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operations research an introduction

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

5 Use Wolfe’s method to solve the following QPP: min z = 2x - x2 s.t. 2x1 - x2 1 x1+x21
6 Use Wolfe’s method to solve the following QPP: min x + 2x s.t. x1 + x2 2 2x1 + x2 3
7 In an electrical network, the power loss incurred when a current of I amperes flows through a resistance of R ohms is I2R watts. In Figure 48, 710 amperes of current must be sent from node 1 to node 4. The current flowing through each node must satisfy conservation of flow. For example, for node
8 Use Wolfe’s method to find the optimal solution to the following QPP: = min z x+x-2x-3x + xx s.t. x1 + 2x2 2 X1, X20
9 (This problem requires some knowledge of regression.)In Table 17, you are given the annual returns on three different types of assets (T-bills, stocks, and gold) (file Invest68.xls) during the years 1968–1988. For example, $1 invested in T-bills at the beginning of 1978 grew to $1.07 by the end
10 (Refer to Problem 9 data.) Suppose that the return on the ith asset may be estimated as mi biM ei, where M is the return on the market. Assume that the ei are independent and that the standard deviation of ei may be estimated by the standard error of the estimate from the regression, with the
11 (Refer to Problem 9 data.) Suppose that you now hold 30% of your investment in stocks, 50% in T-bills, and 20%in gold. Assume that transactions incur costs. Every $100 of stocks traded costs you $1, every $100 of your gold portfolio traded costs you $2, and every $1 of your T-bill portfolio
Oilco must determine how many barrels of oil to extract during each of the next two years.If Oilco extracts x1 million barrels during year 1, each barrel can be sold for $30 - x1.If Oilco extracts x2 million barrels during year 2, each barrel can be sold for $35 - x2.The cost of extracting x1
4 GMCO produces three types of cars: compacts, mid-size, and large. The variable cost per car (in thousands of dollars) and production capacity for each type of car are given in Table 23.The annual demand for each type of car depends on the prices of the three types of cars, given in Table 24. Here
5 Consider the discussion of crashing the length of the Widgetco project given in Section 8.4. For this example, construct a trade-off curve between cost of crashing the project and duration of the project.
12 In time t, a tree can grow to a size F(t), where F (t) 0 and F(t) 0. Assume that for large t, F (t) is near 0. If the tree is cut at time t, then a revenue F(t) is received.Assume that revenues are discounted continuously at a rate r, so $1 received at time t is equivalent to $ert received at
13 Suppose we are hiring a weather forecaster to predict the probability that next summer will be rainy or sunny. The following suggests a method that can be used to ensure that the forecaster is accurate. Suppose that the actual probability of rain next summer is q. For simplicity, we assume that
14 Show that if b a e, then ab ba. Use this result to show that e e. [Hint: Show that max(ln xx) over x a occurs for x a.]
15 Consider the points (0, 0), (1, 1), and (2, 3). Formulate an NLP whose solution will yield the circle of smallest radius enclosing these three points. Use LINGO to solve the NLP.
16 The cost of producing x units of a product during a month is x1/2 dollars. Show that the minimum cost method of producing 40 units during the next two months is to produce all 40 units during a single month. Is it possible to generalize this result to the case where the cost of producing x units
17 Consider the problem max z f (x)s.t. a x b a Suppose f (x) is a convex function that has derivatives for all values of x. Show that x a or x b must be optimal for the NLP. (Draw a picture.)b Suppose f(x) is a convex function for which f (x) may not exist. Show that x a or x b must
18 Reconsider Problem 2. Suppose that the store should now be located to minimize the total distance that customers must walk to the store. Where should the store be located?(Hint: Use Problem 4 and the fact that for any convex function a local minimum will solve the NLP; then show that locating
19† A company uses raw material to produce two products.For c dollars, a unit of raw material can be purchased and processed into k1 units of product 1 and k2 units of product 2. If x1 units of product 1 are produced, they can be sold at p1(x1) dollars per unit. If x2 units of product 2 are
20 The area of a triangle with sides of lengtha, b, and cof the triangle. We have 60 ft of fence and want to fence a triangular-shaped area. Determine how to maximize the fenced area. is Vs(sa)(sb)(sc), where s is half the perimeter
21 The energy used in compressing a gas (in three stages)from an initial pressure I to a final pressure F is given byDetermine how to minimize the energy used in compressing the gas. K P2 ++ VP1 F 3 P2
11 With L labor hours and M machine hours, a company can produce L1/3M2/3 computer disk drives. Each disk drive sells for $150. If labor can be purchased at $50 per hour and machine hours can be purchased at $100 per hour, determine how the company can maximize profits.
