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Supplement A Decision Making Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 01 Decision Making Tools • Break-even analysis – Analysis to compare processes by finding the volume at which two processes have equal total costs. • Preference matrix – Table that allows managers to rate alternatives based on several performance criteria. • Decision theory – Approach when outcomes associated with alternatives are in doubt. • Decision Tree – Model to compare alternatives and their possible consequences. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 02 Break-even analysis notation • Variable cost (c)– The portion of the total cost that varies directly with volume of output. • Fixed cost (F) – – The portion of the total cost that remains constant regardless of changes in levels of output. • Quantity (Q) – – The number of customers served or units produced per year. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 03 Break-Even Analysis Total cost = F + cQ Total revenue = pQ By setting revenue equal to total cost pQ = F + cQ F Q= p-c Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 04 Example A.1 A hospital is considering a new procedure to be offered at $200 per patient. The fixed cost per year would be $100,000 with total variable costs of $100 per patient. What is the break-even quantity for this service? Use both algebraic and graphic approaches to get the answer. The formula for the break-even quantity yields 100,000 F = Q= = 1,000 patients p-c 200 – 100 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 05 Example A.1 The following table shows the results for Q = 0 and Q = 2,000 Quantity (patients) (Q) Total Annual Cost ($) (100,000 + 100Q) Total Annual Revenue ($) (200Q) 0 100,000 0 2,000 300,000 400,000 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 06 Example A.1 Dollars (in thousands) 400 – 300 – (2000, 400) Profits Total annual revenues (2000, 300) Total annual costs Break-even quantity 200 – 100 – Loss 0– Fixed costs | 500 | | 1000 1500 Patients (Q) | 2000 The two lines intersect at 1,000 patients, the break-even quantity Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 07 Application A.1 The Denver Zoo must decide whether to move twin polar bears to Sea World or build a special exhibit for them and the zoo. The expected increase in attendance is 200,000 patrons. The data are: Revenues per Patron for Exhibit Gate receipts $4 Concessions $5 Licensed apparel $15 Estimated Fixed Costs Exhibit construction Salaries Food $2,400,000 $220,000 $30,000 Estimated Variable Costs per Person Concessions $2 Licensed apparel $9 Is the predicted increase in attendance sufficient to break even? Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A -08 Application A.1 TR = pQ $0 $6,000,000 TC = F + cQ $2,650,000 $5,400,000 7– Where p = 4 + 5 + 15 = $24 F = 2,400,000 + 220,000 + 30,000 = $2,650,000 c = 2 + 9 = $11 6– Cost and revenue (millions of dollars) Q 0 250,000 5– 4– 3– 2– 1– 0 |– | | | | | 50 100 150 200 250 Q (thousands of patrons) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 09 Application A.1 Q TR = pQ TC = F + cQ 0 250,000 $0 $6,000,000 $2,650,000 $5,400,000 Where p = 4 + 5 + 15 = $24 F = 2,400,000 + 220,000 + 30,000 = $2,650,000 c = 2 + 9 = $11 Algebraic solution of Denver Zoo problem pQ = F + cQ 24Q = 2,650,000 + 11Q 13Q = 2,650,000 Q = 203,846 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 10 Example A.2 If the most pessimistic sales forecast for the proposed service from Example 1 was 1,500 patients, what would be the procedure’s total contribution to profit and overhead per year? pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)] = $50,000 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 11 Make-or-buy decision notation • Fb – The fixed cost (per year) of the buy option • Fm – The fixed cost of the make option • cb – The variable cost (per unit) of the buy option • cm – The variable cost of the make option Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 12 Make-or-buy decision • Total cost to buy Fb + cbQ • Total cost to make Fm + cmQ Fb + cbQ = Fm + cmQ Fm – Fb Q= c –c b m Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 13 Example A.3 • A fast-food restaurant featuring hamburgers is adding salads to the menu • The price to the customer will be the same • Fixed costs are estimated at $12,000 and variable costs totaling $1.50 per salad • Preassembled salads could be purchased from a local supplier at $2.00 per salad • Preassembled salads would require additional refrigeration with an annual fixed cost of $2,400 • Expected demand is 25,000 salads per year • What is the break-even quantity? Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 14 Example A.3 The formula for the break-even quantity yields the following: Fm – Fb Q= c –c b m = 12,000 – 2,400 = 19,200 salads 2.0 – 1.5 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 15 Application A.2 • At what volume should the Denver Zoo be indifferent between buying special sweatshirts from a supplier or have zoo employees make them? Fixed costs Buy $0 Make $300,000 Variable costs $9 $7 Fm – Fb Q= c –c b m 300,000 – 0 Q= 9–7 Q = 150,000 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 16 Preference Matrix • A Preference Matrix is a table that allows you to rate an alternative according to several performance criteria. – The criteria can be scored on any scale as long as the same scale is applied to all the alternatives being compared. • Each score is weighted according to its perceived importance, with the total weights typically equaling 100. – The total score is the sum of the weighted scores (weight × score) for all the criteria and compared against scores for alternatives. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 17 Example A.4 The following table shows the performance criteria, weights, and scores (1 = worst, 10 = best) for a new thermal storage air conditioner. If management wants to introduce just one new product and the highest total score of any of the other product ideas is 800, should the firm pursue making the air conditioner? Performance Criterion Weight (A) Score (B) Weighted Score (A B) Market potential 30 8 240 Unit profit margin 20 10 200 Operations compatibility 20 6 120 Competitive advantage 15 10 150 Investment requirements 10 2 20 5 4 20 Project risk Weighted score = Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 750 A - 18 Example A.4 Because the sum of the weighted scores is 750, it falls short of the score of 800 for another product. This result is confirmed by the output from OM Explorer’s Preference Matrix Solver below Total Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 750 A - 19 Application A.3 The following table shows the performance criteria, weights, and scores (1 = worst, 10 = best) for a new thermal storage air conditioner. If management wants to introduce just one new product and the highest total score of any of the other product ideas is 800, should the firm pursue making the air conditioner? Performance Criterion Weight (A) Score (B) Weighted Score (A B) Market potential 10 5 50 Unit profit margin 30 8 240 Operations compatibility 20 10 Competitive advantage 25 7 Investment requirements 10 3 30 5 4 20 Project risk No. Because 715 >800 Weighted score = Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 200 175 715 A - 20 Decision Theory Steps • List a reasonable number of feasible alternatives • List the events (states of nature) • Calculate the payoff table showing the payoff for each alternative in each event • Estimate the probability of occurrence for each event • Select the decision rule to evaluate the alternatives Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 21 Example A.5 • A manager is deciding whether to build a small or a large facility • Much depends on the future demand • Demand may be small or large • Payoffs for each alternative are known with certainty • What is the best choice if future demand will be low? Alternative Small facility Large facility Do nothing Possible Future Demand Low High 200 270 160 800 0 0 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 22 Example A.5 • The best choice is the one with the highest payoff • For low future demand, the company should build a small facility and enjoy a payoff of $200,000 • Under these conditions, the larger facility has a payoff of only $160,000 Alternative Small facility Large facility Do nothing Possible Future Demand Low High 200 270 160 800 0 0 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 23 Decision Making under Uncertainty • Maximin • Maximax • Laplace • Minimax Regret Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 24 Example A.6 Reconsider the payoff matrix in Example 5. What is the best alternative for each decision rule? a. Maximin. An alternative’s worst payoff is the lowest number in its row of the payoff matrix, because the payoffs are profits. The worst payoffs ($000) are Alternative Small facility Large facility Worst Payoff 200 160 The best of these worst numbers is $200,000, so the pessimist would build a small facility. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 25 Example A.6 b. Maximax. An alternative’s best payoff ($000) is the highest number in its row of the payoff matrix, or Alternative Best Payoff Small facility 270 Large facility 800 The best of these best numbers is $800,000, so the optimist would build a large facility. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 26 Example A.6 c. Laplace. With two events, we assign each a probability of 0.5. Thus, the weighted payoffs ($000) are Alternative Small facility Large facility Weighted Payoff 0.5(200) + 0.5(270) = 235 0.5(160) + 0.5(800) = 480 The best of these weighted payoffs is $480,000, so the realist would build a large facility. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 27 Example A.6 d. Minimax Regret. If demand turns out to be low, the best alternative is a small facility and its regret is 0 (or 200 – 200). If a large facility is built when demand turns out to be low, the regret is 40 (or 200 – 160). Regret Alternative Low Demand High Demand Small facility 200 – 200 = 0 800 – 270 =530 Large facility 200 – 160 = 40 800 – 800 = 0 Maximum Regret 530 40 The column on the right shows the worst regret for each alternative. To minimize the maximum regret, pick a large facility. The biggest regret is associated with having only a small facility and high demand. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 28 Application A.4 Fletcher (a realist), Cooper (a pessimist), and Wainwright (an optimist) are joint owners in a company. They must decide whether to make Arrows, Barrels, or Wagons. The government is about to issue a policy and recommendation on pioneer travel that depends on whether certain treaties are obtained. The policy is expected to affect demand for the products; however it is impossible at this time to assess the probability of these policy “events.” The following data are available: Payoffs (Profits) Alternative Land Routes No treaty Land Routes Treaty Sea Routes Only Arrows $840,000 $440,000 $190,000 Barrels $370,000 $220,000 $670,000 Wagons $25,000 $1,150,000 ($25,000) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 29 Application A.4 • Which product would be favored by Fletcher (realist)? – Fletcher (realist – Laplace) would choose arrows • Which product would be favored by Cooper (pessimist)? – Cooper (pessimist – Maximin) would choose barrels • Which product would be favored by Wainwright (optimist)? – Wainwright (optimist – Maximax) would choose wagons • What is the minimax regret solution? – The Minimax Regret solution is arrows Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 30 Decision Making Under Risk • Use the expected value rule • Weigh each payoff with associated probability and add the weighted payoff scores. • Choose the alternative with the best expected value. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 31 Example A.7 Reconsider the payoff matrix in Example 5. For the expected value decision rule, which is the best alternative if the probability of small demand is estimated to be 0.4 and the probability of large demand is estimated to be 0.6? The expected value for each alternative is as follows: Alternative Possible Future Demand Alternative Small Large Small facility 200 270 Large facility 160 800 Expected Value Small facility 0.4(200) + 0.6(270) = 242 Large facility 0.4(160) + 0.6(800) = 544 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall The large facility is the best alternative. A - 32 Application A.5 For Fletcher, Cooper, and Wainwright, find the best decision using the expected value rule. The probabilities for the events are given below. What alternative has the best expected results? Land routes, No Treaty (0.50) Land Routes, Treaty Only (0.30) Sea routes, Only (0.20) Arrows 840,000 440,000 190,000 Barrels 370,000 220,000 670,000 Wagons 25,000 1,150,000 -25,000 Alternative Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 33 Application A.5 Land routes, No Treaty (0.50) Land Routes, Treaty Only (0.30) Arrows (.50) * 840,000` + (.30)* 440,000 + (.20) * 190,000 590,000 Barrels (.50) * 370,000` + (.30)* 220,000 + (.20) * 670,000 385,000 Wagons (.50) * 25,000` + Alternative (.30)* 1,150,000 + Sea routes Only (0.20) Expected Value (.20) * -25,000 352,500 Arrows is the best alternative. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 34 Decision Trees E1 & Probability E2 & Probability E3 & Probability Payoff 1 Payoff 2 Payoff 3 Alternative 3 1 2 1st decision Alternative 5 Possible 2nd decision E2 & Probability = Event node Alternative 4 E3 & Probability Payoff 1 Payoff 2 Payoff 3 Payoff 1 Payoff 2 = Decision node Ei = Event i P(Ei) = Probability of event i Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 35 Example A.8 • A retailer will build a small or a large facility at a new location • Demand can be either small or large, with probabilities estimated to be 0.4 and 0.6, respectively • For a small facility and high demand, not expanding will have a payoff of $223,000 and a payoff of $270,000 with expansion • For a small facility and low demand the payoff is $200,000 • For a large facility and low demand, doing nothing has a payoff of $40,000 • The response to advertising may be either modest or sizable, with their probabilities estimated to be 0.3 and 0.7, respectively • For a modest response the payoff is $20,000 and $220,000 if the response is sizable • For a large facility and high demand the payoff is $800,000 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 36 Example A.8 Low demand [0.4] Don’t expand 2 1 Expand $200 $223 $270 Do nothing $40 3 Modest response [0.3] Advertise Sizable response [0.7] $20 $220 High demand [0.6] $800 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 37 Example A.8 Low demand [0.4] Don’t expand 2 1 Expand $200 $223 $270 0.3 x $20 = $6 Do nothing $40 3 Modest response [0.3] $20 Advertise $6 + $154 = $160 Sizable response [0.7] $220 0.7 x $220 = $154 High demand [0.6] $800 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 38 Example A.8 Low demand [0.4] Don’t expand 2 Expand 1 $200 $223 $270 Do nothing $40 3 Modest response [0.3] Advertise $160 $160 Sizable response [0.7] $20 $220 High demand [0.6] $800 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 39 Example A.8 Low demand [0.4] Don’t expand 2 Expand $270 1 $200 $223 $270 Do nothing $40 3 Modest response [0.3] Advertise $160 $160 Sizable response [0.7] $20 $220 High demand [0.6] $800 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 40 Example A.8 Low demand [0.4] $200 x 0.4 = $80 $80 + $162 = $242 Don’t expand 2 Expand $270 1 $223 $270 x 0.6 = $162 Do nothing $40 3 Modest response [0.3] Advertise $160 $160 Sizable response [0.7] $20 $220 High demand [0.6] $800 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 41 Example A.8 Low demand [0.4] $200 $242 Don’t expand 2 Expand $270 1 $223 $270 Do nothing $40 3 Advertise $160 0.4 x $160 = $64 $544 Modest response [0.3] $160 High demand [0.6] $800 Sizable response [0.7] $20 $220 x 0.6 = $480 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 42 Example A.8 Low demand [0.4] $200 $242 Don’t expand 2 Expand $270 1 $223 $270 Do nothing $40 $544 3 Advertise $160 $160 $544 Modest response [0.3] Sizable response [0.7] $20 $220 High demand [0.6] $800 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 43 Application A.6 a. Draw the decision tree for the Fletcher, Cooper, and Wainwright Application 5 b. What is the expected payoff for the best alternative in the decision tree below? Land routes, No Treaty (0.50) Land Routes, Treaty Only (0.30) Arrows 840,000 440,000 190,000 Barrels 370,000 220,000 670,000 Wagons 25,000 1,150,000 -25,000 Alternative Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Sea routes, Only (0.20) A - 44 Application A.6 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 45 Solved Problem 1 • A small manufacturing business has patented a new device for washing dishes and cleaning dirty kitchen sinks • The owner wants reasonable assurance of success • Variable costs are estimated at $7 per unit produced and sold • Fixed costs are about $56,000 per year a. If the selling price is set at $25, how many units must be produced and sold to break even? Use both algebraic and graphic approaches. b. Forecasted sales for the first year are 10,000 units if the price is reduced to $15. With this pricing strategy, what would be the product’s total contribution to profits in the first year? Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 46 Solved Problem 1 a. Beginning with the algebraic approach, we get Q= 56,000 F = p–c 25 – 7 = 3,111 units Using the graphic approach, shown in Figure A.6, we first draw two lines: Total revenue = 25Q Total cost = 56,000 + 7Q The two lines intersect at Q = 3,111 units, the break-even quantity Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 47 Solved Problem 1 250 – Dollars (in thousands) 200 – Total revenues 150 – Break-even quantity 100 – $77.7 Total costs 50 – 3.1 0– | | | | | | | | 1 2 3 4 5 6 7 8 Units (in thousands) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 48 Solved Problem 1 b. Total profit contribution = Total revenue – Total cost = pQ – (F + cQ) = 15(10,000) – [56,000 + 7(10,000)] = $24,000 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 49 Solved Problem 2 Herron Company is screening three new product idea: A, B, and C. Resource constraints allow only one of them to be commercialized. The performance criteria and ratings, on a scale of 1 (worst) to 10 (best), are shown in the following table. The