Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Mathematical Applications for the Management, Life, and Social Sciences
In Problems set up each system of equations and then solve it using inverse matrices.
Bee ancestry Because a female bee comes from a fertilized egg and a male bee comes from an unfertilized egg, the number of ancestors of a male bee generations before the present generation is the sum of the number of ancestors and generations before the present. If the numbers of ancestors of a male bee in a given generation and in the previous generation are given by
then there is a matrix such that the numbers of ancestors in the two generations preceding generation are given by
For a given male bee, the numbers of ancestorsand generations back are given by
Find the numbers of ancestors and generations back by multiplying both sides of
By the inverse of.
CHECKPOINT
The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of unit of product.
Which industry is most dependent on its own goods for its operation?
2CPCHECKPOINT
3. Suppose a primitive economy has wood product industry and a minerals industry, with technology matrix (open Leontief model).
If surpluses of units of wood products and units of minerals are desired, find the gross production of each industry.
4CPThe following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of unit of product.
For each units of manufactured products produced, how many units of fuels are required?
How many units of utilities are required to produce units of agricultural products?
For the economy in Problem 1, what industry is most dependent on utilities?
Problem 1.
The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of unit of product.
For each units of manufactured products produced, how many units of fuels are required?
How many units of utilities are required to produce units of agricultural products?
3E4EThe following technology matrix describes the relation-ship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)
How many units of fuels were required to produce units of manufactured goods?
The following technology matrix describes the relation-ship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)
How many units of fuels were required to produce 1000 units of power (utilities' goods)?
7E8E9E10ESuppose a primitive economy consists of two industries, farm products and farm machinery. Also suppose that its technology matrix is
If surpluses of units of farm products and units of farm machinery are desired, find the gross production of each industry.
Suppose an economy has two industries, agriculture and minerals. Suppose further that the technology matrix for this economy is.
If surpluses of agricultural units and mineral units are desired, find the gross production for each industry
13E14ESuppose the economy of an underdeveloped country has an agricultural industry and an oil industry, with technology matrix .
If surpluses of units of agricultural products and units of oil products are desired, find the gross production of each industry.
Find the additional production needed from each industry for more unit of oil products surplus.
Suppose a simple economy with only an agricultural industry and a steel industry has the following technology matrix.
If surpluses of units of agricultural products and units of steel are desired, find the gross production of each industry.
Find the additional production needed from each industry for more unit of agricultural surplus.
17EAn underdeveloped country has an agricultural industry and a manufacturing industry, with technology matrix .
Surpluses of units of agricultural products and units of manufactured products are desired. Find the gross production of each industry.
Find the additional production needed from each industry for more unit of manufacturing surplus.
A simple economy has an electronic components industry and a computer industry. Each unit of electronic components output requires inputs of unit of electronic components and unit of computers. Each unit of computer industry output requires inputs of unit of electronic components and unit of computers.
Write the technology matrix for this simple economy.
If surpluses of units of electronic components and units of computers are desired, find the gross production of each industry.
20EA primitive economy consists of a fishing industry and an oil industry. A unit of fishing industry output requires unit of fishing industry input and unit of oil industry input. A unit of oil industry output requires inputs of unit of fishing products and unit of oil products.
Write the technology matrix for this primitive economy.
If surpluses of units of fishing products and units of oil products are desired, find the gross production of each industry
An economy has a manufacturing industry and a banking industry. Each unit of manufacturing output requires inputs of unit of manufacturing and unit of banking. Each unit of banking output requires inputs of unit each of manufacturing and banking.
Write the technology matrix for this economy.
If surpluses of units of manufacturing and units of banking are desired, find the gross production of each industry
23E24E25EInterdepartmental costs Within a company is a (micro) economy that is monitored by the accounting procedures. In terms of the accounts, the various departments "producer costs, some of which are internal and some of which are direct costs. Problems 23-26 show how an open Leontief model can be used to determine departmental costs.
The sales department of an auto dealership charges of its total monthly costs to the service department, and the service department charges of its total monthly costs to the sales department. During a given month, the direct costs are for sales and for service. Find the total costs of each department.
Suppose that an economy has three industries (fishing, agriculture, and mining), and that matrix is the technology matrix for this economy.
If surpluses of units of fishing output and units each of agricultural and mining goods are desired, find the gross production of each industry.
28ESuppose that the economy of a small nation has an electronics industry, a steel industry, and an auto industry, with the following technology matrix.
If the nation wants to have surpluses of units of electronics production, units of steel production, and automobiles, find the gross production of each industry
Suppose an economy has the same technology matrix as that in Problem 29. If surpluses of units of electronics,units of steel, and autos are desired, find the gross production for each industry.
Problem 29.
Suppose that the economy of a small nation has an electronics industry, a steel industry, and an auto industry, with the following technology matrix.
If the nation wants to have surpluses of units of electronics production, units of steel production, and automobiles, find the gross production of each industry
Suppose that a simple economy has three industries, (service, manufacturing, and agriculture), and that matrix A is the technology matrix for this economy.
If surpluses of units of service output, units of manufactured goods, and units of agricultural goods are desired, find the gross production of each industry.
32EProblems 33-38 refer to closed Leontief models.
Suppose the technology matrix for a closed model of a simple economy is given by
Find the gross productions for the industries.
Problems 33-38 refer to closed Leontief models.
Suppose the technology matrix for a closed model of a simple economy is given by matrix A. Find the gross productions for the industries.
Problems 33-38 refer to closed Leontief models.
Suppose the technology matrix for a dosed model of a simple economy is given by matrix A. Find the gross productions for the industries.
Problems 33-38 refer to closed Leontief models.
Suppose the technology matrix for a closed model of an economy is given by matrix A. Find the gross productions for the industries.
Problems 33-38 refer to closed Leontief models.
A closed model for an economy has a manufacturing industry, utilities industry, and households industry. Each unit of manufacturing output uses unit of manufacturing input,
unit of utilities input, and unit of households input. Each unit of utilities output requires unit of manufacturing input, unit of utilities input, and unit of households input. Each unit of household output requires unit each of manufacturing and utilities input and unit of house-holds input.
(a) Write the technology matrix for this closed model of the economy.
(b) Find the gross production for each industry
Problems 33-38 refer to closed Leontief models.
