Introduction to Management Science Study Set 3

Business

Quiz 2 :

Linear Programming: Model Formulation and Graphical Solution

Quiz 2 :

Linear Programming: Model Formulation and Graphical Solution

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What would be the effect on the optimal solution in Problem if the cost of rice increased from $0.03 per ounce to $0.06 per ounce Problem The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 2 milligrams of B. An ounce of oats costs $0.05, and an ounce of rice costs $0.03. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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Linear programming is a model that maximizes or minimizes a linear objective function subject to some linear constrains. It is used to develop the relationship between the objective of organization and constraints of resources.
Identify the Decision Variables:
The ingredient, rice and oats provide vitamin A and B. The company wants to know the amount of rice and oats to be included in the cereal box. Hence, the variables are oats ( x 1 ) and rice ( x 2 ).
img The cost of one ounce of oats is $0.05 and cost of one ounce of rice is $0.06 (note that earlier it was $0.03). Formulate the linear programming model, by minimizing the cost, as shown below:
img Calculate the coordinates for the constraints, considering one of the variables as 0:
Calculate the value of x 2 , assuming x 1 is 0 as follows:
img The coordinates are (0, 8).
img img The coordinates are (0, 6).
Calculate the value of objective function as follows:
img Calculate the value of x 1 , assuming x 2 is 0 as follows:
img The coordinates are (6, 0).
img img The coordinates are (12, 0).
Calculate the value of objective function as follows:
img Develop a graphical model using the following coordinates:
O (0,0)
A (0, 8)
D (0, 6)
B (6, 0)
E (12, 0)
Solve the model graphically by plotting the obtained values on graph as shown below:
img The feasible region is represented with the green shaded area in the graph. From the graph, it is seen that the feasible region is unbounded. Furthermore, coordinates (0, 0), (0,6), and (6, 0) are infeasible. Z value for the three points present in the feasible region are as follows:
img From the table, it is evident that the value of objective function (Z value) is minimum at coordinate C (12/5, 24/5) or (2.4, 4.8).
Hence, the optimal solution changes to
img , and
img . The minimum value of objective function is
img .

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The Kalo Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, phosphate, and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of phosphate, while a pound of ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phosphate, and 1 ounce of potassium. Ingredient 1 costs $3 per pound, and ingredient 2 costs $5 per pound. The company wants to know how many pounds of each chemical ingredient to put into a bag of fertilizer to meet the minimum requirements of 20 ounces of nitrogen, 36 ounces of phosphate, and 2 ounces of potassium while minimizing cost. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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Given
A company makes fertilizer by using two ingredients. The ingredients provide some amount of nitrogen, phosphorous and potassium.
The cost of ingredient 1 is $3 and that of ingredient 2 is $5.
It is required that the fertilizer that is made with the help of two ingredients and the must satisfy following requirement:
There should be a minimum amount of 20 pounds of nitrogen in the fertilizer.
There should be a minimum amount of 36 pounds of phosphorous in the fertilizer.
There should be a minimum amount of 2 pounds of potassium in the fertilizer.
a)
Step-1: Decide decision variables:
Let the amount ingredient1 to be used be denoted by "
img " and the amount of ingredient2 to be used be denoted by "
img ".
Step-2: Formulate objective function:
It is given that the cost should be minimized by keeping into consideration various constraints. Thus, the objective function can be formulated as given below:
Objective function
img Step-3: Formulate constrains:
Minimum Composition Constraints:
img The linear programming model is shown below:
img b)
Solve the formulated model using the following methodology as shown below:
Step-1: Plot the constraints on graph as shown below:
For line
img ,
Put
img = 0, we get
img = 2. Thus, one gets a point (2, 0).
Put
img = 0, we get
img = 10. Thus, one gets a point (0, 10).
Plot a line by joining the the points (2, 0) and (0, 10).
Similarly, plot the other constraints as shown in the following graph:
img Step-2: Calculate the value of objective function and the corner points of feasible region A, C and G.
Thus, the corner points of the feasible region are A (0, 160), B (16, 80) and G (64, 32).
img img

