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operations research an introduction

Operations Research An Introduction 11th Edition Hamdy A. Taha - Solutions

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 anywhere between their earliest and latest completion times(a) TF = 10, FF = 10, D = 4(b) TF = 10,
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 combination of its extreme points. Hence conclude that any convex (bounded) solution space is totally defined
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 solutions if B is singular and b is dependent.
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 value. Basic N Z. X3 x2 X1 X1 0 0 0 1 X2 0 0 1 0 x3 0 1 0 0 X4 3 1 1 -1 X5 2 -1 0 1 Solution ? 2 6 2
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 vector representation. From this exercise, state the general conditions for vector
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, CII) and A = (D, I) be the corresponding partitions of C and A, respectively. Show that the matrix
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 nonbasic xj, then the problem has an alternative optimum solution.
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 optimal primal and dual solutions areDetermine the optimal value of the objective function in two
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 one another in an alternative optimum solution.
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 variables are > 0) of the solution space.
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 solution remains feasible even if (B−1Pj)r is negative.
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 time? For continuing to obtain a degenerate solution in the next iteration? For removing degeneracy in
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 assuming no cycling?
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) Study the change in the optimal solution as a function of t ≥ 0. 3x₁ + x₂ + 2x3 = 3 +
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 reducing the resource requirements per unit of xj to 1/α of their original values, α > 1. Since
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 + 2x3 ≥ 6 X₁ + 2x₂ + 5x3 ≤ 4 X1, X₂, X3 ≥ 0
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 produced of each part in the three shifts.Ideally, the number of wheels produced is exactly twice that
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 part:It is desired to balance the two machines by limiting the difference between their total
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 80 and 60 units, respectively. Each machine runs 8 hours a day. Overtime, though not desirable, may
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 male–female ratios for in-state and out-of-state applicants are 1:1 and 3:2, respectively. For international
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, add a proper artificial constraint. x₁ + 2x₂ ≤ 6 2x₁ + x₂ ≤ 8 -x₁ + x₂
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 uses renewable energy sources (windmill, solar, and hydroelectric) all of which are impacted by
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 codes of its three letters. For example, AFT has a score of 1 + 6 + 20 = 27. The goal is to select
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 at least one-ninth. The remainder goes to charity. The will does not specify the total number of
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 parents have stipulated five rules: (a) The selling price is either $1 for 7 apples or $3 for 1
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 captain put aside a certain sum of money in the purser’s office and instructed the first officer
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 lot sizes. The following table summarizes the data of the situation:A raw material, if used, must be
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 grid such that each row, each column, and each subgrid contain distinct digits. Some of the cells
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. The puzzle seeks the smallest number of pieces arranged in sets of 8s (where a same set may repeat
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 total in lots of size 2, 3, 4, 5, or 6, there was always one gadget left over, but no remainder when
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 solution.
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 purchase of one-way tickets. Each airline offers bonus miles for frequent fliers. Eastern gives 1000
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. Additionally, the numbers in all the squares must be distinct. Use ILP to determine the assignment of
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 precincts of the county, limited funds dictate otherwise. The table given below lists the voting
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 with the objective of maximizing expected revenue. The demand for the bikes and the hourly rental
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 in the following table:The shortage costs per unit for items 1, 2, and 3 are $8, $10, and $15,
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 plant is three computers annually at the cost of $5 million each. The actual number of computers
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 from three home sizes: 1000, 1100, and 1200 ft2. Each size requires a certain number of labor
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 cost of the elevated structure on which the tank is installed is proportional to the area of 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 rectangle.
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. Design a GA for the problem starting with a random population of 4 parents and using a 1-point
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 and tomatoes rows are 2 feet wide, and the beans rows are 3 feet wide. I like tomatoes the most and
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, six in year 2, four in year 3, and four in year 4. The corresponding production costs per engine over
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
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 double in size by the end of the year. The farmer plans to sell out completely at the end of n
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 about $1900, and high at about $2500. I advertise the car for up to 3 consecutive days. At the end
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 gross profit is $59.8 million. The final report claims that the company’s average days-in-inventory
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 holiday shopping, ending the year with only 20 units left in stock. The company estimates the total