10 If a company charges a price p for a product and spends$a on advertising, it can sell 10,000 5a 100p units of the product. If the product costs $10 per unit to produce, then how can the company maximize profits?
6 For Example 35 of Section 11.10, construct a trade-off curve between the chosen portfolio’s expected return and variance. This is often called the efficient frontier
1 Show that f (x) ex is a convex function on R1.
2 Five of a store’s major customers are located as in Figure 55. Determine where the store should be located to minimize the sum of the squares of the distances that each customer would have to travel to the store. Can you generalize this result to the case of n customers located at points x1,
3 A company uses a raw material to produce two types of products. When processed, each unit of raw material yields 2 units of product 1 and 1 unit of product 2. If x1 units of product 1 are produced, then each unit can be sold for$49 x1, if x2 units of product 2 are produced, then each unit can
4 Show that f (x) x is a convex function on R1.
5 Use Golden Section Search to locate, within 0.5, the optimal solution to max 3x - x s.t. 0 x 5
6 Perform two iterations of the method of steepest ascent in an attempt to maximize x2)=(x1+x2)-(x1+x2)-x1 f(x1, Begin at the point (0,1).
7 The cost of producing x units of a product during a month is x2 dollars. Find the minimum cost method of producing 60 units during the next three months. Can you generalize this result to the case where the cost of producing x units during a month is an increasing convex function?
8 Solve the following NLP: max z = xyw s.t. 2x+3y+4w 36 =
9 Solve the following NLP 50 20 min z s.t. y +xy x 1, y 1
22 Prove Lemma 1 (use Lagrange multipliers).
1 We are considering investing in three stocks. The random variable Si represents the value one year from now of $1 invested in stock i. We are given that E(S1) 1.15, E(S2) 1.21, E(S3) 1.09; var S1 0.09, var S2 0.04, var S3 0.01; cov(S1, S2) 0.006, cov(S1, S3) 0.004, and cov(S2, S3)
Suppose the grade-point average (GPA) for a student can be accurately predicted from the student’s score on the GMAT (Graduate Management Admissions Test). More specifically, suppose that the ith student observed has a GPA of yi and a GMAT score of xi. How can we use the least squares method to
2 Use the method of steepest ascent to approximate the optimal solution to the following problem: max z (x1 2)2 x1 x2 2. Begin at the point (2.5, 1.5).
3 Use steepest ascent to approximate the optimal solution to the following problem: max z 2x1x2 2x2 x2 1 2x2 2.Begin at the point (0.5, 0.5). Note that at later iterations, successive points are very close together. Variations of steepest ascent have been developed to deal with this problem
4 How would you modify the method of steepest ascent if each variable x1 were constrained to lie in an interval [ai, bi]?
5 Show that at any point x (x1, x2), f (x) is perpendicular to the curve f (x1, x2) f (x1, x2). (Hint: Two vectors are perpendicular if their scalar product equals zero.)
A company is planning to spend $10,000 on advertising. It costs $3,000 per minute to advertise on television and $1,000 per minute to advertise on radio. If the firm buys x minutes of television advertising and y minutes of radio advertising, then its revenue in thousands of dollars is given by f
Given numbers x1, x2, . . . , xn, show thatwith equality holding only if x1 = x2 = ................. =xn. n I=n 1=1 2
1 For Example 30, show that if a dollars are available for advertising, then an extra dollar spent on advertising will increase revenues by approximately 11-a/4.
2 It costs me $2 to purchase an hour of labor and $1 to purchase a unit of capital. If L hours of labor and K units of capital are available, then L2/3K1/3 machines can be produced. If I have $10 to purchase labor and capital, what is the maximum number of machines that can be produced?
3 In Problem 2, what is the minimum cost method of producing 6 machines?
4 A beer company has divided Bloomington into two territories. If x1 dollars are spent on promotion in territory 1, then 6x1/2 1 cases of beer can be sold there; and if x2 dollars are spent on promotion in territory 2, then 4x1/2 2 cases of beer can be sold there. Each case of beer sold in
1 For any vector x, show that the vector x/ x has unit length.
Use the method of steepest ascent to approximate the solution to max z == -(x-3) - (x2-2) = f(x1, x2) s.t. (x1, x2) = R
Find all local maxima, local minima, and saddle points for = f(x1, x2) = x/x2 + x2x1 x1 x2. -
1 A company has n factories. Factory i is located at point(xi, yi), in the x–y plane. The company wants to locate a warehouse at a point (x, y) that minimizesin i1(distance from factory i to warehouse)2 Where should the warehouse be located?