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For Problem: a. How much labor and wood will be unused if the optimal numbers of chairs and tables are produced b. Explain the effect on the optimal solution of changing the profit on a table from $100 to $500. Problem The Pinewood Furniture Company produces chairs and tables from two resources-labor and wood. The company has 80 hours of labor and 36 board-ft. of wood available each day. Demand for chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 board-ft. of wood, whereas a table requires 10 hours of labor and 6 board-ft. of wood. The profit derived from each chair is $400 and from each table, $100. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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Linear programming model
This model is used to solve the maximization and minimization problems, thereby formulating objective function and other constraints.
Given:
It is given that the company has constraint of maximum amount of available labor to be 80 hours and that of available wood to be 36 board ft.
Demand for chair is limited to be 6.
Each chair requires 8 hours of labor and 2 board-ft. of wood.
Each table requires 10 hours of labor and 6 board-ft. of wood.
Profit from each chair is $400 and from each table is $100.
Determination of optimal solution based on the given data
Step 1: Formulation of linear programming model
Let the number of chairs produced each day be denoted by " x 1" and the number of tables produced each day be represented by " x 2".
Therefore, the objective function can be formulated as shown below:
img …… (1)
Subject to constraints
img …… (2)
img …… (3)
img …… (4)
img …… (5)
Thus, the final linear programming model is as follows:
img Step 2: Draw the graph for the above formulated model as follows:
Convert the constraints inequalities into equations follows and plot the graph for the same as follows:
img For random values of X 1 and X 2 , finds the value of objective functions as follows:
img The above given linear programming using graphical method as shown below:
img From the above table of calculations, the value of objective function is maximum at point D. Thus, the optimal solution is achieved at (6, 3.2), which means that 6 chairs and 3 tables should be produced in order to maximize the profit. The maximum value of profit is $2,720.
a.
Compute the unused value of labor and wood when optimal number of chair and tables are produced.
Amount of unused resources are computed or represented by the slack variable.
These unused resources do not have any impact on the optimal solution as these are "unused" or not used during production.
In the present case, optimal solution is achieved when 6 and 3.2 units of chairs and wood, respectively, are used in the production process.
Constraints used for labor is;
img Formulate the equation of labor constraint considering slack variable s 1 as follows:
img To compute the unused amount of labor at optimal solution, substitute the (6, 3.2) in the equation as follows:
img Constraints used for wood is;
img Formulate the equation of wood constraint considering slack variable s 2 as follows:
img To compute the unused amount of labor at optimal solution, substitute the (6, 3.2) in the equation as follows:
img Thus, the unused value of labor and wood when optimal number of chair and tables are produced is 0 and 4.8 respectively.
b.
Effect of change in profit in a table from $100 to $500 on optimal solution
New linear programming will be formulated as follows:
img Draw the graph for the above formulated model as follows:
Convert the constraints inequalities into equations follows and plot the graph for the same as follows:
img For random values of X 1 and X 2 , finds the value of objective functions as follows:
img The above given linear programming using graphical method as shown below:
img From the above table of calculations, the value of objective function is maximum at point C and D.
Thus, the optimal solution is achieved at (4.28, 4.57) and (6, 3.2), where maximum profit is achieved. The maximum value of profit at both the points is $4,000.

