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operations research an introduction

Operations Research: An Introduction 10th Global Edition Hamdy A Taha - Solutions

In the Reddy Mikks model of Example 2.2-1:(a) Determine the range for the ratio of the unit revenue of exterior paint to the unit revenue of interior paint.(b) If the revenue per ton of exterior paint remains constant at $6000 per ton, determine the maximum unit revenue of interior paint that will
Consider Problem 3-63.(a) Determine the optimality condition for cA cB that will keep the optimum unchanged.(b) Determine the optimality ranges for cA and cB, assuming that the other coefficient is kept constant at its present value.(c) If the unit revenues cA and cB are changed simultaneously to
A company produces two products, A and B. The unit revenues are $2 and $3, respectively.Two raw materials, M1 and M2, used in the manufacture of the two products have daily availabilities of 8 and 18 units, respectively. One unit of A uses 2 units of M1 and 2 units of M2, and 1 unit of B uses 3
Consider the LP model Maximize z = 3x1 + 2x1 + 3x3 subject to 2x1 + x2 + x3 … 4 3x1 + 4x2 + 2x3 Ú 16 x1, x2, x3 Ú 0 Use hand computations to show that the optimal solution can include an artificial basic variable at zero level. Does the problem have a feasible optimal solution?
Toolco produces three types of tools, T1, T2, and T3. The tools use two raw materials, M1 and M2, according to the data in the following table:Number of units of raw materials per tool Raw material T1 T2 T3 M1 3 5 6 M2 5 3 4 The available daily quantities of raw materials M1 and M2 are 2000 units
In some ill-constructed LP models, the solution space may be unbounded even though the problem may have a bounded objective value. Such an occurrence points to possible irregularities in the construction of the model. In large problems, it may be difficult to detect “unboundedness” by
Consider the LP:Maximize z = 20x1 + 5x2 + x3 subject to 3x1 + 5x2 - 5x3 … 50 x1 … 10 x1 + 3x2 - 4x3 … 20 x1, x2, x3 Ú 0(a) By inspecting the constraints, determine the direction (x1, x2, or x3) in which the solution space is unbounded.(b) Without further computations, what can you conclude
TORA Experiment. Solve Example 3.5-3 using TORA’s Iterations option and show that even though the solution starts with x1 as the entering variable (per the optimality condition), the simplex algorithm will point eventually to an unbounded solution.
For the following LP, show that the optimal solution is degenerate and that none of the alternative solutions are corner points. You may use TORA for convenience.Maximize z = 3x1 + x2 subject to x1 + 2x2 … 5 x1 + x2 - x3 … 2 7x1 + 3x2 - 5x3 … 20 x1, x2, x3 Ú 0
Solve the following LP:Maximize z = 2x1 - x2 + 3x3 subject to x1 - x2 + 5x3 … 5 2x1 - x2 + 3x3 … 20 x1, x2, x3 Ú 0 From the optimal tableau, show that all the alternative optima are not corner points(i.e., nonbasic). Give a two-dimensional graphical demonstration of the type of solution space
For the following LP, identify three alternative optimal basic solutions, and then write a general expression for all the nonbasic alternative optima comprising these three basic solutions.Maximize z = x1 + 2x2 + 3x3 subject to x1 + 2x2 + 3x3 … 10 x1 + x2 … 5 x1 … 1 x1, x2, x3 Ú 0 Note:
TORA Experiment. Consider the following LP (authored by E.M. Beale to demonstrate cycling):Maximize z = 34 x1 - 20x2 + 12 x3 - 6x4 subject to 14 x1 - 8x2 - x3 + 9x4 … 0 12 x1 - 12x2 - 12 x3 + 3x4 … 0 x3 … 1 x1, x2, x3, x4 Ú 0 From TORA’s SOLVE>MODIFY menu, select Solve 1 Algebraic 1
TORA experiment. Consider the LP in Problem 3-52.(a) Use TORA to generate the simplex iterations. How many iterations are needed to reach the optimum?(b) Interchange constraints (1) and (3) and re-solve the problem with TORA. How many iterations are needed to solve the problem?(c) Explain why the
Consider the following LP:Maximize z = 3x1 + 2x2 subject to 4x1 - x2 … 4 4x1 + 3x2 … 6 4x1 + x2 … 4 x1, x2 Ú 0 (a) Show that the associated simplex iterations are temporarily degenerate (you may use TORA for convenience).(b) Verify the result by solving the problem graphically (TORA’s
Consider the graphical solution space in Figure 3.18. Suppose that the simplex iterations start at A and that the optimum solution occurs at D. Further, assume that the objective function is defined such that at A, x1 enters the solution first.(a) Identify (on the graph) the corner points that
Consider the LP model Minimize z = 2x1 - 4x2 + 3x3 subject to 5x1 - 6x2 + 2x3 Ú 5-x1 + 3x2 + 5x3 Ú 8 2x1 + 5x2 - 4x3 … 4 x1, x2, x3 Ú 0 Show how the inequalities can be modified to a set of equations that requires the use of single artificial variable only (instead of two).
