MAT 540 Entire Course Flashcards
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STRAYER MAT 540 Week 11 Discussion Reflection to Date NEW
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“Reflection to date” Please respond to the following:
• In a paragraph, reflect on what you’ve learned in this course. Identify the most interesting, unexpected, or useful thing you’ve learned, and explain how it can be applied to your work or daily life in some manner.
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STRAYER MAT 540 Week 10 Quiz 5 Set 2 QUESTIONS NEW
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Question 1: If exactly 3 projects are to be selected from a set of 5 projects, this would be written as 3 separate constraints in an integer program.
Question 2: If we are solving a 0-1 integer programming problem with three decision variables, the constraint x1 + x2 + x3 ≤ 3 is a mutually exclusive constraint.
Question 3: Rounding non-integer solution values up to the nearest integer value will result in an infeasible solution to an integer linear programming problem.
Question 4: If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional constraint.
Question 5: If we are solving a 0-1 integer programming problem with three decision variables, the constraint x1 + x2 ≤ 1 is a mutually exclusive constraint.
Question 6: In a problem involving capital budgeting applications, the 0-1 variables designate the acceptance or rejection of the different projects.
Question 7: The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff has 4 different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same.
Write a constraint to ensure that if machine 4 is used, machine 1 will not be used.
Question 8: You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are:
Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7.
Restriction 2. Evaluating sites S2 or
S4 will prevent you from assessing site S5.
Restriction 3. Of all the sites, at least 3 should be assessed.
Assuming that Si is a binary variable, write the constraint(s) for the second restriction
Question 9: In a 0-1 integer programming model, if the constraint x1-x2 = 0, it means when project 1 is selected, project 2 __________ be selected.
Question 10: Max Z = 5x1 + 6x2
Subject to: 17x1 + 8x2 ≤ 136
3x1 + 4x2 ≤ 36
x1, x2 ≥ 0 and integer
What is the optimal solution?
Question 11: If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a __________ constraint.
Question 12: Assume that we are using 0-1 integer programming model to solve a capital budgeting problem and xj = 1 if project j is selected and xj = 0, otherwise.
The constraint (x1 + x2 + x3 + x4 ≤ 2) means that __________ out of the 4 projects must be selected.
Question 13: In a 0-1 integer programming model, if the constraint x1-x2 ≤ 0, it means when project 2 is selected, project 1 __________ be selected.
Question 14: If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a __________ constraint.
Question 15: If the solution values of a linear program are rounded in order to obtain an integer solution, the solution is
Question 16: Binary variables are
Question 17: The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff has 4 different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same.
Write the constraint that indicates they can purchase no more than 3 machines.
Question 18: You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are:
Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7.
Restriction 2. Evaluating sites S2 or
S4 will prevent you from assessing site S5.
Restriction 3. Of all the sites, at least 3 should be assessed.
Assuming that Si is a binary variable, the constraint for the first restriction is
Question 19: Consider the following integer linear programming problem
Max Z = 3x1 + 2x2
Subject to: 3x1 + 5x2 ≤ 30
4x1 + 2x2 ≤ 28
x1 ≤ 8
x1 , x2 ≥ 0 and integer
Find the optimal solution. What is the value of the objective function at the optimal solution. Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty-five) would be written 25
Question 20: Consider the following integer linear programming problem
Max Z = 3x1 + 2x2
Subject to: 3x1 + 5x2 ≤ 30
5x1 + 2x2 ≤ 28
x1 ≤ 8
x1 ,x2 ≥ 0 and integer
Find the optimal solution. What is the value of the objective function at the optimal solution. Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty-five) would be written 25
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STRAYER MAT 540 Week 10 Quiz 5 Set 3 QUESTIONS NEW
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Question 1: In a problem involving capital budgeting applications, the 0-1 variables designate the acceptance or rejection of the different projects.
Question 2: If we are solving a 0-1 integer programming problem with three decision variables, the constraint x1 + x2 + x3 ≤ 3 is a mutually exclusive constraint.
Question 3: If exactly 3 projects are to be selected from a set of 5 projects, this would be written as 3 separate constraints in an integer program.
Question 4: In a 0-1 integer programming problem involving a capital budgeting application (where xj = 1, if project j is selected, xj = 0, otherwise) the constraint x1 - x2 ≤ 0 implies that if project 2 is selected, project 1 can not be selected.
Question 5: In a mixed integer model, some solution values for decision variables are integer and others are only 0 or 1.
Question 6: If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional constraint.
Question 7: You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are:
Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7.
Restriction 2. Evaluating sites S2 or
S4 will prevent you from assessing site S5.
Restriction 3. Of all the sites, at least 3 should be assessed.
Assuming that Si is a binary variable, write the constraint(s) for the second restriction
Question 8: If the solution values of a linear program are rounded in order to obtain an integer solution, the solution is
Question 9: In a capital budgeting problem, if either project 1 or project 2 is selected, then project 5 cannot be selected. Which of the alternatives listed below correctly models this situation?
