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operations research an introduction

Operations Research: An Introduction 10th Global Edition Hamdy A Taha - Solutions

A retailer receives 70% of its batteries from Factory A and 30% from Factory B.The percentages of defectives produced by A and B are known to be 3% and 5%, respectively. A customer has just bought a battery randomly from the retailer.(a) What is the probability that the battery is defective?(b) If
Bayes’ theorem.2 Given the two events A and B, show that P5A0B6 =P5B0A6P5A6 P5B6, P5B6 7 0
Prove that if the probability P5A0B6 = P5A6, then A and B must be independent.
Graduating high school seniors with an ACT score of at least 26 can seek admission in two universities, A and B. The probability of being accepted in A is .4 and in B .25.The chance of being accepted in both universities is only 15%.(a) Determine the probability that the student is accepted in B
The stock of WalMark Stores, Inc. trades on the New York Stock Exchange under the symbol WMS. Historically, the price of WMS goes upward with the Dow 65% of the time and goes downward with the Dow 20% of the time. There is also a 10% chance that WMS will go up when the Dow goes down and 5% that it
In Example 14.1-2, suppose that you are told that the outcome is less than 6.(a) Determine the probability of getting an even number.(b) Determine the probability of getting an odd number larger than one.
Ann, Jim, John, and Nancy are scheduled to compete in a racquetball tournament. Ann is twice as likely to beat Jim, and Jim is at the same level as John. Nancy’s past winning record against John is one out of three. Determine the following:(a) The probability that Jim will win the tournament.(b)
You can toss a fair coin up to 7 times. You will win $100 if three tails appear before a head is encountered. What are your chances of winning?
Two dice are rolled independently and the two numbers that turn up are recorded.Determine the following:(a) The probability that the two numbers are odd with values less than 5.(b) The probability that the sum of the two numbers is 10.(c) The probability that the two numbers differ by at least 3.
A fair 6-faced die is tossed twice. Letting E and F represent the outcomes of the two tosses, compute the following probabilities:(a) The sum of E and F is 10.(b) The sum of E and F is even.(c) The sum of E and F is odd and greater than 3.(d) E is odd less than 6 and F is even greater than 1.(e) E
Answer Problem 14-2 assuming that in a room full of n persons at least one person shares your birthday.
Consider a random gathering of n persons. Determine the smallest n that will make it more likely that two persons or more have the same birthday. (Hint: Assume no leap years and that all days of the year are equally likely to be a person’s birthday.)
In a survey conducted in the State of Arkansas high schools to study the correlation between senior year scores in mathematics and enrollment in engineering colleges, 400 out of 1000 surveyed seniors have studied mathematics. Engineering enrollment shows that, of the 1000 seniors, 150 students have
A small publisher reprints a novel to satisfy the demand over the next 12 months. The demand estimates for the successive months are 100, 120, 50, 70, 90, 105, 115, 95, 80, 85, 100, and 110. The setup cost for reprinting the book is $200, and the holding cost per book per month is $1.20. Determine
The demand for fishing poles is at its minimum during the month of December and reaches its maximum during the month of April. Fishing Hole, Inc., estimates the December demand at 50 poles. It increases by 10 poles a month until it reaches 90 in April. Thereafter, the demand decreases by 5 poles a
Find the optimal inventory policy for the following six-period inventory situation:The unit production cost is $2 for all the periods.Period i Di (units) Ki ($) hi ($)1 10 20 1 2 15 17 1 3 7 10 1 4 20 18 3 5 13 5 1 6 25 50 1
Find the optimal inventory policy for the following five-period model. The unit production cost is $10 for all periods. The unit holding cost is $1 per period.Period i Demand Di (units) Setup cost K1 ($)1 50 80 2 70 70 3 100 60 4 30 80 5 60 60
Solve the following 10-period deterministic inventory model. Assume an initial inventory of 50 units.Period i Demand Di (units)Unit production cost ($)Unit holding cost ($)Setup cost ($)1 150 6 1 100 2 100 6 1 100 3 20 4 2 100 4 40 4 1 200 5 70 6 2 200 6 90 8 3 200 7 130 4 1 300 8 180 4 4 300 9 140
Solve Example 13.4-3, assuming that the initial inventory is 80 units. You may use excelWagnerWhitin.xls to check your calculations.
Develop the backward recursive equation for the model, assuming that the inventoryholding cost is based on the average inventory in the period.
Develop the backward recursive equation for the model, and then use it to solve Example 13.4-2.