Consider the following LP:Maximize z = 3x1 + 2x2 + 3x3 subject to 2x1 + x2 + x3 … 2 3x1 + 4x2 + 2x3 Ú 8 x1, x2, x3 Ú 0 The optimal simplex tableau at the end of Phase I is Basic x1 x2 x3 x4 x5 R Solution r -5 0 -2 -1 -4 0 0 x2 2 1 1 0 1 0 2 R -5 0 -2 -1 -4 1 0 Explain why the nonbasic variables
Consider the following problem:Maximize z = 3x1 + 2x2 + 3x3 subject to 2x1 + x2 + x3 = 4 x1 + 3x2 + x3 = 12 3x1 + 4x2 + 2x3 = 16 x1, x2, x3 Ú 0(a) Show that Phase I terminates with two zero artificial variables in the basic solution(use TORA for convenience).(b) Show that when the procedure of
Consider the following problem:Maximize z = 2x1 + 2x2 + 4x3 subject to 2x1 + x2 + x3 … 2 3x1 + 4x2 + 2x3 Ú 8 x1, x2, x3 Ú 0(a) Show that Phase I will terminate with an artificial basic variable at zero level (you may use TORA for convenience).(b) Remove the zero artificial variable prior to the
Write Phase I for the following problem, and then solve (with TORA for convenience) to show that the problem has no feasible solution.Minimize z = 2x1 + 5x2 subject to 3x1 + 2x2 Ú 12 2x1 + x2 … 4 x1, x2 Ú 0
Solve Problem 3-38, by the two-phase method.
For each case in Problem 3-37, write the corresponding Phase I objective function.
In Phase I, if the LP is of the maximization type, explain why we do not maximize the sum of the artificial variables in Phase I.
Show that the M-method will conclude that the following problem has no feasible solution.Maximize z = 2x1 + 5x2 subject to 3x1 + 2x2 Ú 6 2x1 + x2 … 2 x1, x2 Ú 0
Solve the following problem using x3 and x4 as starting basic feasible variables. As in Problem 3-39, do not use any artificial variables.Minimize z = 3x1 + 2x2 + 3x3 subject to x1 + 4x2 + x3 Ú 14 2x1 + x2 + x4 Ú 20 x1, x2, x3, x4 Ú 0*3-41. Consider the problem Maximize z = x1 + 5x2 + 3x3
Consider the problem Maximize z = 2x1 + 4x2 + 4x3 - 3x4 subject to x1 + x2 + x3 = 4 x1 + 4x2 + x4 = 8 x1, x2, x3, x4 Ú 0 Solve the problem with x3 and x4 as the starting basic variables and without using any artificial variables. (Hint: x3 and x4 play the role of slack variables. The main
Consider the following set of constraints:x1 + x2 + x3 = 7 2x1 - 5x2 + x3 Ú 10 x1, x2, x3 Ú 0 Solve the problem for each of the following objective functions:(a) Maximize z = 2x1 + 3x2 - 5x3.(b) Minimize z = 2x1 + 3x2 - 5x3.(c) Maximize z = x1 + 2x2 + x3.(d) Minimize z = 4x1 - 8x2 + 3x3.
