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operations research an introduction

Questions and Answers of Operations Research An Introduction

The advertising budget is limited to $10,000 a month. Each minute of advertising on radio costs $15 and each minute on TV costs $300. A newspaper ad costs $50. Show &Sell likes to advertise on radio
7.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
4. A company that operates 10 hours a day manufactures three products on three sequential processes. The following table summarizes the data of the problem:Minutes per unit Product Process 1 Process
3. A company produces three products, A, B, and C. The sales volume for A is at least 50% of the total sales of all three products. However, the company cannot sell more than 75 units of A per day.
*2. Consider the TOYCO model.(a) Suppose that any additional time for operation 1 beyond its current capacity of 430 minutes per day must be done on an overtime basis at $50 an hour. The hourly cost
*3. In Problem 2, Set 3.6a:(a) Determine the optimality range for the unit revenue ratio of the two types of hats that will keep the current optimum unchanged.(b) Using the information in (b), will
2. In the Reddy Mikks model of Example 2.2-1;(a) Determine the range for the ratio of the unit revenue of exterior paint to the unit revenue of interior paint.(b) If the revenue per ton of exterior
1. Consider Problem 1, Set 3.6a.(a) Determine the optimality condition for ~ that will keep the optimum unchanged.(b) Determine the optimality ranges for CA and C8, assuming that the other
Question 2. Suppose that the unit revenue of product 2 is fixed at its current value of c2 = $20.00. What is the associated range for Cj, the unit revenue for product 1 that will keep the optimum
Question 1. Suppose that the unit revenues for products 1 and 2 are changed to $35 and $25, respectively.Will the current optimum remain the same?
*2. Wild West produces two types of cowboy hats. A Type 1 hat requires twice as much labor time as a Type 2. If all the available labor time is dedicated to Type 2 alone, the company can produce a
1. A company produces two products, A and B. The unit revenues are $2 and $3, respectively.Two raw materials, M1 and M2, used in the manufacture of the two products have respective daily
Question 5. We know that the change in the optimum objective value equals (dual price X change in resource) so long as the change in the resource is within the feasibility range.What about the
Question 4. Suppose that the capacity of machine 1 is increased to 20 hours, how will this increase impact the optimum revenue?126 Chapter 3 The Simplex Method and Sensitivity Analysis The proposed
Question 3. If the capacity of machine 1 is increased from the present 8 hours to 13 hours, how will this increase impact the optimum revenue?The dual price for machine 1 is $14.00 and is applicable
Question 2. A suggestion is made to increase the capacities of machines 1 and 2 at the additional cost of $10/hr. Is this advisable?For machine 1, the additional net revenue per hour is 14.00 - 10.00
Question 1. If JOBCO can increase the capacity of both machines, which machine should receive higher priority?The dual prices for machines 1 and 2 are $14.00/hr and $2.00/hr. TIlis means that each
*1. Tooleo produces three types of tools, 71, n, and 13.The tools use two raw materials, M1 and M2, according to the data in the following table:Number of units of raw materials per tool Maximize z =
3. In some ill-constructed LP models, the solution space may be unbounded even though the problem may have a bounded objective value. Such an occurrence can point only to irregularities in the
*2. Consider the LP:Maximize z = 20XI + lOx2 + X3 subject to 3Xl - 3X2 + 5X3 :$; 50 Xl + x3::; 10 Xl - X2 + 4x3 ::; 20(a) By inspecting the constraints, determine the direction (XI. X2, or X3) in