Suppose that the inventory-holding cost is based on the average inventory during the period. Develop the corresponding forward recursive equation.
(a) Find the optimal solution for the following four-period inventory model:Period i Demand Di (units)Setup cost Ki ($)Holding cost hi ($)1 5 5 1 2 2 7 1 3 3 9 1 4 3 7 1 The unit production cost is $1 each for the first 6 units and $2 each for additional units.(b) Verify the computations using
Consider Example 13.4-2.(a) Will x4 = 0 in the optimum solution?(b) For each of the following two cases, determine the feasible ranges for z1, z2, z3, x1, x2, and x3. (You will find it helpful to represent each situation as in Figure 13.10.)(i) x1 = 3 and all the remaining data are the same.(ii) x1
The demand for a product over the next five periods may be filled from regular production, overtime production, or subcontracting. Subcontracting may be used only if the overtime capacity has been used. The following table gives the supply, demand, and cost data of the situation:Production capacity
An item is manufactured to meet known demand for four periods according to the following data:Unit production cost ($) for period Production range (units) 1 2 3 4 1–3 1 2 2 3 4–11 1 4 5 4 12–15 2 4 7 5 16–25 5 6 10 7 Unit holding cost to next period ($) .30 .35 .20 .25 Total demand (units)
Solve Example 13.4-1, assuming that the unit production and holding costs are as given in the following table:Period i Regular time unit cost ($)Overtime unit cost ($)Unit holding cost ($)to period i + 1 1 5.00 7.50 .10 2 3.00 4.50 .15 3 4.00 6.00 .12 4 1.00 1.50 .20
In Figure 13.7, determine the combined requirements for subassembly S in each of the following cases:(a) Lead time for M1 is only one period.(b) Lead time for M1 is three periods.
The following data describe four inventory items:Item i Ki ($) Di (units per day) hi ($)1 100 10 .1 2 50 20 .2 3 90 5 .2 4 20 10 .1 The company wishes to determine the economic order quantity for each of the four items such that the total number of orders per 365-day year is at most 150. Formulate
In Problem 13-19, assume that the only restriction is a limit of $1000 on the amount of capital that can be invested in inventory. The purchase costs per unit of items 1, 2, and 3 are $100, $55, and $100, respectively. Determine the optimum solution.
Solve the model of Example 13.3-3, assuming that we require the sum of the average inventories for all the items to be less than 25 units.
The following data describe five inventory items:11 Item i Ki ($) Di (units per day) hi ($) ai (ft2)1 35 22 0.35 1.0 2 28 34 0.15 0.8 3 30 14 0.28 1.1 4 25 21 0.30 0.5 5 20 26 0.42 1.2 Total available storage area = 22 ft2 Determine the optimal order quantities.
In the inventory model discussed in Section 13.3.2, suppose that the holding cost per unit per unit time is h1 for quantities below q and h2 otherwise, h1 7 h2. Show how the economic lot size is determined.
In Problem 13-15, determine the range on the price discount percentage that, when offered for lots of size 150 units or more, will not result in any financial advantage to the company.
An item sells for $30 a unit, but a 10% discount is offered for lots of 200 units or more.A company uses this item at the rate of 20 units per day. The setup cost for ordering a lot is $50, and the holding cost per unit per day is $.30. The lead time is 15 days. Should the company take advantage of
An item is consumed at the rate of 30 items per day. The holding cost per unit per day is $.05, and the setup cost is $100. Suppose that no shortage is allowed and that the purchasing cost per unit is $10 for any quantity not exceeding 500 units and $8 otherwise.The lead time is 21 days. Determine
The normal charge for washing a soiled towel is $.60, but the laundry service will charge only $.45 if the hotel delivers them in lots of at least 2600 towels. Should the hotel take advantage of the discount?