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A jewelry store makes necklaces and bracelets from gold and platinum. The store has 18 ounces of gold and 20 ounces of platinum. Each necklace requires 3 ounces of gold and 2 ounces of platinum, whereas each bracelet requires 2 ounces of gold and 4 ounces of platinum. The demand for bracelets is no more than four. A necklace earns $300 in profit and a bracelet, $400. The store wants to determine the number of necklaces and bracelets to make in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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A clothier makes coats and slacks. The two resources required are wool cloth and labor. The clothier has 150 square yards of wool and 200 hours of labor available. Each coat requires 3 square yards of wool and 10 hours of labor, whereas each pair of slacks requires 5 square yards of wool and 4 hours of labor. The profit for a coat is $50, and the profit for slacks is $40. The clothier wants to determine the number of coats and pairs of slacks to make so that profit will be maximized. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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The Crumb and Custard Bakery makes coffee cakes and Danish pastries in large pans. The main ingredients are flour and sugar. There are 25 pounds of flour and 16 pounds of sugar available, and the demand for coffee cakes is 5. Five pounds of flour and 2 pounds of sugar are required to make a pan of coffee cakes, and 5 pounds of flour and 4 pounds of sugar are required to make a pan of Danish. A pan of coffee cakes has a profit of $1, and a pan of Danish has a profit of $5. Determine the number of pans of cakes and Danish to produce each day so that profit will be maximized. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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In Problem, explain the effect on the optimal solution of increasing the profit on a bracelet from $400 to $600. What will be the effect of changing the platinum requirement for a necklace from 2 ounces to 3 ounces Problem A jewelry store makes necklaces and bracelets from gold and platinum. The store has 18 ounces of gold and 20 ounces of platinum. Each necklace requires 3 ounces of gold and 2 ounces of platinum, whereas each bracelet requires 2 ounces of gold and 4 ounces of platinum. The demand for bracelets is no more than four. A necklace earns $300 in profit and a bracelet, $400. The store wants to determine the number of necklaces and bracelets to make in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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In Problem 37 in Chapter, when Tracy McCoy wakes up Saturday morning, she remembersthat she promised the PTA she would make some cakes and/or homemade bread for its bake salethat afternoon In Problem in Chapter, when Tracy McCoy wakes up Saturday morning, she remembers that she promised the PTA she would make some cakes and/or homemade bread for its bake sale that afternoon. However, she does not have time to go to the store to get ingredients, and she has only a short time to bake things in her oven. Because cakes and breads require different baking temperatures, she cannot bake them simultaneously, and she has only 3 hours available to bake. A cake requires 3 cups of flour, and a loaf of bread requires 8 cups; Tracy has 20 cups of flour. A cake requires 45 minutes to bake, and a loaf of bread requires 30 minutes. The PTA will sell a cake for $10 and a loaf of bread for $6. Tracy wants to decide how many cakes and loaves of bread she should make. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis. Problem Assume that the objective function in Problem has been changed from Z = 30 x 1 + 70 x 2 to Z = 90 x 1 + 70 x 2. Determine the slope of each objective function and discuss what effect these slopes have on the optimal solution. Problem A manufacturing firm produces two products. Each product must undergo an assembly process and a finishing process. It is then transferred to the warehouse, which has space for only a limited number of items. The firm has 80 hours available for assembly and 112 hours for finishing, and it can store a maximum of 10 units in the warehouse. Each unit of product 1 has a profit of $30 and requires 4 hours to assemble and 14 hours to finish. Each unit of product 2 has a profit of $70 and requires 10 hours to assemble and 8 hours to finish. The firm wants to determine the quantity of each product to produce in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis
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The Weemow Lawn Service wants to start doing snow removal in the winter when there are no lawns to maintain. Jeff and Julie Weems, who own the service, are trying to determine how much equipment they need to purchase, based on the various job types they have. They plan to work themselves and hire some local college students on a per-job basis. Based on historical weather data, they estimate that there will be six major snowfalls next winter. Virtually all customers want their snow removed no more than 2 days after the snow stops falling. Working 10 hours per day (into the night), Jeff and Julie can remove the snow from a normal driveway in about 1 hour, and it takes about 4 hours to remove the snow from a business parking lot and sidewalk. The variable cost (mainly for labor and gas) per job is $12 for a driveway and $47 for a parking lot. Using their lawn service customer base as a guideline, they believe they will have demand of no more than 40 homeowners and 25 businesses. They plan to charge $35 for a home driveway and $120 for a business parking lot, which is slightly less than the going rate. They want to know how many jobs of each type will maximize their profit. a. Formulate a linear programming model for this problem. b. Solve this model graphically.