A closed model for an economy identifies government, the profit sector, the nonprofit sector, and households as its industries. Each unit of government output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Each unit of profit sector output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and 0.4 unit of households input. Each unit of nonprofit sector output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Each unit of households output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input.
(a) Write the technology matrix for this closed model of the economy.
(b) Find the gross production for each industry.
39ECard tables are made by joining legs and a top using bolts. Each leg is made from a steel rod. The top has a frame made from steel rods. A cover and special damps that brace the top and hold the legs are joined to the frame using a total of bolts. The parts listing matrix for the card table assembly is given by
If an order is received forcard tables, legs, top, cover, clamps, and bolts, how many of each primary assembly item are required to fill the order?
41EA log carrier has a body made from a -ft length of reinforced cloth having a patch on each side and a dowel slid through each end to act as handles. The parts-listing matrix for the log carrier is given by
How many of each primary assembly item are required to fill an order forlog carriers andhandles?
In Problems 1-4, use the given feasible region determined by the constraint inequalities to find the maximum and minimum of the given objective function (if they exist).
2RE3RE4REIn Problems 5-8, a function and the graph of a feasible region are given. In each case, find both the maximum and minimum values of the function, if they exist, and the point at which each occurs
5.
w
6RE7REIn Problems 5-8, a function and the graph of a feasible region are given. In each case, find both the maximum and minimum values of the function, if they exist, and the point at which each occurs.
8.
9RE10RE11REIn Problems 9-15, solve the linear programming problems using graphical methods. Restrict and .
12. Minimize subject to
13REIn Problems 9-15, solve the linear programming problems using graphical methods. Restrict and .
14. Maximize subject to
15REIn Problems 16-19, use the simplex method to solve the linear programming problems. Assume that all variables are nonnegative.
16. Maximize subject to the conditions in Problem 10.
17RE18RE19RE20RE21RE22RE23RE24REIn Problems 22-25, form the dual and use the simplex method to solve the minimization problem. Assume that all variables are nonnegative.
25. Minimize subject to
26RE27RE28REIn Problems 28-35, use the simplex method. Assume that all variables are nonnegative.
30RE31RE32RE33RE34RE In Problems 28-35, use the simplex method. Assume that all variables are nonnegative.
35. Minimize subject to
36. Manufacturing A company manufactures backyard swing sets of two different sizes. The larger set requires 5 hours of labor to complete, the smaller set requires 2 hours, and there are 700 hours of labor available each week. The packaging department can package at most 185 swing sets per week. If the profit is $200 on each larger set and $100 on each smaller set, how many of each should be produced to yield the maximum profit? What is the maximum profit? Use graphical methods.
37. Production A company produces two different grades of steel, A and B, at two different factories, 1 and 2. The following table summarizes the production capabilities of the factories, the cost per day, and the number of units of each grade of steel that is required to fill orders.
Factory 1 Factory 2 Required
Grade A steel 1 unit 2 units 80 units
Grade B steel 3 units 2 units 140 units
Cost per day $5000 $6000
How many days should each factory operate to fill the orders at minimum cost? What is the minimum cost? Use graphical methods.
38. Manufacturing Chairco manufactures two types of chairs, standard and plush. Standard chairs require 2 hours to construct and finish, and plush chairs require 3 hours to construct and finish. Upholstering takes 1 hour for standard chairs and 3 hours for plush chairs. There are 240 hours per day available for construction and finishing, and 150 hours per day are available for upholstering. If the profit for standard chairs is $89 and for plush chairs is $133.50, how many of each type should be produced each day to maximize profit?
39RE40. Production Pinocchio Crafts makes two types of wooden crafts: Jacob’s ladders and locomotive engines. The manufacture of these crafts requires both carpentry and finishing. Each Jacob’s ladder requires 1 hour of finishing and hour of carpentry. Each locomotive engine requires 1 hour of finishing and 1 hour of carpentry. Pinocchio Crafts can obtain all the necessary raw materials, but only 120 finishing hours and 75 carpentry hours per week are available. Also, demand for Jacob’s ladders is limited to at most 100 per week. If Pinocchio Crafts makes a profit of $9 on each Jacob’s ladder and $15 on each locomotive engine, how many of each should it produce each week to maximize profits? What is the maximum profit?
Profit At its Jacksonville factory, Nolmaur Electronics manufactures 4 models of TV sets: LCD models in 27-, 32-, and 42-in. sizes and a 42-in. plasma model. The manufacturing and testing hours required for each model and available at the factory each week, as well as each models profit, are shown in the following table. 27-in. LCD 32-in. LCD 42-in. LCD 42-in. Plasma Available Hours Manufacturing (hr) 8 10 12 15 1870 Testing (hr) 2 4 4 4 530 Profit $80 $120 $160 $200 In addition, the supplier of the amplifier units can provide at most 200 units per week with at most 100 of these for both types of 42-in. models. The weekly demand for the 32-in. sets is at most 120. Nolmaur wants to determine the number of each type of set that should be produced each week to obtain maximum profit. (a) Carefully identify the variables for Nolmaurs linear programming problem. (b) Carefully state Nolmaurs linear programming problem. (c) Solve this linear programming problem to determine Nolmaurs manufacturing plan and maximum profit.42. Nutrition A nutritionist wants to find the least expensive combination of two foods that meet minimum daily vitamin requirements, which are 5 units of A and 30 units of B. Each ounce of food I provides 2 units of A and 1 unit of B, and each ounce of food II provides 10 units of A and 10 units of B. If food I costs 30 cents per ounce and food II costs 20 cents per ounce, find the number of ounces of each food that will provide the required vitamins and minimize the cost.
43. Nutrition A laboratory technician wants to purchase two different feeds, A and B, for its animals. The following
table summarizes the nutritional contents of the feeds, the required amounts of each ingredient, and the cost of each type of feed.
Feed A Feed B Required
Carbohydrates 1 unit/lb 4 units/lb 40 units
Protein 2 units/lb 1 unit/lb 80 units
Cost 14C/lb 16C/lb
How many pounds of each type of feed should the laboratory technician buy to satisfy its needs at minimum cost?
