Questions and Answers of Operations Research An Introduction

Chicken feed is transported by trucks from three silos to four farms. Some of the silos cannot ship directly to some of the farms. The capacities of the other routes are limited by the number of
A parent has five (teenage) children and five household chores to assign to them. Past experience has shown that forcing chores on a child is counterproductive. With this in mind, the children are
Four factories are engaged in the production of four types of toys. The following table lists the toys that can be produced by each factory.All toys require approximately the same per-unit labor and
In Example 6.3-6, use LP to determine the shortest routes between the following pairs of nodes:(a) Node 1 to node 5.(b) Node 2 to node 5.Example 6.3-6In the network of Example 6.3-4, determine the
The academic council at the U of A is seeking representation from among six students who are affiliated with four honor societies. The academic council representation includes three areas:
Develop the linear program and the associated minimum-cost flow network for the employment scheduling model of Example 6.6-3 using the following 5-month demand data. The per-worker costs of
Use vectors to determine graphically the type of solution for each of the sets of equations below: unique solution, an infinite number of solutions, or no solution. For the cases of unique solutions,
In the solution space in Figure 7.6 (drawn to scale), express the interior point (3, 1) as a convex combination of the extreme points A, B, C, and D by determining the weights associated with each
In the following LP, compute the entire simplex tableau associated with XB = (x1, x2, x3)T.Minimize z = 2x1 + x2subject to 3x1 + X2 - Х3 4x1 + 3x2 x1 + 2x2 -Х4 = 3 = 6 +X; = 3 х1, х2, Х3, X4, X5
An opinion survey involves designing and printing questionnaires, hiring and training personnel, selecting participants, mailing questionnaires, and analyzing the data. Construct the project network,
For each of the following activities, determine the maximum delay in the starting time relative to its earliest start time that will allow all the immediately succeeding activities to be scheduled
Determine graphically the extreme points of the following convex set:Q = { x1 , x2|x1 + x2 + 2, x1 ≥ 0, x2 ≥ 0}Show that the entire feasible solution space can be determined as a convex
True or False?(a) The system BX = b has a unique solution if B is nonsingular.(b) The system BX = b has no solution if B is singular and b is independent of B.(c) The system BX = b has an infinity of
The following is an optimal LP tableau:The variables x3, x4, and x5 are slacks in the original problem. Use matrix manipulations to reconstruct the original LP, and then compute the optimum objective
In the following sets of equations, (a) and (b) have unique (basic) solutions, (c) has an infinite number of solutions, and (d) has no solution. Show how these results can be verified using graphical
In the matrix simplex tableau, suppose that X = (XI, XII)T, where XII corresponds to a typical starting basic solution (consisting of slack and/or artificial variables) with B = I, and let C = (CI,
Prove that, in any simplex iteration, zj − cj = 0 for all the associated basic variables.
(a) Prove that if zj − cj > 0 (<0) for all the nonbasic variables xj of a maximization (minimization) LP problem, then the optimum is unique. (b) Else, if zj – cj equals zero for a
An LP model includes two variables x1 and x2 and three constraints of the type ≤. The associated slacks are x3, x4, and x5. Suppose that the optimal basis is B = (P1, P2, P3), and its inverse isThe
Consider an LP in which the variable xk is unrestricted in sign. Prove that by substituting xk = xk− − xk+, where xk− and xk+ are nonnegative, it is impossible that the two variables replace
Given the general LP in equation form with m equations and n unknowns, determine the maximum number of adjacent extreme points that can be reached from a nondegenerate extreme point (all basic
In applying the feasibility condition of the simplex method, suppose that xr = 0 is a basic variable and that xj is the entering variable with (B−1Pj)r ≠ 0. Prove that the resulting basic
In the implementation of the feasibility condition of the simplex method, specify the mathematical conditions for encountering a degenerate solution (at least one basic variable = 0) for the first
What are the relationships between extreme points and basic solutions under degeneracy and nondegeneracy? What is the maximum number of iterations that can be performed at a given extreme point
Considered the following parameterized LP, given t ≥ 0.Minimize z = 4x1 + x2 + 2x3 subject to(a) Solve the LP at t = 0 using TORA Iterations module, from which one can determine XB and B0-1.(b)
Consider the LP Maximizez = CX subjectto AX ≤ b, X ≥ 0, where b ≥ 0 After obtaining the optimum solution, it is suggested that a nonbasic variable xj can be made basic (profitable) by
Prove that the dual of the dual is the primal.
Define the dual problem given the primal is min z = {CX AX ≥ b, X ≥ 0}.