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The Pinewood Furniture Company produces chairs and tables from two resources-labor and wood. The company has 80 hours of labor and 36 board-ft. of wood available each day. Demand for chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 board-ft. of wood, whereas a table requires 10 hours of labor and 6 board-ft. of wood. The profit derived from each chair is $400 and from each table, $100. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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In Problem, how much flour and sugar will be left unused if the optimal numbers of cakes and Danish are baked Problem The Crumb and Custard Bakery makes coffee cakes and Danish pastries in large pans. The main ingredients are flour and sugar. There are 25 pounds of flour and 16 pounds of sugar available, and the demand for coffee cakes is 5. Five pounds of flour and 2 pounds of sugar are required to make a pan of coffee cakes, and 5 pounds of flour and 4 pounds of sugar are required to make a pan of Danish. A pan of coffee cakes has a profit of $1, and a pan of Danish has a profit of $5. Determine the number of pans of cakes and Danish to produce each day so that profit will be maximized. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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In Problem: a. If Jeff and Julie pay $3,700 for snow removal equipment, will they make any money b. If Jeff and Julie reduce their prices to $30 for a driveway and $100 for a parking lot, they will increase demand to 55 for driveways and 32 for businesses. Will this affect their possible profit c. Alternatively, hiring additional people on a per-job basis will increase Jeff and Julie's variable cost to $16 for a driveway and $53 for a parking lot, but it will lower the time it takes to clear a driveway to 40 minutes and a parking lot to 3 hours. Will this affect their profit d. If Jeff and Julie combine the two alternatives suggested in (b) and (c), will this affect their profit Problem The Weemow Lawn Service wants to start doing snow removal in the winter when there are no lawns to maintain. Jeff and Julie Weems, who own the service, are trying to determine how much equipment they need to purchase, based on the various job types they have. They plan to work themselves and hire some local college students on a per-job basis. Based on historical weather data, they estimate that there will be six major snowfalls next winter. Virtually all customers want their snow removed no more than 2 days after the snow stops falling. Working 10 hours per day (into the night), Jeff and Julie can remove the snow from a normal driveway in about 1 hour, and it takes about 4 hours to remove the snow from a business parking lot and sidewalk. The variable cost (mainly for labor and gas) per job is $12 for a driveway and $47 for a parking lot. Using their lawn service customer base as a guideline, they believe they will have demand of no more than 40 homeowners and 25 businesses. They plan to charge $35 for a home driveway and $120 for a business parking lot, which is slightly less than the going rate. They want to know how many jobs of each type will maximize their profit. a. Formulate a linear programming model for this problem. b. Solve this model graphically.
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Angela Fox and Zooey Caulfield were food and nutrition majors at State University, as well as close friends and roommates. Upon graduation Angela and Zooey decided to open a French restaurant in Draperton, the small town where the university was located. There were no other French restaurants in Draperton, and the possibility of doing something new and somewhat risky intrigued the two friends. They purchased an old Victorian home just off Main Street for their new restaurant, which they named "The Possibility." Angela and Zooey knew in advance that at least initially they could not offer a full, varied menu of dishes. They had no idea what their local customers' tastes in French cuisine would be, so they decided to serve only two full-course meals each night, one with beef and the other with fish. Their chef, Pierre, was confident he could make each dish so exciting and unique that two meals would be sufficient, at least until they could assess which menu items were most popular. Pierre indicated that with each meal he could experiment with different appetizers, soups, salads, vegetable dishes, and desserts until they were able to identify a full selection of menu items. The next problem for Angela and Zooey was to determine how many meals to prepare for each night so they could shop for ingredients and set up the work schedule. They could not afford too much waste. They estimated that they would sell a maximum of 60 meals each night. Each fish dinner, including all accompaniments, requires 15 minutes to prepare, and each beef dinner takes twice as long. There is a total of 20 hours of kitchen staff labor available each day. Angela and Zooey believe that because of the health consciousness of their potential clientele, they will sell at least three fish dinners for every two beef dinners. However, they also believe that at least 10% of their customers will order beef dinners. The profit from each fish dinner will be approximately $12, and the profit from a beef dinner will be about $16. Formulate a linear programming model for Angela and Zooey that will help them estimate the number of meals they should prepare each night and solve this model graphically. If Angela and Zooey increased the menu price on the fish dinner so that the profit for both dinners was the same, what effect would that have on their solution Suppose Angela and Zooey reconsidered the demand for beef dinners and decided that at least 20% of their customers would purchase beef dinners. What effect would this have on their meal preparation plan
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The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 2 milligrams of B. An ounce of oats costs $0.05, and an ounce of rice costs $0.03. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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The Metropolitan Police Department was recently criticized in the local media for not responding to police calls in the downtown area rapidly enough. In several recent cases, alarms had sounded for break-ins, but by the time the police car arrived, the perpetrators had left, and in one instance a store owner had been shot. Sergeant Joe Davis was assigned by the chief as head of a task force to find a way to determine the optimal patrol area (dimensions) for their cars that would minimize the average time it took to respond to a call in the downtown area. Sergeant Davis solicited help from Angela Maris, an analyst in the operations area for the police department. Together they began to work through the problem. Joe noted to Angela that normal patrol sectors are laid out in rectangles, with each rectangle including a number of city blocks. For illustrative purposes he defined the dimensions of the sector as x in the horizontal direction and as y in the vertical direction. He explained to Angela that cars traveled in straight lines either horizontally or vertically and turned at right angles. Travel in a horizontal direction must be accompanied by travel in a vertical direction, and the total distance traveled is the sum of the horizontal and vertical segments. He further noted that past research on police patrolling in urban areas had shown that