44. Production A company makes three products (I, II, and III) at three different factories. At factory A, it can make 10 units of each product per day. At factory B, it can make 20 units of II and 20 units of III per day. At factory C, it can make 20 units of 1,20 units of II, and 10 units of III per day. The company has orders for 200 units of 1,500 units of II, and 300 units of III. If the daily costs are $2000 at A, $3000 at B, and $5000 at C, find the number of days each factory should operate to fill the company’s orders at minimum cost. Find the minimum cost.
45. Profit A company makes pancake mix and cake mix. Each pound of pancake mix uses 0.6 lb of flour and 0.1 lb of shortening. Each pound of cake mix uses 0.4 lb of flour, 0.1 lb of shortening, and 0.4 lb of sugar. Suppliers can deliver at most 6000 lb of flour, at least 500 lb of shortening, and at most 1200 lb of sugar.
If the profit per pound is $0.35 for pancake mix and $0.25 for cake mix, how many pounds of each mix should be made to earn maximum profit? What is the maximum profit?
46RE47RE1T2T3T4T5T6T7T8T9T10T11T12. River Brewery is a microbrewery that produces an amber lager and an ale. Producing a barrel of lager requires 3 lb of corn and 2 lb of hops. Producing a barrel of ale requires 2 lb each of corn and hops. Profits are $35 from each barrel of lager and $30 from each barrel of ale. Suppliers can provide at most 1200 lb of corn and at most 1000 lb of hops per month. Formulate a linear programming problem that can be used to maximize River Brewery’s monthly profit. Then solve it using the simplex method.
13. A marketing research group conducting a telephone survey must contact at least 150 wives and 120 husbands. It costs $3 to make a daytime call and (because of higher labor costs) $4 to make an evening call. On average, daytime calls reach wives 30% of the time, husbands 10% of the time, and neither of them 60% of the time, whereas evening calls reach wives 30% of the time, husbands 30% of the time, and neither of them 40% of the time. Staffing considerations mean that daytime calls must be less than or equal to half of the total calls made. Formulate a linear programming problem that can be used to minimize the cost of completing the survey. Then solve it using any method from this chapter.
14. Lawn Rich, Inc., makes four different lawn care products that have fixed percents of phosphate, nitrogen, potash, and other minerals. The following table shows the percent of each of the four ingredients in each of Lawn Rich’s products and the profit per ton for each product.
Product 1 Product 2 Product 3 Product 4
% Phosphate 60 30 20 10
% Nitrogen 20 20 20 20
% Potash 9 6 15 18
% Other minerals 8 3 2 1
Profit/ton $120 $150 $180 $210
A lawn care company buys in bulk from Lawn Rich and then blends the products for its own purposes. For the blend it desires, the lawn care company wants its Lawn Rich order to contain at least 35 tons of phosphate, at most 20 tons of nitrogen, at most 9 tons of potash, and at most 4 tons of other minerals. How many tons of each product should Lawn Rich produce to fulfill the customers needs and maximize its own profit? What is the maximum profit?
1. Graph the region determined by the inequalities
2. Determine the corners of the region.
In Problems 1-6, graph each inequality.
In Problems 1-6, graph each inequality.
2.
In Problems 1-6, graph each inequality.
3.
In Problems 1-6, graph each inequality.
4.
In Problems 1-6, graph each inequality.
5.
In Problems 1-6, graph each inequality.
6.
In Problems 7-12, the graph of the boundary equations for each system of inequalities is shown with that system. Locate the solution region and find the corners.
7.
In Problems 7-12, the graph of the boundary equations for each system of inequalities is shown with that system. Locate the solution region and find the corners.
8.
In Problems 7-12, the graph of the boundary equations for each system of inequalities is shown with that system. Locate the solution region and find the corners.
9.
In Problems 7-12, the graph of the boundary equations for each system of inequalities is shown with that system. Locate the solution region and find the corners.
10.
In Problems 7-12, the graph of the boundary equations for each system of inequalities is shown with that system. Locate the solution region and find the corners.
11.
In Problems 7-12, the graph of the boundary equations for each system of inequalities is shown with that system. Locate the solution region and find the corners.
12.
In Problems 13-26, graph the solution of each system of inequalities.
13.
In Problems 13-26, graph the solution of each system of inequalities.
14.
15EIn Problems 13-26, graph the solution of each system of inequalities.
16.
17E18EIn Problems 13-26, graph the solution of each system of inequalities.
19.
20E21EIn Problems 13-26, graph the solution of each system of inequalities.
22.
23E24E25EIn Problems 13-26, graph the solution of each system of inequalities.
26.
27. Management The Wellbuilt Company produces two types of wood chippers, economy and deluxe. The deluxe model requires 3 hours to assemble and 1 /2 hour to paint, and the economy model requires 2 hours to assemble and 1 hour to paint. The maximum number of assembly hours available is 24 per day, and the maximum number of painting hours available is 8 per day.
(a) Write the system of inequalities that describes the constraints on the number of each type of wood chipper produced. Begin by identifying what x and y represent.
(b) Graph the solution of the system of inequalities and find the corners of the solution region.
28E29. Manufacturing A company manufactures two types of electric hedge trimmers, one of which is cordless. The cord-type trimmer requires 2 hours to make, and the cordless model requires 4 hours. The company has only 800 work hours to use in manufacturing each day, and the packaging department can package only 300 trimmers per day.
(a) Write the inequalities that describe the constraints on the number of each type of hedge trimmer produced. Begin by identifying what x and y represent.
(b) Graph the region determined by these constraint inequalities.
30. Manufacturing Sierra Wood Products manufactures two products, rockers and bookshelf units. Its profit is $30 per rocker and $42 per bookshelf unit. Next week's production will be constrained by two limited resources, labor and wood. The labor available next week is expected to be at most 930 hours, and the amount of wood available is expected to be at most 2400 board feet. Each rocker requires 4 labor hours and 8 board feet of wood. Each bookshelf unit requires 3 labor hours and 12 board feet of wood.
(a) Write the inequalities that describe the constraints on the number of each product produced next week. Begin by identifying what x and y represent.
(b) Graph the region determined by these constraint inequalities.
31. Advertising Apex Motors manufactures luxury cars and sport-utility vehicles. The most likely customers are high-income men and women, and company managers want to initiate an advertising campaign targeting these groups. They plan to run 1 -minute spots on business/investment programs, where they can reach 7 million women and 4 million men from their target groups. They also plan 1-minute spots during sporting events, where they can reach 2 million women and 12 million men from their target groups. Apex believes that the ads must reach at least 30 million women and at least 28 million men who are prospective customers.