1. TORA Experiment. Solve Example 3.5-3 using TORA's Iterations option and show that even though the solution starts with Xl as the entering variable (per the optimality condition), the simplex
3. For the following LP, show that the optimal solution is degenerate and that none of the alternative solutions are corner points (you may use TORA for convenience).Maximize z = 3XI + Xl subject to
2. Solve the following LP:Maximize z = 2Xl - X2 + 3X3 subject to Xl - X2 + 5X3 :%: 10 2Xl - X2 + 3x3 :%: 40 From the optimal tableau, show that all the alternative optima are not corner points(i.e.,
*1. For the following LP, identify three alternative optimal basic solutions, and then write a general expression for all the nonbasic alternative optima comprising these three basic
3. TORA experiment. Consider the LP in Problem 2.(a) Use TORA to generate the simplex iterations. How many iterations are needed to reach the optimum?(b) Interchange constraints (1) and (3) and
2. Consider the following LP:Maximize z = 3Xl + 2X2 subject to 1II 4xI - X2 ::; 8 4xI + 3X2 ::; 12 4xl + X2 ::; 8"--------------- Xl X2(a) Show that the associated simplex iterations are temporarily
*1. Consider the graphical solution space in Figure 3.8. Suppose that the simplex iterations start at A and that the optimum solution occurs at D. Further, assume that the objective function is
*7. Consider the following LP:Maximize z = 3x\ + 2x2 + 3X3 subject to 2x\ + X2 + x3:5 2 3x] + 4X2 + 2X3 2: 8 The optimal simplex tableau at the end of Phase I is given as Basic Xl X2 X3 X4 Xs R
6. Consider the following problem:Maximize z = 3Xl + 2X2 + 3X3 subject to 2Xl + X2 + X3 = 2 Xl + 3X2 + X3 = 6 3Xl + 4X2 + 2x3 = 8 Xl> X2, X3 2: 0(a) Show that Phase I terminates with two zero
5. Consider the following problem:Maximize z = 2xl + 2x2 + 4x3 subject to 2Xl + X2 + X3:5 2 3xl + 4X2 + 2x3 ~ 8(a) Show that Phase I will terminate with an artificial basic variable at zero level
4. Write Phase I for the following problem, and then solve (withTORA for convenience)to show that the problem has no feasible solution.Maximize z = 2Xl + 5X2 subject to
3. Solve Problem 5, Set 3.4a, by the two-phase method.
2. For each case in Problem 4, Set 3.4a, write the corresponding Phase I objective function.
*1. 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.
9. Show how the M-method will indicate that the following problem has no feasible solution.Maximize z = 2x1 + 5X2 subject to 3XI + 2x2 ~ 6 2x1 + x2:5 2 XI> X2 ~ 0
8. Consider the problem Maximize z = Xl + 5X2 + 3X3 subject to Xl + 2X2 + X3 = 3 2Xi - X2 = 4 The variable X3 plays the role of a slack. Thus, no artificial variable is needed in the first
7. Solve the following problem using X3 and X4 as starting basic feasible variables. As in Problem 6, do not use any artificial variables.Minimize z = 3xI + 2X2 + 3X3 subject to XI + 4X2 + X3 ~ 7 2x1
4. Consider the following set of constraints:-2XI + 3X2 = 3 (1)4Xl + 5X2 ;::: 10 (2)Xl + 2X2:::::: 5 (3)6Xl+7x2~3 (4)4XI + 8X2 ;::: 5 (5)Xl + X2 + X3 = 4 Xl + 4x2 + X4 = 8 Solve the problem for each
3. 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
2. TORA experiment. Generate the simplex iterations of Example 3.4-1 using TORA's.iterations => M-fuethbd module (file toraEx3.4-l.txt). Compare the effect of using M = 1, M == 10, and M = 1000 on
13. TORA experiment. In Problem 12, use TORA to find the next-best optimal solution.1. Use hand computations to complete the simplex iteration of Example 3.4-1 and obtain the optimum solution.
12. TORA experiment. Consider the following LP:Maximize z = XI + X2 + 3X3 + 2X4 subject to 1-1-l I, fi f, 1.....2Xj + 3X2 - 2x3 + 3X4 :s; 3- XI + xJ + 2X4 :s; 0(a) Use TORA's iterations option to
10. Can you extend the procedure in Problem 9 to determine the third best optimal value. of z?1L The Gutchi Company manufactures purses, shaving bags, and backpacks. The construction includes leather
*9. In Example 3.3-1, show how the second best optima) value of z can be determined from the optimal tableau.