the average distance traveled by a patrol car responding to a call in either direction was one-third of the dimensions of the sector, or x 3 and y 3. He also explained that the travel time it took to respond to a call (assuming that a car left immediately on receiving the call) is simply the average distance traveled divided by the average travel speed. Angela told Joe that now that she understood how average travel time to a call was determined, she could see that it was closely related to the size of the patrol area. She asked Joe if there were any restrictions on the size of the area sectors that cars patrolled. He responded that for their city, the department believed that the perimeter of a patrol sector should not be less than 5 miles or exceed 12 miles. He noted several policy issues and staffing constraints that required these specifications. Angela wanted to know if any additional restrictions existed, and Joe indicated that the distance in the vertical direction must be at least 50% more than the horizontal distance for the sector. He explained that laying out sectors in that manner meant that the patrol areas would have a greater tendency to overlap different residential, income, and retail areas than if they ran the other way. He said that these areas were layered from north to south in the city, so if a sector area was laid out east to west, all of it would tend to be in one demographic layer. Angela indicated that she had almost enough information to develop a model, except that she also needed to know the average travel speed the patrol cars could travel. Joe told her that cars moving vertically traveled an average of 15 miles per hour, whereas cars traveled horizontally an average of 20 miles per hour. He said that the difference was due to different traffic flows. Develop a linear programming model for this problem and solve it by using the graphical method.
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The Elixer Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics, in different proportions. One gram of ingredient 1 contributes 3 units, and 1 gram of ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required, and the ingredients contribute 1 unit each per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is $80, and the cost for a gram of ingredient 2 is $50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at the minimum cost. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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Solve the following linear programming model graphically: img
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Annabelle Sizemore has cashed in some treasury bonds and a life insurance policy that her parents had accumulated over the years for her. She has also saved some money in certificates of deposit and savings bonds during the 10 years since she graduated from college. As a result, she has $120,000 available to invest. Given the recent rise in the stock market, she feels that she should invest all of this amount there. She has researched the market and has decided that she wants to invest in an index fund tied to S P stocks and in an Internet stock fund. However, she is very concerned about the volatility of Internet stocks. Therefore, she wants to balance her risk to some degree. She has decided to select an index fund from Shield Securities and an Internet stock fund from Madison Funds, Inc. She has also decided that the proportion of the dollar amount she invests in the index fund relative to the Internet fund should be at least one-third but that she should not invest more than twice the amount in the Internet fund that she invests in the index fund. The price per share of the index fund is $175, whereas the price per share of the Internet fund is $208. The average annual return during the last 3 years for the index fund has been 17%, and for the Internet stock fund it has been 28%. She anticipates that both mutual funds will realize the same average returns for the coming year that they have in the recent past; however, at the end of the year she is likely to reevaluate her investment strategy anyway. Thus, she wants to develop an investment strategy that will maximize her return for the coming year. Formulate a linear programming model for Annabelle that will indicate how much money she should invest in each fund and solve this model by using the graphical method. Suppose Annabelle decides to change her riskbalancing formula by eliminating the restriction that the proportion of the amount she invests in the index fund to the amount that she invests in the Internet fund must be at least one-third. What will the effect be on her solution Suppose instead that she eliminates the restriction that the proportion of money she invests in the Internet fund relative to the stock fund not exceed a ratio of 2 to 1. How will this affect her solution If Annabelle can get $1 more to invest, how will that affect her solution $2 more $3 more What can you say about her return on her investment strategy, given these successive changes
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In Problem, what would be the effect on the optimal solution if the available labor were increased from 200 to 240 hours Problem A clothier makes coats and slacks. The two resources required are wool cloth and labor. The clothier has 150 square yards of wool and 200 hours of labor available. Each coat requires 3 square yards of wool and 10 hours of labor, whereas each pair of slacks requires 5 square yards of wool and 4 hours of labor. The profit for a coat is $50, and the profit for slacks is $40. The clothier wants to determine the number of coats and pairs of slacks to make so that profit will be maximized. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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In Problem: a. The maximum demand for bracelets is 4. If the store produces the optimal number of bracelets and necklaces, will the maximum demand for bracelets be met If not, by how much will it be missed b. What profit for a necklace would result in no bracelets being produced, and what would be the optimal solution for this profit Problem A jewelry store makes necklaces and bracelets from gold and platinum. The store has 18 ounces of gold and 20 ounces of platinum. Each necklace requires 3 ounces of gold and 2 ounces of platinum, whereas each bracelet requires 2 ounces of gold and 4 ounces of platinum. The demand for bracelets is no more than four. A necklace earns $300 in profit and a bracelet, $400. The store wants to determine the number of necklaces and bracelets to make in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
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