(a) Write the inequalities that describe the constraints on the number of each type of 1-minute spots needed to reach these target groups.
(b) Graph the region determined by these constraint inequalities.
32E33. Politics A candidate wants to use a combination of radio and television advertisements in her campaign. Research has shown that each 1-minute spot on television reaches 0.09 million people and each 1-minute spot on radio reaches 0.006 million. The candidate believes she must reach at least 2.16 million people, and she must buy a total of at least 80 minutes of advertisements.
(a) Write the inequalities that relate the number of each type of advertising to her needs.
(b) Graph the region determined by these constraint inequalities.
34E35E36. Manufacturing A cereal manufacturer makes two different kinds of cereal, Senior Citizens Feast and Kids Go. Each pound of Senior Citizens Feast requires 0.6 lb of wheat and 0.2 lb of vitamin-enriched syrup, and each pound of Kids Go requires 0.4 lb of wheat, 0.2 lb of sugar, and 0.2 lb of vitamin -enriched syrup. Suppliers can deliver at most 2800 lb of wheat, at most 800 lb of sugar, and at least 1000 lb of the vitamin-enriched syrup.
(a) Write the inequalities that describe how many pounds of each type of cereal can be made.
(b) Graph the region determined by these constraint inequalities.
1. Find the maximum and minimum values of the objective function on the shaded region in the Figure, determined by the following constraints.
2. Find the maximum and minimum values (if they exist) of the objective function subject to the following constraints.
In Problems 1-4, use the given feasible region determined by the constraint inequalities to find the maximum and minimum of the given objective function (if they exist).
In Problems 1-4, use the given feasible region determined by the constraint inequalities to find the maximum and minimum of the given objective function (if they exist).
2.
In Problems 1-4, use the given feasible region determined by the constraint inequalities to find the maximum and minimum of the given objective function (if they exist).
3.
In Problems 1-4, use the given feasible region determined by the constraint inequalities to find the maximum and minimum of the given objective function (if they exist).
4.
In Problems 5-8, the graph of the feasible region is shown. Find the corners of each feasible region and then find the maximum and minimum of the given objective function (if they exist).
5.
In Problems 5-8, the graph of the feasible region is shown. Find the corners of each feasible region and then find the maximum and minimum of the given objective function (if they exist).
6.
In Problems 5-8, the graph of the feasible region is shown. Find the corners of each feasible region and then find the maximum and minimum of the given objective function (if they exist).
7.
In Problems 5-8, the graph of the feasible region is shown. Find the corners of each feasible region and then find the maximum and minimum of the given objective function (if they exist).
8.
In Problems 9-12, find the indicated maximum or minimum value of the objective function in the linear programming problem. Note that the feasible regions for these problems are found in the answers to Problems 19, 20, 23, and 24 in the Section 4.1 Exercises.
9. Minimize subject to
10EIn Problems 9-12, find the indicated maximum or minimum value of the objective function in the linear programming problem. Note that the feasible regions for these problems are found in the answers to Problems 19, 20, 23, and 24 in the Section 4.1 Exercises.
11. Minimize subject to
In Problems 9-12, find the indicated maximum or minimum value of the objective function in the linear programming problem. Note that the feasible regions for these problems are found in the answers to Problems 19, 20, 23, and 24 in the Section 4.1 Exercises.
12. Minimize subject to
In Problems 13-24, solve the following linear programming problems. Restrict
13. Maximize subject to
14EIn Problems 13-24, solve the following linear programming problems. Restrict
15. Maximize subject to
16EIn Problems 13-24, solve the following linear programming problems. Restrict and .
17. Minimize subject to
18E19E20E21EIn Problems 13-24, solve the following linear programming problems. Restrict and .
22. Minimize subject to
In Problems 13-24, solve the following linear programming problems. Restrict and .
23. Minimize subject to
In Problems 13-24, solve the following linear programming problems. Restrict and .
24. Minimize subject to
25. Manufacturing The Wellbuilt Company produces two types of wood chippers, economy and deluxe. The deluxe model requires 3 hours to assemble and 1/2 hour to paint, and the economy model requires 2 hours to assemble and 1 hour to paint. The maximum number of assembly hours available is 24 per day, and the maximum number of painting hours available is 8 per day. If the profit on the deluxe model is $90 per unit and the profit on the economy model is $72 per unit, how many units of each model will maximize profit? (See Problem 27 in the Section 4.1 Exercises.)
26E27. Manufacturing A company manufactures two types of electric hedge trimmers, one of which is cordless. The cord-type trimmer requires 2 hours to make, and the cordless model requires 4 hours. The company has only 800 work hours to use in manufacturing each day, and the packaging department can package only 300 trimmers per day. If the company profits are $22.50 for the cord-type model and $45.00 for the cordless model, how many of each type should the company produce per day to maximize profits?
28. Manufacturing Sierra Wood Products manufactures two products, rockers and bookshelf units. Its profit is $30 per rocker and $42 per bookshelf unit. Next week's production will be constrained by two limited resources, labor and wood. The labor available next week is expected to be at most 930 hours, and the amount of wood available is expected to be at most 2400 board feet. Each rocker requires 4 labor hours and 8 board feet of wood. Each bookshelf unit requires 3 labor hours and 12 board feet of wood. Find how many rockers and bookshelf units should be produced next week to maximize Sierra’s profit. Find the maximum profit.
29. Politics A candidate wants to use a combination of radio and television advertisements in her campaign. Research has shown that each 1-minute spot on television reaches 0.09 million people and that each 1-minute spot on radio reaches 0.006 million. The candidate believes she must reach at least 2.16 million people, and she must buy a total of at least 80 minutes of advertisements. How many minutes of each medium should be used to minimize costs if television costs $500/minute and radio costs $100/minute?
30. Nutrition In a hospital ward, the patients can be grouped into two general categories depending on their condition and the amount of solid foods they require in their diet. A combination of two diets is used for solid foods because they supply essential nutrients for recovery, but each diet has an amount of a substance deemed detrimental. The following table summarizes the patient group, minimum diet requirements, and the amount of the detrimental substance. How many servings from each diet should be given each day to minimize the intake of this detrimental substance?