2 A company can sell all it produces of a given output for$2/unit. The output is produced by combining two inputs. If q1 units of input 1 and q2 units of input 2 are used, then the company can produce q1/3 1 q2/3 2 units of the output. If it costs$1 to purchase a unit of input 1 and $1.50 to
3 (Collusive Duopoly Model) There are two firms producing widgets. It costs the first firm q1 dollars to produce q1 widgets and the second firm 0.5q2 2 dollars to produce q2 widgets. If a total of q widgets are produced, consumers will pay $200 q for each widget. If the two manufacturers want to
4 It costs a company $6/unit to produce a product. If it charges a price p and spends a dollars on advertising, it can sell 10,000p2a1/6 units of the product. Find the price and advertising level that will maximize the company’s profits.
5 A company manufactures two products. If it charges a price pi for product i, it can sell qi units of product i, where q1 60 3p1 p2 and q2 80 2p2 p1. It costs $25 to produce a unit of product 1 and $72 to produce a unit of product 2. How many units of each product should be produced to
6 Find all local maxima, local minima, and saddle points for f (x1, x2) x3 1 3x1x2 2 x4 2.
7 Find all local maxima, local minima, and saddle points for f (x1, x2) x1x2 x2x3 x1x3.
8 (Cournot Duopoly Model) Let’s reconsider Problem 3.The Cournot solution to this situation is obtained as follows:Firm i will produce qi, where if firm 1 changes its production level from q1 (and firm 2 still produces q2), then firm 1’s profit will decrease. Also, if firm 2 changes its
9 In the Bloomington Girls Club basketball league, the following games have been played: team A beat team B by 7 points, team C beat team A by 8 points, team B beat team C by 6 points, and team B beat team C by 9 points. Let A, B, and C represent “ratings” for each team in the sense that if,
5 We must invest all our money in two stocks: x and y.The variance of the annual return on one share of stock x is var x, and the variance of the annual return on one share of stock y is var y. Assume that the covariance between the annual return for one share of x and one share of y is cov(x, y).
I have $1,000 to invest in three stocks. Let Si be the random variable representing the annual return on $1 invested in stock i. Thus, if Si 0.12, $1 invested in stock i at the beginning of a year was worth $1.12 at the end of the year. We are given the following information:E(S1) = 0.14, E(S2)
7 For Example 31, explain why 1 10 and 2 0.(Hint: Think about the economic principle that for each product produced, marginal revenue must equal marginal cost.)
8 Use the K–T conditions to find the optimal solution to the following NLP: max z s.t. -x-x+4x1 + 6x2 6 x + x2 6 x1 3
9 Use the K–T conditions to find the optimal solution to the following NLP min s.t. zee-2x2 x + x2 1 x1, x20
10 Use the K–T conditions to find the optimal solution to the following NLP: min z = (x-3) + (x2-5) s.t. | x1 + x2 7 x1, x20
11 Solve Problem 7 of Section 11.2.
12 Solve Problem 8 of Section 11.2.
13 Solve Problem 11 of Section 11.2.
14 Solve Problem 15 of Section 11.2.
15 Solve Problem 16 of Section 11.2
16 We must determine the percentage of our money to be invested in stocks x and y. Let a percentage of money invested in x and b 1 a percentage of money invested in y. A choice of a and b is called a portfolio. A portfolio is efficient if there exists no other portfolio whose return has a
6 Use the K–T conditions to find the optimal solution to the following NLP: min z s.t. = (x11)+(22) -x1+x21 x1 + x2 2 x1, x20
5 A total of 160 hours of labor are available each week at$15/hour. Additional labor can be purchased at $25/hour.Capital can be purchased in unlimited quantities at a cost of$5/unit of capital. If K units of capital and L units of labor are available during a week, then L1/2K1/3 machines can be
6 As in Problem 5, assume that we must determine the percentage of our money that is invested in stocks x and y. A choice of a and b is called a portfolio. A portfolio is efficient if there exists no other portfolio whose return has a higher mean return and lower variance, or a higher mean return
7 Suppose product i (i = 1, 2) costs $ci per unit. If xi(i 1, 2) units of products 1 and 2 are purchased, then a utility many of each type should be purchased?b Show that an increase in the cost of product i decreases the number of units of product i that should be purchased.c Show that an
8 Suppose that a cylindrical soda can must have a volume of 26 cu in. If the soda company wants to minimize the surface area of the soda can, what should be the ratio of the height of the can to the radius of the can? (Hint: The volume of a right circular cylinder is pr2h, and the surface area of a
9 Show that if the right-hand side of the ith constraint is increased by a small amount bi (in either a maximization or minimization problem), then the optimal z-value for (11)will increase by approximately (.).
