New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
business
operations research an introduction

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

3 Consider the following LP and its optimal tableau(Table 52):a Find the dual to this LP and its optimal solution.b Find the range of values of c1 for which the current basis remains optimal.c Find the range of values of c2 for which the current basis remains optimal. s.t. max z = 5x1 + x2 + 2x3 x1
2 For the LP in Problem 1, graphically determine the range of values on c1 for which the current basis remains optimal. (Hint: The feasible region is a line segment.)
1 Consider the following LP and its optimal tableau(Table 51):a Find the dual of this LP and its optimal solution.b Find the range of values of b3 for which the current basis remains optimal. If b3 11, what would be the new optimal solution? max z 4x + x2 = 4x+x s.t. x1 + 2x2 = 6 x1 x2 = 3 2x1 +
7 a Explain why the amount of each input used by the composite hospital is at most (efficiency of hospital 2) TABLE 50 Support Faculty Staff Supply Budget (in Millions) (in Thousands) Credit Hours Research Publications Business 150 70 5 15 225 Education 60 20 3 5.4 70 Arts and Sciences 800 140 20
6 Explain why the dual price for the 8w1 + 15w2 = 1 constraint must equal the optimal z-value for the hospital 2 LP.
5 Explain why the amount of each output produced by the composite hospital obtained by averaging hospitals 1 and 3(with the absolute value of the dual prices as weights) is at least as large as the amount of the corresponding output produced by hospital 2. (Hint: Price out variables t1, t2, and t3,
4 You have been assigned by Indiana University to evaluate the relative efficiency of four degree-granting units: Business;Education; Arts and Sciences; and Health, Physical Education, and Recreation (HPER). You are given the information in Table 50. Use DEA to find all inefficient units. Comment
2 Pine Valley Bank has three branches. You have been assigned to evaluate the efficiency of each. The following inputs and outputs are to be used for the study.Input 1 = labor hours used (hundreds per month)Input 2 = space used (in hundreds of square feet)Input 3 = supplies used per month (in
1 The Salem Board of Education wants to evaluate the efficiency of the town’s four elementary schools. The three outputs of the schools are defined to be Output 1 = average reading score Output 2 = average mathematics score Output 3 = average self-esteem score The three inputs to the schools are
4 Find the new optimal solution to the Dakota problem if 15 carpentry hours are available.
3 Find the new optimal solution to the Dakota problem if only 20 board ft of lumber are available.
2 In solving the following LP, we obtain the optimal tableau shown in Table 45.a Find the optimal solution to this LP if we add the constraint 3x1 + x2 b Find the optimal solution if we add the constraint x1 - x2 > 6.c Find the optimal solution if we add the constraint 8x1 + x2 12. max z = 6x1
1 Use the dual simplex method to solve the following LP: max z = -2x1 - x3 s.t. x1 + x2 x3 5 X35 x12x2 + 4x3 8 x1, x2, x30
4 Let x [x1 x2 x3 s1 s2 s3] be a primal feasible point for the Dakota problem and y [ y1 y2 y3 e1 e2 e3] be a dual feasible point.a Multiply the ith constraint (in standard form) of the primal by yi and sum the resulting constraints.b Multiply the jth dual constraint (in standard form)by xj and
3 Consider the following LP:Graphically solve the dual of this LP. Then use complementary slackness to solve the max problem. max z=5x+3x2 + x3 s.t. 2x1 + x2+x3 6 x1 + 2x2 + x3 7 x1, x2, x30
2 Use the Theorem of Complementary Slackness to show that in the LINDO output, the SLACK or SURPLUS and DUAL PRICE entries for any row cannot both be positive.