Herron managers give equal weights to the performance criteria. Which is the best alternative, as indicated by the preference matrix method? Performance Criteria 1. Demand uncertainty and project risk 2. Similarity to present products 3. Expected return on investment (ROI) 4. Compatibility with current manufacturing process 5. Competitive Strategy Product A 3 7 10 4 Rating Product B 9 8 4 7 Product C 2 6 8 6 4 6 5 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 50 Solved Problem 2 Each of the five criteria receives a weight of 1/5 or 0.20 Product Calculation Total Score A (0.20 × 3) + (0.20 × 7) + (0.20 × 10) + (0.20 × 4) + (0.20 × 4) = 5.6 B (0.20 × 9) + (0.20 × 8) + (0.20 × 4) + (0.20 × 7) + (0.20 × 6) = 6.8 C (0.20 × 2) + (0.20 × 6) + (0.20 × 8) + (0.20 × 6) + (0.20 × 5) = 5.4 The best choice is product B as Products A and C are well behind in terms of total weighted score Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 51 Solved Problem 3 Adele Weiss manages the campus flower shop. Flowers must be ordered three days in advance from her supplier in Mexico. Although Valentine’s Day is fast approaching, sales are almost entirely last-minute, impulse purchases. Advance sales are so small that Weiss has no way to estimate the probability of low (25 dozen), medium (60 dozen), or high (130 dozen) demand for red roses on the big day. She buys roses for $15 per dozen and sells them for $40 per dozen. Construct a payoff table. Which decision is indicated by each of the following decision criteria? a. Maximin b. Maximax c. Laplace d. Minimax regret Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 52 Solved Problem 3 The payoff table for this problem is Demand for Red Roses Low (25 dozen) Alternative Medium (60 dozen) High (130 dozen) Order 25 dozen $625 $625 $625 Order 60 dozen $100 $1,500 $1,500 ($950) $450 $3,250 $0 $0 $0 Order 130 dozen Do nothing Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 53 Solved Problem 3 a. Under the Maximin criteria, Weiss should order 25 dozen, because if demand is low, Weiss’s profits are $625, the best of the worst payoffs. b. Under the Maximax criteria, Weiss should order 130 dozen. The greatest possible payoff, $3,250, is associated with the largest order. c. Under the Laplace criteria, Weiss should order 60 dozen. Equally weighted payoffs for ordering 25, 60, and 130 dozen are about $625, $1,033, and $917, respectively. d. Under the Minimax regret criteria, Weiss should order 130 dozen. The maximum regret of ordering 25 dozen occurs if demand is high: $3,250 – $625 = $2,625. The maximum regret of ordering 60 dozen occurs if demand is high: $3,250 – $1,500 = $1,750. The maximum regret of ordering 130 dozen occurs if demand is low: $625 – (–$950) = $1,575. A - 54 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Solved Problem 4 White Valley Ski Resort is planning the ski lift operation for its new ski resort and wants to determine if one or two lifts will be necessary. Each lift can accommodate 250 people per day and skiing occurs 7 days per week in the 14-week season and lift tickets cost $20 per customer per day. The table below shows all the costs and probabilities for each alternative and condition. Should the resort purchase one lift or two? Alternatives Conditions One lift Two lifts Utilization Installation Operation Bad times (0.3) 0.9 $50,000 $200,000 Normal times (0.5) 1.0 $50,000 $200,000 Good times (0.2) 1.0 $50,000 $200,000 Bad times (0.3) 0.9 $90,000 $200,000 Normal times (0.5) 1.5 $90,000 $400,000 Good times (0.2) 1.9 $90,000 $400,000 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 55 Solved Problem 4 The decision tree is shown on the following slide. The payoff ($000) for each alternative-event branch is shown in the following table. The total revenues from one lift operating at 100 percent capacity are $490,000 (or 250 customers × 98 days × $20/customer-day). Alternatives Economic Conditions One lift Bad times 0.9(490) – (50 + 200) = 191 Normal times 1.0(490) – (50 + 200) = 240 Good times 1.0(490) – (50 + 200) = 240 Bad times 0.9(490) – (90 + 200) = 151 Normal times 1.5(490) – (90 + 400) = 245 Good times 1.9(490) – (90 + 400) = 441 Two lifts Payoff Calculation (Revenue – Cost) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 56 Solved Problem 4 Bad times [0.3] 0.3(191) + 0.5(240) + 0.2(240) = 225.3 Normal times [0.5] $191 $240 One lift $225.3 $256.0 Good times [0.2] Bad times [0.3] Two lifts Normal times [0.5] $256.0 0.3(151) + 0.5(245) + 0.2(441) = 256.0 Good times [0.2] Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall $240 $151 $245 $441 A - 57 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall A - 58