Describe the optimal solution to max f(x) s.t. axb
A monopolist can purchase up to 17.25 oz of a chemical for $10/oz. At a cost of $3/oz, the chemical can be processed into an ounce of product 1; or, at a cost of $5/oz, the chemical can be processed into an ounce of product 2. If x1 oz of product 1 are produced, it sells for a price of $30 - x1
Show that the Kuhn–Tucker conditions fail to hold at the optimal solution to the following NLP: max z=x1 s.t. x2-(1-x)0 x0,x20
1 A power company faces demands during both peak and off-peak times. If a price of p1 dollars per kilowatt-hour is charged during the peak time, customers will demand 60 0.5 p1 kwh of power. If a price of p2 dollars is charged during the off-peak time, then customers will demand 40 p2 kwh. The
2 Use the K–T conditions to find the optimal solution to the following NLP: max zx1 = x2 s.t. x + x 1
3 Consider the Giapetto problem of Section 3.1:Find the K–T conditions for this problem and discuss their relation to the dual of the Giapetto LP and the complementary slackness conditions for the LP. max z=3x+2x2 s.t. 2x + x2 100 x1 + x2 80 x1 40
4 If the feasible region for (26) is bounded and contains its boundary points, then it can be shown that (26) has an optimal solution. Suppose that the regularity conditions are valid but that the hypotheses of Theorems 11 and 11 are not valid. If we can prove that only one point satisfies the
14 Solve Review Problem 24 of Chapter 3 on LINDO and answer the following questions:a For which type of DRGs should the hospital seek to increase demand?b What resources are in excess supply? Which resources should the hospital expand?c What is the most the hospital should be willing to pay
12 Machinco produces four products, requiring time on two machines and two types (skilled and unskilled) of labor.The amount of machine time and labor (in hours) used by each product and the sales prices are given in Table 13.Each month, 700 hours are available on machine 1 and 500 hours on machine
11 Autoco has three assembly plants located in various parts of the country. The first plant (built in 1937 and located in Norwood, Ohio) requires 2 hours of labor and 1 hour of machine time to assemble one automobile. The second plant(built in 1958 and located in Bakersfield, California) requires
10 Use LINDO to solve the Sailco problem of Section 3.10, then use the output to answer the following questions:a If month 1 demand decreased to 35 sailboats, what would be the total cost of satisfying the demands during the next four months?b If the cost of producing a sailboat with regular-time
2 Vivian’s Gem Company produces two types of gems:Types 1 and 2. Each Type 1 gem contains 2 rubies and 4 diamonds. A Type 1 gem sells for $10 and costs $5 to produce. Each Type 2 gem contains 1 ruby and 1 diamond.A Type 2 gem sells for $6 and costs $4 to produce. A total of 30 rubies and 50
1 HAL produces two types of computers: PCs and VAXes.The computers are produced in two locations: New York and Los Angeles. New York can produce up to 800 computers and Los Angeles up to 1,000 computers. HAL can sell up to 900 PCs and 900 VAXes. The profit associated with each production site and
7 Consider the Sailco problem (Example 12 in Chapter 3).Suppose we want to consider how profit will be affected if we change the number of sailboats that can be produced each month with regular-time labor. How can we use the PARA command to answer this question? (Hint: Let c change in number of
6 For Example 2, suppose that we increase the cost of producing a type of car. Show that in the new optimal solution to the LP, the number of cars produced of that type cannot increase.
5 For Example 1, suppose that we increase the sales price of a product. Show that in the new optimal solution, the amount produced of that product cannot decrease.
4 For the Dorian Auto example (Example 2 in Chapter 3), let c1 be the objective function coefficient of x1. Determine the optimal z-value as a function of c1.
3 For the Giapetto example of Section 3.1, graph the optimal z-value as a function of x2’s objective function coefficient. Also graph the optimal z-value as a function of b1, b2, and b3.
2 Use the PARA command to graph the optimal z-value for Example 2 as a function of b1. Then answer the same questions for b2, b3, and b4, respectively.
In what follows, bi represents the right-hand side of an LP’s ith constraint.1 Use the LINDO PARA command to graph the optimal z-value for Example 1 as a function of b4.
11 In solving part (c) of Example 8, a manager reasons as follows: The average cost of producing a car is $11,600 up to 1,000 cars. Therefore, if a customer is willing to pay me$25,000 for a car, I should certainly fill his order. What is wrong with this reasoning?
10 In Problem 7 of Section 5.2, what is the most the company would be willing to pay for having one more worker at the beginning of month 1?
9 In Problem 7 of Section 5.2, suppose that a new customer wishes to buy a pair of shoes during month 1 for $70.Should Shoeco oblige him?
8 In Problem 6 of Section 5.2, what is the most that Steelco should be willing to pay for an extra hour of labor?
7 In Problem 6 of Section 5.2, what is the most that Steelco should be willing to pay for an extra ton of iron?

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