1 Glassco manufactures glasses: wine, beer, champagne, and whiskey. Each type of glass requires time in the molding shop, time in the packaging shop, and a certain amount of glass. The resources required to make each type of glass are given in Table 32. Currently, 600 minutes of molding time, 400
3 Suppose, in the Dakota problem, a desk still sells for$60 but now uses 8 board ft of lumber, 4 finishing hours, and 15 carpentry hours. Determine whether the current basis remains optimal. What is wrong with the following reasoning?The change in the column for desks leaves the second and third
2 The following questions refer to the Sugarco problem(Problem 6 of Section 6.3):a For what values of profit on a Type 1 candy bar does the current basis remain optimal?b If a Type 1 candy bar used 0.5 oz of sugar and 0.75 oz of chocolate, would the current basis remain optimal?c A Type 4 candy bar
1 For the Dakota problem, suppose computer tables sell for $35 and use 6 board feet of lumber, 2 hours of finishing time, and 1 hour of carpentry time. Is the current basis still optimal? Interpret this result in terms of shadow prices.
We want to add a new activity. Suppose Dakota is considering manufacturing footstools(x4). A footstool sells for $15 and uses 1 board foot of lumber, 1 finishing hour, and 1 carpentry hour. Does the current basis remain optimal?
We want to change the column for a nonbasic activity. Suppose a table sells for $43 and uses 5 board feet of lumber, 2 finishing hours, and 2 carpentry hours. Does the current basis remain optimal?
We want to change the objective function coefficient of a nonbasic variable. Let c2 be the coefficient of x2 (tables) in the Dakota objective function. In other words, c2 is the price at which a table is sold. For what values of c2 will the current basis remain optimal?
11 Consider the LP:a Solve this LP with LINDO and use your output to show that the optimal solution is degenerate.b Use your LINDO output to find an example of Oddities 1–3. max z = 9x1 + 8x2 + 5x3 + 4x4 s.t. + x 200 x2+x3 150 x1 + x2 + x3 350 2x1 + x2 + x3 + x4 = 550 X1, X2, X3, X40
10 For the diet problem, suppose at least 8 oz of chocolate and at least 9 oz of sugar are required (with other requirements remaining the same). What is the new optimal z-value?
9 For the Dakota problem, suppose that 22 finishing hours and 9 carpentry hours are available. What would be the new optimal z-value? [Hint: Use the 100% Rule to show that the current basis remains optimal, and mimic (34)–(36).]
8 If it seems difficult to believe that the shadow price of an equality constraint should be urs, try this problem.Consider the following two LPs:In which LP will the constraint have a positive shadow price? Which will have a negative shadow price? max z = x2 x1 + x2 = 2 X1, X20 (LP 1) s.t. max z =
7 For the Dorian problem (see Problem 8 of Section 6.3), answer the following questions:a What would Dorian’s cost be if 40 million HIW exposures were required?b What would Dorian’s cost be if only 20 million HIM exposures were required?
6 Suppose that the company in Problem 5 owns no labor and raw material but can purchase them at the following prices: as many as 100 hours of skilled labor at $3/hour, 70 hours of unskilled labor at $2/hour, and 30 units of raw material at $1 per unit of raw material. If the company’s goal is to
5 A company manufactures two products (1 and 2). Each unit of product 1 can be sold for $15, and each unit of product 2 for $25. Each product requires raw material and two types of labor (skilled and unskilled) (see Table 29).Currently, the company has available 100 hours of skilled labor, 70 hours
4 Suppose we are working with a min problem and increase the right-hand side of a constraint. What can happen to the optimal z-value?
3 Suppose we are working with a min problem and increase the right-hand side of a constraint. What can happen to the optimal z-value?
2 The following questions refer to the Sugarco problem(Problem 6 of Section 6.3):a Find the shadow prices for the Sugarco problem.b If 60 oz of sugar were available, what would be Sugarco’s profit?c How about 40 oz of sugar?d How about 30 oz of sugar? TABLE 29 Product 2 Resource Skilled labor
1 Use the Dual Theorem to prove (37).
Steelco has received an order for 100 tons of steel. The order must contain at least 3.5 tons of nickel, at most 3 tons of carbon, and exactly 4 tons of manganese. Steelco receives$20/ton for the order. To fill the order, Steelco can combine four alloys, whose chemical composition is given in Table
Leatherco manufactures belts and shoes. A belt requires 2 square yards of leather and 1 hour of skilled labor. A pair of shoes requires 3 sq yd of leather and 2 hours of skilled labor.As many as 25 sq yd of leather and 15 hours of skilled labor can be purchased at a price of $5/sq yd of leather and
For the Dakota problem:1 Find and interpret the shadow prices 2 If 18 finishing hours were available, what would be Dakota’s revenue? (It can be shown by the methods of Section 6.3 that if 16 finishing hours 24, the current basis remains optimal.)3 If 9 carpentry hours were available, what
9 Use the information given in Problem 8 of Section 6.3 to determine the dual of the Dorian Auto problem and its optimal solution.
