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operations research an introduction

Operations Research An Introduction 8th Edition Hamdy A. Taha - Solutions

*5. In the project of Example 6.5-2 (Figure 6.42), assume that the durations of activities B and F are changed from 6 and 11 days to 20 and 25 days, respectively.(a) Determine the critical path.
(b) Determine the total and free floats for the network, and identify the red-flagged activities.
6. Compute the floats and identify the red-flagged activities for the projects (a) and (b) in Figure 6.44, then develop the time schedules under the following conditions:Project (a)(i) Activity (1,5) cannot start any earlier than time 14.
(ii) Activities (5,6) and (5,7) use the same equipment, of which only one unit is available.
(iii) All other activities start as early as possible.Project (b)(i) Activity (1,3) must be scheduled at its earliest start time while accounting for the requirement that (1,2), (1, 3), and (1,6) use a special piece of equipment, of which 1 unit only is available.(ii) All other activities start as
1. Consider Problem 2, Set 6.Sb. The estimates (a, m,b) are listed below. Determine the probabilities that the different nodes of the project will be realized without delay.Project (a) Project (b)Activity (a, m,b) Activity (a, m,b) Activity (a,m,b) Activity (a, m, b)1-2 (5,6,8) 3-6 (3,4,5) 1-2
10. Water Quality Management. (Stark and Nicholes, 1972) Four cities discharge waste water into the same stream. City 1 is upstream, followed downstream by city 2, then city 3, then city 4. Measured alongside the stream, the cities are approximately 15 miles apart. A measure of the amount of
9. Military Planning. (Shepard and Associates, 1988) The Red Army (R) is trying to invade the territory defended by the Blue Army (B). Blue has three defense lines and 200 regular combat units and can draw also on a reserve pool of 200 units. Red plans to attack on two fronts, north and south, and
8. Leveling the Terrain for a New Highway. (Stark and Nicholes, 1972) The Arkansas Highway Department is planning a new 10-mile highway on uneven terrain as shown by the profile in Figure 2.10.The width of the construction terrain is approximately 50 yards.To simplify the situation, the terrain
7. Filling a Straight Line into Empirical Data (Regression). In a lO-week typing class for beginners, the average speed per student (in words per minute) as a function of the number of weeks in class is given in the following table.Week,x Words per minute, y 15 29 315 419 521 624 726 830 931 10 35
*6. Traffic Light Control. (Stark and Nicholes, 1972) Automobile traffic from three highways, HI, H2, and H3, must stop and wait for a green light before exiting to a toll road. TIle tolls are $3, $4, and $5 for cars exiting from HI, H2, and H3, respectively.The flow rates from HI, H2, and H3 are
5. Pollution Control. Three types of coal, C1, C2, and C3, are pulverized and mixed together to produce 50 tons per hour needed to power a plant for generating electricity. The burning of coal emits sulfur oxide (in parts per million) which must meet the Environmental Protection Agency (EPA)
4. Assembly-Line Balancing. A product is assembled from three different parts. The parts are manufactured by two departments at different production rates as given in the following table:Production rate (units/hr)Capacity Department (hr/wk) Part] Part 2 Part]1 100 8 5 10 2 80 6 12 4 Determine the
3. Voting on Issues. In a particular county in the State of Arkansas, four election issues are on the ballot: Build new highways, increase gun control, increase farm subsidies, and increase gasoline tax. The county includes 100,000 urban voters, 250,000 suburban voters, and 50,000 rural volers, all
2. Shelf Space Allocation. A grocery store must decide on the shelf space to be allocated to each of five types of breakfast cereals.lhe maximum daily demand is 100,85,140,80, and 90 boxes, respectively. The shelf space in square inches for the respective boxes is 16,24, 18,22, and 20. The total
1. Consider the trim-loss model of Example 2.3-9.(a) If we slit 200 rolls using setting 1 and 100 rolls using setting 3, compute the associated trim-loss area.(b) Suppose that the only available standard roll is 15 feet wide. Generate all possible knife settings for producing 5-, 7-, and 9-foot
6. A large department store operates 7 days a week. The manager estimates that the minimum number of salespersons required to provide prompt service is 12 for Monday, 18 for Tuesday, 20 for Wednesday, 28 for Thursday, 32 for Friday, and 40 for each of Saturday'and Sunday. Each salesperson works 5
*5. On most university campuses students are contracted by academic departments to do errands, such as answering the phone and typing. The need for such service fluctuates during work hours (8:00 A.M. to 5:00 P.M.). In the IE department, the minimum number of students needed is 2 between 8:00 A.M.
