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operations research an introduction

Questions and Answers of Operations Research An Introduction

22 The national economy of Seriland manufactures three products: steel, cars, and machines. (1) To produce $1 of steel requires 30¢ of steel, 15¢ of cars, and 40¢ of machines.(2) To produce $1 of
5 Find the inverse of the matrix 23
6 The grades of two students during their last semester at S.U. are shown in Table 2.Courses 1 and 2 are four-credit courses, and courses 3 and 4 are three-credit courses. Let GPAi be the semester
7 Use the Gauss–Jordan method to find all solutions to the following linear system: 2x1 + x2 = 3 3x1 + x2 = 4 x1 x2 = 0
8 Find the inverse of the matrix [23] [3 5]
9 Let Ct = number of children in Indiana at the beginning of year t, and At = number of adults in Indiana at the beginning of year t. During any given year, 5% of all children become adults, and 1%
10 Use the Gauss–Jordan method to find all solutions to the following linear equation system: x1 - x3 = 4 2+x3 = 2 x1 + x2 25 = 5
11 Use the Gauss–Jordan method to find the inverse of the matrix [10 0 27 0 1 0 0 1 1
12 During any given year, 10% of all rural residents move to the city, and 20% of all city residents move to a rural area(all other people stay put!). Let Rt be the number of rural residents at the
13 Determine whether the set V = {[1 2 1],[2 0 0]} is a linearly independent set of vectors.
14 Determine whether the set V = {[1 0 0], [0 1 0],[-1 -1 0]} is a linearly independent set of vectors.
15 Leta For what values ofa, b,c, and d will A-1 exist?b If A-1 exists, then find it. a 00 0 0 0 b 00 A = 00 LO 0 0 0 d
16 Show that the following linear system has an infinite number of solutions: [1 1 0 0x 001 12 1 0 1 0 01 0x3 Lo 1 0 10 1x4 234
17 Before paying employee bonuses and state and federal taxes, a company earns profits of $60,000. The company pays employees a bonus equal to 5% of after-tax profits.State tax is 5% of profits
18 Find the determinant of the matrix [246] A = 1 0 0 LO O 0 1.
19 Show that any 2 * 2 matrix A that does not have an inverse will have det A= 0.
20 Let A be an m * m matrix.a Show that if rank A = m, then Ax = 0 has a unique solution. What is the unique solution?b Show that if rank A
21 Consider the following linear systemIf the sum of each row of the P matrix equals 1, then use Problem 20 to show that this linear system has an infinite number of solutions. [x1 x2 x] = [x1 x2
Use the Gauss–Jordan method to determine all solutions to the following linear system 2x + 3x2 = 3 x1 + x2 = 1 x1 + 2x2 = 2
3 Each year, 20% of all untenured State University faculty become tenured, 5% quit, and 75% remain untenured. Each year, 90% of all tenured S.U. faculty remain tenured and 10% quit. Let Ut be the
2 Find the inverse of the matrix 2 1
6 Use the answer to Problem 2 to solve the following linear system: x + x3 = 4 4x1+x22x3 = 0 3x1 + x2 x3 = 2
5 Use the answer to Problem 1 to solve the following linear system:x1 + 3x2 = 4 2x1 + 5x2 = 7
Find A-1 (if it exists) for the following matrice 4 1 2 224 2 1 20 4 1
Find A-1 (if it exists) for the following matrice 3 0 1 1 1 12 21
Find A-1 (if it exists) for the following matrice 10 2 4 4 3 1-2 1
Find A-1 (if it exists) for the following matrice 1 3 25
9 Show that a set of vectors V (not containing the 0 vector)is linearly dependent if and only if there exists some vector in V that can be written as a nontrivial linear combination of other vectors
8 Suppose there is a collection of three or more twodimensional vectors. Provide an argument showing that the collection must be linearly dependent.
7 Show that a square matrix has an inverse if and only if its rows form a linearly independent set of vectors.
8 Consider a square matrix B whose inverse is given by B-1.a In terms of B-1, what is the inverse of the matrix 100B?b Let B be the matrix obtained from B by doubling every entry in row 1 of B.
1 Find all solutions to the following linear system: x1 + x2 = 2 x2 + x3 = 3 X3 35 x1 + 2x2 + x3 = 5
4 a Show that for any 1* 1 and 3 *3 matrix, det -A = det A.b Show that for any 2 * 2 and 4 * 4 matrix, det -A = det A.c Generalize the results of parts (a) and (b).