8 Following along the lines of Problem 7, use weak duality to prove Lemma 4.
7 In this problem, we use weak duality to prove Lemma 3.a Show that Lemma 3 is equivalent to the following:If the dual is feasible, then the primal is bounded. (Hint:Do you remember, from plane geometry, what the contrapositive is?)b Use weak duality to show the validity of the form of Lemma 3
6 Show that (for a max problem) if the ith primal constraint is a >= constraint, then the optimal value of the ith dual variable may be written as (coefficient of ai in optimal row 0) - M.
5 Consider the following linear programming problem:Suppose that in solving this problem, row 0 of the optimal tableau is found to be z + 2x2 + s2 = 20/3 . Use the Dual Theorem to prove that the computations must be incorrect. max z=4x1 + x2 s.t. 3x1 + 2x2 6 6x + 3x2 10 X1, X20
4 The following questions refer to the Bevco problem of Section 4.10.a Find the dual of the Bevco problem.b Use the optimal tableau for the Bevco problem that is given in Section 4.10 to find the optimal solution to the dual. Verify that the Dual Theorem holds in this instance.
3 For the following LP, max z = -x + 5x2 s.t. x + 2x2 0.5 -x1 + 3x2 0.5 x1, x20 X2 row 0 of the optimal tableau is z + 0.4s + 1.4s = ? De- termine the optimal z-value for the given LP.
2 Consider the following LP:a Find the dual of this LP.b After adding a slack variable s1, subtracting an excess variable e2, and adding artificial variables a2 and a3, row 0 of the LP’s optimal tableau is found to beFind the optimal solution to the dual of this LP. = -2xx+x3 max z = -2x s.t. x1
1 The following questions refer to the Giapetto problem(see Problem 7 of Section 6.3).a Find the dual of the Giapetto problem.b Use the optimal tableau of the Giapetto problem to determine the optimal dual solution.c Verify that the Dual Theorem holds in this instance.
2 Find the dual of Example 2 in Chapter 3 (Dorian Auto)and give an economic interpretation of the dual problem
1 Find the dual of Example 3 in Chapter 3 (an auto company) and give an economic interpretation of the dual problem.
6 This problem shows why a dual variable yi corresponding to a >= constraint in a max problem must satisfy yi 0.a Using the rules given in the text, find the dual ofb Transform the LP of part (a) into a normal max problem. Now use (16) and (17) to find the dual of the transformed LP. Let y2 be
5 This problem shows why the dual variable for an equality constraint should be urs.a Use the rules given in the text to find the dual ofb Now transform the LP in part (a) to the normal form.Using (16) and (17), take the dual of the transformed LP. Use y2 and y2 as the dual variables for the two
Find the duals of the following LPs: 4 min w4y, +2y2 - y3 s.t. y + 2y 6 y1 y2+2y3 = 8 y, y2 0, y3 urs
Find the duals of the following LPs: 3 max max z = 4x1 - x2 + 2x3 s.t. x1 + x2 5 2x1 + x2 7 2x2 + x3 6 x1 +x3 = 4 x10, x2, x3 urs
Find the duals of the following LPs: 2 min wy - y2 s.t. 2y + y2 4 y + y21 y + 22 3 yi, y2 0
Find the duals of the following LPs: 1 max z=2x1 + x2 s.t. - x1 + x2] x1 + x2 3 x12x2 4 x1, x20
13 In this problem, we sketch a proof of the 100% Rule for right-hand sides. Consider an LP with two constraints and right-hand sides b1 and b2. Suppose that if only one right-hand side is changed, the current basis remains optimal for L1 b1 U1 and L2 b2 U2. Suppose we change the right-hand
12 Prove the Case 1 result for right-hand sides. Use the fact that if a constraint is nonbinding in the optimal solution, then its slack or excess variable is in the optimal basis, and the corresponding column of B1 will have a single 1 and all other elements equal to 0.