Consider the following set of constraints:-2x1 + 3x2 = 3 112 4x1 + 5x2 Ú 10 122 x1 + 2x2 … 5 132 6x1 + 7x2 … 3 142 4x1 + 8x2 Ú 5 152 x1, x2 Ú 0 For each of the following problems, develop the z-row after substituting out the artificial variables:(a) Maximize z = 5x1 + 6x2 subject to (1),
In Example 3.4-1, identify the starting tableau for each of the following (independent)cases, and develop the associated z-row after substituting out all the artificial variables:*(a) The third constraint is x1 + 2x2 Ú 4.*(b) The second constraint is 4x1 + 3x2 … 6.(c) The second constraint is
TORA experiment. Generate the simplex iterations of Example 3.4-1 using TORA’s Iterations 1 M@method module (file toraEx3.4-1.txt). Compare the effect of using M = 1, M = 10, and M = 1000 on the solution. What conclusion can be drawn from this experiment?
Use hand computations to complete the simplex iteration of Example 3.4-1 and obtain the optimum solution.
TORA experiment. In Problem 3-32, use TORA to find the next-best optimal solution.
TORA experiment. Consider the following LP:Maximize z = x1 + x2 + 3x3 + 2x4 subject to x1 + 2x2 - 3x3 + 5x4 … 4 5x1 - 2x2 + 6x4 … 8 2x1 + 3x2 - 2x3 + 3x4 … 3 -x1 + x3 + 2x4 … 0 x1, x2, x3, x4 Ú 0 (a) Use TORA’s iterations option to determine the optimum tableau.(b) Select any nonbasic
The Gutchi Company manufactures purses, shaving bags, and backpacks. The construction includes leather and synthetics, leather being the scarce raw material. The production process requires two types of skilled labor: sewing and finishing. The following table gives the availability of the
Can you extend the procedure in Problem 3-9 to determine the third-best optimal value of z?
In Example 3.3-1, show how the second-best optimal value of z can be determined from the optimal tableau.
Consider the following LP:Maximize z = 16x1 + 15x2 subject to 40x1 + 31x2 … 124-x1 + x2 … 1 x1 … 3 x1, x2 Ú 0(a) Solve the problem by the simplex method, where the entering variable is the nonbasic variable with the most negative z-row coefficient.(b) Resolve the problem by the simplex
Consider the two-dimensional solution space in Figure 3.17.(a) Suppose that the objective function is given as Maximize z = 6x1 + 3x2 If the simplex iterations start at point A, identify the path to the optimum point D.(b) Determine the entering variable, the corresponding ratios of the feasibility
The following tableau represents a specific simplex iteration. All variables are nonnegative.The tableau is not optimal for either maximization or minimization. Thus, when a nonbasic variable enters the solution, it can either increase or decrease z or leave it unchanged, depending on the
Solve the following problem by inspection, and justify the method of solution in terms of the basic solutions of the simplex method.Maximize z = 5x1 - 6x2 + 3x3 - 5x4 + 12x5 subject to x1 + 3x2 + 5x3 + 6x4 + 3x5 … 30 x1, x2, x3, x4, x5 Ú 0(Hint: A basic solution consists of one variable only.)