4. In an LTL (less-than-truckload) trucking company, terminal docks include casual workers who are hired temporarily to account for peak loads. At the Omaha, Nebraska, dock, the minimum demand for casual workers during the seven days of the week (starting on Monday) is 20, 14, 10, 15, 18, 10, 12
3. In Problem 2, suppose that no volunteers will start at noon or 6:00 P.M. to allow for lunch and dinner. Determine the optimal schedule.
2. A hospital employs volunteers to staff the reception desk between 8:00 A.M. and 10:00 P.M.Each volunteer works three consecutive hours except for those starting at 8:00 P.M. who work for two hours only. The minimum need for volunteers is approximated by a step function over 2-hour intervals
*1. In the bus scheduling example suppose that buses can run either 8- or 12-hour shi-fts. If a bus runs for 12 hours, the driver must be paid for the extra hours at 150% of the regular hourly pay. Do you recommend the use of 12-hour shifts?60 Chapter 2 Modeling with Linear Programming
10. Two alloys, A and B, are made from fOUf metals, I, II, III, and IV, according to the following specifications:Alloy AB Specifications At most 80% of I At most 30% of II At least 50% of IV Between 40% and 60% of II At least 30% of III At most 70% of IV Selling price ($)200 300 The fOUf metals,
9. A foundry smelts steel, aluminum, and cast iron scraps to produce two types of metal ingots, I and II, with specific limits on the aluminum, graphite and silicon contents. Aluminum and silicon briquettes may be used in the smelting process to meet the desired specifications. The following tables
7. Hawaii Sugar Company produces brown sugar, processed (white) sugar, powdered sugar, and molasses from sugar cane syrup. The company purchases 4000 tons of syrup weekly and is contracted to deliver at least 25 tons weekly of each type of sugar. The production process starts by manufacturing brown
6. In the refinery situation of Problem 5, suppose that the distillation unit actually produces the intermediate products naphtha and light oil. One bbl of crude A produces .35 bbl of naphtha and .6 bbl of light oil, and one bbl of crude B produces .45 bbl of naphtha and.5 bbl of light oil. Naphtha
5. An oil company distills two types of crude oil,A and B, to produce regular and premium gasoline and jet fuel. There are limits on the daily availability of crude oil and the minimum demand for the final products. If the production is not sufficient to cover demand, the shortage must be made up
4. A refinery manufactures two grades of jet fuel, f1 and Fl, by blending four types of gasoline, A, B, C, and D. Fuel F1 uses gasolines A, B, C, and D in the ratio 1:1:2:4, and fuel Fl uses the ratio 2:2:1:3. The supply limits for A, B, C, and Dare 1000, 1200,900, and 1500 bbllday,
3. All-Natural Coop makes three breakfast cereals, A, B, and C, from four ingredients:rolled oats, raisins, shredded coconuts, and slivered almonds. The daily availabilities of the ingredients are 5 tons, 2 tons, 1 ton, and 1 ton, respectively. The corresponding costs per ton are $100, $120, $110,
*2. A hardware store packages handyman bags of screws, bolts, nuts, and washers. Screws come in 100-lb boxes and cost $110 each, bolts come in 1oo-lb boxes and cost $150 each, nuts come in 80-lb boxes and cost $70 each, and washers come in 30-lb boxes and cost $20 each. The handyman package weighs
1. Hi-V produces three types of canned juice drinks, A, B, and C, using fresh strawberries, grapes, and apples. TIle daily supply is limited to 200 tons of strawberries, 100 tons of grapes, and 150 tons of apples. The cost per ton of strawberries, grapes, and apples is $200,$100, and $90,
8. Two products are manufactured sequentially on two machines. The time available on each machine is 8 hours per day and may be increased by up to 4 hours of overtime, if necessary, at an additional cost of $100 per hour. The table below gives the production rate on the two machines as well as the
*7. The manufacturing process of a product consists of two successive operations, I and II. The following table provides the pertinent data over the months of June, July, and August:,,.,, Finished product demand (units)Capacity of operation I (hr)Capacity of operation II (hr)June 500 800 1000 July
2. Four products are processed sequentially on three machines. The following table gives the pertinent data of the problem.Manufacturing time (hr) per unit Machine Cost per hr ($) Productl Product 2 Product 3 Product 4 Capacity (hr)1 10 2 3 4 2 500 2 5 3 2 1 2 380 3 4 7 3 2 1 450 Unit selling price
1. Tooleo has contracted with AutoMate to supply their automotive discount stores with wrenches and chisels. AutoMate's weekly demand consists of at least 1500 wrenches and 1200 chisels. Tooleo cannot produce all the requested units with its present one-shift capacity and must use overtime and
7. (Lewis, 1996) Monthly bills in a household are received monthly (e.g., utilities and home mortgage), quarterly (e.g., estimated tax payment), semiannually (e.g., insurance) ,or annually(e.g., subscription renewals and dues). The following table provides the monthly bills for next year.Month Jan.
