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operations research an introduction

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

15 A function f (x1, x2, . . . , xn) is quasi-concave on a convex set S Rn if x S, x S, and 0 c 1 implies f [cx (1 c)x] min[ f (x ), f (x)]Show that if f is concave on R1, then f is quasi-concave.Which of the functions in Figure 19 is quasi-concave? Is a quasi-concave function
On the given set S, determine whether each function is convex, concave, or neither 5 f(x) = In x; S = (0, )
On the given set S, determine whether each function is convex, concave, or neither 6 f(x1, x2) = x + 3xx2 + x; S = R
On the given set S, determine whether each function is convex, concave, or neither 7 f(x1, x2) = x + x; S = R
On the given set S, determine whether each function is convex, concave, or neither 8 f(x1, x2) = x - x1x2 - 2x; S = R -x-xx-2x;
On the given set S, determine whether each function is convex, concave, or neither 9 f(x1, x2, x3) = x - x - 2x + .5x1x2; S = R
10 For what values ofa, b, and c will be a convex function on R2? A concave function on R2? ax + bx1x2 + cx
11 Prove Theorem 1 .
12 Show that if f (x1, x2, . . . , xn) and g(x1, x2, . . . , xn) are convex functions on a convex set S, then h(x1, x2, . . . , xn) f (x1, x2, . . . , xn) g(x1, x2, . . . , xn) is a convex function on S.
13 If f(x1, x2, . . . , xn) is a convex function on a convex set S, show that for c 0, g(x, x2, . . . , xn) cf(x1, x2, . . . , xn)is a convex function on S, and for c 0, g(x1, x2, . . . , xn) cf(x1, x2, . . . , xn) is a concave function on S.
14 Show that if y f (x) is a concave function on R1, then z f (1 x) is a convex function [assume that f (x) 0].
The drug taxoprol costs $60 to produce. Currently, the exchange rate is .667 $/mark, and we are charging 150 marks for taxoprol. Current demand for taxoprol is 100 units, and it is estimated that the elasticity for taxoprol is 2.5. Assuming a linear demand curve, determine how the price (in marks)
A candy bar costs 55 cents to produce. We are considering charging a price of between$1.10 and $1.50 for this candy bar. For a price of $1.10, $1.30, and $1.50, the marketing department estimates the demand for the candy bar in the three regions where the candy bar will be sold (see Table 10). What
13 It costs a company c(x) dollars to produce x units. The curve y c (x) is called the firm’s marginal cost curve.(Why?) The firm’s average cost curve is given by z c(x x).Let x* be the production level that minimizes the company’s average cost. Give conditions under which the marginal
14 When a machine is t years old, it earns revenue at a rate of et dollars per year. After t years of use, the machine can be sold for t 11 dollars.a When should the machine be sold to maximize total revenue?b If revenue is discounted continuously (so that $1 of revenue received t years from now
15 Suppose a company must service customers lying in an area of A sq mi with n warehouses. Kolesar and Blum have shown that the average distance between a warehouse and a customer isA nAssume that it costs the company $60,000 per year to maintain a warehouse and $400,000 to build a
16 Prove Theorem 4.
17 Prove Theorem 5
Use Golden Section Search to findwith the final interval of uncertainty having a length less than 1/4 max -x21 s.t. -1 x 0.75
1 Use Golden Section Search to determine (within an interval of 0.8) the optimal solution to max x + 2x s.t. -3 x 5
2 Use Golden Section Search to determine (within an interval of 0.6) the optimal solution to max x e s.t. -1 x 3
3 Consider a line segment [0, 1] that is divided into two parts (Figure 39). The line segment is said to be divided into the Golden Section ifShow that for the line segment to be divided into the Golden Section, Length of whole line Length of larger part of line length of larger part of line
4 Hughesco is interested in determining how cutting fluid jet pressure (p) affects the useful life of a machine tool (t), using the data in Table 11. Pressure p is constrained to be between 0 and 600 pounds per square inch (psi). Use Golden Section Search to estimate (within 50 units) the value of
12 You are the publisher of a new magazine. The variable cost of printing and distributing each weekly copy of the magazine is $0.25. You are thinking of charging between$0.50 and $1.30 per week for the magazine. The estimated numbers of subscribers (in millions) for weekly prices of$0.50, $0.80,
11 It costs $250 to produce an X-Box. We are trying to determine the selling price for the X-Box. Prices between$200 and $400 are under consideration, with demand for prices of $200, $250, $350, and $400 given below. Suppose MSFT earns $10 in profit for each game that an X-Box owner purchases.