Consider the following LP:Maximize z = x1 subject to 5x1 + x2 = 4 6x1 + x3 = 8 3x1 + x4 = 3 x1, x2, x3, x4 Ú 0(a) Solve the problem by inspection (do not use the Gauss-Jordan row operations), and justify the answer in terms of the basic solutions of the simplex method.(b) Repeat (a) assuming that
Consider the following system of equations:x1 + 2x2 - 3x3 + 5x4 + x5 = 8 5x1 - 2x2 + 6x4 + x6 = 16 2x1 + 3x2 - 2x3 + 3x4 + x7 = 6-x1 + x3 - 2x4 + x8 = 0 x1, x2,c, x8 Ú 0 Let x5, x6, . . . , and x8 be a given initial basic feasible solution. Suppose that x1 becomes basic. Which of the given basic
Consider the following set of constraints:x1 + 2x2 + 2x3 + 4x4 … 40 2x1 - x2 + x3 + 2x4 … 8 4x1 - 2x2 + x3 - x4 … 10 x1, x2, x3, x4 Ú 0 Solve the problem for each of the following objective functions.(a) Maximize z = 2x1 + x2 - 3x3 + 5x4.(b) Maximize z = 8x1 + 6x2 + 3x3 - 2x4.(c) Maximize z
This problem is designed to reinforce your understanding of the simplex feasibility condition. In the first tableau in Example 3.3-1, we used the minimum (nonnegative)ratio test to determine the leaving variable. The condition guarantees feasibility (all the new values of the basic variables remain
For each of the given objective functions and the solution space in Figure 3.16, select the nonbasic variable that leads to the next simplex corner point, and determine the associated improvement in z.*(a) Maximize z = x1 - 2x2 + 3x3(b) Maximize z = 5x1 + 2x2 + 4x3(c) Maximize z = -2x1 + 7x2 +
For the solution space in Figure 3.16, all the constraints are of the type … and all the variables x1, x2, and x3 are nonnegative. Suppose that s1, s2, s3, and s4 1Ú 02 are the slacks associated with constraints represented by the planes CEIJF, BEIHG, DFJHG, and IJH, respectively. Identify the
Consider the three-dimensional LP solution space in Figure 3.16, whose feasible extreme points are A, B, . . . , and J.(a) Which of the following pairs of corner points cannot represent successive simplex iterations: (A, B), (H, I ), (E, H ), and (A, I )? Explain why.(b) Suppose that the simplex
Consider the graphical solution of the Reddy Mikks model given in Figure 2.2. Identify the path of the simplex method and the basic and nonbasic variables that define this path.
Consider the following LP:Maximize z = x1 + 3x2 subject to x1 + x2 … 2-x1 + x2 … 4 x1 unrestricted x2 Ú 0(a) Determine all the basic feasible solutions of the problem.(b) Use direct substitution in the objective function to determine the best basic solution.(c) Solve the problem graphically,
Consider the following LP:Maximize z = 2x1 + 3x2 + 5x3 subject to-6x1 + 7x2 - 9x3 Ú 4 x1 + x2 + 4x3 = 10 x1, x3 Ú 0 x2 unrestricted Conversion to the equation form involves using the substitution x2 = x2- - x2+. Show that a basic solution cannot include both x2- and x2+ simultaneously.
Show algebraically that all the basic solutions of the following LP are infeasible.Maximize z = x1 + x2 subject to x1 + 2x2 … 3 2x1 + x2 Ú 8 x1, x2 Ú 0
Determine the optimum solution for each of the following LPs by enumerating all the basic solutions.(a) Maximize z = 2x1 - 4x2 + 5x3 - 6x4 subject to x1 + 4x2 - 2x3 + 8x4 … 2-x1 + 2x2 + 3x3 + 4x4 … 1 x1, x2, x3, x4 Ú 0(b) Minimize z = x1 + 2x2 - 3x3 - 2x4 subject to x1 + 2x2 - 3x3 + x4 = 4 x1
Consider the following LP:Maximize z = 2x1 + 3x2 subject to x1 + 3x2 … 12 3x1 + 2x2 … 12 x1, x2 Ú 0(a) Express the problem in equation form.(b) Determine all the basic solutions of the problem, and classify them as feasible and infeasible.*(c) Use direct substitution in the objective function
In an LP in which there are several unrestricted variables, a transformation of the type xj = xj- - xj+, xj-, xj+ Ú 0 will double the corresponding number of nonnegative variables.We can, instead, replace k unrestricted variables with exactly k + 1 nonnegative variables by using the substitution
JoShop manufactures three products whose unit profits are $2, $5, and $3, respectively.The company has budgeted 80 hrs of labor time and 65 hrs of machine time for the production of the three products. The labor requirements per unit of products 1, 2, and 3 are 2, 1, and 2 hrs, respectively. The
Two products are manufactured in a machining center. The production times per unit of products 1 and 2 are 10 and 12 minutes, respectively. The total regular machine time is 2400 minutes per day. The daily production is between 150 and 200 units of product 1 and no more than 45 units of product 2.