6. A gambler plays a game that requires dividing bet money among four choices. The game has three outcomes. The following table gives the corresponding gain or loss per dollar for the different options of the game.Return per dollar deposited in choice Outcome 1 2 3 4 1 -3 4 -7 15 2 5 -3 9 4 3 3 -9
*5. A business executive has the option to invest money in t\\'o plans: Plan A guarantees that each dollar invested will earn $.70 a year later, and plan B guarantees that each dollar invested will earn $2 after 2 years. In plan A, investments can be made annually, and in plan B, investments are
4. In anticipation of the immense college expenses, a couple have started an annual investment program on their child's eighth birthday that will last until the eighteenth birthday.The couple estimate that they will be able to invest the following amounts at the beginning of each year:Year 1 Amount
3. HiRise Construction can bid on two I-year projects. TIle following table provides the quarterly cash flow (in milIions of dollars) for the two projects.Cash flow (in millions of $) at Project III 111/08-1.0-3.0 4/1/08-3.1-2.5 7/1/08-1.5 1.5 10/l/08 1.8 1.8 12/31/08 5.0 2.8 HiRise has cash funds
*2. Investor Doe has $10,000 to invest in four projects. TIle following table gives the cash flow for the four investments.Cash flow ($1000) at the start of Project Year 1 Year 2 Year 3 Year 4 Year 5 1 -1.00 0.50 0.30 1.80 1.20 2 -1.00 0.60 0.20 1.50 1.30 3 0.00 -l.00 0.80 1.90 0.80 4 -1.00 0.40
(d) Suppose in the original model that the yearly funds available for any year can be exceeded, if necessary, by borrowing from other financial activities within the company.Ignoring the time value of money, reformulate the LP model, and find the optimum solution. Would the new solution require
(c) In the original model, suppose that any funds left at the end of a year are used in the next year. Find the new optimal solution, and determine how much each year "borrows"from the preceding year. For simplicity, ignore the time value of money.
1. Fox Enterprises is considering six projects for possible construction over the next four years. The expected (present value) returns and cash outlays for the projects are given below. Fox can undertake any of the projects partially or completely. A partial undertaking of a project will prorate
4. Suppose that the company wishes to buy $6 million. The transaction limits for different currencies are the same as in the original problem. Devise a buying schedule for this transaction, given that mix may not include more than €3 million, £2 million, and KD2 million.5. Suppose that the
3. Suppose the initial amount I = $7 million and that the company wants to convert it optimally to a combination of euros, pounds, and yen. TIle final mix may not include more than €2 million, £3 million, and ¥200 million. Modify the original model to determine the optimal buying mix of the
*2. Suppose that the company is willing to convert the initial $5 million to any other currency that will provide the highest rate of return. Modify the original model to determine which currency is the best.
1. Modify the arbitrage model to account for a commission that amounts to.1% of any currency buy. Assume that the commission does not affect the circulating funds and that it is collected after the entire order is executed. How does the solution compare with that of the original model?
6. Consider the Realco model of Problem 5. Suppose that an additional 100 acres of land can be purchased for $450,000, which will increase the total acreage to 900 acres. Is this a profitable deal for Realco?