Diet A Diet B Daily Requirement
Group 1 4 oz per 1 oz per 26 oz
serving serving
Group 2 2 oz per 1 oz per 18 oz
serving serving
Detrimental 0.18 oz per 0.09 oz per
substance serving serving
31. Production scheduling Newjet, Inc. manufactures inkjet printers and laser printers. The company has the capacity to make 70 printers per day, and it has 120 hours of labor per day available. It takes 1 hour to make an inkjet printer and 3 hours to make a laser printer. The profits are $40 per inkjet printer and $60 per laser printer. Find the number of each type of printer that should be made to give maximum profit and find the maximum profit.
32. Production scheduling At one of its factories, a jeans manufacturer makes two styles: #891 and #917. Each pair of style 891 takes 10 minutes to cut out and 20 minutes to assemble and finish. Each pair of style 917 takes 10 minutes to cut out and 30 minutes to assemble and finish. The plant has enough workers to provide at most 7500 minutes per day for cutting and at most 19,500 minutes per day for assembly and finishing. The profit on each pair of style 891 is $18.00, and the profit on each pair of style 917 is $22.50. How many pairs of each style should be produced per day to obtain maximum profit? Find the maximum daily profit.
33. Nutrition A privately owned lake contains two types of game fish, bass and trout. The owner provides two types of food, A and B, for these fish. Bass require 2 units of food A and 4 units of food B, and trout require 5 units of food A and 2 units of food B. If the owner has 800 units of each food, find the maximum number of fish the lake can support.
34. Nutrition In a zoo, there is a natural habitat containing several feeding areas. One of these areas serves as a feeding area for two species, I and II, and is supplied each day with 120 pounds of food A, 110 pounds of food B, and 57 pounds of food C. Each individual of species I requires 5 lb of A, 5 lb of B, and 2 lb of C, and each individual of species II requires 6 lb of A, 4 lb of B, and 3 lb of C. Find the maximum number of the two species that can be supported.
Shadow prices—land management For Problems 35 and 36, refer to the farm co-op application in Example 2(a). Rework that linear programming problem with the indicated changes in one constraint and answer the questions.
35. (a) If one more acre of land became available (for a total of 6001 acres), how would this change the co-op’s planting strategy and its maximum profit?
(b) Repeat part (a) if 8 more acres of land were available.
(c) Based on parts (a) and (b), what is the profit value of each additional acre of land? This value is called the shadow price of an acre of land.
Shadow prices—land management For Problems 35 and 36, refer to the farm co-op application in Example 2(a). Rework that linear programming problem with the indicated changes in one constraint and answer the questions.
36. (a) If one more gallon of fertilizer/herbicide became available (for a total of 40,301 gallons), how would this change the co-op’s planting strategy and its maximum profit?
(b) Repeat part (a) if 8 more gallons of fertilizer/herbicide were available.
(c) Based on parts (a) and (b), what is the profit value of each additional gallon of fertilizer/herbicide (that is, the shadow price of a gallon of fertilizer/ herbicide)?
37. Manufacturing Two factories produce three different types of kitchen appliances. The following table summarizes the production capacity, the number of each type of appliance ordered, and the daily operating costs for the factories. How many days should each factory operate to fill the orders at minimum cost? Find the minimum cost.
38. Nutrition In a laboratory experiment, two separate foods are given to experimental animals. Each food contains essential ingredients, A and B, for which the animals have a minimum requirement; each food also has an ingredient C, which can be harmful to the animals. The following table summarizes this information.
Food 1 Food 2 Required
Ingredient A 10 units/g 3 units/g 49 units
Ingredient B 6 units/g 12 units/g 60 units
Ingredient C 3 units/g 1 unit/g
How many grams of foods 1 and 2 should be given to the animals to satisfy the requirements for A and B while minimizing the amount of ingredient C ingested?
39. Manufacturing The Janie Gioffre Drapery Company makes three types of draperies at two different locations. At location I. it can make 10 pairs of deluxe drapes, 20 pairs of better drapes, and 13 pairs of standard drapes per day. At location II, it can make 20 pairs of deluxe, 50 pairs of better, and 6 pairs of standard per day. The company has orders for 2000 pairs of deluxe drapes, 4200 pairs of better drapes, and 1200 pairs of standard drapes. If the daily costs are $500 per day at location I and $800 per day at location II, how many days should Janie schedule at each location to fill the orders at minimum cost? Find the minimum cost.
40. Nutrition Two foods contain proteins, carbohydrates, and fats. Food A costs $ 1 per pound and contains 30% protein, 10% fat, and 50% carbohydrates. Food B costs $1.50 per pound and contains 20% protein, 4% fat, and 75% carbohydrates. What combination of these two foods provides at least 1 pound of protein, pounds of carbohydrates, and pound of fat at the lowest cost?
41. Manufacturing A sausage company makes two different kinds of hot dogs, regular and all beef. Each pound of all-beef hot dogs requires 0.75 lb of beef and 0.2 lb of spices, and each pound of regular hot dogs requires 0.18 lb of beef, 0.3 lb of pork, and 0.2 lb of spices. Suppliers can deliver at most 1020 lb of beef, at most 600 lb of pork, and at least 500 lb of spices. If the profit is $1.50 on each pound of all-beef hot dogs and $1.00 on each pound of regular hot dogs, how many pounds of each should be produced to obtain maximum profit? What is the maximum profit?
42. Manufacturing A cereal manufacturer makes two different kinds of cereal, Senior Citizens Feast and Kids Go. Each pound of Senior Citizen’s Feast requires 0.6 lb of wheat and 0.2 lb of vitamin-enriched syrup, and each pound of Kids Go requires 0.4 lb of wheat, 0.2 lb of sugar, and 0.2 lb of vitamin-enriched syrup. Suppliers can deliver at most 2800 lb of wheat, at most 800 lb of sugar, and at least 1000 lb of the vitamin- enriched syrup. If the profit is $0.90 on each pound of Senior Citizens Feast and $1.00 on each pound of Kids Go, find the number of pounds of each cereal that should be produced to obtain maximum profit. Find the maximum profit.