McBurger fast-food restaurant sells quarter-pounders and cheeseburgers. A quarterpounder uses a quarter of a pound of meat, and a cheeseburger uses only .2 lb. The restaurant starts the day with 250 lb of meat but may order more at an additional cost of 28 cents per pound to cover the delivery
Show that the m equations an j=1 aijxj = bi, i = 1, 2,c, m are equivalent to the following m + 1 inequalities:a nj=1 aijxj … bi, i = 1, 2,c, m an j=1 aa mi=1 aijbxj Ú a mi=1 bi
Show how the following objective function can be presented in equation form:Minimize z = max 5|x1 - x2 + 3x3|, | -x1 + 3x2 - x3|6 x1, x2, x3 Ú 0(Hint: 0 a 0 … b is equivalent to a … b and a Ú -b.)
Two different products, P1 and P2, can be manufactured by one or both of two different machines, M1 and M2. The unit processing time of either product on either machine is the same. The daily capacity of machine M1 is 200 units (of either P1 or P2, or a mix of both), and the daily capacity of
Consider the following inequality 22x1 - 4x2 Ú -7 Show that multiplying both sides of the inequality by -1 and then converting the resulting inequality into an equation is the same as converting it first to an equation and then multiplying both sides by -1.
In the diet model (Example 2.2-2), determine the surplus amount of feed consisting of 525 lb of corn and 425 lb of soybean meal.
In the Reddy Mikks model (Example 2.2-1), consider the feasible solution x1 = 2 tons and x2 = 2 tons. Determine the value of the associated slacks for raw materials M1 and M2.
Allocation of Aircraft to Routes. Consider the problem of assigning aircraft to four routes according to the following data:Aircraft type Capacity(passengers)Number of aircraft Number of daily trips on route 1 2 3 4 1 50 5 3 2 2 1 2 30 8 4 3 3 2 3 20 10 5 5 4 2 Daily number of customers 1000 2000
Loading Structure, Stark and Nichole (1972). The overhead crane in Figure 2.14 with two lifting yokes is used to transport mixed concrete to a yard for casting concrete barriers.The concrete bucket hangs at midpoint from the yoke. The crane end rails can support a maximum of 25 kip each, and the
Water Quality Management, Stark and Nicholes (1972). Four cities discharge wastewater into the same stream. City 1 is upstream, followed downstream by city 2, then city 3, and then city 4. Measured alongside the stream, the cities are approximately 15 miles apart.A measure of the amount of
Military Planning, Shepard and Associates (1988). The Red Army (R) is trying to invade the territory defended by the Blue Army (B). Blue has three defense lines and 200 regular combat units and can draw also on a reserve pool of 200 units. Red plans to attack on two fronts, north and south. Blue
Leveling the Terrain for a New Highway, Stark and Nicholes (1972). The Arkansas Highway Department is planning a new 10-mile highway on uneven terrain as shown by the profile in Figure 2.13. The width of the construction terrain is approximately 50 yards. To simplify the situation, the terrain
Fitting a Straight Line into Empirical Data (Regression). In a 10-week typing class for beginners, the average speed per student (in words per minute) as a function of the number of weeks in class is given in the following table.Week, x 1 2 3 4 5 6 7 8 9 10 Words per minute, y 5 9 15 19 21 24 26 30
Traffic Light Control, Stark and Nicholes (1972). Automobile traffic from three highways, H1, H2, and H3, must stop and wait for a green light before exiting to a toll road.The tolls are $4, $5, and $6 for cars exiting from H1, H2, and H3, respectively. The flow rates from H1, H2, and H3 are 550,
Pollution Control. Three types of coal, C1, C2, and C3, are pulverized and mixed together to produce 50 tons per hour needed to power a plant for generating electricity. The burning of coal emits sulfur oxide (in parts per million) which must meet the EPA specifications of no more than 2000 parts
