obtain the tableau in Table 22. Although x1 can enter the basis, this LP is unbounded.Why?
7 Explain why the set of optimal solutions to an LP is a convex set.
6 Suppose you have found this optimal tableau (Table 17) for a maximization problem. Use the fact that each nonbasic variable has a strictly positive coefficient in row 0 to show that x1 = 4, x2 =
1 Show that if a toy soldier sold for $28, then the Giapetto problem would have alternative optimal solutions.
7 It has been suggested that at each iteration of the simplex algorithm, the entering variable should be (in a maximization problem) the variable that would bring about the greatest increase in the
4 Suppose you want to solve the Dorian problem (Example 2 in Chapter 3) by the simplex algorithm. What difficulty would occur?
1 Use the simplex algorithm to solve the Giapetto problem(Example 1 in Chapter 3).
7 Recall that Example 5 of Chapter 3 is an unbounded LP. Find a direction of unboundedness along which we can move for which the objective function becomes arbitrarily large.
6 For an LP in standard form with constraints Ax = b and x >= 0, show that d is a direction of unboundedness if and only if Ad = 0 and d >= 0.
2 For the Dorian problem (Example 2 in Chapter 3), show how the basic feasible solutions to the LP in standard form correspond to the extreme points of the feasible region.
1 For the Giapetto problem (Example 1 in Chapter 3), show how the basic feasible solutions to the LP in standard form correspond to the extreme points of the feasible region.
Why Does an LP Have an Optimal bfs?
2 Convert the Dorian problem (Example 2 in Chapter 3)to standard form.
1 Convert the Giapetto problem (Example 1 in Chapter 3)to standard form.
How to Convert an LP to Standard Form
8. 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
*7. 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,
6. 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
*5. OilCo is building a refinery to produce four products: diesel, gasoline, lubricants, and jet fuel. The minimum demand (in bblJday) for each of these products is 14,000,30,000, 10,000, and 8,000,
4. John must work at least 20 hours a week to supplement his income while attending school. He has the opportunity to work in two retail stores. In store 1, he can work between 5 and 12 hours a week,
3. For the diet model, what type of optimum solution would the model yield if the feed mix should not exceed 800 Ib a day? Does the solution make sense?
2. For the diet model, suppose that the daily availability of corn is limited to 450 lb. Identify the new solution space, and determine the new optimum solution.
1. Identify the direction of decrease in z in each of the following cases:*(a) Minimize z = 4xI - 2x2'(b) Minimize z = -3xl + X2'(c) Minimize z = -Xl - 2X2'
22. TORA Experiment. In the Reddy Mikks model, suppose that the following constraint is added to the problem.X2 ~ 3 Use TORA to show that the resulting model has conflicting constraints that cannot
21. TORA Experiment. In the Reddy Mikks model, use TORA to show that the removal of the raw material constraints (constraints 1 and 2) would result in an unbounded solution space. What can be said in
20. TORA Experiment. Consider the following LP model:Maximize z = 5xI + 4X2 subject to 6xI + 4X2 ::; 24 6x1 + 3X2 ::; 22.5 XI + X2::; 5 XI + 2X2::; 6- XI + X2::; 1 xz::; 2 XJ, X2 ~ 0 In LP, a
*18. An assembly line consisting of three consecutive stations produces two radio models: HiFi1 and HiFi-2. The following table provides the assembly times for the three workstations.Minutes per unit
17. A furniture company manufactures desks and chairs. The sawing department cuts the lumber for both products, which is then sent to separate assembly departments. Assembled items are sent for
16. The Burroughs Garment Company manufactures men's shirts and women's blouses for Walmark Discount Stores. Walmark will accept all the production supplied by Burroughs.The production process
15. Top Toys is planning a new radio and TV advertising campaign. A radio commercial costs$300 and a TV ad cosls $2000.A total budget of $20,000 is allocated to the campaign.However, to ensure that
*14. Wyoming Electric Coop owns a steam-turbine power-generating plant. Because Wyoming is rich in coal deposits, the plant generates its steam from coal. 111is, however, may result in emission that
13. Show & Sell can advertise its products on local radio and television (TV). The advertising budget is limited to $10,000 a month. Each minute of radio advertising costs $15 and each minute ofTY
12. Wild West produces two types of cowboy hats. A type 1 hat requires twice as much labor time as a type 2. If the all available labor time is dedicated to Type 2 alone, the company can produce a
11. Jack is an aspiring freshman at Diem University. He realizes that "all work and no play make Jack a dull boy." As a result, Jack wants to apportion his available time of about 10 hours a day
10. In the Ma-and-Pa grocery store, shelf space is limited and must be used effectively to increase profit. Two cereal items, Grano and Wheatie, compete for a total shelf space of 60 ft2. A box of
9. ChemLabs uses raw materials I and II to produce two domestic cleaning solutions, A and B. The daily availabilities of raw materials I and II are 150 and 145 units, respectively.One unit of
8. The Continuing Education Division at the Ozark Community College offers a total of 30 courses each semester. The courses offered are usually of two types: practical, such as woodworking, word
*7. An individual wishes to invest $5000 over the next year in two types of investment: Investment A yields 5% and investment B yields 8%. Market research recommends an allocation of at least 25% in
6. Alumco manufactures aluminum sheets and aluminum bars. The maximum production capacity is estimated at either 800 sheets or 600 bars per day. The maximum daily demand is 550 sheets and 580 bars.