43. Shipping costs TV Circuit has 30 large-screen televisions in a warehouse in Erie and 60 large-screen televisions in a warehouse in Pittsburgh. Thirty-five are needed in a store in Blairsville, and 40 are needed in a store in Youngstown. It costs $18 to ship from Pittsburgh to Blairsville and $22 to ship from Pittsburgh to Youngstown, whereas it costs $20 to ship from Erie to Blairsville and $25 to ship from Erie to Youngstown. How many televisions should be shipped from each warehouse to each store to minimize the shipping cost? Hint: If the number shipped from Pittsburgh to Blairsville is represented by x, then the number shipped from Erie to Blairsville is represented by
44. Construction A contractor builds two types of homes. The Carolina requires one lot, $160,000 capital, and 160 worker-days of labor, whereas the Savannah requires one lot, $240,000 capital, and 160 worker-days of labor. The contractor owns 300 lots and has $48,000,000 available capital and 43,200 worker-days of labor. The profit on the Carolina is $40,000 and the profit on the Savannah is $50,000. Find how many of each type of home should be built to maximize profit. Find the maximum profit.
Management A bank has two types of branches. A satellite branch employs 3 people, requires $100,000 to construct and open, and generates an average daily revenue of $10,000. A full-service branch employs 6 people, requires $140,000 to construct and open, and generates an average daily revenue of $18,000. The bank has up to $2.98 million available to open new branches, and has decided to limit the new branches to a maximum of 25 and to hire at most 120 new employees. (a) How many branches of each type should the bank open in order to maximize the average daily revenue? Find the maximum average daily revenue. (b) At the optimal solution from part (a), analyze the banks constraints (number of new branches, number of new employees, and budget) to determine the Amount Available, Amount Used, and Amount Not Used (Slack). (c) Obtaining additional quantities of which constraint items would have the potential to increase the banks average daily revenue? Explain. (d) Obtaining more of which constraint item would not increase average daily revenue? Explain.46. Manufacturing A company manufactures two different sizes of boat lifts. Each size requires some time in the welding and assembly department and some time in the parts and packaging department. The smaller lift requires hour in welding and assembly and if hours in parts and packaging. The larger lift requires hours in welding and assembly and 1 hour in parts and packaging. The factory has 156 hours/day available in welding and assembly and 174 hours/day available in parts and packaging. Furthermore, daily demand for the lifts is at most 90 large and at most 100 small, and profits are $200 for each large lift and $120 for each small lift.
1. Write the following constraints as equations by using slack variables.
2CP3CP4CP5CP6CP1E2E3E4E5E6E7E8EIn Problems 7-10, a simplex matrix is given in which the solution is complete. Identify the maximum value off and a set of values of the variables that gives this maximum value.
10E11E12E13E14E15EIn Problems 11-18, a simplex matrix for a standard maximization problem is given.
(a) Write the values of all the variables and of the objective function f.
(b) Indicate whether the solution from part (a) is complete (optimal).
(c) If the solution is not complete, find the next pivot and state all row operations with that pivot (that is, row operations that make that pivot equal to 1, and then make other entries in the pivot column equal to 0). Do not perform the row operations.
17E18E19E20E21E22E23E24E25E26E27E28. Maximize subject to
29E30EProblems 31-38 involve linear programming problems that have nonunique solutions.
In Problems 31-34, a simplex matrix is given for which the associated maximization problem either has no solution or has multiple solutions. If there is no solution, explain how you know this. If there are multiple solutions, give one solution and explain how you would find another.
Problems 31-38 involve linear programming problems that have nonunique solutions.
In Problems 31-34, a simplex matrix is given for which the associated maximization problem either has no solution or has multiple solutions. If there is no solution, explain how you know this. If there are multiple solutions, give one solution and explain how you would find another.
33E34E35EIn Problems 35-38, use the simplex method to maximize each function (whenever possible) subject to the given constraints. If there is no solution, indicate this; if multiple solutions exist, find two of them.
36. Maximize
37E38EManufacturing Newjet Inc. manufactures two types of printers, an inkjet printer and a laser printer. The company can make a total of 60 printers per day, and it has 120 labor-hours per day available. It takes 1 labor- hour to make an inkjet printer and 3 labor-hours to make a laser printer. The profits are $40 per inkjet printer and $60 per laser printer. (a) Write the simplex matrix to maximize the daily profit. (b) Find the maximum profit and the number of each type of printer that will give the maximum profit.40. Construction A contractor builds two types of homes. The Carolina requires one lot, $160,000 capital, and 160 worker-days of labor, and the Savannah requires one lot, $240,000 capital, and 160 worker-days of labor. The contractor owns 300 lots and has $48,000,000 available capital and 43,200 worker-days of labor. The profit on the Carolina is $40,000, and the profit on the Savannah is $50,000.
(a) Write the simplex matrix to maximize the profit.
(b) Use the simplex method to find how many of each type of home should be built to maximize profit, and find the maximum possible profit.
41E42. Budget utilization A car rental agency has a budget of $1.8 million to purchase at most 100 new cars. The agency will purchase either subcompact cars at $15,000 each or midsized cars at $30,000 each. From past rental patterns, the agency decides to purchase at most 50 midsized cars and expects an annual profit of $7500 per subcompact car and $11,000 per midsized car. How many of each type of car should be purchased to obtain the maximum profit while satisfying budgetary and other planning constraints? Find the maximum profit.
43. Production scheduling Happy Valley Ice Cream Company is planning its production for next week. Demand for Happy Valley’s premium and light ice creams continues to outpace the company’s production capacities. Two resources used in ice cream production are in short supply for next week. The mixing machines will be available for only 140 hours, and only 28,000 gallons of high-grade milk will be available. One hundred gallons of premium ice cream requires 0.3 hour of mixing and 90 gallons of milk. One hundred gallons of
light ice cream requires 0.5 hour of mixing and 70 gallons of milk. If Happy Valley earns a profit of $100 per hundred gallons on both of its ice creams, how many hundreds of gallons of premium and of light ice cream should Happy Valley produce next week to maximize profit? How much profit will result?
44. Experimentation An experiment involves placing the males and females of a laboratory animal species in two separate controlled environments. There is a limited time available in these environments, and the experimenter wants to maximize the number of animals subject to the constraints described.
