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business
operations research an introduction
Optimization In Operations Research 2nd Edition Ronald Rardin - Solutions
Do Exercise 17-36 using squared penalty functions. Stop the search in part (f) when the total constraint violation is … 0.2.
Consider the NLP min 21x1 - 322 - x1x2 + 1x2 - 522 s.t. 1x122 + 1x222 … 4 0 … x1 … 2, x2 Ú 0 with optimal solution x* = 11.088, 1.6782.(a) Use unsquared penalty functions to reduce this problem to an unconstrained penalty model.(b) Explain why local minima of the unconstrained model in part
Do Exercise 17-34 using squared penalty functions.
Use absolute value (unsquared) penalty functions to reduce each NLP of Exercise 17-30 to an unconstrained penalty model.
Do Exercise 17-32 for NLP max 2 ln1x12 + 8 ln1x22 s.t. 4x1 + x2 = 8 x1, x2 Ú 1 with nonoptimal point x = 11, 42, improving feasible direction x = 1 -1, 42, and global optimum x* = 11, 42.
Consider the NLP min 151x122 + 41x222 s.t. 3x1 + 2x2 = 8 x1, x2 Ú 0(a) State the KKT optimality conditions for this model.(b) Verify that at solution x = 10, 42 there exists an improving feasible direction x = 12, -32.(c) Confirm that KKT conditions have no solution for the nonoptimal x of part
For each mathematical program in Exercise 17-30, determine whether principle 17.26 assures that a KKT point is a global optimum.
State the Karush–Kuhn–Tucker optimality conditions for each of the following mathematical programs.(a) min 141x1 - 922 + 31x2 - 522 + 1x3 - 1122 s.t. 2x1 + 18x2 - x3 = 19 6x1 + 8x2 + 3x3 … 20 x1, x2 Ú 0(b) max 6x1 + 40x2 + 5x3 s.t. x1 sin1x22 + 9x3 Ú 2 e18x1 + 3x2 + 14x3 … 50 x2, x3 Ú
Do Exercise 17-28 for the NLP max 300 - 51x1 - 2022 - 41x2 - 622 s.t. x1 + x2 = 8 and part (d) extra constraint x2 Ú 0.
Consider the nonlinear program min 81x1 - 222 + 21x2 - 122 s.t. 32x1 + 12x2 = 126(a) Form the Lagrangian function for this model.(b) Write stationary conditions for the Lagrangian.(c) Solve your stationary conditions for x1 and x2, and explain why your answers are optimal in the original model.(d)
Demonstrate that each of the following NLPs is a posynomial geometric program by placing the model in standard form and detailing the sets Ki, and associated coefficients dk and ak,j.(a) min 13x1x2>x3 + 91x1x3 s.t. 3x1 + 8x2 … x3 20> 1x324 … 4 x1, x2, x3 7 0(b) min 40>x1 + x2>1x3 s.t. x1x2 …
Determine whether each of the following is a posynomial.(a) 23x1 - 34x2 + 60x3(b) 54x1 + 89x2 + 52x3(c) 7x1x2> 1x322.3 + 41x1(d) 44x1> ln 1x22 + e-x3
Determine whether each of the following NLPs is a quadratic program, and if so, identify the c and Q of matrix objective function form c # x + xQx.(a) min x1x2 + 134>x3 + ln1x12 s.t. x1 + 4x2 - x3 … 7 14x1 + 2x3 = 16 x1, x2, x3 Ú 0(b) min 61x122 + 34x1x2 + 51x222-12x1 + 19x2 s.t. 7x1 + 3x2 Ú 15
Determine which of the NLPs in Exercise 17-23 are separable programs.
Determine whether each of the following NLP’s is a convex program.(a) max l n1x12 + 3x2 s.t. x1 Ú 1 2x1 + 3x2 = 1 1x122 + 1x222 … 9(b) min x1 + x2 s.t. x1, x2 … 9-5 … x1 … 5-5 … x2 … 5(c) max x1 + 6>x1 + 51x222 s.t. 4x1 + 6x2 … 35 x1 Ú 5, x2 Ú 0(d) min 14x1 + 9x2 - 7x3 s.t. 6x1 +
The commander of a battlefront11 must plan how to employ his f frontline and r reserve firepower to minimize the advance achieved over days t = 1,c, 14 by an attack of opposing forces with firepowera. Intelligence and battle simulations predict that each surviving unit of firepower in the attacking
Three urban neighborhoods are mutually connected by freeways admitting traffic in both directions. Net output bi,k (per hour) at each neighborhood k of vehicles originating at i can be estimated from known patterns 1bi,k = -aki bi,k2. The delay vehicles experience on any arc (i, j) is an
Each day qi tons of freight arrive by sea10 in Japan bound for in-country regions i = 1,c, 50.These goods may arrive at any of the major ports j = 1,c, 17, but the internal transportation cost per ton ci,j varies by port and destination.The government plans a capital investment program in port
A stirred tank reactor9 is a tank equipped with a large stirring device that is used in the chemical and biochemical industry to produce chemical reactions. A series of 5 such tanks will be used to lower the concentration of toxic chemical from c0 at input for tank 1, to no more that cQ on exit
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 tons per month of the additive, and suppliers i can provide up to si metric tons. The transportation
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 sales per month at a profit margin of pj per unit. However, extensive discussion and surveying has
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 sides and a stacker crane moving in the middle which can carry a pallet to/from any location in the
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. The laser printer market is very competitive, so the price that can be charged for any model is a
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 following table show the index values for each ingredient, along with the minimum and maximum
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 of each index for the 5 ingredients, along with lower and upper limits for the index of the blend
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 performance to guide his decisions. The following table shows the average return per acre and the
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 periods when the others decrease. Specifically, his analysis of recent experience produced the following
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 shows the number of (equally) qualified workers available in each community and community location
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 4 30 20 16 N-S coordinate 0 30 8 17 15 Each site will be able to handle up to 200 tons per day, and
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 following table shows the weekly demand and the unit volume (cubic feet) for each product.Product 1 2
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 advancing the tool at a feed rate of f inches per revolution, classic empirical relationships
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 estimated annual income (in thousands of dollars)that each can generate when new. However, as 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 for manufacturing, the unit volume, the unit annual inventory holding cost, and the estimated
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 exposed above ground should not exceed 2250 square meters. Within these limits, the designer wishes
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 designer wishes to choose box dimensions that maximize volume.(a) Formulate this design problem as a
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 transportation total ci,p,k per square foot to make tile of family i at plant p and ship to SDP k.