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) Determine the inventory cost per week of the present ordering policy.(b) Determine the optimal
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) Determine the optimum inventory policy, assuming a lead time of 1 week.(b) Determine the optimum number of
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
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 during the replenishment period, it is necessary that a > D. The setup cost is K per order, and 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
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
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
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 that 10 additional units are needed in period 4. Where should they be produced? Production range
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 $1 per period. Period i 12345 Demand D, (units) 50 70 100 30 60 Setup cost K₁ ($) 80 70 60 80 60
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.The unit production
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; ($) 20 1 17 1 10 18 5 50 1 3 1 1
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
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 of file excelWagnerWhitin.xls instead. The goal is to facilitate an understanding the big picture of
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 that the demand occurs instantaneously at the start of the period. The demand pdf for the period is:
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
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 $200.00, and the holding cost per book per month is $1.20. Apply
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 is a 7-day lead time for delivery. Determine the store’s optimal inventory policy given that the
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 shop’s inventory policy is to order 150 rolls whenever the inventory level drops to 80 units.
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 cost parameters, the order quantity is determined such that the probability of either surplus or
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, uniformly distributed. A copy costs $10 to produce, and it sells for $25. The students purchase their
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 a.m., the donuts sell for 5 cents apiece. The number of customers buying donuts between 6:00 and 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 during the winter season is more than 20 but less than or equal to 30, all with equal probabilities.
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 optimal order quantity.
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 cost parameters are given asUnit selling price = $2Unit purchase price = $1Unit holding cost per
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 following table provides the average quarterly trip fares for 2009, the year following the housing
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 cost per month = $1Unit penalty cost per month= $10Discount factor = .9Determine the optimal
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 14.2-1The Q-P demand curve for a flight between two destinations is estimated as Q = 250 − 25P.The
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 reservations have been processed: 10 seats for fare 1, 3 seats for fare 3, 15 seats for fare 7, 10
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 14.2-4Historical data for the number of business travelers on the 125-seat daily flight between XNA and LAX are
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 of fare 1, 4 seats of fare 1, and 2 seats of fare 2.Example 14.2-3Consider a 50 seat flight operating
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 in paperback.(b) In the film industry, new movies are released in theatres and after several months
The use of price differentiation to increase revenue in airlines need not be limited to two-segmentation fare classes only (as evident by the seemingly run-out-of-alphabet letters used presently by airlines).(a) Segment the optimum number of seats (= 125 ) in Example 14.2-2 to three fares with seat
Consider the hypothetical Q-P demand curve of Problem 14-3, Q = 100 / P . The revenue at any point is R = PQ at any point (Q-P) is the same and equals 100. Show that the four-fare Q-segmentation (0, 30, 65, 80, 100) will still produce a higher revenue.Problem 14-3Suppose that the Q-P demand curve
Prove that unit elasticity satisfies the necessary condition for determining the maximum revenue.
Categorize the demand for each of the following goods as likely elastic or likely inelastic: Gasoline, soft drinks, cars, medication, computers, home utilities (water, electric, and gas), air travel, restaurant meals, homes, paintings.
The daily number of meals, Q, in a popular 24-hour fast food restaurant is estimated as a function of the price per meal by the demand curve Q = 5000 − 250P.(a) The restaurant currently charges $8.00 per meal. Compute the daily revenue for the restaurant.(b) In terms of revenue, is the current
Consider the two-hierarchal data of Problem 15-1. Copy the weights in a logical order into the solution summary section of the spreadsheet excelAHP.xls, then develop the formula for evaluating the first alternative, UA, and copy it to evaluate the remaining two alternatives.Problem 15-1Suppose that
The personnel department at C&H has narrowed the search for a new hire to three candidates: Steve (S), Jane (J), and Maisa (M). The final selection is based on three criteria: personal interview (I), experience (E), and references (R). The department uses matrix A (given below) to establish the
Kevin and June Park (K and J) are in the process of buying a new house. Three houses, A, B, and C, are available. The Parks have agreed on two criteria for the selection of the house—amount of yard work (Y) and proximity to place of work (W)—and have developed the following comparison matrices.
An individual is in the process of buying a car and has narrowed the choices to three models: M1, M2, and M3. The deciding factors include purchase price (PP), maintenance cost (MC), cost of city driving (CD), and cost of rural driving (RD). The following table provides the relevant data for 3
A new author sets three criteria for selecting a publisher for an OR textbook: royalty percentage (R), marketing (M), and advance payment (A). Two publishers, H and P, have expressed interest in the book. Using the following comparison matrices, rank the two publishers and assess the consistency of

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