Question 10: In a 0-1 integer programming model, if the constraint x1-x2 ≤ 0, it means when project 2 is selected, project 1 __________ be selected.
Question 11: In a __________ integer model, some solution values for decision variables are integers and others can be non-integer.
Question 12: Assume that we are using 0-1 integer programming model to solve a capital budgeting problem and xj = 1 if project j is selected and xj = 0, otherwise.
The constraint (x1 + x2 + x3 + x4 ≤ 2) means that __________ out of the 4 projects must be selected.
Question 13: If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a __________ constraint.
Question 14: If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a __________ constraint.
Question 15: You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are:
Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7.
Restriction 2. Evaluating sites S2 or
S4 will prevent you from assessing site S5.
Restriction 3. Of all the sites, at least 3 should be assessed.
Assuming that Si is a binary variable, the constraint for the first restriction is
Question 16: Binary variables are
Question 17: Max Z = 5x1 + 6x2
Subject to: 17x1 + 8x2 ≤ 136
3x1 + 4x2 ≤ 36
x1, x2 ≥ 0 and integer
What is the optimal solution?
Question 18: If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a __________ constraint.
Question 19: Max Z = 3x1 + 5x2
Subject to: 7x1 + 12x2 ≤ 136
3x1 + 5x2 ≤ 36
x1, x2 ≥ 0 and integer
Find the optimal solution. What is the value of the objective function at the optimal solution. Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty-five) would be written 25
Question 20: Consider the following integer linear programming problem
Max Z = 3x1 + 2x2
Subject to: 3x1 + 5x2 ≤ 30
4x1 + 2x2 ≤ 28
x1 ≤ 8
x1 , x2 ≥ 0 and integer
Find the optimal solution. What is the value of the objective function at the optimal solution. Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty-five) would be written 25
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Question 1: Rounding non-integer solution values up to the nearest integer value will result in an infeasible solution to an integer linear programming problem.
Question 2: The solution to the LP relaxation of a maximization integer linear program provides an upper bound for the value of the objective function.
Question 3: If we are solving a 0-1 integer programming problem with three decision variables, the constraint x1 + x2 ≤ 1 is a mutually exclusive constraint.
Question 4: A conditional constraint specifies the conditions under which variables are integers or real variables.
Question 5: If we are solving a 0-1 integer programming problem with three decision variables, the constraint x1 + x2 + x3 ≤ 3 is a mutually exclusive constraint.
Question 6: If exactly 3 projects are to be selected from a set of 5 projects, this would be written as 3 separate constraints in an integer program.
Question 7: The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff has 4 different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same.
Write the constraint that indicates they can purchase no more than 3 machines.
Question 8: In a 0-1 integer programming model, if the constraint x1-x2 = 0, it means when project 1 is selected, project 2 \_\_\_\_\_\_\_\_\_\_ be selected.
Question 9: Assume that we are using 0-1 integer programming model to solve a capital budgeting problem and xj = 1 if project j is selected and xj = 0, otherwise.
The constraint (x1 + x2 + x3 + x4 ≤ 2) means that \_\_\_\_\_\_\_\_\_\_ out of the 4 projects must be selected.
Question 10: If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a \_\_\_\_\_\_\_\_\_\_ constraint.
Question 11: If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a \_\_\_\_\_\_\_\_\_\_ constraint.
Question 12: If the solution values of a linear program are rounded in order to obtain an integer solution, the solution is
Question 13: If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a \_\_\_\_\_\_\_\_\_\_ constraint.
Question 14: If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a \_\_\_\_\_\_\_\_\_\_ constraint.
Question 15: Binary variables are
Question 16: In a 0-1 integer programming model, if the constraint x1-x2 ≤ 0, it means when project 2 is selected, project 1 \_\_\_\_\_\_\_\_\_\_ be selected.
Question 17: The solution to the linear programming relaxation of a minimization problem will always be \_\_\_\_\_\_\_\_\_\_ the value of the integer programming minimization problem.
Question 18: In a \_\_\_\_\_\_\_\_\_\_ integer model, some solution values for decision variables are integers and others can be non-integer.