Males Females Time Available
Environment A 20 min 25 min 800 min
Environment B 20 min 15 min 600 min
How many males and how many females will maximize the total number of animals?
Problems 45-48 involve three variables. Solve them with the simplex method, Excel, or some other technology.
45. Construction A contractor builds three types of houses: the Aries, the Belfair, and the Wexford. Each house requires one lot, and the following table gives the number of labor-hours and the amount of capital needed for each type of house, as well as the profit on the sale of each house. There are 12 lots, 47,500 labor-hours, and $3,413,000 available for the contractor’s use.
(a) Building how many of each type of house will maximize his profit?
(b) What is the maximum possible profit?
Aries Belfair Wexford
Labor-hours 3,000 3,700 5,000
Capital $205,000 $279,600 $350,000
Profit $20,000 $25,000 $30,000
Problems 45-48 involve three variables. Solve them with the simplex method, Excel, or some other technology.
46. Medicine A medical clinic performs three types of medical tests that use the same machines. Tests A, B, and C take 15 minutes, 30 minutes, and 1 hour, respectively, with respective profits of $150, $250, and $500. The clinic has four machines available. One person is qualified to do test A, two to do test B, and one to do test C. If the clinic has a rush of customers for these tests, how many of each type should it schedule in a 12-hour day to maximize its profit?
Problems 45-48 involve three variables. Solve them with the simplex method, Excel, or some other technology.
47. Revenue A woman has a building with 26 one-bed- room apartments, 40 two-bedroom apartments, and 60 three-bedroom apartments available to rent to students. She has set the rent at $500 per month for the one-bedroom units, $800 per month for the two-bed- room units, and $1150 per month for the three-bed- room units. She must rent to one student per bedroom, and zoning laws limit her to at most 250 students in this building. There are enough students available to rent all the apartments.
(a) How many of each type of apartment should she rent to maximize her revenue?
(b) What is the maximum possible revenue?
Problems 45-48 involve three variables. Solve them with the simplex method, Excel, or some other technology.
48. Manufacturing Patio Iron makes wrought iron outdoor dining tables, chairs, and stools. Each table uses 8 feet of a standard width wrought iron, 4 hours of labor for cutting and assembly, and 2 hours of labor for detail and finishing work. Each chair uses 6 feet of the wrought iron, 2 hours of cutting and assembly labor, and 1.5 hours of detail and finishing labor. Each stool uses 1 foot of the wrought iron, 1.5 hours for cutting and assembly, and 0.5 hour for detail and finishing work, and the daily demand for stools is at most 16. Each day Patio Iron has available at most 108 feet of wrought iron, 50 hours for cutting and assembly, and 40 hours for detail and finishing. If the profits are $60 for each dining table, $48 for each chair, and $36 for each stool, how many of each item should be made each day to maximize profit? Find the maximum profit.
In Problems 49-54, use Excel to solve each linear programming problem.
49. Advertising Tire Corral has $6000 available per month for advertising. Newspaper ads cost $100 each and can occur a maximum of 21 times per month. Radio ads cost $300 each and can occur a maximum of 28 times per month at this price. Each newspaper ad reaches 6000 men over 20 years of age, and each radio ad reaches 8000 of these men. The company wants to maximize the number of ad exposures to this group. How many of each ad should it purchase? What is the maximum possible number of exposures?
In Problems 49-54, use Excel to solve each linear programming problem.
50. Manufacturing A bicycle manufacturer makes mountain bikes and road bikes. Each mountain bike requires 2 units of steel and 6 units of aluminum in its frame and 12 special components for the hub, sprocket, and gear assembly. Each road bike requires 5 units each of steel and aluminum for its frame and 5 of the special components. Shipments are such that steel is limited to 100 units per day, aluminum is limited to 120 units per day, and the special components are limited to 180 units per day. If the profits are $300 on each mountain bike and $200 on each road bike, how many of each should be produced to yield the maximum profit? What is the maximum profit?
51E In Problems 49-54, use Excel to solve each linear programming problem.
52. Advertising The Laposata Pasta Company has $12,000 available for advertising. The following table gives the costs per ad and the numbers of people exposed to its ads for three different media (with numbers in thousands).
Ad Packages Newspaper Radio TV
Cost 2 2 4
Total audience 30 21 54
Working mothers 6 12 8
If the total available audience is 420,000 and the company wants to maximize the number of exposures to working mothers, how many ads of each type should it purchase?
In Problems 49-54, use Excel to solve each linear programming problem.
53. Manufacturing At one of its factories, a manufacturer of televisions makes one or more of four models of HD units (without cases): a 20-in. LCD, a 42-in. LCD, a 42-in. plasma, and a 50-in. plasma. The assembly and testing time requirements for each model are shown in the following table, together with the maximum amounts of time available per week for assembly and testing. In addition to these constraints, the supplier of cases indicated that it would supply no more than 200 cases per week and that of these, no more than 40 could be for the 20-in. LCD model.
20-in. LCD 42-in. LCD 42-in. Plasma 50-in. Total Plasma Available
Assembly time (hours) 7 10 12 15 2000
Test time (hours) 2 2 4 5 500
Profit (dollars) 46 60 75 100
Use the profit for each television shown in the table to find the number of completed models of each type that should be produced to obtain the maximum profit for the week. Find the maximum profit.
In Problems 49-54, use Excel to solve each linear programming problem.
54. Investment analysis Conglomo Corporation is considering investing in other smaller companies, and it can purchase any fraction of each company. Each investment by Conglomo requires a partial payment now
and a final payment 1 year from now. For each investment, the following table summarizes the amount of each payment (in millions of dollars) and the projected 5-year profit (also in millions of dollars).
Company Company Company Company Company
1 2 3 4 5
Paid now Paid in 13.2 63.6 6.0 6.0 34.8
1 year Projected 3.6 7.2 6.0 1.2 40.8
profit 15.6 19.2 19.2 16.8 46.8
This table may be interpreted as follows: If Conglomo purchases one-fifth of Company 4, then it pays million now and million = $240,000 after 1 year, and the one-fifth share has a projected 5-year profit of million. Conglomo has $48 million available for investment now and $24 million available for investment 1 year from now. What fraction of each smaller company should Conglomo purchase to maximize the projected 5-year profit? What is the maximum profit? Hint: If represents the fraction of Company 1 that is purchased, then is a constraint.