8. Consider the following LP:Maximize z = 16xI + 15x2 subject to 40XI + 31x2 :5 124-xl + X2:5 1 Xl :5 3 XI> X2 ;::: 0(a) Solve the problem by the simplex method, where the entering variable is the
7. Consider the two-dimensional solution space in Figure 3.6.(3) Suppose that the objective function is given as Maximize z = 3XI + 6x2 If the simplex iterations start at point A, identify the path
6. The following tableau represents a specific simplex iteration. All variables are nonnegative.The tableau is not optimal for either a maximization or a minimization problem.ll1Us, when a nonbasic
5. Solve the following problem by inspection, and justify the method of solution in terms of the basic solutions of the simplex method.Maximize z = 5x] - 6X2 + 3X3 - 5X4 + 12xs subject to(Hint: A
4. Consider the following LP:Maximize z = Xl subject to=4=8(a) Solve the problem by inspection (do not use the Gauss-Jordan row operations), and justify the answer in terms of the basic solutions of
*3. Consider the following system of equations:Xl + 2X2 - 3X3 + 5X4 + Xs =4=8=3+ Xs = 0 Let xs, x6,' .. , and Xg be a given initial basic feasible solution. Suppose that Xl becomes basic. Which of
2. Consider the following set of constraints:Xl + 2X2 + 2X3 + 4X4 :5 40 2x1 - x2 + X3 + 2X4 :5 8 4x1 - 2X2 + X3 - X4:5 10 Solve the problem for each of the following objective functions.(a) Maximize
1. 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
5. Consider the solution space in Figure 3.4, where the simplex algorithm starts at point A.Determine the entering variable in the first iteration together with its value and the improvement in z for
4. For the solution space in Figure 3.4, all the constraints are of the type :5 and all the variables XI> X2, and X3 are nonnegative. Suppose that SJ, S2, s3, and S4 (~ 0) are the slacks associated
*3. Consider the three-dimensional LP solution space in Figure 3.4, whose feasible extreme points are A, B, ... , and 1.(a) Which of the following pairs of corner points cannot represent successive
2. 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.
1. In Figure 3.3, suppose that the objective function is changed to Maximize z = 8x[ + 4X2 Identify the path of the simplex method and the basic and nonbasic variables that define this path.
5. Consider the following LP:Maximize z = XI + 3X2 subject to-XI + x2 s4 Xl unrestricted X2 ?:: 0(a) Determine all the basic feasible solutions of the problem.(b) Use direct substitution in the
4. Consider the following LP:Maximize z = 2xI + 3X2 + 5x3 subject to-6xI + 7X2 - 9x3 ?:: 4 Xl + X2 + 4x3 = 10 X2 unrestricted Conversion to the equation form involves using the substitution X2 = xi -
*3. Show algebraically that all the basic solutions of the following LP are infeasible.Maximize z = Xl + X2 subject to XI + 2x2 :5 6 2xI + x2:5 16 90 Chapter 3 The Simplex Method and Sensitivity
1. Consider the following LP:Maximize z = 2xI + 3X2 subject to Xl + 3X2 :5 6 3x[ + 2x2 :5 6 1S~S~1 1)2.s(a) Express the problem in equation form.(b) Determine all the basic solutions of the problem,
4. In an LP in which there are several unrestricted variables, a transformation of the type Xj = xj - xj, xj, xj ;::: 0 will double the corresponding number of nonnegative variables.We can, instead,
*3. JoShop manufactures three products whose unit profits are $2, $5, and $3, respectively.The company has budgeted 80 hours of labor time and 65 hours of machine time for the production of three
2. Two products are manufactured in a machining center. The productions times per unit of products 1 and 2 are 10 and 12 minutes, respectively. The total regular machine time is 2500 minutes per day.
1. 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
5. Show how the following objective function can be presented in equation form:Minimize z = max{!xl - X2 + 3x31, I-Xl + 3X2 -:- X31}(Hint: lal :5 b is equivalent to a :5 b and a ~ -b.)
*1. In the Reddy Mikks model (Example 2.2-1), consider the feasible solution Xl = 3 tons and X2 = 1 ton. Determine the value of the associated slacks for raw materials M1 and M2.In the diet model
2. Develop AMPL models for the following problems:(a) The diet problem of Example 2.2-2 and find the optimum solution.(b) Problem 4, Set 2.3b.*(c) Problem 7, Set 2.3d.(d) Problem 7, Set 2.3g.(e)
1. 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 .5 and.75 ton, respectively.The daily demand for
2. Develop the Excel Solver model for the following problems:(a) The diet model of Example 2.2-2.(b) Problem 16, Set 2.2a(c) The urban renewal model of Example 2.3-1.*(d) The currency arbitrage model
1. Modify the Reddy Mikks Solver model of Figure 2.12 to account for a third type of paint named "marine." Requirements per ton of raw materials 1 and 2 are .5 and .75 ton, respectively.The daily
Daily demands for loaves of bread at a grocery store are specified by the following probability distribution:The store buys a loaf for 55 cents and sells it for $1.20 each. Any unsold loaves at the
In RM4.txt, suppose the statementsare replaced withExplain why AMPL will not execute properly with the proposed change. read table RM4profit; read table RM4rhs; read table RM4aij;
An appliance store can place orders for refrigerators at the beginning of each month for immediate delivery. A fixed cost of $100 is incurred every time an order is placed. The storage cost per
The following data represent the interarrival time (in minutes) at a service facility:(a) Use Excel to develop three histograms for the data based on bin widths of .5, 1, and 1.5 minutes,
In Problem 2, show that AB ≠ BAProblem 2Givenfind(a) A + 7B(b) 2A − 3B(c) (A + 2B)T A = 1 9 2 5-8, B = 37 2 4 7 9 3 -1 4 6 2 8 10
The number of air conditioning units sold over the past 24 months is given in Table F.4. Analyze the data from the standpoint of the applicability of the moving average technique. TABLE F.4 Data
Consider a random gathering of n persons. Determine the smallest n that will make it more likely that two persons or more have the same birthday.