Question 19: Max Z = 3x1 + 5x2
Subject to: 7x1 + 12x2 ≤ 136
3x1 + 5x2 ≤ 36
x1, x2 ≥ 0 and integer
Find the optimal solution. What is the value of the objective function at the optimal solution. Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty-five) would be written 25
Question 20: Consider the following integer linear programming problem
Max Z = 3x1 + 2x2
Subject to: 3x1 + 5x2 ≤ 30
5x1 + 2x2 ≤ 28
x1 ≤ 8
x1 ,x2 ≥ 0 and integer
Find the optimal solution. What is the value of the objective function at the optimal solution. Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty-five) would be written 25
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- Consider the following transportation problem:
From To (Cost) Supply 1 2 3 A 6 5 5 150 B 11 8 9 85 C 4 10 7 125 Demand 70 100 80
Formulate this problem as a linear programming model and solve it by the using the computer. - Consider the following transportation problem:
From To (Cost) Supply 1 2 3 A 8 14 8 120 B 6 17 7 80 C 9 24 10 150 Demand 110 140 100 Solve it by using the computer. - World foods, Inc. imports food products such as meats, cheeses, and pastries to the United States from warehouses at ports in Hamburg, Marseilles and Liverpool. Ships from these ports deliver the products to Norfolk, New York and Savannah, where they are stored in company warehouses before being shipped to distribution centers in Dallas, St. Louis and Chicago. The products are then distributed to specialty foods stores and sold through catalogs. The shipping costs ($/1,000 lb.) from the European ports to the U.S. cities and the available supplies (1000 lb.) at the European ports are provided in the following table: From To (Cost) Supply 4. Norfolk 5. New York 6. Savannah 1. Hamburg 320 280 555 75 2. Marseilles 410 470 365 85 3. Liverpool 550 355 525 40
The transportation costs ($/1000 lb.) from each U.S. city of the three distribution centers and the demands (1000 lb.) at the distribution centers are as follows:
Warehouse Distribution Center 7. Dallas 8. St. Louis 9. Chicago 4. Norfolk 80 78 85 5. New York 100 120 95 6. Savannah 65 75 90 Demand 85 70 65 Determine the optimal shipments between the European ports and the warehouses and the distribution centers to minimize total transportation costs. - The Omega Pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions. Given their various previous contacts, the sales persons are able to cover the regions in different amounts of time. The amount of time (days) required by each salesperson to cover each city is shown in the following table:
Salesperson Region (days) A B C D E 1 20 10 12 10 22 2 14 10 18 11 15 3 12 13 19 11 14 4 16 12 14 22 16 5 12 15 19 26 23 Which salesperson should be assigned to each region to minimize total time? Identify the optimal assignments and compute total minimum time.
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STRAYER MAT 540 Week 10 Discussion Transshipment problems NEW
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For more classes visit http://www.uopassignments.com Discussion assignment and transshipment problems Select one (1) of the following topics for your primary discussion posting: • Explain the assignment model and how it facilitates in solving transportation problems. Determine the benefits to be gained from using this model. • Identify any challenges you have in setting up an transshipment model in Excel, and solving it with Solver. Explain exactly what the challenges are and why they are challenging. Identify resources that can help you with that.
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STRAYER MAT 540 Week 9 Quiz 4 Set 3 QUESTIONS NEW
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Question 1: A constraint for a linear programming problem can never have a zero as its right-hand-side value.
Question 2: In a balanced transportation model, supply equals demand such that all constraints can be treated as equalities.
Question 3: In a media selection problem, instead of having an objective of maximizing profit or minimizing cost, generally the objective is to maximize the audience exposure.
Question 4: The standard form for the computer solution of a linear programming problem requires all variables to be to the right and all numerical values to be to the left of the inequality or equality sign
Question 5: When using a linear programming model to solve the “diet” problem, the objective is generally to maximize profit.
Question 6: Fractional relationships between variables are permitted in the standard form of a linear program.
Question 7: Assume that x2, x7 and x8 are the dollars invested in three different common stocks from New York stock exchange. In order to diversify the investments, the investing company requires that no more than 60% of the dollars invested can be in “stock two”. The constraint for this requirement can be written as:
Question 8: A systematic approach to model formulation is to first
Question 9: In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The stockbroker suggests limiting the investments so that no more than $10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed 350, whichever is more restrictive. How would this be formulated as a linear programming constraint?
Question 10: Balanced transportation problems have the following type of constraints:
Question 11: Compared to blending and product mix problems, transportation problems are unique because
Question 12: A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today’s production run. Bear claw profits are 20 cents each, and almond filled croissant profits are 30 cents each. What is the optimal daily profit?
Question 13: The following types of constraints are ones that might be found in linear programming formulations:
1. ≤
2. =
3. >
Question 14: In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor stipulates that stock 1 must not account for more than 35% of the number of shares purchased. Which constraint is correct?
Question 15: The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint?
Question 16: Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8,000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the supply constraint for component 1.
Question 17: When systematically formulating a linear program, the first step is
Question 18: In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The expected returns on investment of the three stocks are 6%, 8%, and 11%. An appropriate objective function is
Question 19: Quickbrush Paint Company makes a profit of $2 per gallon on its oil-base paint and $3 per gallon on its water-base paint. Both paints contain two ingredients, A and B. The oil-base paint contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. The company wishes to use linear programming to determine the appropriate mix of oil-base and water-base paint to produce to maximize its total profit. How many gallons of oil based paint should the Quickbrush make? Note: Please express your answer as a whole number, rounding the nearest whole number, if appropriate.