Perform the following steps to begin the process of finding the minimum value of subject to the following constraints.
1. Form the matrix associated with the minimization problem.
2CP3CP4CP5CP1E2E3E4EIn Problems 5 and 6, suppose a primal minimization problem and its dual maximization problem were solved using the simplex method on the dual problem and the final simplex matrix is given.
(a) Find the solution of the minimization problem. Use as the variables and g as the function.
(b) Find the solution of the maximization problem. Use as the variables and f as the function.
6EIn Problems 7-10, write the dual maximization problem and then solve both the primal and dual problems with the simplex method.
7. Minimize subject to
8E9EIn Problems 7-10, write the dual maximization problem and then solve both the primal and dual problems with the simplex method.
10. Minimize subject to
11E12E13E14E15E16E17EIn Problems 17-20, use Excel or some other technology.
18. Minimize subject to
19E20E21. Production scheduling CDF Appliances has assembly plants in Atlanta and Fort Worth where it produces a variety of kitchen appliances, including a 12-cup coffee- maker and a cappuccino machine. In each hour at the Atlanta plant, 160 of the coffeemakers and 200 of the cappuccino machines can be assembled, and the hourly cost is $700. In each hour at the Fort Worth plant, 800 of the coffeemakers and 200 of the cappuccino machines can be assembled, and the hourly cost is $2100. CDF Appliances expects orders each week for at least 64,000 of the coffeemakers and at least 40,000 of the cappuccino machines. How many hours per week should each plant be operated to provide inventory for the orders at minimum cost? Find the minimum cost.
Manufacturing Nekita Corporation assembles cell phones and smart phones at two different factories within the same city. During each hour at the first factory, 15 cell phones and 30 smart phones can be assembled at a cost of $100/hour. During each hour at the second factory, 10 cell phones and 60 smart phones can be assembled at a cost of $150/hour. If Nekita expects weekly orders for at least 15,000 cell phones and at least 45,000 smart phones, how many hours per week should it schedule at each location to be able to fill the orders at minimum cost? What is the minimum cost?Manufacturing The Video Star Company makes two different models of DVD players, which are assembled on two different assembly lines. Line 1 can assemble 30 units of the Star model and 40 units of the Prostar model per hour, and line 2 can assemble 150 units of the Star model and 40 units of the Prostar model per hour. The company needs to produce at least 270 units of the Star model and 200 units of the Prostar model to fill an order. If it costs $200 per hour to run line and $400 per hour to run line 2, how many hours should each line be run to fill the order at the minimum cost? What is the minimum cost?24E25. Production A small company produces two products, I and II, at three facilities, A, B, and C. It has orders for 2000 of product I and 1200 of product II. The production capacities and costs per week to operate the three facilities are summarized in the following table.
A B C
1 200 200 400
II 100 200 100
Cost/week $1000 $3000 $4000
How many weeks should each facility operate to fill the company’s orders at a minimum cost, and what is the minimum cost?
26. Nutrition In a hospital ward, the patients can be grouped into two general categories depending on their conditions and the amounts of solid foods they require in their diet. A combination of two diets is used for solid foods because they supply essential nutrients for recovery, but each diet has an amount of a substance deemed detrimental. For each patient group, the following table summarizes the diet requirements and the amounts of the detrimental substance. How many servings from each diet should be given each day to minimize the intake of this detrimental substance?
Diet A Diet B Required Daily
Group 1 4 oz per 1 oz per 26 oz
serving serving
Group 2 2 oz per 1 oz per 18 oz
serving serving
Detrimental 0.18 oz per 0.07 oz per
substance serving serving
27. Production Two factories produce three different types of kitchen appliances. The following table summarizes the production capacities, the numbers of each type of appliance ordered, and the daily operating costs for the factories. How many days should each factory operate to fill the orders at minimum cost?
Factory 1 Factory 2 Number Ordered
Appliance 1 80/day 20/day 1600
Appliance 2 10/day 10/day 500
Appliance 3 50/day 20/day 1900
Daily cost $10,000 $20,000
28. Nutrition In a laboratory experiment, two separate foods are given to experimental animals. Each food contains essential ingredients, A and B, for which the animals have minimum requirements, and each food also has an ingredient C, which can be harmful to the animals. The following table summarizes this information.
Food 1 Food 2 Required
Ingredient A 10 units/g 3 units/g 49 units
Ingredient B 6 units/g 12 units/g 60 units
Ingredient C 3 units/g 1 unit/g
How many grams of foods 1 and 2 will satisfy the requirements for A and B and minimize the amount of ingredient C that is ingested?
29E30E31E In Problems 31-36, use the simplex method, Excel, or some other technology.
32. Dieting A dieting company offers three foods (A, B, and C) and groups its customers into two groups according to their nutritional needs. The following table gives the percent of the daily nutritional requirements that a serving of each food provides and the number of ounces of detrimental substances in each food. Determine the combination of food types that will provide at least 100% of the daily requirements and will minimize the detrimental substances. What is the minimum amount of the detrimental substance?
Food A Food B Food C Daily Requirement
Group 1 30% per serving 10% per serving 20% per serving 100%
Group II 10% per serving 20% per serving 40% per serving 100%
Detrimental 0.1 oz 0.2 oz 0.25 oz
substances per serving per serving per serving
33E In Problems 31-36, use the simplex method, Excel, or some other technology.
34. Pollution Three factories each dump waste water containing three different types of pollutants into a river. State regulations require the factories to treat their waste to reduce pollution levels. The following table shows the possible percent reduction of each pollutant at each site and the cost per ton to process the waste.
Factory 1 Factory 2 Factory 3
Pollutant 1 75% 45% 20%
Pollutant 2 65% 30% 15%
Pollutant 3 10% 15% 5%
Cost/ton $50 $20 $8
If the state requires a reduction of at least 65 tons per day of pollutant 1, at least 40 tons per day of pollutant 2, and at least 20 tons per day of pollutant 3, find the number of tons of waste that must be treated each day at each site so that the states requirements are satisfied and the treatment costs are minimized. Find the mini
mum cost.
1CP2. Using the matrix from Problem 1, find the maximum value of subject to the constraints.
1E2E3E4E5E