3 A matrix is said to be upper triangular if for i > j, aij =0. Show that the determinant of any upper triangular 3 3 matrix is equal to the product of the matrix’s diagonal elements. (This
2 Find det 0 0 0 0 200 003 0 0 0 0 5
1 Verify that det by using expansions by row 2 and row 3 cofactors. 1 2 23] [1] 4 5 6 = 0 7 8 9
11 A square matrix A is orthogonal if AAT = I. What properties must be possessed by the columns of an orthogonal matrix?
10 Suppose A has an inverse. Show that (AT)-1 = (A-1)T.(Hint: Use the fact that AA-1 = I, and take the transpose of both sides.)
9 Suppose that A and B both have inverses. Find the inverse of the matrix AB.
(b) Find the probability that x is an even value. FIGURE 12.1 CDF and pdf for rolling a fair die - 56463/6 2/6 6 0 CDF, P(x) 1 2 3 4 5 pdf, p(x)
1. The joint pdf of Xl and X2, p(xr. Xl)' is*(a) Find the marginal pdfs Pl(Xl) and pz(xz).*(b) Are Xl and Xz independent?(c) Compute E{XI + xd.(d) Compute cov{Xl' xz}.(e) Compute var{5xl - 6xz}. x =
2. Consider the following function: k f(x): = ,10 x 20 *(a) Find the value of the constant k that will make f(x) a pdf. (b) Determine the CDF, and find the probability that x is (i) larger than 12,
3. Consider the infinite-horizon inventory situation with zero delivery lag and backlogged demand.Develop the optimal inventory policy based on the minimization of cost given that Holding cost for z
*2. The pdf of the demand per period in an infinite-horizon inventory model is given as feD) = .08D,O :5 D:5 5 The unit cost parameters are Unit selling price = $10 Unit purchase price = $8 Unit
1. Consider a two-period probabilistic inventory model in which the demand is backlogged, and orders are received with zero delivery lag. The demand pdf per period is uniform between 0 and 10, and
3. Work Problem 5, Set 14.2a, assuming that there is a fixed cost of $10 associated with the delivery of donuts.
2. In the single-period model in Section 14.2.1, suppose instead that the model maximizes profit and that a setup cost K is incurred. Given that r is the unit selling price and using the information
*1. Determine the optimal inventory policy for the situation in Example 14.2-2, assuming that the setup cost is $5.
8. Solve Example 14.2-1, assuming that the demand is continuous and uniform during the period and that the pdf of demand is uniform between 0 and 100. (Hint: Use the results of Problem 7.)
7. For the single-period model, suppose that the item is consumed uniformly during the period(rather than instantaneously at the start of the period). Develop the associated cost model, and find the
*6. Colony Shop is stocking heavy coats for next winter. Colony pays $50 for a coat and sells it for $110. At the end of the winter season, Colony offers the coats at $55 each. The demand for coats
5. QuickStop provides its customers with coffee and donuts at 6:00 A.M. each day. The convenience store buys the donuts for 7 cents apiece and sells them for 25 cents apiece until 8:00 A.M. After
4. The U ofA Bookstore offers a program of reproducing class notes for participating professors.Professor Yataha teaches a freshmen-level class, where an enrollment of between 200 and 250 students,
*3. The unit holding cost in a single-period inventory situation is $1. If the order quantity is 4 units, find the permissible range of the unit penalty cost implied by the optimal conditions.Assume
2. The demand for an item during a single period occurs instantaneously at the start of the period. The associated pdf is uniform between 10 and 15 units. Because of the difficulty in estimating the
1. For the single-period model, show that for the discrete demand the optimal order quantity is determined from P{D :$ y* - I} :$ _P- :$ P{D :$ y*}p+h
4. Find the optimal solution for Example 14.1-2, assuming that the demand during lead time is N(lOO, 2). Assume that D = 10,000 gallons per month, h = $2 per gallon per month, p = $4 per gallon, and
*3. In Example 14.1-2, suppose that the demand during lead time is uniform between 40 and 60 gallons. Compare the solution with that obtained in Example 14.1-2, and interpret the results. (Hint: In
*2. Solve Example 14.1-2, assuming that the demand during lead time is unifonn between 0 and 50 gallons.
1. For the data given in Example 14.1-2, determine the following:(a) The approximate number of orders per month.(b) The expected monthly setup cost.(c) The expected holding cost per month.(d) The
(b) Given the data of the situation, recommend an inventory policy for the shop, assuming that the probability of running out of films during the lead time does not exceed .10.
3. The daily demand for camera films at a gift shop in a resort area is normally distributed with mean 300 rolls and standard deviation 5 rolls. The cost of holding a roll in the shop is$.02. A fixed
2. A music store sells a best-selling compact disc. The daily demand (in number of units) for the disc is approximately normally distributed with mean 200 discs and standard deviation 20 discs. The
1. In Example 14.1-1, determine the optimal inventory policy for each of the following cases:*(a) Lead time = 15 days.(b) Lead time = 23 days.(c) Lead time = 8 days.(d) Lead time = 10 days.