11 To illustrate the validity of the 100% Rule for objective function coefficients, consider an LP with four decision variables (x1, x2, x3, and x4) and two constraints in which x1 and x2 are basic variables in the optimal basis. Suppose (if only a single objective function coefficient is changed)
10 Prove the Case 1 result for the objective function coefficients.
9 Suppose 40 board ft of lumber, 21 finishing hours, and 8.5 carpentry hours are available. Show that the current basis remains optimal.
8 Suppose that 60 board ft of lumber and 23 finishing hours are available. Show that the current basis remains optimal.
7 Suppose that the price of a desk is $65, a table is $25, and a chair is $18. Show that the current basis remains optimal. What is the new optimal z-value?
6 If 8 oz of chocolate and 60 calories are required, show that the current basis remains optimal.The following questions refer to the Dakota problem.
5 If the price of a bottle of soda is 15¢ and a piece of cheesecake is 60¢, show that the current basis remains optimal.What will be the new optimal solution to the diet problem?
4 If the fat requirement is 6 oz and the calorie requirement is 600 calories, does the current basis remain optimal?
3 If the fat requirement is reduced to 3 oz and the calorie requirement is increased to 800 calories, does the current basis remain optimal?
2 If the cost of a brownie is 20¢ and a piece of cheesecake is $1, does the current basis remain optimal?
1 If the cost of a brownie is 70¢ and a piece of cheesecake costs 60¢, does the current basis remain optimal?
In the diet problem, suppose the chocolate requirement is increased to 8 oz and the sugar requirement is reduced to 7 oz. Does the current basis remain optimal?
In the Dakota problem, suppose 22 finishing hours and 9 carpentry hours are available.Does the current basis remain optimal?
Suppose the calorie requirement is decreased to 400 calories and the fat requirement is increased to 15 oz. Is the current basis still optimal?
Suppose the calorie requirement is decreased to 400 calories and the fat requirement is increased to 10 oz. Does the current basis remain optimal? What is the new optimal solution?
Show that if the price of tables increases to $33 and desk prices decrease to $58, the 100%Rule does not tell us whether the current basis is still optimal.
Suppose the desk price increases to $70 and chairs decrease to $18. Does the current basis remain optimal? What is the new optimal z-value?
If prices drop to 40¢ for a brownie and 25¢ for a piece of pineapple cheesecake, is the current basis still optimal?
Suppose the price of a brownie increases to 60¢ and a piece of pineapple cheesecake decreases to 50¢. Does the current basis remain optimal? What would be the new optimal solution?
8 Consider the Dorian Auto problem (Example 2 of Chapter 3):(x1 = number of comedy ads, and x2 = number of football ads). The optimal tableau is given in Table 13. Remember that for a min problem, a tableau is optimal if and only if each variable has a nonpositive coefficient in row 0 and the
7 The following questions refer to the Giapetto problem(Section 3.1). Giapetto’s LP wasa Show that as long as soldiers (x1) contribute between $2 and $4 to profit, the current basis remains optimal. If soldiers contribute $3.50 to profit, find the new optimal solution to the Giapetto problem.b
6 Sugarco can manufacture three types of candy bar. Each candy bar consists totally of sugar and chocolate. The compositions of each type of candy bar and the profit earned from each candy bar are shown in Table 10. Fifty oz of sugar and 100 oz of chocolate are available. After defining xi to be
5 Dakota Furniture is considering manufacturing home computer tables. A home computer table sells for $36 and uses 6 board ft of lumber, 2 finishing hours, and 2 carpentry hours.Should the company manufacture any home computer tables?
4 Show that if tables sell for $50 and use 1 board ft of lumber, 3 finishing hours, and 1.5 carpentry hours, the current basis for the Dakota problem will no longer be optimal. Find the new optimal solution.