1 It costs a company $100 in variable costs to produce an air conditioner, plus a fixed cost of $5,000 if any air conditioners are produced. If the company spends x dollars on advertising, then it can sell x1/2 air conditioners at $300 each. How can the company maximize its profit? If the fixed
2 If a monopolist produces q units, she can charge 100 4q dollars/unit. The fixed cost of production is $50, and the variable per-unit cost is $2. How can the monopolist maximize profits? If a sales tax of $2/unit must be paid by the monopolist, then would she increase or decrease production?
3 Show that for allmin f (x)s.t. x R occurs for x 0.] x, ex+1. [Hint: Let f(x) = e* - x-1. Show that
4 Suppose that in n “at bats,” a baseball player gets x hits.Suppose we want to estimate the player’s probability (p) of getting a hit on each “at bat.” The method of maximum likelihood estimates p by ˆp, where ˆp maximizes the probability of observing x hits in n “at bats.” Show
5 Find the optimal solution to maxx s.t. -1 x 1
6 Find the optimal solution to min x-3x+2x-1 s.t. -2 x 4
7 During the Reagan administration, economist Arthur Laffer became famous for his Laffer curve, which implied that an increase in the tax rate might decrease tax revenues, while a decrease in the tax rate might increase tax revenues.This problem illustrates the idea behind the Laffer curve.Suppose
8 The cost per day of running a hospital is 200,000.002x2 dollars, where x patients served per day. What size hospital minimizes the per-patient cost of running the hospital?
9 Each morning during rush hour, 10,000 people want to travel from New Jersey to New York City. If a person takes the subway, the trip lasts 40 minutes. If x thousand people per morning drive to New York, it takes 20 5x minutes to make the trip. This problem illustrates a basic fact of life: If
10 Currently, the exchange rate is 100 yen per dollar. In Japan, we sell a product that costs $5 to produce for 700 yen. The product has an elasticity of 3. For exchange rates varying from 70 to 130 yen per dollar, determine the optimal product price in Japan and the profit in dollars. Assume a
A monopolist producing a single product has two types of customers. If q1 units are produced for customer 1, then customer 1 is willing to pay a price of 70 - 4q1 dollars. If q2units are produced for customer 2, then customer 2 is willing to pay a price of 150 15q2 dollars. For q > 0, the cost of
On the given set S, determine whether each function is convex, concave, or neither 3 f(x) = ;S = (0, )
1 Show that = x-2 lim x-2x=2 = 2(2) = 0. -
7 Suppose that if x dollars are spent on advertising during a given year, k(1 ecx) customers will purchase a product(c 0).a As x grows large, the number of customers purchasing the product approaches a limit. Find this limit.b Can you give an interpretation for k?c Show that the sales response
8 Let the total cost of producing x units, c(x), be given by c(x) = kx1-b (0 < b
9 If a company has m hours of machine time and w hours of labor, it can produce 3m1/3w2/3 units of a product.Currently, the company has 216 hours of machine time and 1,000 hours of labor. An extra hour of machine time costs$100, and an extra hour of labor costs $50. If the company has $100 to
It costs a company c dollars per unit to manufacture a product. If the company charges p dollars per unit for the product, customers demand D( p) units. To maximize profits, what price should the firm charge?
If K units of capital and L units of labor are used, a company can produce KL units of a manufactured good. Capital can be purchased at $4/unit and labor can be purchased at$1/unit. A total of $8 is available to purchase capital and labor. How can the firm maximize the quantity of the good that can
Oilco produces three types of gasoline: regular, unleaded, and premium. All three are produced by combining lead and crude oil brought in from Alaska and Texas. The required sulphur content, octane levels, minimum daily demand (in gallons), and sales price per gallon of each type of gasoline are
Truckco is trying to determine where it should locate a single warehouse. The positions in the x–y plane (in miles) of four customers and the number of shipments made annually to each customer are given in Table 5. Truckco wants to locate the warehouse to minimize the total distance trucks must
Firerock produces rubber used for tires by combining three ingredients: rubber, oil, and carbon black. The cost in cents per pound of each ingredient is given in Table 6.The rubber used in automobile tires must have a hardness of between 25 and 35, an elasticity of at least 16, and a tensile
1 Q & H Company advertises on soap operas and football games. Each soap opera ad costs $50,000, and each football game ad costs $100,000. Giving all figures in millions of viewers, if S soap opera ads are bought, they will be seen by 5S men and 20S women. If F football ads are bought, they will
2 The area of a triangle with sides of lengtha, b, and c iss(s - a)(s -b)(s -c), where s is half the perimeter of the triangle. We have 60 ft of fence and want to fence a triangular-shaped area. Formulate an NLP that will enable us to maximize the fenced area.
6 Let q=f ( p) be the demand for a product when the price is p. For a given price p, the price elasticity E of the product is defined byIf the change in price (p) is small, this formula reduces toa Would you expect f ( p) to be positive or negative?b Show that if E 1, a small decrease in price will
5 Find the second-order Taylor series expansion of ln x about x = 1.
2 Show that lim does not exist. x 0x
Bakeco orders sugar from Sugarco. The per-pound purchase price of the sugar depends on the size of the order (see Table 1). Let x = number of pounds of sugar purchased by Bakeco f (x) = cost of ordering x pounds of sugar ThenFor all values of x, determine if x is continuous or discontinuous.
If a company charges a price p for a product, then it can sell 3e-p thousand units of the product. Then, f (p) = 3,000pep is the company’s revenue if it charges a price p.1 For what values of p is f (p) decreasing? For what values of p is f (p) increasing?2 Suppose the current price is $4 and
Find the first-order Taylor series expansion of e-x about x=0.
The demand f (p,a) =30,000p-2a1/6 for a product depends on p=product price (in dollars)and a = dollars spent advertising the product. Is demand an increasing or decreasing function of price? Is demand an increasing or decreasing function of advertising expenditure?If p = 10 and a = 1,000,000,
For f ( p,a) = 30,000p-2a1/6, find all second-order partial derivatives.
1 Find limo 3h+ h h
2 It costs Sugarco 25¢/lb to purchase the first 100 lb of sugar, 20¢/lb to purchase the next 100 lb, and 15¢ to buy each additional pound. Let f (x) be the cost of purchasing x pounds of sugar. Is f (x) continuous at all points? Are there any points where f (x) has no derivative?
3 Find f (x) for each of the following functions a xe b x+1 cet C d (3x+2)-2 e Inx
4 Find all first- and second-order partial derivatives for f(x1, x2) = xe.
3 The energy used in compressing a gas (in three stages)from an initial pressure I to a final pressure F is given byFormulate an NLP whose solution describes how to minimize the energy used in compressing the gas. F P2 K + + -31 P VP2
4 Use LINGO to solve Problem 1.
A linear function of the form f (x) = ax+b is both a convex and a concave function.This follows fromBoth (3) and (4) hold with equality, so f (x) = ax+b is both a convex and a concave function. f[cx' (1c)x"] = a[cx' + (1 - c)x"] + b = c(ax' + b) + (1 - c)(ax" + b) = cf(x)+(1- cf(x) + (1 c)f(x")
1 Show that f (x) = x2 is a convex function on S = R1.
2 Show that f (x) = ex is a convex function on S = R1.
3 Show that f (x) = x1/2 is a concave function on S = (0, ∞).
4 Show that f (x) = ax+ b is both a convex and a concave function on S = R1.
Show that f (x1, x2) = is a convex function on S=R2. x+2x+x+x
Show that is a concave function on R2. f(x1, x2) = x x1x22x1 -x
Show that for is not a convex or a concave function. S = R, f(x1, x2) = x - 3x1x2 + 2x
Show that for is a convex function. S = R, f(x1, x2, x3) = x + x + 2x-xx-xx X1X3
On the given set S, determine whether each function is convex, concave, or neither 1 f(x) = x; S = [0, )
Because the line segment AB lies below y f (x) and the line segment BC lies above y f (x), f (x) as pictured in Figure 18 is not a convex or a concave function. y B y=flx)
It can be shown (see Problem 12 at the end of this section) that the sum of two convex functions is convex and the sum of two concave functions is concave. Thus, f (x) = x2+ex is a convex function.
5 Use LINGO to solve Problem 2.
6 Use LINGO to solve Problem 3. Use I 64 and F 1,000.
7 For Example 6 of Chapter 8, let A number of days duration of A is reduced, B number of days duration of B is reduced, and so on. Suppose that the cost of crashing each activity is as follows:A, 5A2; B, 20B2; C, 2C2; D, 20D2; E, 10E2; F, 15F2 and that each activity may be “crashed” to a
8 Beerco has $100,000 to spend on advertising in four markets. The sales revenue (in thousands of dollars) that can be created in each market by spending xi thousand dollars in market i is given in Table 7. To maximize sales revenue, how much money should be spent in each market?