5. Realco owns 800 acres of undeveloped land on a scenic lake in the heart of the Ozark Mountains. In the past, little or no regulation was imposed upon new developments around the lake. The lake shores are now dotted with vacation homes, and septic tanks, most of them improperly installed, are in
4. The city of Fayetteville is embarking on an urban renewal project that will include lowerand middle-income row housing, upper-income luxury apartments, and public housing.The project also includes a public elementary school and retail facilities. The size of the elementary school (number of
3, the associated income over the five-year planning horizon is .4 X $50,000(for year 2) + .4 X $50,000 (for year 3) + (.4 + .6) X $50,000 (for year 4) +(.4 + .6) X $50,000 (for year 5) = (4 X .4 + 2 X .6) X $50,000. Determine the optimal schedule for the projects that will maximize the total
*3. A city will undertake five urban renewal housing projects over the next five years. Each project has a different starting year and a different duration. The following table provides the basic data of the situation:Cost Annual income Year 1 Year 2 Year 3 Year 4 Year 5 (million $) (million
2. TIle city council of Fayetteville is in the process of approving the construction of a new 200,000-ft2 convention center. Two sites have been proposed, and both require exercising the "eminent domain" law to acquire the property. The following table provides data about proposed (contiguous)
1. A realtor is developing a rental housing and retail area. TIle housing area consists of efficiency apartments, duplexes, and single-family homes. Maximum demand by potential renters is estimated to be 500 efficiency apartments, 300 duplexes, and 250 single-family homes, but the number of
4. In the Reddy Mikks model, the company is considering the production of a cheaper brand of exterior paint whose input requirements per ton include .75 ton of each of raw materials Ml and M2. Market conditions still dictate that the excess of interior paint over the production of both types of
3. In the TOYCO model, suppose that a new toy (fire engine) requires 3, 2, 4 minutes, respectively, on operations 1,2, and 3. Determine the optimal solution when the revenue per unit is given by*(a) $5.(b) $10._,f ;.~~-.:...• _.lSt I), a inS lp-:..,t:~-.:...•..References 191
*1. In the originalTOYCO model, toy trains are not part of the optimal product mix. The company recognizes that market competition will not allow raising the unit price of the toy. Instead, the company wants to concentrate on improving the assembly operation itself.This entails reducing the
2. Investigate the optimality of the Reddy Mikks solution (Example 4.3-1) for each of the following objective functions. If the solution changes, use post-optimal analysis to determine the new optimum. (The optimal tableau of the model is given in Example 3.3-1.)*(a) z = 3x} + 2X2(b) z = 8x} +
2. Secondary Constraints. Instead of solving a problem using all of its constraints, we can start by identifying the so-called secondary constraints. These are the constraints that we
5. Show that the 100% feasibility rule in Problem 12, Set 3.6c (Chapter 3) is based on the condition(?Ptimum)(Origi~alright-hand) ;=: 0 mverse side vector 6. Post-optimal Analysis for Cases Affecting Both Optimality and Feasibility. Suppose that you are given the following simultaneous changes in
3. Consider the Reddy Mikks model of Example 2.1-1. Its optimal tableau is given in Example 3.3-1. If the daily availabilities of raw materials Ml and M2 are increased to 28 and 8 tons, respectively, use post-optimal analysis to determine the new optimal solution.*4. The Ozark Farm has 20,000
1. In the TOYCO model listed at the start of Section 4.5, would it be more advantageous to assign the 20-minute excess capacity of operation 3 to operation 2 instead of operation I?2. Suppose that TOYCO wants to change the capacities of the three operations according to the following cases:(460)(a)
5. Solve the following LP in three different ways (useTORA for convenience). Which method appears to be the most efficient computationally?subject to 5XI + 6X2 - 3X3 + 4X4 ;::: 12 X2 - 5X3 - 6X4 ;::: 10 2XI + 5X2 + x3 + x4;::: 8
*(c) Minimize z = -xl + X2 subject to Xl - 4X2 ;::: 5 Xl - 3X2 :5 1 2xI - 5X2 ;::: 1(d) Maximize z = 2X3 subject to
4. Using the artificial constraint procedure introduced in Problem 3, solve the following problems by the dual simplex method. In each case, indicate whether the resulting solution is feasible, infeasible, or unbounded.(a) Maximize z = 2X3 subject to(b) Maximize z := Xl - 3x2 subject to
(c) Minimize z == 4Xl + 2X2 subject to Xl + X2 == 1 3Xl - X2 =::: 2 Xl> X2 =::: 0
(b) Minimize z == 5Xl + 6X2 subject to Xl + X2 2: 2 4xl + X2 2: 4 Xl, X2 2: 0
2. Generate the dual simplex iterations for the following problems (usingTORA for convenience), and trace the path of the algorithm on the graphical solution space.(a) Minimize z = 2xt + 3X2
*3. JoShop uses lathes and drill presses to produce four types of machine parts, PPl, PPl, PP3, and PP4. The table below summarizes the pertinent data.Machining time in minutes per unit of Machine PPI PP2 PP3 PP4 Capacity (minutes)Lathes 2 5 3 4 5300 Drill presses 3 4 6 4 5300 Unit revenue ($) 3 6
if we increase the capacity of soldering by 1O%?