3 In the Dakota problem, show that if the amount of lumber(board ft) available (b1) satisfies b1 >= 24, the current basis remains optimal. If b1 = 30, find the new optimal solution.
2 If c1 = 55 in the Dakota problem, show that the new optimal solution does not produce any desks.
1 In the Dakota problem, show that the current basis remains optimal if c3, the price of chairs, satisfies 15
2 For the following LP, x2 and s1 are basic variables in the optimal tableau. Use the formulas of this section to determine the optimal tableau. max z=-x1 + x2 s.t. 2x1 + x2 4 x1 + x2 2 x1, x2 0
1 For the following LP, x1 and x2 are basic variables in the optimal tableau. Use the formulas of this section to determine the optimal tableau max z = 3x1 3x1 + x2 s.t. 2x1 x22 -x1 + x2 4 x1, x20
For the following LP, the optimal basis is BV = {x2, s2}. Compute the optimal tableau. max z=x+4x2 s.t. x1 + 2x2 6 2x1 + x2 8 X1, X2 8
5 Radioco manufactures two types of radios. The only scarce resource needed to produce radios is labor. The company now has two laborers. Laborer 1 is willing to work as many as 40 hours per week and is paid $5 per hour.Laborer 2 is willing to work up to 50 hours per week and is paid $6 per hour.
4 For the Dorian Auto problem (Example 2 in Chapter 3), a Find the range of values of the cost of a comedy ad for which the current basis remains optimal.b Find the range of values of the cost of a football ad for which the current basis remains optimal c Find the range of values of required HIW
3 Show that if the weekly demand for soldiers is at least 20, the current basis remains optimal, and Giapetto should still produce 20 soldiers and 60 trains.
2 Show that if available carpentry hours remain between 60 and 100, the current basis remains optimal. If between 60 and 100 carpentry hours are available, then would Giapetto still produce 20 soldiers and 60 trains?
1 Show that if the contribution to profit for trains is between $1.50 and $3, the current basis remains optimal. If the contribution to profit for trains is $2.50, what would be the new optimal solution?
5 Find a minimum spanning tree for the network in Figure 68.
4 During the next three months, Shoemakers, Inc. must meet(on time) the following demands for shoes: month 1, 1,000 pairs; month 2, 1,500 pairs; month 3, 1,800 pairs. It takes 1 hour of labor to produce a pair of shoes. During each of the next three months, the following number of regular-time
1 Consider the problem of finding the shortest path from node 1 to node 6 in Figure 2.a Formulate this problem as an MCNFP.b Find a bfs in which x12, x24, and x46 are positive.(Hint: A degenerate bfs will be obtained.)c Use the network simplex to find the shortest path from node 1 to node 6.
Use the network simplex to solve the MCNFP in Figure 56.
4 a Three cities are at the vertices of an equilateral triangle of unit length. Flying Lion Airlines needs to supply connecting service between these three cities.What is the minimum length of the two routes needed to supply the connecting service?b Now suppose Flying Lion Airlines adds a hub at
2 The city of Smalltown consists of five subdivisions.Mayor John Lion wants to build telephone lines to ensure that all the subdivisions can communicate with each other.The distances between the subdivisions are given in Figure 50. What is the minimum length of telephone line required?Assume that
1 The distances (in miles) between the Indiana cities of Gary, Fort Wayne, Evansville, Terre Haute, and South Bend are shown in Table 38. It is necessary to build a state road system that connects all these cities. Assume that for political reasons no road can be built connecting Gary and Fort
The State University campus has five minicomputers. The distance between each pair of computers (in city blocks) is given in Figure 48. The computers must be interconnected by underground cable. What is the minimum length of cable required? Note that if no arc is drawn connecting a pair of nodes,
7 Braneast Airlines must determine how many airplanes should serve the Boston–New York–Washington air corridor and which flights to fly. Braneast may fly any of the daily flights shown in Table 36. The fixed cost of operating an airplane is $800/day. Formulate an MCNFP that can be used to

Showing 2500 - 2600 of 4739

First
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved