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business
operations research an introduction

Questions and Answers of Operations Research An Introduction

A major oil company8 manufactures petroleum lubricants at sites j = 1,c, 10 using a critical additive purchased from suppliers i = 1,c, 15 all over the world. Manufacturing site j requires dj metric
Assume that Syntex Laboratories7 is reexamining the distribution of its sales force across major pharmaceutical products j = 1,c, 7.Present force levels ej are expected to produce sj units of product
A new automatic storage and retrieval(ASAR)6 area is being added to an existing warehouse on land already owned by the company. It will have n Ú 1 aisles, each with pallet storage cells on both
A laser printer manufacturer can make models i = 1,c, 6 at any of plants j = 1,c, 4.The fraction of plant j capacity required per unit of printer i has been estimated for each combination at fi,j.
A chemical manufacturer needs to produce 1250 barrels of a special industrial cleaning fluid by blending 5 available ingredients. The quality of the result is measured by 3 quantitative indices. The
A new premium whiskey will be produced by blending up to 5 different distilling products, and the quality of the results will be measured by 3 performance indices. The following table shows the value
A farmer wants to allocate between 10 and 60% of his available acreage to each of corn, soybeans, and sunflowers. With markets varying wildly from year to year, he has done some research on past
An investor has decided to divide his $1.5 million portfolio among government bonds, interestsensitive stocks, and technology stocks because some of these categories tend to increase return in
A light manufacturing firm is planning a new factory in a rural part of the western United States. A total of 100 employees are to be hired from the 5 surrounding communities. The following table
A solid waste company must locate 2 disposal sites to service the demand (tons per day) of the 5 communities detailed in the following table.Community 1 2 3 4 5 Demand 60 90 35 85 70 E-W coordinate 0
A warehousing firm services orders for its 5 products from an automatic storage and retrieval(ASAR) system, refilling storage from backup areas whenever the ASAR stock of any item reaches zero. The
A machinist will remove excess metal from a rotary (round) machine part by passing the cutting tool of a lathe along 42 inches of the part length. For a lathe turning at N revolutions per minute and
A print shop plans to maintain 5 different presses, replacing each every few years on a regular cycle. The following table shows the replacement cost (in thousands of dollars) of each press, and the
A company maintains inventories of its 5 prod ucts, replenishing the stock of an item whenever it reaches zero by manufacturing a fixed lot size of new units. The following table shows the setup cost
A partially buried, rectangular office building is to be constructed with a volume of at least 50,000 cubic meters. To minimize energy for heating and cooling, the exterior roof and sidewall surface
A lidless, rectangular box is to be manufactured from 30- by 40-inch cardboard stock sheets by cutting squares from the four corners, folding up ends and sides, and joining with heavy tape.The
American Olean15 makes product families i = 1,c, 10 of tile at plants p = 1,c, 4 to meet demands di,k (square feet) at sales distribution points (SDPs) k = 1,c, 120. Variable costs of production and
As commercial airliner14 makes stops j = 1,c, n of its daily routine and returns to where it started it takes on fuel for the next leg.Fuel is added at stop j to assure that the plane will arrive at
One of the ways that airlines operating through major hubs can improve their service is to allow as many transferring passengers as possible who land at one of the scheduled arrival–departure peaks
Major league baseball umpires12 work in crews that move among league cities to officiate series of 2–4 games. After every series, all crews move on to another that must involve different teams. To
KS brand tires11 are shaped in changeable molds type i = 1,c, m, installed in the company’s 40 presses. Production plans dictate the minimum ri, t and maximum ri, t numbers of molds i that should
A substantial part of United Parcel Service’s10 freight traffic moves as trailer-on-flatcars(i.e., with truck trailers traveling most of the way on railroad flatcars). The required number of
Freight trains9 run a regular weekly schedule in both the forward and reverse directions of a railroad’s main line through section boundary points i = 1,c, 22. Dividing the week into hourly time
To estimate the impact of proposed tax changes, the U.S. Department of the Treasury8 maintains two data files of records statistically characterizing the taxpayer population. Each of the i = 1,c,
A new highway7 is being built through terrain points i = 1,c, 40. The distance from i to i + 1 is di. To level the route, net earth deficits bi truckloads must be corrected at all nodes(bi 6 0 if
Forest fire control organizations in Canada’s provinces6 must reposition the numbers of observation aircraft available at stations i = 1,c, 11 on a daily basis to adjust for changing fire
Hilltop University (HU) is building a new campus on one of the highest hills in its hometown.The following table shows details about 6 key buildings under construction at the new site.No. Name Coords
A new grocery store has 3 weeks to train its full staff of 39 employees. There are 5 employees now. At least 2 employees must work on preparing the store during the next week, 5 employees the week
Maine Miracle’s 2 restaurants sell lobsters obtained from 3 fisherman. A total of 350 lobsters per day are served at the first restaurant, and 275 at the second. Each fisherman can ship up to 300
The Wonder Waste disposal company has 5 truckloads of nuclear waste and 5 truckloads of hazardous chemical wastes that must be moved from its current cleanup site to nuclear and chemical disposal
Three workstations are located on the circular conveyor of a manufacturing facility. Most of the flow between them moves from one station to the next on the conveyor. However, 7 units per minute must
Although Dynamic Programming methods of Chapter 9 are usually more efficient, the problem of finding a shortest path from a given origin s to a destination t in a graph with no negative dicycles can
Do Exercise 10-42(a)–(b) for each of the following maximum flow models:(a) Maximum flow Exercise 10-38(b) Maximum flow Exercise 10-39(c) Maximum flow Exercise 10-41
Consider the digraph below. Numbers on arcs are capacities.Do (a)–(e) of Exercise 10-42.
The capital city Capria of the remote Republic of Democracio (ROD) is under growing attack from murderous terrorist forces. To defend themselves the ROD forces in Capria have an urgent need for
Formulate as a minimum cost network flow problem, and solve by inspection, the problem of finding a maximum flow from the specified source to the specified sink in each of the following networks. Use
Makers of the new Ditti Doll are urgently trying to get as many to market as possible because a craze has created almost unlimited demand.One plant can supply up to 8 thousand per week to its
A relief agency is urgently trying to get the maximum possible quantity of supplies from its base at Alto to the volcano-ravaged city of Epi.One available road goes via Billi. The agency estimates
Do Exercise 10-36 on each of the following assignment models.(a) The digraph of Exercise 10-34(c).(b) The digraph of Exercise 10-35(c).
The following table shows the weights for assigning rows i to columns j in a maximum total weight assignment problem.j = 4 5 6 i = 1 25 13 22 2 21 14 19 3 20 25 29(a) Formulate the model as a linear
Paltry Properties has just acquired four rental homes. Paltry wishes to have the houses painted within the next week so that all can be available for the prime rental season. This means that each
Maize Mills has 800 thousand, 740 thousand, and 460 thousand bushels of corn stored at its three rural elevators. Its three processing plants will soon require 220 thousand, 1060 thousand, and 720
The following digraph depicts a partially solved minimum cost flow problem with labels on the nodes indicating net demand and those on the arcs showing unit cost, capacity, and current flow.(a)
Apply network simplex Algorithm 10C to compute an optimal flow in each of the following networks. Start from the flow given in the figure, using the basis specified below.(a) The network of Exercise
Do Exercise 10-25 on the problem.and basis 511, 22, 13, 12, 13, 426.
The following digraph depicts a partially solved minimum cost flow problem with labels on nodes indicating net demand and those on arcs showing unit cost, capacity, and current flow.(a) Verify that
Return to the minimum cost network flow instance of Exercise 10-20. The figure below depicts the same instance with different values for the current flow, including many marked ‘?’ for
The following depicts a network flow problem, with labels on nodes indicating net demand and those on arcs showing capacity.For each of the following lists of possible nonbasic arcs, either compute
Demonstrate that columns of the node–arc incidence matrix corresponding to arcs in each of the following cycles of the digraph in Exercise 10-1 form a linearly dependent set.(a) (2, 5), (5, 2)(b)
Do Exercise 10-20(b) on each of the following networks.(a) The network of Exercise 10-13(b) The network of Exercise 10-14(c) The network of Exercise 10-25(d) The network of Exercise 10-26
The digraph below shows a minimum cost network flow instance. Labels on arcs are (cost, capacity, current flow. Labels on the nodes show net demand.(a) Establish that the given flow is feasible.(b)
Do Exercise 10-18 on the network of Exercise 10-26.
Refer to the partially solved minimum cost network flow problem of Exercise 10-25.(a) Verify that the given flow is feasible.(b) Construct the residual digraph corresponding to the current flow.(c)
Add an artificial node and artificial arcs to prepare each of the following networks for twophase or big-M solution starting with a zero flow on all original arcs. Show the starting flow, and verify
The digraph below shows an instance of the minimum cost network flow problem. Labels on arcs are (cost, capacity, current flow), and those on nodes are net demand.(a) Show the corresponding
Solve each of the following by rudimentary cycle direction Algorithm 10A starting from the solution given in the figure. Find needed cycle directions by inspection.(a) The network in Exercise
Do Exercise 10-13 for the network
The following digraph shows a partially solved minimum cost network flow problem with node labels indicating net demand and arcs labels showing unit cost, capacity, and current flow.(a) Verify that
Determine whether each of the following sequences is a chain, a path, a cycle and/or a dicycle of the network in Exercise 10-2.(a) 2–3–4–2(b) 1–3–5–4(c) 3–4–2(d) 2–3–4–5
Determine whether each of the following sequences is a chain, a path, a cycle, and/or a dicycle of the network in Exercise 10-1.(a) 1–2–4–1(b) 3–4–2(c) 3–4–1(d) 3–4–5–3
Return to the Crazy Crude problem of Exercise 10-8, and suppose now that we wish to plan over a 2-day time horizon. Refinery demand remains 2000 on the first day but will be 3000 on the second. There
Return to the Super Sleep problem of Exercise 10-7, and suppose now that we wish to plan over a 2-week time horizon. Customer demands remains 160 and 700 in the first week, but they are predicted to
Crazy Crude oil company can produce 1500 barrels per day from one of its fields and 1210 from the other. From there the crude oil can be piped to either of two tank farms, one at Axel and the other
Super Sleep is a company making matresses for king-size beds. Matresses can be shipped directly from either of its plants to retail store customers, or they may be transshipped through the
Do Exercise 10-5 for the network
The following digraph represents a network flow problem with values at nodes showing net demand.1 2 3-100-50 80(a) Verify that total supply exceeds total demand.(b) Add a new sink to create an
Do Exercise 10-3 for the matrix§-1 -1 0 0 0 1 0 1 1 0 0 1 -1 0 1 0 0 0 -1 -1¥
Consider the matrix§-1 0 0 1 0 1 -1 -1 0 0 0 1 0 -1 -1 0 0 1 0 1¥(a) Explain why it is a node–arc incidence matrix.(b) Draw the corresponding digraph.
Do Exercise 10-1 for the network.
The figure that follows depicts a minimum cost network flow problem. Numbers on the nodes show net demand, while those on arcs show unit cost and capacity.(a) Identify the node set V and the arc set
Elite Air (EA)2 is a business-class only airline advertising complete meals for all its passengers.EA must choose, then update the number to order qe, at epochs e = 4,c, 0, which is initially set to
Mini Job (MJ) is a small job shop manufacturer with a contract to stamp 200 copies per day for the next 5 days of a metal door panel needed by an automobile manufacturer. MJ’s machining center that
Mindy is playing a gambling game of 3 rounds. She will start with 4 chips, and she can wager any number of chips she has on hand at each round. With probability 0.45, she will win the bet and receive
Repeat Exercise 9-37, increasing the KP instance on the right-hand side to 19.
Consider solving the following Knapsack ILP by Discrete Dynamic Programming (DDP):min 18x1 + 13x2 + 20x3 + 12x4 s.t. 2x1 + 6x2 + 4x3 + 3x4 Ú 14 x1, x2, x3, x4 [0, 2] and integer(a) Define the
Do Exercise 9-35 with all parameters the same except a weight limit of 40 pounds.
A copy machine repairman has four pieces of test equipment for which he estimates 25%, 30%, 55%, and 15% chances of using them at his next stop. However, the devices weigh 20, 30, 40, and 20 pounds,
Do Exercise 9-33 with all parameters the same except a holding cost of $1000.