(c) Does the minimum production requirements for the four cables represent an advantage or a disadvantage for NWAC Electronics? Provide an explanation based on the dual prices.(d) Can the present unit contribution to revenue as specified by the dual price be guaranteed
(b) Based on the dual prices, do you recommend making increases in the daily capacities of any of the four operations? Explain.
*2. NWAC Electronics manufactures four types of simple cables for a defense contractor.Each cable must go through four sequential operations: splicing, soldering, sleeving, and inspection. The following table gives the pertinent data of the situation.Minutes per unit Cable Splicing Soldering
8. Consider the following LP:Maximize z = 2x] + 4x2 + 4xJ - 3X4 subject to Xl + X2 + XJ = 4 x I + 4X2 + X4 = 8 Use the dual problem to show that the basic solution (Xl, X2) is not optimal.
*7. TIle following is the optimal tableau for a maximization LP model with three (:s;) constraints and all nonnegative variables. The variables X3, X4, and Xs are the slacks associated with the three constraints. Determine the associated optimal objective value in two different ways by using the
Determine the following:(a) The right-hand-side values, bi and &2'(b) The optimal dual solution,(c) The elementsa, b, C,d, e.
6. Consider the following LP model:Maximize z = 5XI + 2X2 + 3X3 subject to Xl + 5x2 + 2x3 :s bi Xl - 5X2 - 6X3 :s b2 Xl> X2, X3 ~ 0 The following optimal tableau corresponds to specific values of b} and ~:Basic Xl X2 X3 X4 Xs Solution z 0 a 7 d e 150 Xl 1 b 2 1 0 30 Xs 0 c -8 -1 1 10
5. Consider the following LP model:Maximize z = 5XI + 12x2 + 4x~subject to Xl + 2X2 + X3 + X4 == 10 2XI - X2 + 3X3 == 2(a) Identify the best solution from among the following basic feasible solutions:((i) Basic variables = (X4. X3), Inverse = 01 --3il)(~ ---~~) (ii) Basic variables = (X2, Xl)'
3. Consider the following LP model:Maximize z = 3x) + 2X2 + 5x3 subject to 2.·1= 30+ X5 = 60+ x6 = 20 c;-Check the optimality and feasibility of the following basic solutions:Basic variables ~ (x" XJ, x,), Inverse ~G 1 D-2(a) 120 1 1 Basic v"iables -~) ~ (X2, XJ, x,), Invme ~ ( i -8(b) _!4~r -1 12
2. Consider the following LP model:Maximize z = 4x) + 14x2 subject to 2x] + txz + X3 = 21 7x) + 2X2 + X4 = 21 xl> x2, x3' X4 ~ 0 4.2 Primal-Dual Relationships 167 Check the optimality and feasibility of each of the following basic solutions.*(a) Basic variables = (X2, X4), Inverse = (_~ ~)7(b)
9. In Problem 7(a}, let Yl and Y2 be the dual variables. Determine whether the following pairs of primal-dual solutions are optimal:*(a) (Xl = 3,X2 = l;Yl = 4'Y2 = 1)(b) (XI = 4, x2 = 1; YI = 1, Y2 = 0)(c) (XI = 3, X2 = 0; YI = 5, Y:z = 0)
(d) Maximize z = 3xj + 2X2 subject to 2xI + X2:::; 3 3x} + 4x2 :::; 12 4.2 Primal-Dual Relationships 165
(c) Maximize z = 2xI + X2 subject to XI - X2 :$ 10 2Xl :$ 40
(b) Maximize z = Xl + 5X2 + 3x)subject to Xl + 2X2 + x3 = 3 2xI - X2 = 4
8. Estimate a range for the optimal objective value for the following LPs:*(a) Minimize z = 5XI + 2X2 subject to Xl - X2 ~ 3 2x} + 3X2 ~ 5
7. Consider the following set of inequalities:2xI + 3X2:::; 12-3Xl + 2x2 :::; -4 3Xl - 5x2:::; 2 Xl unrestricted X2 ~ 0 A feasible solution can be found by augmenting the trivial objective function Maximize Z = XI + X2 and then solving the problem. Another way is to solve the dual; from which a
