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operations research an introduction

Questions and Answers of Operations Research An Introduction

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
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
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
An item is manufactured to meet known demand for four periods according to the following data:Unit production cost ($) for period Production range (units) 1 2 3 4 1–3 1 2 2 3 4–11 1 4 5 4 12–15
Solve Example 13.4-1, assuming that the unit production and holding costs are as given in the following table:Period i Regular time unit cost ($)Overtime unit cost ($)Unit holding cost ($)to period i
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
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, $55, and
Solve the model of Example 13.3-3, assuming that we require the sum of the average inventories for all the items to be less than 25 units.
The following data describe five inventory items:11 Item i Ki ($) 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
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
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
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
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
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
A company can produce an item or buy it from a contractor. If it is produced, it will cost $20 each time the machines are set up. The production rate is 100 units per day. If it is bought from a
Consider the inventory situation in which the stock is replenished uniformly (rather than instantaneously) at the ratea. Consumption occurs at the constant rate D. Because consumption also occurs
Lewis (1996). An employee of a multinational company is on loan from the United States to the company’s subsidiary in Europe. During the year, the employee’s financial obligations in the United
A hotel uses an external laundry service to provide clean towels. The hotel generates 600 soiled towels a day. The laundry service picks up the soiled towels and replaces them with clean ones at
Walmark Store compresses and palletizes empty merchandise cartons for recycling. The store generates five pallets a day. The cost of storing a pallet in the store’s back lot is$.10 per day. The
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
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)
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)
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)
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
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
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,
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
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
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
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
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
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
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
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
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.
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
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
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)
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
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
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
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
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
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
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 =
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
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 =
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
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
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
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
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
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
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
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.
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.
Solve Problem 11-7, by B&B.
Solve Problem 11-2, by B&B.
Solve Example 11.3-1 using subtour 2-5-4-2 to start the branching process at node 1, using the following sequences for exploring the nodes:(a) Explore all the subproblems horizontally from left to
Warehouse order picking, Ratliff and Rosenthal (1983). In a rectangular warehouse, a stacker overhead crane is used to pick and deliver orders between specified locations in the warehouse. The tasks
Wallpaper cutting, Garfinkel (1977). Covering the walls of a room usually requires cutting sheets of different lengths to account for doors and windows, and the like. The sheets are cut from a single
Automatic guided vehicle. An AGV makes a round-trip (starting and ending at the mailroom)to deliver mail to 5 departments on a factory floor. Using the mailroom as the origin (0, 0), the (x, y)
The U.S. space agency, NASA, uses satellites for imaging celestial objects. The amount of fuel needed to reposition the satellites is a function of the sequence in which the objects are imaged. The
DNA sequencing. In genetic engineering, a collection of DNA strings, each of length 10 ft, is concatenated to form one universal string. The genes of individual DNA strings may overlap, thus
(Integrated circuit boards) Circuit boards (such as those used in PCs) are drilled with holes for mounting different electronic components. The boards are fed one at a time under a moving drill. The
Meals-on-Wheels is a charity service that prepares meals in its central facility for delivery to people who qualify for the service. Ideally, all meals should be delivered within 20 min from the time
A manager has a total of 10 employees working on six projects. Projects are reviewed weekly with each employee. A project may employ more than one employee resulting in assignment overlaps, as the
A tourist in New York City uses local transportation to visit 8 sites. The start and end and the order in which the sites are visited are unimportant. What is important is to spend the least amount
Proteins clustering. Proteins are clustered using an overall measure of similarity based on protein–protein interaction information. Clustering information is used to predict unknown protein
A baseball fan wishes to visit eight major league parks in (1) Seattle, (2) San Francisco,(3) Los Angeles, (4) Phoenix, (5) Denver, (6) Dallas, (7) Chicago, and (8)Tampa before returning home to
Seers Service Center schedules its daily repair visits to customers. The matrix ‘Tij ‘ below gives the travel time (in minutes) between the service center (row 1 and column 1)and seven jobs. The
A book salesperson who lives in Basin must call once a month on four customers located in Wald, Bon, Mena, and Kiln before returning home to Basin. The following table gives the distances in miles
In each of the following instances, describe the data elements (cities and distances)needed to model the problem as a TSP.(a) Seers Service Center schedules its daily repair visits to customers. The
Repeat Problem 10-41 using the variable y.
Construct the search tree in Figure 10.3 using the variable x to initiate the search.Compare the resulting amount of computations with that in Figure 10.3.
Excel experiment. Apply excelIPHeuristicGA.xls to Problem 10-37.

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