A pharmaceutical manufacturer must supply 30 batches of its new medication in the next quarter, then 25, 10, and 35 in successive quarters.Each quarter in which the company makes product requires a
The campus shuttle bus begins running at 7:00 p.m. and continues until 2 a.m. Several drivers will be engaged, but only one should be on duty at any time. If a shift starts at or before 9:00 p.m., a
The table that follows shows the activities required to construct a new computer laboratory, along with their estimated durations (in weeks)and predecessor activities.Activity Time Predecessors Order
Determine whether acyclic shortest path Algorithm 9D could be applied to compute shortest paths from node 1 to all other nodes in each digraph of Exercise 9-17. If so, explain whether it would be
Do Exercise 9-15 on the graph of Exercise 9-2.
Return to the graph of Exercise 9-1, and suppose that we seek shortest paths from node 1 to all other nodes.(a) Explain why Dijkstra Algorithm 9C can be employed to compute the required shortest
Use Floyd–Warshall Algorithm 9B to identify a negative dicycle in each of the following graphs.(a) The digraph of Exercise 9-7(b) The digraph of Exercise 9-8
Do Exercise 9-12 on the graph of Exercise 9-6.
Return to the graph of Exercise 9-5, and suppose that we seek shortest paths from all nodes to all other nodes.(a) Explain why Floyd–Warshall Algorithm 9B can be employed to compute the required
Use Bellman–Ford Algorithm 9A to identify a negative dicycle in each of the following graphs.(a) The digraph of Exercise 9-7(b) The digraph of Exercise 9-8
Do Exercise 9-9 on the graph of Exercise 9-2.
Return to the graph of Exercise 9-1, and suppose that we seek shortest paths from node 1 to all other nodes.(a) Explain why Bellman–Ford Algorithm 9A can be employed to compute the required
Do Exercise 9-3 for the problem of Exercise 9-2, verifying the optimal 1 to 3 path in part (b).
Return to the problem of Exercise 9-1.(a) Find (by inspection) shortest paths from node 1 to all other nodes.(b) Verify that every subpath of the optimal 1 to 2 path in part (a) is itself optimal.(c)
Consider the linear constraints-w1 + w2 … 1 w2 … 3 w1, w2 Ú 0(a) Sketch the feasible space in a 2-dimensional plot.(b) Determine geometrically whether each of the following solutions are
Do Exercise 5-1 for the LP 3w1 + 5w2 … 15 5w1 + 3w2 … 15 w1, w2 Ú 0 and points w112 = 10, 02, w122 = 11, 12, w132 =12, 02, w142 = 13, 32 and w152 = 15, 02.
Place each of the following LPs in standard form and identify the corresponding A,b, and c of definition 5.4 .(a) min 4x1 + 2x2 - 33x3 s.t. x1 - 4x3 + x3 … 12 9x1 + 6x3 = 15-5x1 + 9x2 Ú 3 x1, x2,
Consider the linear constraints-y1 + y2 … 2 5y1 … 10 y1, y2 Ú 0(a) Sketch the feasible set in a 2-dimensional plot.(b) Add slacks y3 and y4 to place constraints in LP standard form.(c) Determine
Do Exercise 5-4 for the LP y1 + 2y2 … 6 y2 … 2 y1, y2 Ú 0 and possible basic sets 5y1, y26,5y2, y36,5y16, 5y2, y46,5y2, y3, y46,5y1, y36.
Write all conditions that a feasible directionw must satisfy, at the solution w indicated, to each of the following standard-form systems of LP constraints.(a) 5w1 + 1w2 - 1w3 = 9 at w = 12, 0, 12
Following is a maximizing, standard-form linear program and a classification of variables as basic and nonbasic.x1 x2 x3 x4 max c 10 1 0 0 b-1 1 4 21 13 2 6 0 -2 2 B N B B(a) Compute the current
Do Exercise 5-7 for minimizing the standard form LP x1 x2 x3 x4 min c 8 -5 0 1 b 13 2 3 1 7-4 1 0 -1 -1 N N B B

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