*6. Consider the following LP:Maximize z = XI + 5X2 + 3X3 subject to XI + 2X2 + X3 = 3 2xI - X2 = 4 Xl, X2, X3 ;::: 0 164 Chapter 4 Duality and Post-Optimal Analysis The starting solution consists of x) in the first constraint and an artificial X4 in the second constraint with M = 100. The optimal
5. Consider the following LP:Maximize z = 2xI + 4X2 + 4x) - 3X4 subject to Using X) and X4 as starting variables, the optimal tableau is given as Basic Xl Z 2 XJ .75 X2 .25 oo 1o 1o 3-.25.25 Solution 16 22.es It .j tL.··.·: .' ~_...Write the associated dual problem and determine its optimal
4. Consider the following LP:Minimize z = 4x, + X2 subject to 3Xl + X2 = 3 4X1 + 3X2 ;::: 6 XI + 2X2 :0:; 4 The starting solution consists of artificial X4 and Xs for the first and second constraints and slack X6 for the third constraint. Using M = 100 for the artificial variables, the optimal
*3. Consider the following LP:Maximize z = 5Xl + 2X2 + 3X3 subject to Xl + 5X2 + 2X3 = 30 Xl - 5X2 - 6X3 S; 40 Given that the artificial variable X4 and the slack variable X5 form the starting basic variables and that M was set equal to 100 when solving the problem, the optimal tableau is given as
2. Solve the dual of the following problem, then find its optimal solution from the solution of the dual. Does the solution of the dual offer computational advantages over solving the primal directly?Minimize z = 5Xl + 6X2 + 3X3 subject to 5Xl + 5X2 + 3X3;Z: 50 Xl + X2 - x3;Z: 20 7Xl + 6x2 - 9X3 2:
1. Find the optimal value of the objective function for the following problem by inspecting only its dual. (Do not solve the dual by the simplex method.)Minimize z = lOx) + 4X2 + 5X3 subject to
*5. Consider Example 4.1-1. The application of the simplex method to the primal requires the use of an artificial variable in the second constraint of the standard primal to secure a starting basic solution. Show that the presence of an artificial primal in equation form variable does not affect
4. Write the dual for each of the following primal problems:(a) Maximize z = -5x[ + 2x2 subject to-XI + x2:5-2 2Xl + 3X2:5 5(b) Minimize z = 6x} + 3X2 subject to 6Xl - 3X2 + X3 ;;:: 2 3x[ + 4X2 + X3 2: 5*(c) Maximize z == Xl + X2 subject to 2x[ + X2 = 5 3Xl - X2 = 6 Xl> X2 unrestricted
*2. In Example 4.1-2, derive the associated dual problem given that the primal problem is augmented with a third constraint, 3Xl + X2 = 4.
10. Consider Problem 10, Set 2.3e (Chapter 2).(a) Which of the specification constraints impacts the optimum solution adversely?(b) What is the most the company should pay per ton of each ore?
*9. Consider Problem 3, Set 2.3d (Chapter 2).(a) Suppose that the manufacturer can purchase additional units of raw material A at$12 per unit. Would it be advisable to do so?(b) Would you recommend that the manufacturer purchase additional units of raw material B at $5 per unit?
8. Consider Problem 2, Set 2.3d (Chapter 2). Suppose that any additional capacity of machines 1 and 2 can be acquired only by using overtime. What is the maximum cost per hour the company should be willing to incur for either machine?
7. Consider Problem 1, Set 2.3d (Chapter 2). Relate the dual prices to the unit production costs of the model.

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