4. In the two-player, two-finger Morra game, each player shows one or two fingers, and simultaneously guesses the number of fingers the opponent will show. The player making the correct guess wins an
*(a) Formulate the problem as a two-person zero-sum game, and solve by linear programming.(b) Which army will win the battle?
3. Colonel Blotto's army is fighting for the control of two strategic locations. Blotto has two regiments and the enemy has three. A location will fall to the army that attacks with more regiments.
(b) Determine the optimal strategy and the value of the game.
*(a) Set up the problem as a two-person zero-sum game.
1. On a picnic outing, 2 two-person teams are playing hide-and-seek. There are four hiding locations (A, B, C, and D), and the two members of the hiding team can hide separately in any two of the
*2. Robin, who travels frequently between two cities, has two route options: Route A is a fast four-lane highway, and route B is a long winding road. The highway patrol has a limited police force. If
1. Solve the coin-tossing game of Example 13.4-2 graphically.
3. 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
(b) Suppose that Hank is more interested in the letter grade he will get. The dividing scores for the passing letter grades A to Dare 90,80,70, and 60, respectively. Would this attitude toward grades
(d) Repeat part (c) for investor B.
(c) Suppose that investor A has the chance to invest in one of two ventures, I or II. Venture I can produce a net return of $3000 with probability .4 or a net loss of $1000 with probability.6.
3. An investment of $10,000 in a high-risk venture has a 50-50 chance over the next year of increasing to $14,000 or decreasing to $8000. Thus the net return can be either $4000 or -$2000.(a)
*2. The Golden family have just moved to a location where earthquakes are known to occur.TIley must decide whether they should build their house according to the high-standard earthquake code. TIle
(c) Based on the utility function you developed in (b), would you be tempted to participate in the poker game?
(b) Based on your ardent desire to see the game, translate the actual money into a utility function.
*1. You are a student at the University of Arkansas and desperately want to attend the next Razorbacks basketball game. The problem is that the admission ticket costs $10, and you have only $5. You
*(b) Suppose that the manufacturer ships batches to two customers, A and B. The contracts specify that the defectives for A and B should not exceed 5% and 8%, respectively.A penalty of $100 is
7. Historical data at Acme Manufacturing estimate a 5% chance that a manufactured batch of widgets will be unacceptable (bad). A bad batch has 15% defective items, and a good batch includes only 4%
(b) Determine the best course of action for AFe.
(b) Suppose that you contract a literary agent to conduct a survey concerning the potential success of the novel. From past experience, the agent advises you that when a novel is successful, the
*4. You are the author of what promises to be a successful novel. You have the option to either publish the novel yourself or through a publisher. 'The publisher is offering you $20,000 for signing
3. In Example 13.2-2, suppose that you have the additional option of investing the original$10,000 in a safe certificate of deposit that yields 8% interest. Your friend's advice applies to investing
*2. Elektra receives 75% of its electronic components from vendor A and the remaining 25%from vendor B.lhe percentage of defectives from vendors A and Bare 1% and 2%, respectively.When a random
1. Data in a community college show that 75% of new students who took calculus in high school do well, compared with 50% of those who did not take calculus. Admissions for the current academic year
*19. Aspiration Level Criterion. Acme Manufacturing uses an industrial chemical in one of its processes. The shelf life of the chemical is 1 month, following which any amount left is destroyed.The
(b) The manufacturer can invest $1000 to obtain additional information about whether or not the price will increase. This information says that there is a 58% chance that the probability of price
18. (Rappaport, 1967) A manufacturer has used linear programming to determine the optimum production mix of the variousTV models it produces. Recent information received by the manufacturer indicates
(b) Study the sensitivity of the solution to changes in the probability of good weather.
17. (Cohan and Associates, 1984) Modern forest management uses controlled fires to reduce fire hazards and to stimulate new forest growth. Management has the option to postpone or plan a burning. In
16. The outer diameter,d, of a cylinder is processed on an automatic machine with upper and lower tolerance limits of d + tv and d - tL . The production process follows a normal distribution with
(b) Determine the value of a that maximizes the expected profit.
*15. An automatic machine produces a (thousands of) units of a product per day. As a increases, the proportion of defectives,p, goes up according to the following probability density function Each
14. In Problem 13, suppose that the store wishes to extend the decision problem to a 2-day horizon. The alternatives for the second day depend on the demand in the first day. If demand on day 1
*12. Sunray Electric Coop uses a fleet of 20 trucks to service its electric network. The company wants to develop a schedule of periodic preventive maintenance for the fleet. The probability of a
10. Rework Problem 9, assuming that the annual interest rate is 10% and that the decisions are made considering the time value of money. [Note: You need compound interest tables to solve this

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