Consider the hotel laundry service situation in Problem
In Problem 13-10, suppose that shortage is allowed at a penalty cost of p per unit per unit time.(a) If w is the maximum shortage during the inventory cycle, show that(b) Show that the EOQ results in Section 13.3.1 can be derived from the general formulas in (a). KD h{y(1)- w} + pw TCU (y, w) + y
A company can produce an item or buy it from a contractor. If it is produced, it will cost $20 each time the machines are set up. The production rate is 100 units per day. If it is bought from a contractor, it will cost $15 each time an order is placed. The cost of maintaining the item in stock,
Consider the inventory situation in which the stock is replenished uniformly (rather than instantaneously) at the ratea. Consumption occurs at the constant rate D. Because consumption also occurs during the replenishment period, it is necessary that a 7 D.The setup cost is K per order, and the
Lewis (1996). An employee of a multinational company is on loan from the United States to the company’s subsidiary in Europe. During the year, the employee’s financial obligations in the United States (e.g., mortgage and insurance premium payments)amount to $12,000, distributed evenly over the
A hotel uses an external laundry service to provide clean towels. The hotel generates 600 soiled towels a day. The laundry service picks up the soiled towels and replaces them with clean ones at regular intervals. There is a fixed charge of $81 per pickup and delivery service, in addition to the
Walmark Store compresses and palletizes empty merchandise cartons for recycling. The store generates five pallets a day. The cost of storing a pallet in the store’s back lot is$.10 per day. The company that moves the pallets to the recycling center charges a flat fee of $100 for the rental of its
Two inventory policies have been suggested by the purchasing department of a company:Policy 1. Order 150 units. The reorder point is 50 units, and the time between placing and receiving an order is 10 days.Policy 2. Order 200 units. The reorder point is 75 units, and the time between placing and
A company stocks an item that is consumed at the rate of 60 units per day. It costs the company $25 each time an order is placed. An inventory unit held in stock for a week will cost $.36.(a) Determine the optimum inventory policy, assuming a lead time of 2 weeks.(b) Determine the optimum number of
McBurger orders ground meat at the start of each week to cover the week’s demand of 300 lb. The fixed cost per order is $20. It costs about $.03 per lb per day to refrigerate and store the meat.(a) Determine the inventory cost per week of the present ordering policy.(b) Determine the optimal
In each of the following cases, no shortage is allowed, and the lead time between placing and receiving an order is 35 days. Determine the optimal inventory policy and the associated cost per day.(a) K = $120, h = $.04, D = 25 units per day(b) K = $80, h = $.03, D = 35 units per day(c) K = $100, h
A small business financial data show that its inventory level of an item held steady at 1000 units during the first 9 months of the year. Sales accelerated during the last quarter in time for Christmas shopping, ending the year with only 20 units left in stock. The company estimates the total
The current-year balance sheet of a company shows a beginning and end inventories of $90.4 million and $20.2 million, respectively. The net revenue from sales for the year is $210.3 million and the gross profit is $30.4 million. The final report claims that the company’s average days-in-inventory
In the n-item knapsack problem of Example 12.3-1, suppose that the weight and volume limitations are W and V, respectively. Given that w i, v i, and ri are the weight, value, and revenue per unit, respectively, of item i, write the DP backward recursive equation for the problem.
Solve the following problems by DP.(a) Maximize z = 4x1 + 14x2 subject to 2x1 + 7x2 … 21 7x1 + 2x2 … 21 x1, x2 Ú 0(b) Maximize z = 8x1 + 7x2 subject to 2x1 + x2 … 8 5x1 + 2x2 … 15 x1, x2 Ú 0 and integer(c) Maximize z = 7x1 2 + 6x1 + 5x2 2 subject to x1 + 2x2 … 10 x1 - 3x2 … 9 x1, x2
A farmer owns k sheep. At the end of each year, a decision is made as to how many to sell or keep. The profit from selling a sheep in year i is pi. The sheep kept in year i will double in number in year i + 1. The farmer plans to sell out completely at the end of n years.(a) Derive the general
An investor with an initial capital of $10,000 must decide at the end of each year how much to spend and how much to invest in a savings account. Each dollar invested returns a = $1.09 at the end of the year. The satisfaction derived from spending $y in any one year is quantified monetarily as $1y.
Solve Example 12.3-4, assuming that r1 = .085 and r2 = .08. Additionally, assume that P1 = $5000, P2 = $4000, P3 = $3000, and P4 = $2000.
Solve Problem 12-26, assuming that the equipment is 1 year old and that n = 4, c = $6000, and r1t2 = n 1 + t.
Consider the equipment replacement problem over a period of n years. A new piece of equipment costs c dollars, and its resale value after t years in operation is s1t2 = n - t for n 7 t and zero otherwise. The annual revenue is a function of the age t and is given by r1t2 = n2 - t2 for n 7 t and
Circle Farms wants to develop a replacement policy for its 2-year-old tractor over the next 5 years. A tractor must be kept in service for at least 3 years, but must be disposed of after 5 years. The current purchase price of a tractor is $40,000 and increases by 10%a year. The salvage value of a
My son, age 13, has a lawn-mowing business with 10 customers. For each customer, he cuts the grass 3 times a year, which earns him $50 for each mowing. He has just paid$200 for a new mower. The maintenance and operating cost of the mower is $120 for the first year in service and increases by 20% a
In each of the following cases, develop the network, and find the optimal solution for the model in Example 12.3-3:(a) The machine is 2 years old at the start of year 1.(b) The machine is 1 year old at the start of year 1.(c) The machine is bought new at the start of year 1.