Question 20: Kitty Kennels provides overnight lodging for a variety of pets. An attractive feature is the quality of care the pets receive, including well balanced nutrition. The kennel’s cat food is made by mixing two types of cat food to obtain the “nutritionally balanced cat diet.” The data for the two cat foods are as follows:
Kitty Kennels wants to be sure that the cats receive at least 5 ounces of protein and at least 3 ounces of fat per day. What is the cost of this plan? Express your answer with two places to the right of the decimal point. For instance, $9.32 (nine dollars and thirty-two cents) would be written as 9.32
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Question 1: In a transportation problem, a demand constraint (the amount of product demanded at a given destination) is a less-than-or equal-to constraint (≤).
Question 2: A constraint for a linear programming problem can never have a zero as its right-hand-side value.
Question 3: Product mix problems cannot have “greater than or equal to” (≥) constraints.
Question 4: In formulating a typical diet problem using a linear programming model, we would expect most of the constraints to be related to calories.
Question 5: In a balanced transportation model, supply equals demand such that all constraints can be treated as equalities.
Question 6: When using a linear programming model to solve the “diet” problem, the objective is generally to maximize profit.
Question 7: A systematic approach to model formulation is to first
Question 8: Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8,000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the supply constraint for component 1.
Question 9: A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today’s production run. Bear claw profits are 20 cents each, and almond filled croissant profits are 30 cents each. What is the optimal daily profit?
Question 10: The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint?
Question 11: Balanced transportation problems have the following type of constraints:
Question 12: In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The expected returns on investment of the three stocks are 6%, 8%, and 11%. An appropriate objective function is
Question 13: The following types of constraints are ones that might be found in linear programming formulations:
1. ≤
2. =
3. >
Question 14: In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The stockbroker suggests limiting the investments so that no more than $10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed 350, whichever is more restrictive. How would this be formulated as a linear programming constraint?
Question 15: The owner of Black Angus Ranch is trying to determine the correct mix of two types of beef feed, A and B which cost 50 cents and 75 cents per pound, respectively. Five essential ingredients are contained in the feed, shown in the table below. The table also shows the minimum daily requirements of each ingredient.
Ingredient Percent per pound in Feed A Percent per pound in Feed B Minimum daily requirement (pounds)
1 20 24 30
2 30 10 50
3 0 30 20
4 24 15 60
5 10 20 40
The constraint for ingredient 3 is:
Question 16: In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor stipulates that stock 1 must not account for more than 35% of the number of shares purchased. Which constraint is correct?
Question 17: The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are constraint production time (8 hours = 480 minutes per day) and syrup (1 of her ingredient) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the optimal daily profit?
Question 18: If Xij = the production of product i in period j, write an expression to indicate that the limit on production of the company’s 3 products in period 2 is equal to 400.
Question 19: Kitty Kennels provides overnight lodging for a variety of pets. An attractive feature is the quality of care the pets receive, including well balanced nutrition. The kennel’s cat food is made by mixing two types of cat food to obtain the “nutritionally balanced cat diet.” The data for the two cat foods are as follows:
Kitty Kennels wants to be sure that the cats receive at least 5 ounces of protein and at least 3 ounces of fat per day. What is the cost of this plan? Express your answer with two places to the right of the decimal point. For instance, $9.32 (nine dollars and thirty-two cents) would be written as 9.32
Question 20: Quickbrush Paint Company makes a profit of $2 per gallon on its oil-base paint and $3 per gallon on its water-base paint. Both paints contain two ingredients, A and B. The oil-base paint contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. The company wishes to use linear programming to determine the appropriate mix of oil-base and water-base paint to produce to maximize its total profit. How many gallons of oil based paint should the Quickbrush make? Note: Please express your answer as a whole number, rounding the nearest whole number, if appropriate.
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STRAYER MAT 540 Week 9 Quiz 4 Set 1 QUESTIONS NEW
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Question 1: When using a linear programming model to solve the “diet” problem, the objective is generally to maximize profit.
Question 2: In a balanced transportation model, supply equals demand such that all constraints can be treated as equalities.
Question 3: In a transportation problem, a demand constraint (the amount of product demanded at a given destination) is a less-than-or equal-to constraint (≤).
Question 4: Fractional relationships between variables are permitted in the standard form of a linear program.
Question 5: The standard form for the computer solution of a linear programming problem requires all variables to be to the right and all numerical values to be to the left of the inequality or equality sign
Question 6: A systematic approach to model formulation is to first construct the objective function before determining the decision variables.
Question 7: A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today’s production run. Bear claw profits are 20 cents each, and almond filled croissant profits are 30 cents each. What is the optimal daily profit?
Question 8: In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor stipulates that stock 1 must not account for more than 35% of the number of shares purchased. Which constraint is correct?
Question 9: Balanced transportation problems have the following type of constraints:
Question 10: The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint?