Study the change in the optimal solution of the following parameterized LP, given t ≥ 0.Minimize z = (4 − t )x1 + (1 − 3t )x2 + (2 − 2t )x3 subject to 3x₁ + x₂ + 2x3 = 3 4x₁ + 3x₂ +
Mantel produces a toy carriage, whose final assembly must include four wheels and two seats. The factory producing the parts operates three shifts a day. The following table provides the amounts
Write the dual of max z = {CX AX = b, X unrestricted}.
Camyo Manufacturing produces four parts that require the use of a lathe and a drill press. The two machines operate 10 hours a day. The following table provides the time in minutes required by each
Show that the dual of max z = {CX AX ≤ b, 0 < L ≤ X ≤ U} always has a feasible solution.
Formulate the Fairville tax problem, assuming that the town council is specifying an additional goal, G5, that requires gasoline tax to equal at least 10% of the total tax bill.
Two products are manufactured on two sequential machines. The following table gives the machining times in minutes per unit for the two products.The daily production quotas for the two products are
Ozark University admission office is processing new applications for the upcoming academic year. The applications fall into three categories: in-state, out-of-state, and international. The
In each of the following cases, determine the feasible extreme points graphically, and express the feasible solution space as a function of these extreme points. If the solution space is unbounded,
The annual demand of electricity in small town is estimated at 120MWH annually received primarily its Co-op but can be supplemented from the grid network at $160/ MWH in case of shortages. The Co-op
Consider the following three-letter words: AFT, FAR, TVA, ADV, JOE, FIN, OSF, and KEN. A letter is assigned a code starting with A = 1 and ending with Z = 26. Each word is scored by adding numeric
A parent left a collection of paintings of equal values to be distributed among the three children: Tarek receives at least one-half of the paintings, Sharif gets at least one-third, and Maisa gets
The three children of a farm couple are sent to the market to sell 90 apples. Karen, the oldest, carries 50 apples; Bill, the middle one, carries 30; and John, the youngest, carries only 10. The
Once upon a time, there was a captain of a merchant ship who wanted to reward three crew members for their valiant effort in saving the ship’s cargo during an unexpected storm in the high seas. The
A manufacturing process uses four interchangeable raw materials. The raw materials differ in properties, which leads to different output units per unit of raw material. They also differ in cost and
The world-renowned logic puzzle, Sudoku, deals with a 9 × 9 grid subdivided into 9 nonoverlapping 3 × 3 subgrids. The puzzle calls for assigning the numerical digits 1 through 9 to the cells of the
You have access to any number of toy molds of the number 8. The pieces can be stacked contiguously in sets of one or more, to form an all-8 algebraic number. For example, a 3-peice set reads as 888.
You have a 4 × 4 grid and a total of 10 tokens. Use ILP to place the tokens on the grid such that each row and each column will have an even number of tokens.
A street vendor selling electronic gadgets was robbed of all possessions. When reporting the matter to the police, the vendor could not recall the number of gadgets but stated that when dividing the
A widely circulated puzzle requires assigning a single distinct digit (0 through 9) to each letter in the equation SEND + MORE = MONEY. Formulate the problem as an integer program, and find the
Professor Yataha needs to schedule six round-trips between Boston and Washington, D.C. The route is served by three airlines—Eastern, US Air, and Continental—and there is no penalty for the
A game board has 3 × 3 equal squares. You are required to fill each square with a number between 1 and 9 such that the sum of the numbers in each row, each column, and each diagonal equals 15.
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 four
PackRat Outdoors owns three sports centers in downtown Highfill. On Easter Day, bicycle riding is a desirable outdoors activity. A total of eight rental bikes is to be allocated to the three centers
A 10-m3 compartment is available for storing three items. The volumes needed to store 1 unit of items 1, 2, and 3 are 2, 1, and 3m3, respectively. The probabilistic demand for the items is specified
HiTec has just started to produce supercomputers for a limited period of 4 years. The annual demand, D, for the new computer is described by the following distribution:The production capacity of the
Suppose that the product quantity zw occurs in a constraint, where z and w are binary variables. Show how this term can be linearized.
Habitat for Humanity is a nonprofit organization that helps families build and improve places to call home. They rely on volunteer labor and donated building materials. An eligible family can choose
The height of a cylindrical water tank must be at least twice as much as its base diameter. Neither the diameter nor the height can exceed 10 ft. The volume of the tank must be at least 300 ft3. The
You have a piece of wire whose length is L = 107.1 inches and you would like to shape it into a rectangular frame. Use the GA to determine the width and height that will yield the maximum area of the
Consider the following problem Maximize f(x, y) = xsin (4x) + 1.1sin(2y), x = 0, 1, 2,…,10, y = 0, 1, 2,…,10 Carry out five GA iterations to estimate the optimum solution.
In the game of chess, queens move horizontally, vertically, or along a (45°) diagonal path. It is necessary to position N queens in (N × N ) grid so that no queen can “take” any other queen.