GECO is contracted for the next 4 years to supply aircraft engines at the rate of four engines a year. Available production capacity and production costs vary from year to year. GECO can produce five engines in year 1, six in year 2, three in year 3, and five in year 4. The corresponding production
Luxor Travel arranges 1-week tours to southern Egypt. The agency provides 7, 4, 7, and 8 rental cars over the next 4 weeks. Luxor Travel subcontracts with a local car dealer to supply rental needs. The dealer charges a rental fee of $220 per car per week, plus a flat fee of $500 for any rental
In Example 12.3-2, if a severance pay of $100 is incurred for each fired worker, determine the optimum solution.
Solve Example 12.3.2 for each of the following minimum labor requirements:(a) b1 = 6, b2 = 5, b3 = 3, b4 = 6, b5 = 8(b) b1 = 6, b2 = 4, b3 = 7, b4 = 8, b5 = 2
Solve the following problem by DP:Minimize z = max 5f1y12, f1y22,c, f1yn26 subject to y1 + y2 + c + yn = c yi Ú 0, i = 1, 2,c, n Provide the solution for the special case of n = 3, c = 10, and f1y12 = y1 + 5, f1y22 = 5y2 + 3, and f1y32 = y3 - 2.
Solve the following problem by DP:Maximize z = 1y1 + 222 + y2 y3 + 1y4 - 522 subject to y1 + y2 + y3 + y4 … 5 yi Ú 0 and integer, i = 1, 2, 3, 4
Solve the following problem by DP:Minimize z = y1 2 + y2 2 + c + yn 2 subject to qn i=1 yi = c yi 7 0, i = 1, 2,c, n
Solve the following model by DP:Maximize z = q ni=1 yi subject to y1 + y2 + c + yn = c yj Ú 0, j = 1, 2,c, n(Hint: This problem is similar to Problem 12-14, except that the variable yj is continuous.)
An electronic device consists of three components. The three components are in series so that the failure of one component causes the failure of the device. The reliability(probability of no failure) of the device can be improved by installing one or two standby units in each component. The table
Sheriff Bassam is up for reelection in Washington County. The funds available for the campaign are about $10,000. Although the reelection committee would like to launch the campaign in all five precincts of the county, limited funds dictate otherwise. The table given below lists the voting
Habitat for Humanity is a wonderful (U.S.-based) international charity organization that builds homes for needy families using volunteer labor and donated building materials. An eligible family can choose from three home sizes: 1000, 1100, and 1200 ft2. Each size requires a certain number of labor
I have a small backyard garden that measures 10 * 20 ft. This spring I plan to plant three types of vegetables: tomatoes, green beans, and corn. The garden is organized in 10-foot rows. The corn and tomatoes rows are 2 ft wide, and the beans rows are 3 ft wide. I like tomatoes the most and beans
A student must select 10 electives from four different departments, with at least one course from each department. The 10 courses are allocated to the four departments in a manner that maximizes “knowledge.” The student measures knowledge on a 100-point scale and comes up with the following
A wilderness hiker must pack three items: food, first-aid kits, and clothes. The backpack has a capacity of 3 ft3. Each unit of food takes 1 ft3. A first-aid kit occupies 14 ft3, and each piece of cloth takes about 12 ft3. The hiker assigns the priority weights 3, 4, and 5 to food, first aid, and
In the cargo-loading model of Example 12.3-1, suppose that the revenue per item includes a constant amount that is realized only if the item is chosen, as the following table shows:Find the optimal solution using DP. (Hint: You can use the Excel file excelSetupKnapsack.xls to check your
Solve the cargo-loading problem of Example 12.3-1 for each of the following sets of data:(a) w 1 = 4, r1 = 70, w 2 = 1, r2 = 20, w 3 = 2, r3 = 40, W = 6(b) w 1 = 1, r1 = 15, w 2 = 2, r2 = 30, w 3 = 3, r3 = 40, W = 4
In Example 12.3-1, determine the optimum solution, assuming that the maximum weight capacity of the vessel is 2 tons. Repeat the question for a weight capacity of 5 tons.8
For the network in Figure 12.9, it is desired to determine the shortest route between cities 1 to 7. Define the stages and the states using backward recursion, and then solve the problem.