Question 11: A systematic approach to model formulation is to first
Question 12: The following types of constraints are ones that might be found in linear programming formulations:
1. ≤
2. =
3. >
Question 13: The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of Lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of Vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of Lime chips are $0.40, and for a bag of Vinegar chips $0.50.
What is the constraint for salt?
Question 14: Small motors for garden equipment is produced at 4 manufacturing facilities and needs to be shipped to 3 plants that produce different garden items (lawn mowers, rototillers, leaf blowers). The company wants to minimize the cost of transporting items between the facilities, taking into account the demand at the 3 different plants, and the supply at each manufacturing site. The table below shows the cost to ship one unit between each manufacturing facility and each plant, as well as the demand at each plant and the supply at each manufacturing facility.
What is the demand constraint for plant B?
Question 15: Compared to blending and product mix problems, transportation problems are unique because
Question 16: Assume that x2, x7 and x8 are the dollars invested in three different common stocks from New York stock exchange. In order to diversify the investments, the investing company requires that no more than 60% of the dollars invested can be in “stock two”. The constraint for this requirement can be written as:
Question 17: The owner of Black Angus Ranch is trying to determine the correct mix of two types of beef feed, A and B which cost 50 cents and 75 cents per pound, respectively. Five essential ingredients are contained in the feed, shown in the table below. The table also shows the minimum daily requirements of each ingredient.
Ingredient Percent per pound in Feed A Percent per pound in Feed B Minimum daily requirement (pounds) 1 20 24 30 2 30 10 50 3 0 30 20 4 24 15 60 5 10 20 40
The constraint for ingredient 3 is:
Question 18: Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8,000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the supply constraint for component 1.
Question 19: Quickbrush Paint Company makes a profit of $2 per gallon on its oil-base paint and $3 per gallon on its water-base paint. Both paints contain two ingredients, A and B. The oil-base paint contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. The company wishes to use linear programming to determine the appropriate mix of oil-base and water-base paint to produce to maximize its total profit. How many gallons of water based paint should the Quickbrush make? Note: Please express your answer as a whole number, rounding the nearest whole number, if appropriate.
Question 20: Kitty Kennels provides overnight lodging for a variety of pets. An attractive feature is the quality of care the pets receive, including well balanced nutrition. The kennel’s cat food is made by mixing two types of cat food to obtain the “nutritionally balanced cat diet.” The data for the two cat foods are as follows:
Kitty Kennels wants to be sure that the cats receive at least 5 ounces of protein and at least 3 ounces of fat per day. What is the cost of this plan? Express your a
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STRAYER MAT 540 Week 9 Homework Chapter 5 NEW
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1. Rowntown Cab Company has 70 drivers that it must schedule in three 8-hour shifts. However, the demand for cabs in the metropolitan area varies dramatically according to time of the day. The slowest period is between midnight and 4:00 A.M. the dispatcher receives few calls, and the calls that are received have the smallest fares of the day. Very few people are going to the airport at that time of the night or taking other long distance trips. It is estimated that a driver will average $80 in fares during that period. The largest fares result from the airport runs in the morning. Thus, the drivers who sart their shift during the period from 4:00 A.M. to 8:00 A.M. average $500 in total fares, and drivers who start at 8:00 A.M. average $420. Drivers who start at noon average $300, and drivers who start at 4:00 P.M. average $270. Drivers who start at the beginning of the 8:00 P.M. to midnight period earn an average of $210 in fares during their 8-hour shift.
To retain customers and acquire new ones, Rowntown must maintain a high customer service level. To do so, it has determined the minimum number of drivers it needs working during every 4-hour time segment- 10 from midnight to 4:00 A.M. 12 from 4:00 to 8:00 A.M. 20 from 8:00 A.M. to noon, 25 from noon to 4:00 P.M., 32 from 4:00 to 8:00 P.M., and 18 from 8:00 P.M. to midnight.
a. Formulate and solve an integer programming model to help Rowntown Cab schedule its drivers.
b. If Rowntown has a maximum of only 15 drivers who will work the late shift from midnight to 8:00 A.M., reformulate the model to reflect this complication and solve it
c. All the drivers like to work the day shift from 8:00 A.M. to 4:00 P.M., so the company has decided to limit the number of drivers who work this 8-hour shift to 20. Reformulate the model in (b) to reflect this restriction and solve it.
2.
2. Juan Hernandez, a Cuban athlete who visits the United States and Europe frequently, is allowed to return with a limited number of consumer items not generally available in Cuba. The items, which are carried in a duffel bag, cannot exceed a weight of 5 pounds. Once Juan is in Cuba, he sells the items at highly inflated prices. The weight and profit (in U.S. dollars) of each item are as follows: Item Weight (lb.) Profit Denim jeans 2 $90 CD players 3 150 Compact discs 1 30 Juan wants to determine the combination of items he should pack in his duffel bag to maximize his profit. This problem is an example of a type of integer programming problem known as a
“knapsack” problem. Formulate and solve the problem.