Assembly-Line Balancing. A product is assembled from three different parts. The parts are manufactured by two departments at different production rates as given in the following table:Capacity(hr/wk)Production rate (units/hr)Department Part 1 Part 2 Part 3 1 100 6 8 12 2 90 6 12 4 Determine the
Voting on Issues. In a particular county in the State of Arkansas, four election issues are on the ballot: Build new highways, increase gun control, increase farm subsidies, and increase gasoline tax. The county includes 100,000 urban voters, 250,000 suburban voters, and 50,000 rural voters, all
Shelf Space Allocation. A grocery store must decide on the shelf space to be allocated to each of five types of breakfast cereals. The maximum daily demand is 110, 80, 150, 85, and 100 boxes, respectively. The shelf space in square inches for the respective boxes is 15, 25, 16, 20, and 22. The
Two alloys, A and B, are made from four metals, I, II, III, and IV, according to the following specifications:Alloy Specifications Selling price ($)A At most 80% of I 200 At most 30% of II At least 50% of IV B Between 40% and 60% of II 300 At least 30% of III At most 70% of IV The four metals are
A foundry smelts steel, aluminum, and cast iron scraps to produce two types of metal ingots, I and II, with specific limits on the aluminum, graphite, and silicon contents.Aluminum and silicon briquettes may be used in the smelting process to meet the desired specifications. The following tables
Shale Oil refinery blends two petroleum stocks, A and B, to produce two high-octane gasoline products, I and II. Stocks A and B are produced at the maximum rates of 450 and 700 bbl/hr, respectively. The corresponding octane numbers are 98 and 89, and the vapor pressures are 10 and 8 lb/in2.
Hawaii Sugar Company produces brown sugar, processed (white) sugar, powdered sugar, and molasses from sugarcane syrup. The company purchases 4000 tons of syrup weekly and is contracted to deliver at least 25 tons weekly of each type of sugar. The production process starts by manufacturing brown
In the refinery situation of Problem 2-71, suppose that the distillation unit actually produces the intermediate products naphtha and light oil. One bbl of crude A produces.35 bbl of naphtha and .6 bbl of light oil, and one bbl of crude B produces .45 bbl of naphtha and .5 bbl of light oil. Naphtha
An oil company distills two types of crude oil, A and B, to produce regular and premium gasoline and jet fuel. There are limits on the daily availability of crude oil and the minimum demand for the final products. If the production is not sufficient to cover demand, the shortage must be made up
A refinery manufactures two grades of jet fuel, F1 and F2, by blending four types of gasoline, A, B, C, and D. Fuel F1 uses gasolines A, B, C, and D in the ratio 1:1:2:4, and fuel F2 uses the ratio 2:2:1:3. The supply limits for A, B, C, and D are 1000, 1200, 900, and 1500 bbl/day, respectively.
All-Natural Coop makes three breakfast cereals, A, B, and C, from four ingredients:rolled oats, raisins, shredded coconuts, and slivered almonds. The daily availabilities of the ingredients are 5 tons, 2 tons, 1 ton, and 1 ton, respectively. The corresponding costs per ton are $100, $120, $110, and
A hardware store packages handyman bags of screws, bolts, nuts, and washers. Screws come in 100-lb boxes and cost $120 each, bolts come in 100-lb boxes and cost $175 each, nuts come in 80-lb boxes and cost $75 each, and washers come in 30-lb boxes and cost $25 each.The handyman package weighs at
Hi-V produces three types of canned juice drinks, A, B, and C, using fresh strawberries, grapes, and apples. The daily supply is limited to 200 tons of strawberries, 90 tons of grapes, and 150 tons of apples. The cost per ton of strawberries, grapes, and apples is $210,$110, and $100, respectively.