*5. A company produces two products, A and B. The sales volume for A is at least 80% of the total sales of both A and B. However, the company cannot sell more than 100 units of A per day. Both
4. A company that operates 10 hours a day manufactures two products on three sequential processes. TIle following table summarizes the data of the problem:Minutes per unit Product 12 Process 1 10
3. Determine the solution space and the optimum solution of the Reddy Mikks model for each of the following independent changes:(a) The maximum daily demand for exterior paint is at most 2.5 tons.(b)
2. Identify the direction of increase in z in each of the following cases:*(a) Maximize z = Xl - X2'(b) Maximize z = - 5xI - 6X2'(c) Maximize z = -Xl + 2X2'*(d) Maximize z = -3XI + X2'
1. Determine the feasible space for each of the following independent constraints, given that Xl, X2 :::: O.*(a) - 3XI + X2 5; 6.(b) Xl - 2X2 :::: 5.(c) 2Xl - 3X2 5; 12.*(d) XI - X2 5; O.(e) -Xl + X2
4. Suppose that Reddy Mikks sells its exterior paint to a single wholesaler at a quantity discount.1l1e profit per ton is $5000 if the contractor buys no more than 2 tons daily and $4500 otherwise.
*3. For the feasible solution XI = 2, x2 = 2 of the Reddy Mikks model, determine the unused amounts of raw materials Ml and M2.
2. Determine the best feasible solution among the following (feasible and infeasible) solutions of the Reddy Mikks model:(a) XI = 1, X2 = 4.(b) Xl = 2, X2 = 2.(c) XI = 3, x2 = 1.5.(d) X I = 2, X2 =
1. For the Reddy Mikks model, construct each of the following constraints and express it with a linear left-hand side and a constant right-hand side:*(a) The daily demand for interior paint exceeds
What Is Operations Research?
6. During the construction of a house, six joists of 24 feet each must be trimmed to the correct length of 23 feet. The operations for cutting a joist involve the following sequence:1.~Operation 1.
*5. In a baseball game, Jim is the pitcher and Joe is the batter. Suppose that Jim can throw either a fast or a curve ball at random. If Joe correctly predicts a curve ball, he can maintain a .500
*(C)1 What is the smallest time for moving all four people to the other side of the river?
(b) Define the criterion for evaluating the alternatives.
(a) Identify at least two feasible plans for crossing the river (remember, the canoe is the only mode of transportation and it cannot be shuttled empty).
5, and 10 minutes, respectively. If two people are in the canoe, the slower person dictates the crossing time. The objective is for all four people to be on the other side of the river in the
4. Amy, Jim, John, and Kelly are standing on the east bank of a river and wish to croSs to the west side using a canoe. The canoe can hold at most two people at a time. Amy, being the most athletic,
3. Determine the optimal solution of the rectangle problem. (Hint: Use the constraint to express the objective function in terms of one variable, then use differential
2. In the rectangle problem, identify two feasible solutions and determine which one is better.
L In the tickets example, identify a fourth feasible alternative.
1 Use the linear congruential generator to obtain a sequence of 10 random numbers, given that a 17, c 43, m 100, and x0 31.
7 We all hate to bring small change to the store. Using random numbers, we can eliminate the need for change and give the store and the customer a fair shake.a Suppose you buy something that costs
6 Show that on any iteration of the acceptance–rejection method, there is a probability 1/M(b-a) that a value of the random variable is generated.