- The Texas Consolidated Electronics Company is contemplating a research and development program encompassing eight research projects. The company is constrained from embarking on all projects by the number of available management scientists (40) and the budget available for R&D projects ($300,000). Further, if project 2 is selected, project 5 must also be selected (but not vice versa). Following are the resources requirement and the estimated profit for each project.
Project Expense Management Estimated Profit ($1,000s) Scientists required (1,000,000s) 1 50 6 0.30 2 105 8 0.85 3 56 9 0.20 4 45 3 0.15 5 90 7 0.50 6 80 5 0.45 7 78 8 0.55 8 60 5 0.40 Formulate the integer programming model for this problem and solve it using the computer.
Corsouth Mortgage Associates is a large home mortgage firm in the southeast. It has a poll of permanent and temporary computer operators who process mortgage accounts, including posting payments and updating escrow accounts for insurance and taxes. A permanent operator can process 220 accounts per day, and a temporary operator can process 140 accounts per day. On average, the firm must process and update at least 6,300 accounts daily. The company has 32 computer workstations available. Permanent and temporary operators work 8 hours per day. A permanent operator averages about 0.4 error per day, whereas a temporary operator averages 0.9 error per day. The company wants to limit errors to 15 per day. A permanent operator is paid $120 per day wheras a temporary operator is paid $75 per day. Corsouth wants to determine the number of permanent and temporary operators it needs to minimize cost. Formulate, and solve an integer programming model for this problem and compare this solution to the non-integer solution.
5. Globex Investment Capital Corporation owns six companies that have the following estimated returns (in millions of dollars) if sold in one of the next 3 years: Year Sold (estimated returns, $1,000,000s) Company 1 2 3 1 $14 $18 $232 9 11 153 18 23 274 16 21 255 12 16 226 21 23 28 To generate operating funds, the company must sell at least $20 million worth of assets in year 1, $25 million in year 2, and $35 million in year 3. Globex wants to develop a plan for selling these companies during the next 3 years to maximize return. Formulate an integer programming model for this problem and solve it by using the computer.
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STRAYER MAT 540 Week 9 Discussion Application of Integer Programming NEW
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Explain how the applications of Integer programming differ from those of linear programming. Give specific instances in which you would use an integer programming model rather than an LP model. Provide real-world examples.
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STRAYER MAT 540 Week 8 Homework Chapter 4 NEW
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1. Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday, and she must determine how much beer to stock. Betty stocks three brands of beer- Yodel, Shotz, and Rainwater. The cost per gallon (to the tavern owner) of each brand is as follows:
Brand………………..Cost/Gallon
Yodel…………………$1.50
Shotz…………………. 0.90
Rainwater…………… 0.50
The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel, 500 gallons of shotz, and 300 gallons of Rainwater. The tavern has the capacity to stock 1,000 gallons of beer; Betty wants to stock up completely. Betty wants to determine the number of gallons of each brand of beer to order so as to maximize profit.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
2. As result of a recently passed bill, a congressman’s district has been allocated $3 million for programs and projects. It is up to the congressman to decide how to distribute the money. The congressman has decide to allocate the money to four ongoing programs because of their importance to his district- a job training program, a parks project, a sanitation project, and a mobile library. However, the congressman wants to distribute the money in a manner that will please the most voters, or, in other words, gain him the most votes in the upcoming election. His staff’s estimates of the number of votes gained per dollar spent for the various programs are as follows.
Program………………..Votes/Dollar
Job training…………….0.03
Parks………………………..0.08
Sanitation………………..0.05
Mobile library………….0.03
In order also to satisfy several local influential citizens who financed his election, he is obligated to observe the following guidelines:
• None of the programs can receive more than 30% of the total allocation
• The amount allocated to parks cannot exceed the total allocated to both the sanitation project and the mobile library.
• The amount allocated to job training must at least equal the amount spent on the sanitation project. Any money not spent in the district will be returned to the government; therefore, the congressman wants to spend it all. Thee congressman wants to know the amount to allocate to each program to maximize his votes.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
3. Anna Broderick is the dietician for the State University football team, and she is attempting to determine a nutritious lunch menu for the team. She has set the following nutritional guidelines for each lunch serving:
• Between 1,300 and 2,100 calories
• At least 4 mg of iron
• At least 15 but no more than 55g of fat
• At least 30g of protein
• At least 60g of carbohydrates
• No more than 35 mg of cholesterol
She selects the menu from seven basic food items, as follows, with the nutritional contributions per pound and the cost as given:
………… Calories……Iron….Protein …Carbohydrates….Fat….Cholesterol……Cost
…………. (Per lb)…….(mg/lb)…..(g/lb)……..(g/lb)…………..(g/lb)………(mg/lb)……….($/lb)
Chicken 500 4.2 17 0 30 180 0.85
Fish 480 3.1 85 0 5 90 3.35
Ground beef 840 0.25 82 0 75 350 2.45
Dried beans 590 3.2 10 30 3 0 0.85
Lettuce 40 0.4 6 0 0 0 0.70
Potatoes 450 2.25 10 70 0 0 0.45
Milk (2%) 220 0.2 16 22 10 20 0.82
The dietician wants to select a menu to meet the nutritional guidelines while minimizing the total cost per serving.
a. Formulate a linear programming model for this problem and solve.
b. If a serving of each of the food items (other than milk) was limited to no more than a half pound, what effect would this have on the solution?