9 Widgetco produces widgets at plant 1 and plant 2. It costs 20x1/2 to produce x units at plant 1 and 40x1/3 to produce x units at plant 2. Each plant can produce as many as 70 units. Each unit produced can be sold for $10. At most, 120 widgets can be sold. Formulate an NLP whose solution will tell
10 Three cities are located at the vertices of an equilateral triangle. An airport is to be built at a location that minimizes the total distance from the airport to the three cities.Formulate an NLP whose solution will tell us where to build the airport. Then solve your NLP on LINGO.
11 The yield of a chemical process depends on the length of time T (in minutes) that the process is run and the temperature TEMP (in degrees centigrade) at which the process is operated.This dependence is described by the equationT must be between 60 and 120 minutes, while TEMP must be between 100
12 Consider Problem 5 of Section 3.8 with the following modification: Suppose that we can add a chemical called Superquality (SQ) to improve the quality level of gasoline and heating oil. If we add an amount x of SQ to each barrel of gasoline we improve its quality level by x.5 over what it would
13 A salesperson for Fuller Brush has three options: quit, put forth a low-effort level, or put forth a high-effort level.Suppose for simplicity that each salesperson will either sell$0, $5,000, or $50,000 worth of brushes. The probability of each sales amount depends on the effort level in the
For x >= 0, f (x)= x2 and f (x) = ex are convex functions and f (x) = x1/2 is a concave function.These facts are evident from Figure 17. f(x) (x) = x a Convex f(x) Concave f(x) = xin b Convex f(x) f(x) =
On the given set S, determine whether each function is convex, concave, or neither 2 f(x) = x; S R =
2 Show that the objective function for Example 35 is convex [it can be shown that the variance of any portfolio is a convex function of (x1, x2, . . . , xn)].
3 This problem will give you an idea why the restricted entry rule is unnecessary when (for a maximization problem)each fj(xj) is concave and each gij(xj) is convex. Consider the Oilco example. When we solve the approximating problem by the simplex, show that a solution that violates the adjacency
4 Suppose an NLP appears to be separable except for the fact that a term of the form xixj appears in the objective function or constraints. Show that an NLP of this type can be made into a separable programming problem by defining two new variables yi and yj by xi 1 2(yi yj) and xj 1 2(yi yj).
Perform two iterations of the feasible directions method on the following NLP:Begin at the point (0,0). max z = s.t. = f(x, y) = 2xy + 4x + 6y 2x - 212 x + y 2 x, y 0
Perform two iterations of the method of feasible directions for each of the following NLPs. 1 max z = s.t. =4x+6y - 2x-2xy - 2y x + 2y 2 Begin at the point ( x, y0
Perform two iterations of the method of feasible directions for each of the following NLPs. 2 max z s.t. Begin at the point (1, 0). = 3xy-x-y2 3x + y 4 x, y 0
Chemco is considering producing three products. The per-unit contribution to profit, labor requirements, raw material used per unit produced, and pollution produced per unit of product are given in Table 18. Currently, 1,300 labor hours and 1,000 units of raw material are available. Chemco’s two
Proctor and Ramble places ads on football games and soap operas. If F one-minute ads are placed on football games and S one-minute ads are placed on soap operas, then the number of men and women reached (in millions) and the cost (in thousands) of the ads are given in Table 19. P & R has a $1
1 Widgetco produces two types of widgets. Each widget is made of steel and aluminum and is assembled with skilled labor. The resources used and the per-unit profit contribution(ignoring cost of overtime labor purchased) for each type of widget are given in Table 20. Currently, 200 units of steel
2 Plantco produces three products. Three workers work for Plantco, and the company must determine which product(s) each worker should produce. The number of units each worker would produce if he or she spent the whole day producing each type of product are given in Table 21.The company is also
3 If a company spends $a on advertising and charges aprice of $p per unit, then it sells 1,000 100p 20a1/2 units of the product. The per-unit cost of producing the product is $6. Construct a trade-off curve between the objectives of profit and units sold
Set up an approximating problem for the following separable programming problems = max zx5x1 + x2-5x2-x3 s.t. x1+x+x34 x-x2 3 x1, x2, x30
Set up an approximating problem for the following separable programming problems min z = x + x s.t. x + 2x 4 x + x 6 X1 X20
3 In Figure 45, interpret the entries in the PRICE column for rows 2 and 3.
4 Fruit Computer Company produces Pear and Apricot computers. If the company charges a price p1 for Pear computers and p2 for Apricot computers, it can sell q1 Pear and q2 Apricot computers, where q1 4,000 10p1 p2, and q2 2,000 9p2 0.8p1. Manufacturing a Pear computer requires 2 hours of

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