4 For Problem 2, develop a computer program for the process generator. Generate 100 random variates and compare the mean and variance of this sample against the theoretical mean and variance of this
2 Perform the simulation for Pierre’s Bakery for 25 more days (days 16 through 40) for policy C in Table 11. Compare the answer with the results in the table. Use the random numbers in Table 5 to
1 Simulate the single-server queuing system described in Section 21.2 for the first 25 departures from the system to develop an estimate for the expected time in the waiting line. Is this a
Pierre’s Bakery bakes and sells french bread. Each morning, the bakery satisfies the demand for the day using freshly baked bread. Pierre’s can bake the bread only in batches of a dozen loaves
1 For the years 1961–1970, the annual return on General Motors stock and the return on the Standard and Poor’s market index were as given in Table 20 (file Beta.xls).a Let Y return on General
4 To determine how price influences sales, a company changed the price of a product over a 20-week period. The price charged each week and the number of units sold are given in Table 23 (file
5 Confederate Express Service is attempting to determine how its shipping costs for a month depend on the number of units shipped during a month. For the last 15 months, the number of units shipped
6 In Example 1, we ran a regression with only x1 (miles driven) as an independent variable. We found thecoefficient of x1 in this regression to be 51.68. This appears to indicate (contrary to what we
2 We are to predict sales for a motel chain based on the information in Table 26 (file Motel.xls).a Use this data and multiple regression to make predictions for the motel chain’s sales during the
7 Suppose we are trying to fit a curve to data, and part (i)of Figure 17 is relevant. Explain why the points of the form (x 1i , ln yi) should, when plotted, indicate a straight-line relationship.
8 Consider the regression in which we estimated cost of running an insurance company as a function of the number of home and car insurance policies. If there were a 1%increase in the number of car
9 In the example in which we predicted the number of customers to enter the credit union, suppose that we had used five (instead of four) dummy variables to represent the days of the week. What
4 Using x as the independent variable and y as the dependent variable, find the least squares line for the following three data points:x y 1 2 4 5 7 2
5 We are trying to predict the number of uses of automatic bank teller machines as a function of time. The data are given in Table 29. Here, x number of years after 1980 and y number of monthly
6 Carboco puts metal coatings on jet propeller blades. The harder the coating, the higher the quality of the coating. The coating is shot onto the blade using pressurized gas contained in an F-gun.
7 We have been assigned to determine how the total weekly production cost for Widgetco depends on the number of widgets produced during the week. The following model has been proposed:Y b0 b1X
8 Let Yt sales during month t (in thousands of dollars)for a photography studio (SALES in Table 31) and Pt price charged for portraits during month t (PRICE). Use a computer to fit the following
3 Quarterly sales for a department store over a six-year period are given in Table 22 (file Sales.xls).a Use multiple regression to develop a model that canbe used to predict future quarterly sales.
2 We are trying to determine the number of labor hours required to produce a unit of a product. We are given the information in Table 21 (file Learn.xls). For example, the 2nd unit produced required
8 The Holt method (exponential smoothing with trend and without seasonality) is being used to forecast weekly car sales at TOD Ford. Currently, the base is estimated to be 50 cars per week, and the
We want to predict maintenance expense ( y) for a truck during the current year, from the independent variables x1 = miles driven (in thousands) during the current year and x2 =age of the truck (in
3 Suppose that the number of customers entering the bank is growing at around 20% per year. How could we incorporate this fact into the forecasting procedure described in this section?
2 Suppose again that the bank is a college credit union, but now the staff gets paid every other Friday. Again, bank traffic is much higher than usual on staff paydays. How could we incorporate this
1 Suppose the bank is a college credit union and that on days when the college’s professors get paid, bank traffic is much higher than usual. Assuming that college professors are paid on the first
16 A version of simple exponential smoothing can be used to predict the outcome of sporting events. To illustrate, consider pro football. We first assume that all games are played on a neutral field.
15 Holt’s method assumes an additive trend. For example, a trend of 5 means that the base will increase by 5 units per period. Suppose there is actually a multiplicative trend.Thus, if the current
14 Winter’s method assumes a multiplicative seasonality but an additive trend. For example, a trend of 5 means that the base will increase by 5 units per period. Suppose there is actually a
13 In our discussion of Winter’s method, a monthly seasonality of (say) 0.80 for January means that during January, air conditioner sales are expected to be 80% of the sales during an average
12 Simple exponential smoothing with a 0.3 is being used to predict sales of radios at Lowland Appliance.Predictions are made on a monthly basis. After observing August radio sales, the forecast
11 Winter’s method is being used to forecast quarterly U.S. retail sales (in billions of dollars). At the end of the first quarter, Lt 300, Tt 30, and the seasonal indexes are as follows:
10 We have been assigned to forecast the number of aircraft engines ordered each month by Engine Company.At the end of February, the forecast is that 100 engines will be ordered during April. During

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