4. Dr. Maureen Becker, the head administrator at Jefferson County Regional Hospital, must determine a schedule for nurses to make sure there are enough of them on duty throughout the day. During the day, the demand for nurses varies. Maureen has broken the day in to twelve 2- hour periods. The slowest time of the day encompasses the three periods from 12:00 A.M. to 6:00 A.M., which beginning at midnight; require a minimum of 30, 20, and 40 nurses, respectively. The demand for nurses steadily increases during the next four daytime periods. Beginning with the 6:00 A.M.- 8:00 A.M. period, a minimum of 50, 60, 80, and 80 nurses are required for these four periods, respectively. After 2:00 P.M. the demand for nurses decreases during the afternoon and evening hours. For the five 2-hour periods beginning at 2:00 P.M. and ending midnight, 70, 70, 60, 50, and 50 nurses are required, respectively. A nurse reports for duty at the beginning of one of the 2-hour periods and works 8 consecutive hours (which is required in the nurses’ contract). Dr. Becker wants to determine a nursing schedule that will meet the hospital’s minimum requirement throughout the day while using the minimum number of nurses.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
5. The production manager of Videotechnics Company is attempting to determine the upcoming 5-month production schedule for video recorders. Past production records indicate that 2,000 recorders can be produced per month. An additional 600 recorders can be produced monthly on an overtime basis. Unit cost is $10 for recorders produced during regular working hours and $15 for those produced on an overtime basis. Contracted sales per month are as follows: Month Contracted Sales (units) 1 1200 2 2100 3 4 5 2400 3000 4000 Inventory carrying costs are $2 per recorder per month. The manager does not want any inventory carried over past the fifth month. The manager wants to know the monthly production that will minimize total production and inventory costs. a. Formulate a linear programming model for this problem. b. Solve the model by using the computer.
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STRAYER MAT 540 Week 8 Discussion Practice setting up linear programming models for business applications NEW
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Practice setting up linear programming models for business applications
Select an even-numbered LP problem from the text, excluding 14, 20, 22, 36 (which are part of your homework assignment). Formulate a linear programming model for the problem you select.
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STRAYER MAT 540 Week 8 Assignment Linear Programming Case Study You are a portfolio manager for the XYZ investment fund NEW
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Week 8 Project
You are a portfolio manager for the XYZ investment fund. The objective for the fund is to maximize your portfolio returns from the investments on four alternatives. The investments include (1) stocks, (2) real estate, (3) bonds, and (4) certificate of deposit (CD). Your total investment portfolio is $1,000,000.
Investment Returns
Based on the returns from the past five years, you concluded that the investment annual returns on stocks are 10%, on real estates are 7% on bonds are 4% and on CD is 1%.
Risk Constraints
However, you also have to analyze the risks associate with each investment category. A wildly used risk measurement parameter is called Value at Risk (VaR). (Note: VaR measures the risk of loss on a specific portfolio of financial assets.) For example, given a million dollar stock investment, if a portfolio of stocks has a one-day 4% VaR, there is a 5% probability that the stock portfolio will fall in value by more than 1,000,000 * 0.004 = $4,000 over a one day period. In the portfolio, the VaR for stock investments is 6%. Similarly, the VaR for real estate investment is 2% and the VaR for bond investment is 1% and the VaR for investment in CD is 0%. To manage the portfolio, you decided that at 5% probability, your VaR for stocks cannot exceed $25,000, VaR for real estate cannot exceed $15,000, VaR for bonds cannot exceed $2,500 and the VaR for CD investment is $0.
Diversification and Liquidity Constraints
As a diversified investment portfolio, you also decided that each investment category must hold at least $50,000 of the total investment assets. In addition, you must hold combined CD and bond investment no less than $200,000 in order to meet liquidity requirement.
The total amount of real estate holding shall not exceed 30% of the portfolio assets.
A. As a portfolio manager, please formulate and solve the investment portfolio problem using linear programming technique. What are the amounts invest in (1) stocks, (2) real estate, (3) bonds and (4) CD?
B. If $500,000 additional investments are available to you in your portfolio, how would you invest the capital?
C. Would you maintain the portfolio investment if stock yields lowered to 6%? How would you re-distribute your investment portfolio?
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STRAYER MAT 540 Week 7 Quiz 3 Set 2 QUESTIONS NEW
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MAT 540 Week 7 Quiz 3 Set 3 QUESTIONS NEW
Question 1: A feasible solution violates at least one of the constraints.