Suppose that an additional 100 acres of land can be purchased for $450,000, which will increase the total acreage to 900 acres.Is this a profitable deal for Realco?
Consider the Realco model of Problem
Realco owns 900 acres of undeveloped land on a scenic lake in the heart of the Ozark Mountains. In the past, little or no regulation was imposed upon new developments around the lake. The lake shores are now dotted with vacation homes, and septic tanks are in extensive use, most of them improperly
The city of Fayetteville is embarking on an urban renewal project that will include lowerand middle-income row housing, upper-income luxury apartments, and public housing.The project also includes a public elementary school and retail facilities. The size of the elementary school (number of
A city will undertake five urban renewal housing projects over the next 5 years. Each project has a different starting year and a different duration. The following table provides the basic data of the situation:Year 1 Year 2 Year 3 Year 4 Year 5 Cost(million $)Annual income(million $)Project 1
The city council of Fayetteville is in the process of approving the construction of a new 180,000-ft2 convention center. Two sites have been proposed, and both require exercising the “eminent domain” law to acquire the property. The following table provides data about proposed (contiguous)
A realtor is developing a rental housing and retail area. The housing area consists of efficiency apartments, duplexes, and single-family homes. Maximum demand by potential renters is estimated to be 500 efficiency apartments, 300 duplexes, and 250 single-family homes, but the number of duplexes
A large department store operates 7 days a week. The manager estimates that the minimum number of salespersons required to provide prompt service is 12 for Monday, 18 for Tuesday, 20 for Wednesday, 28 for Thursday, 32 for Friday, and 40 for each of Saturday and Sunday. Each salesperson works 5 days
In an LTL (less-than-truckload) trucking company, terminal docks include casual workers who are hired temporarily to account for peak loads. At the Omaha, Nebraska dock, the minimum demand for casual workers during the seven days of the week (starting on Monday) is 12, 20, 14, 10, 15, 18, and 10
In Problem 2-56, suppose that no volunteers will start at 2:00 p.m. or 7:00 p.m. to allow for lunch and dinner. Develop the LP, and determine the optimal schedule using AMPL, Solver, or TORA.
A hospital employs volunteers to staff the reception desk between 8:00 a.m. and 10:00 p.m.Each volunteer works three consecutive hours except for those starting at 8:00 p.m. who work for two hours only. The minimum need for volunteers is approximated by a step function over 2-hour intervals
In the bus scheduling example suppose that buses can run either 8- or 12-hr shifts. If a bus runs for 12 hr, the driver must be paid for the extra hours at 150% of the regular hourly pay. Do you recommend the use of 12-hr shifts? Solve the new model using AMPL, Solver, or TORA.
Two products are manufactured sequentially on two machines. The time available on each machine is 8 hours per day and may be increased by up to 4 hours of overtime, if necessary, at an additional cost of $110 per hour. The table below gives the production rate on the two machines as well as the
The manufacturing process of a product consists of two successive operations, I and II.The following table provides the pertinent data over the months of June, July, and August:June July August Finished product demand (units) 500 450 600 Capacity of operation I (hr) 800 700 550 Capacity of
A company has contracted to produce two products, A and B, over the months of June, July, and August. The total production capacity (expressed in hours) varies monthly.The following table provides the basic data of the situation:June July August Demand for A (units) 500 5000 750 Demand for B
The demand for an item over the next four quarters is 280, 400, 450, and 300 units, respectively. The price per unit starts at $20 in the first quarter and increases by $1 each quarter thereafter. The supplier can provide no more than 400 units in any one quarter.Although we can take advantage of

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