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 stop j + 1 with at least the required safety reserve rj + 1. Fuel unit costs cj (dollars per pound)
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 to carry on with their next flight on the same plane.13 Such through-flight connections must
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 provide adequate travel time, any crew that works a series ending with a night game must also not go
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 be operational during time period t = 1,c, n, and the planning period begins t = 0 with numbers of
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 truckloads di,j to be shipped between points i, j = 1,c, n in this way is known, but UPS can use either
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 blocks t = 1,c, 168 (with t = 1 following t = 168), a train leaving i bound for j 7 i advances through
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, 10,000 records in the first file represents a known number of families ai and describes corresponding
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 surplus, ai bi = 0). This will be done by moving truckloads of earth from surplus to deficit points
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 threats.The required number ri and the present number pi are known for all stations, along with the cost ci,j
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 x y Altitude 1 Administration 100 100 300 2 Library 52 55 210 3 Student Union 151 125 204 4
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 after, and 10 in the final week before opening. Employees assigned to these duties earn $300 per 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 per day, but not all arrive suitable for serving. The following table show the cost(including
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 facilities, respectively. The following table shows that many of the available roads are restricted for
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 move from each station to the station after the next. As much as possible of this 2-step flow
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 easily be represented as a minimum cost flow problem. Using arc lengths as costs, it is only
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 rocket propelled grenades (RPGs)which they use in close combat. There is a supply of 400 RPGs in the ROD
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 capacities specified on the original digraph.(a) Source 3, sink 2 in the network of Exercise
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 distribution center, but that center can then get only 3 thousand per week to east region customers, and 1
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 that the Alto-to-Billi part of that road can carry 500 tons per day, and the Billi-to-Epi segment, 320
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 assignment model.(b) Construct the starting dual solution and equality subgraph of Hungarian
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 house will have to be painted by a different contractor. The following table shows the bids (thousands
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 thousand respectively, to make cornstarch. The following table shows the cost per thousand bushels of
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) Verify that the given solution is a basic feasible solution for basis 511, 22, 11, 32, 13, 426.(b)
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 10-13 with basis 511, 22, 12, 42, 13, 426(b) The network of Exercise 10-14 with basis 511, 22, 12,
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 the given solution is a basic feasible solution for basis 511, 22, 11, 32, 12, 426.(b) Compute all
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 unknown.(a) Demonstrate that arcs with flow indicated as ‘?” form a basis of the corresponding LP.(b)
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 the corresponding basic solution and indicate whether it is basic feasible, or apply principle 10.33
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) (4, 2), (5, 2), (4, 5)(c) (1, 2), (4, 2), (1, 4)(d) (1, 4), (4, 5), (5, 3), (3, 1)
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) Start from the given solution and solve the instance to optimality by the Cycle Cancelling Algorithm
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) Use Floyd-Warshall Algorithm 9B on your residual digraph to identify an improving feasible cycle
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 that it balances at the artificial node.(a) The network of Exercise 10-1(b) The network of Exercise
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 Node–Arc Incidence Matrix.(b) Start from the given solution and solve the instance to optimality by the
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 10-13(b) The network in Exercise 10-14
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 the current flow is feasible.(b) Generate the six possible cycle directions of flow change.(c) Verify
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 are no initial inventories, but either tank farm can hold over petroleum at $0.05 per barrel per
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 be 300 and 810 in the second. There is no initial inventory, but matresses may be held in the
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 at Bull. Axel then trucks oil on to the Crazy Crude refinery at $0.40 per barrel to help meet its
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 company’s single warehouse. The table that follows shows the unit cost of shipping matresses from the plants
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 equivalent network with total supply equal to total demand.
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 A of the network.(b) Classify all nodes as source, sink, or transshipment.(c) Verify that total
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 the number of booked and standby passengers b4 known at that time. Values of both qe and be can
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 does the stamping can meet that demand with regular 40-hour shifts, costing a total of $3000 in
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 a number of additional chips equal to her wager. Otherwise, with probability 0.55 she will lose all
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 states and stages of the DDP to compute an optimum.(b) Draw a digraph from which an optimum can be
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, respectively, and he can carry no more than 60 pounds. The repairman seeks a maximum utility
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 $100,000 setup, plus $3000 per batch produced. There is no limit on production capacity.Batches can
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 regular driver can be engaged for a 4-hour shift at cost $50. Otherwise, part-time drivers will be
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 furniture (OF) 1 None Order computers (OC) 1 None Order software (OS) 1 OC Furniture delivery (FD)
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