Question 2: A linear programming model consists of only decision variables and constraints.
Question 3: If the objective function is parallel to a constraint, the constraint is infeasible.
Question 4: In minimization LP problems the feasible region is always below the resource constraints.
Question 5: Surplus variables are only associated with minimization problems.
Question 6: If the objective function is parallel to a constraint, the constraint is infeasible.
Question 7: A linear programming problem may have more than one set of solutions.
Question 8: The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular (R) and diet(D). Two of the limited resources are production time (8 hours = 480 minutes per day) and syrup limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint?
Question 9: The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. For the production combination of 135 cases of regular and 0 cases of diet soft drink, which resources will not be completely used?
Question 10: In a linear programming problem, the binding constraints for the optimal solution are:
5x1 + 3x2 ≤ 30
2x1 + 5x2 ≤ 20
Which of these objective functions will lead to the same optimal solution?
Question 11: Which of the following statements is not true?
Question 12: The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular (R) and diet (D). Two of her limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the objective function?
Question 13: The following is a graph of a linear programming problem. The feasible solution space is shaded, and the optimal solution is at the point labeled Z.
Which of the following points are not feasible?
Question 14: Decision variables
Question 15: Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the objective function?
Question 16: Which of the following could be a linear programming objective function?
Question 17: The following is a graph of a linear programming problem. The feasible solution space is shaded, and the optimal solution is at the point labeled Z.
Which of the following constraints has a surplus greater than 0?
Question 18: A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function. What would be the new slope of the objective function if multiple optimal solutions occurred along line segment AB? Write your answer in decimal notation.
Question 19: Consider the following linear programming problem:
Max Z = $15x + $20y
Subject to: 8x + 5y ≤ 40
0.4x + y ≥ 4
x, y ≥ 0
At the optimal solution, what is the amount of slack associated with the first constraint?
Question 20: Consider the following minimization problem:
Min z = x1 + 2x2
s.t. x1 + x2 ≥ 300
2x1 + x2 ≥ 400
2x1 + 5x2 ≤ 750
x1, x2 ≥ 0
Find the optimal solution. What is the value of the objective function at the optimal solution? Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty five) would be written 25
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STRAYER MAT 540 Week 7 Quiz 3 Set 1 QUESTIONS NEW
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Question 1: Graphical solutions to linear programming problems have an infinite number of possible objective function lines.
Question 2: The following inequality represents a resource constraint for a maximization problem:
X + Y ≥ 20
Question 3: In minimization LP problems the feasible region is always below the resource constraints.
Question 4: In a linear programming problem, all model parameters are assumed to be known with certainty.
Question 5: A feasible solution violates at least one of the constraints.
Question 6: If the objective function is parallel to a constraint, the constraint is infeasible.
Question 7: If the objective function is parallel to a constraint, the constraint is infeasible.
Question 8: Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the maximum profit?
Question 9: The following is a graph of a linear programming problem. The feasible solution space is shaded, and the optimal solution is at the point labeled Z.
Which of the following points are not feasible?
Question 10: The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular (R) and diet(D). Two of the limited resources are production time (8 hours = 480 minutes per day) and syrup limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint?
Question 11: The following is a graph of a linear programming problem. The feasible solution space is shaded, and the optimal solution is at the point labeled Z.
The equation for constraint DH is:
Question 12: Which of the following statements is not true?
Question 13: In a linear programming problem, a valid objective function can be represented as
Question 14: The linear programming problem:
MIN Z = 2x1 + 3x2
Subject to: x1 + 2x2 ≤ 20
5x1 + x2 ≤ 40
4x1 +6x2 ≤ 60
x1 , x2 ≥ 0 ,
Question 15: The following is a graph of a linear programming problem. The feasible solution space is shaded, and the optimal solution is at the point labeled Z*.
This linear programming problem is a:
Question 16: A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function.
If this is maximization, which extreme point is the optimal solution?
Question 17: Which of the following could be a linear programming objective function?
Question 18: Solve the following graphically
Max z = 3x1 +4x2
s.t. x1 + 2x2 ≤ 16
2x1 + 3x2 ≤ 18
x1 ≥ 2
x2 ≤ 10
x1, x2 ≥ 0
Find the optimal solution. What is the value of the objective function at the optimal solution? Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty five) would be written 25
Question 19: Consider the following linear programming problem:
Max Z = $15x + $20y
Subject to: 8x + 5y ≤ 40
0.4x + y ≥ 4
x, y ≥ 0
At the optimal solution, what is the amount of slack associated with the first constraint?
Question 20: Max Z = $3x + $9y
Subject to: 20x + 32y ≤ 1600
4x + 2y ≤ 240
y ≤ 40
x, y ≥ 0
At the optimal solution, what is the amount of slack associated with the second constraint?