I have a small backyard garden that measures 10 × 20 feet. 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
GECO is contracted for the next 4 years to supply four aircraft engines per year. Available production capacity and production costs vary from year to year. GECO can produce five engines in year 1,
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
A farmer owns a herd of K sheep now. At the end of each year, the farmer decides how many to sell or keep. The profit from selling a sheep in year i is pi. The sheep kept at the star of year i will
I would like to sell my used car to the highest bidder. Studying the market, I have concluded that I am likely to receive three types of offers with equal probabilities: low at about $1050, medium at
Current-year balance sheet of a company shows beginning and end inventories of $100.3M million and $71.2M million, respectively. The net revenue from sales for the year is $230.3M million and the
small business financial data shows that its inventory level of an item is held steady at 1000 units during the first 9 months of the year. Sales accelerated during the last quarter in time for
McBurger orders ground meat at the start of each week to cover the weeks 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)
A company stocks an item that is consumed at the rate of 50 units per day. It costs the company $20 each time an order is placed. An inventory unit held in stock for a week will cost $.35.(a)
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
Consider the inventory situation in which the stock is replenished uniformly (rather than instantaneously) at the rate a. Consumption occurs at the constant rate D. Because consumption also occurs
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
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
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
An item is manufactured to meet known demand for four periods according to the following data:(a) Find the optimal solution, indicating the number of units to be produced in each period.(b) Suppose
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 12345 Demand D,
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
Find the optimal inventory policy for the following six-period inventory situation: The unit production cost is $2 for all the periods. Period I 123456 6 D; (units) 10 15 7 20 13 25 K; ($) h;
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
Solve the following 10-period deterministic inventory model. Assume an initial inventory of 50 units.This problem is purposely designed too big for hand computations, primarily to encourage the use
Suppose that the inventory-holding cost is based on the average inventory during the period. Develop the corresponding forward recursive equation.
The unit holding cost in a single-period inventory situation is $1. If the order quantity is 4 units, find the permissible range of the unit penalty cost implied by the optimal conditions. Assume
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
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
The daily demand for a popular CD in a music store is approximately N(200, 20). The cost of keeping the CD on the shelves is $.04 per disc per day. It costs the store $100 to place a new order. There
The daily demand for printer cartridges in at an office supply store is N(30, 5). The cost of holding a roll in the store is $.02 per day, and the fixed cost of placing a replenishment order is $30.
The demand for an item during a single period occurs instantaneously at the start of the period. The associated pdf is uniform between 10 and 15 units. Because of the difficulty in estimating the
The U of A Bookstore offers a program of reproducing class notes for participating professors. Professor Yataha teaches a freshmen-level class with an enrollment of between 200 and 250 students,
QuickStop provides its customers with coffee and donuts at 6:00 a.m. each day. The convenience store buys the donuts for 7 cents apiece and sells them for 25 cents apiece until 8:00 a.m. After 8:00
Colony Shop is stocking heavy coats for next winter. Colony pays $50 for a coat and sells it for $110. At the end of the winter season, Colony offers the coats at $55 each. The demand for coats
For the single-period model, suppose that the item is consumed uniformly during the period (rather than instantaneously at the start of the period). Develop the associated cost model, and find the
Consider a two-period probabilistic inventory model in which the demand is backlogged, and orders are received with zero delivery lag. The demand pdf per period is uniform between 0 and 10, and the
Suppose that the Q-P demand curve for a commodity is Q = 100/P.(a) Graph the Q-P curve.(b) Determine the optimum revenue.
The quarterly average yield for the airlines industry is computed asDomestic U.S. flight’s average trip miles per passenger per quarter is approximately 2900 for 2009, 3300 miles for 2014. The
The pdf of the demand per period in an infinite-horizon inventory model is given as f(D) = .08D,0 ≤ D ≤ 5The unit cost parameters areUnit selling price = $10Unit purchase price = $8Unit holding
In Example 14.2-1 suppose that the overhead cost associated with each ticket is $100. Reproduce the graph in Figure 14.1 to provide the net revenue. Does the optimum price per ticket change?Example
Consider a flight with a 90-seat capacity operating on a route with the following fares:(a) Determine the booking limits for all fares.(b) Determine the booking limits after the following sequential
For the situation in Example 14.2-4, determine the optimum business protection level given business demand is discrete uniform in the range [30, 49] with f = $750 and d = $350.Example
In Example 14.2-3, given the initial booking limits (50, 25, 13, 9, 3), determine the booking limits resulting from the following sequential requests: 15 seats of fare 2, 4 seats of fare 3, 5 seats
Consider the following marketing strategies designed to increase revenue:(a) In the book publishing business, publishers print new titles in hardcover, and at a later date the same title is reprinted

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