For Problem 12-2, develop the backward recursive equation, and use it to find the optimum solution.
For Problem 12-1, develop the backward recursive equation, and use it to find the optimum solution.
I am an avid hiker. Last summer, my friend G. Don and I went on a 5-day hike-andcamp trip in the beautiful White Mountains in New Hampshire. We decided to limit our hiking to an area comprising three well-known peaks: Mounts Washington, Jefferson, and Adams. Mount Washington has a 6-mile
Solve Example 12.1-1, assuming the following routes are used:d11, 22 = 5, d11, 32 = 9, d11, 42 = 8 d12, 52 = 10, d12, 62 = 17 d13, 52 = 4, d13, 62 = 10 d14, 52 = 9, d14, 62 = 9 d15, 72 = 19 d16, 72 = 9
Excel–AMPL Experiment. The matrix below provides the distances among 10 cities (all off-diagonal missing entries = ). (For convenience, file prob11-35.txt gives the distance data in AMPL format.)1 2 3 4 5 6 7 8 9 10 1 100 2 11 80 5 39 95 28 2 17 42 33 21 59 46 79 29 3 63 57 92 55 68 52 4 36 27
Apply the genetic metaheuristic to the following problems starting with best nearestneighbor tour:(a) The paint sequencing problem of Example 11.1-1.(b) Problem 11-2..(c) Problem 11-5..(d) Problem 11-6..
Carry out iterations 3 and 4 in Example 11.5-3.
Excel–AMPL Experiment. The matrix below provides the distances among 10 cities (all off-diagonal missing entries = ). (For convenience, file prob11-32.txt gives the distance data in AMPL format.)1 2 3 4 5 6 7 8 9 10 1 100 2 11 80 5 39 95 28 2 17 42 33 21 59 46 79 29 3 63 57 92 55 68 52 4 36 27
Apply simulated annealing to the following problems starting with best nearestneighbor tour:(a) The paint sequencing problem of Example 11.1-1.(b) Problem 11-2..(c) Problem 11-5..(d) Problem 11-6..
Carry out three more iterations of Example 11.5-2.
Excel–AMPL Experiment. The matrix below provides the distances among 10 cities(all off-diagonal missing entries = ). (For convenience, file prob11-29.txt gives the distance data in AMPL format.)Use file ExcelTabuTSP.xls starting with the following:(a) A random tour.(b) Tour
Apply tabu to the following problems starting with best nearest-neighbor tour:(a) The paint sequencing problem of Example 11.1-1.(b) Problem 11-2..(c) Problem 11-5..(d) Problem 11-6..
Carry out three more iterations of Example 11.5-1.
Excel–AMPL Experiment. The matrix below provides the distances among 10 cities(all missing entries = ). (For convenience, file Prob.txt gives the distance matrix in AMPL format.)Use file excelReversalTSP.xls to implement the following situations:(a) Use the nearest-neighbor heuristic to determine
Apply the reversal heuristic to the following problems starting with best nearestneighbor tour:(a) The paint sequencing problem of Example 11.1-1..(b) Problem 11-2..(c) Problem 11-5..(d) Problem 11-6..
In Table 11.5 of Example 11.4-2, use the infinite-length disconnected tour 3-2-5-4-1-3(i.e., a tour missing at least one leg) as a starting tour to demonstrate that the subtour reversal heuristic can still lead to a solution that is just as good as when the heuristic starts with a connected tour.
In Table 11.5 of Example 11.4-2, specify the deleted and added legs associated with each of the two-at-a-time reversals.
AMPL experiment. In the circuit board model of Problem 11-9, the input data are usually given in terms of the (x, y)-coordinates of the holes rather than the distance between the respective holes. Specifically, consider the following (x, y) coordinates for a 9-hole board:Hole (x, y) in mm 1 (1, 2)2
AMPL experiment. Use AMPL to solve the following TSP problem by the cutting-plane algorithm:(a) Problem 11-3..(b) Problem 11-4..(c) Problem 11-12..
Write down the cuts associated with the following TSP:‘dij ‘ = • 43 21 20 10 12 9 22 30 20 10 5 13 14 30 42 20 44 7 9 10 μ
AMPL experiment. Use AMPL files amplAssign.txt and solutionAssign.txt to solve Problem 11-6, by B&B.
Solve Problem 11-9, by B&B.

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