Table F.5 gives the number of individuals who visit a tourist area by car and air over a 10-year period. Analyze the data from the standpoint of the applicability of the moving average technique.
The U of A offers off-campus courses at five different locations around the state. Table F.7 summarizes the enrollment data over a 6-year period. The data for each year are categorized by semester:
Table F.6 gives the sales in millions of dollars for a department store. Analyze the data from the standpoint of the applicability of the moving average technique. TABLE F.6 Data for Problem F-4 1980
In linear regression, prove that the sum of differences between the predicted and estimated values over all the data points always equals zero—that is, n Σ(; – y ) = 0 - ni) 0 i=1
For the single-period model, show that for the discrete demand the optimal order quantity is determined from Р p+h P{D ≤ y" − 1} ≤ _P - ≤ P{D ≤ y*}
Two identical balls made of a tough alloy are tested for hardness. The test fails if a ball dropped from a floor in a 100-story building is dented upon impact. A ball can be reused in fresh drops
One of N machines must be selected for manufacturing Q units of a specific product. The minimum and maximum demands for the product are Q* and Q**, respectively. The total production cost for Q items
A town is served by taxi services: Green Company and Blue Company. Statistics shows that Green controls 90% of the market. A hit-and-run night accident was reported by a witness who claimed it
You have the chance to play the following game in a gambling casino. A fair die is rolled twice, leading to four outcomes: (1) Both rolls show the same even number, (2) Both rolls show the
A test is available for detecting a certain genetic defect that is known to exist in 2% of the population. The test is false positive 10% of the time but detects true positives with probability .96.
Sunray Electric Coop uses a fleet of 20 trucks to service its electric network. The company wants to develop a preventive maintenance schedule for the fleet. The probability of a breakdown in year 1
Classify the states of the following Markov chains. If a state is periodic, determine its period:(a)(b)(c)(d) 01 10 0 0 1 100
Consider Problem 16-3. Suppose that Bank1 currently has $500,000 worth of outstanding loans. Of these, $100,000 have just been paid, $50,000 are 1 quarter late, $150,000 are 2 quarters late, $100,000
Consider Problem 16-4.(a) For a patient who is currently on dialysis, what is the probability of receiving a transplant in 2 years?(b) For a patient who is currently a more-than-1-year survivor, what
A die-rolling game uses a 4-square grid. The squares are designated clockwise as A, B, C, and D with monetary rewards of $4, –$2, –$6, and $9, respectively. Starting at square A, roll the die to
A game involves four balls and two urns. A ball in either urn has probability 50:50 chance of being transferred to the other urn. Represent the game as a Markov chain, and show that its states are
On a sunny day, MiniGolf can gross $2000 in revenues. If the day is cloudy, revenues drop by 20%. A rainy day will reduce revenues by 80%. If today’s weather is sunny, there is an 80% chance it
A museum has six rooms of equal sizes arranged in the form of a grid with three rows and two columns. Each interior wall has a door that connects adjacent rooms. Museum guards move about the rooms
The following table provides weather data in Fayetteville, Arkansas:Based on these records, use a Markov chain to determine the probability that a typical day in Fayetteville will be cloudy, sunny,
A car rental agency has offices in Phoenix, Denver, Chicago, and Atlanta. The agency allows one- and two-way rentals so that cars rented in one location may end up in another. Statistics show that at
A company reviews the state of one of its important products annually and decides whether it is successful (state 1) or unsuccessful (state 2). The company must decide whether to advertise the
A company can advertise through radio, TV, or newspaper. The weekly costs of advertising on the three media are estimated at $200, $900, and $300, respectively. The company can classify its sales

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