6. The Livewright Medical Supplies Company has a total of 12 salespeople it wants to assign to three regions - the South, the East, and the Midwest. A salesperson in the South earns $600 in profit per month of the company, a salesperson in the East earns $540, and a salesperson in the Midwest earns $375. The southern region can have a maximum assignment of 5 salespeople. The company has a total of $750 per day available for expenses for all 12 salespeople. A salesperson in the South has average expenses of $80 per day, a salesperson in the East has average expenses of $70 per day, and a salesperson in the Midwest has average daily expenses of $50. The company wants to determine the number of salespeople to assign to each region to maximize profit. Formulate an integer programming model for this problem b. Solve this model by using the computer.
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- This problem is based on Motorolas online method for choosing suppliers. Suppose Motorola solicits bids from five suppliers for eight products. The list price for each product and the quantity of each product that Motorola needs to purchase during the next year are listed in the file P06_93.xlsx. Each supplier has submitted the percentage discount it will offer on each product. These percentages are also listed in the file. For example, supplier 1 offers a 7% discount on product 1 and a 30% discount on product 2. The following considerations also apply: There is an administrative cost of 5000 associated with setting up a suppliers account. For example, if Motorola uses three suppliers, it incurs an administrative cost of 15,000. To ensure reliability, no supplier can supply more than 80% of Motorolas demand for any product. A supplier must supply an integer amount of each product it supplies. Develop a linear integer model to help Motorola minimize the sum of its purchase and administrative costs.A beer company has divided Bloomington into two territories. If the company spends x1 dollars on promotion in territory 1, it can sell 60x11/2 cases of beer there; and if it spends x2 dollars on promotionin territory 2, it can sell 40x2 1/2 cases of beer there. Each case of beer sold in territory 1 sells for $10 and incurs $5 in shipping and production costs. Each case of beer sold in territory 2 sells for $9 and incurs$4 in shipping and production costs. A total of $5000 is available for promotion.a. How can the company maximize its profit?b. If an extra dollar could be spent on promotion, by approximately how much would the company’s profit increase? By how much would its revenue increase?c. Use SolverTable to see how a change in the price of beer 1 affects the optimal solution. Do the samefor a change in the price of beer 2.2. 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. 8. In Problem 2, how much flour and sugar will be left unused if the optimal numbers of cakes and Danish are baked?
- 2 ABCD company is an office equipment company that produces two types of desks: standard and deluxe. Deluxe desks have oak tops, more expensive hardware, and require additional time for finishing and polishing. Standard desks require 80 square feet of pine and 10 hours of labor, while deluxe desks require 60 square feet of pine, 18 square feet of oak, and 16 hours of labor. For the next week, the company has 5,000 square feet of pine, 750 square feet of oak, and 400 hours of labor available. Standard desks net a profit of $150, while deluxe desks net a profit of $320. All desks can be sold to national chains shops. Instructions: If the spreadsheet and Excel Solver results are given to you as follows (see Table 1 below), address the following questions: What are your Binding Constraints? What is your Optimum profit [hint: maximized profit]? What are your slack values for Pine, Oak, and Labor? Looking at the Shadow prices, how much would the objective profit increase by if you were to…Consider the following primal LP problem:Maximize X1 + 2X2 – 9X3 + 8X4 – 36X5Subject to 2X2 – X3 + X4 – 3X5 ≤ 40 X1 – X2 + 2X4 – 2X5 ≤ 10 X1 ≥ 0, X2 ≤ 0, X3 ≥ 0, X4 ≥ 0, X5 ≥ 0 (a) Write the dual of the LP above, using variables Y1, Y2, etc.(b) Sketch the feasible region of the dual LP in 2 dimensions, and use the graphical method to find the dual optimalsolution (Plot an isovalue line corresponding to the feasible solution, move the line in improving direction, findthe last one touching the feasible region, and any point(s) on the intersection of the last isovalue line andfeasible region are optimal solutions)(c) Using complementary slackness conditions, - write equations which must be satisfied by the optimal primal solution X* - which primal variables must be zero?(d) Using the information in (c), determine the optimal primal solution X* (e) Compare the optimal objective values of the primal and dual solutionsAMCHEM Chemical Company produces three products: A, B, and C. Each product requires labor to produce it, and production of each product creates pollutants. By law the firm is not allowed to produce more than the following pollutants per day: 200 pounds of sulfur dioxide, 300 pounds of carbon monoxide, 150 pounds of hydrogen sulfide, and 50 pounds of benzene. The total number of person-hours of labor available per day is 6000. In addition, the total output per day of products A and B combined cannot be more than the output of product C. Each pound of product A generates a profit of $5, each pound of B generates $7, and each pound of C generates $4. Pollutant and labor rates per hundred pounds of product are given here. a. Formulate this problem as a linear program to maximize daily profit.b. Solve the problem using a computer.
- 1 ABCD company is an office equipment company that produces two types of desks: standard and deluxe. Deluxe desks have oak tops, more expensive hardware and require additional time for finishing and polishing. Standard desks require 80 square feet of pine and 10 hours of labor, while deluxe desks require 60 square feet of pine, 18 square feet of oak, and 16 hours of labor. For the next week, the company has 5,000 square feet of pine, 750 square feet of oak, and 400 hours of labor available. Standard desks net a profit of $150, while deluxe desks net a profit of $320. All desks can be sold to national chains shops. Instructions: Develop a linear optimization model to determine how many of each type of desk the ABCD company should make next week to maximize profit contribution [Assume: S = units of standard desk produced and D = units of deluxe desk produced. No need to solve the model, neither graphically nor by Solver: just develop the mathematical model for this part].The Pine Valley Board of Education must hire teachers for the coming school year. The types of teachersand the salaries that must be paid are as follows:Teacher can teach Number applying Annual salaryHistory, science 20 $21, 000History, math 15 $22, 000English, science 12 $23, 000English, math 14 $24, 000English, history 13 $25, 000Science, math 12 $26, 000For example, 20 teachers who are qualified to teach history and science have applied for jobs, and eachof these teachers must be paid an annual salary of $21, 000. Each teacher who is hired teaches the twosubjects he or she is qualified to teach. Pine Valley needs to hire 35 teachers qualified to teach history,30 teachers qualified to teach science, 40 teachers qualified to teach math, and 32 teachers qualified toteach English. The board has $1.4 million to spend on teachers’ salaries. A penalty cost of $1 is incurredfor each dollar the board goes over budget. For each teacher by which Pine Valley’s goals are unmet,the following…A refinery manufactures two grades of jet fuel, Fl and F2, by blending four types of gasoline, A. B, C, and D. Fuel Fl uses gasolines A. B. C, and D in the ratio 1:1:2:4, and fucl F2 uses the ratio 2:2:1:3. The supply limits for A, B.C, and D are 1000, 1200, 900, and 1500 bbl/day, respectively. The costs per bbl for gasolines A, B, C, and D are $120, $90, $100, and $150, respectively. Fucls Fl and F2 sell for $200 and $250 per bbl, respectively. The minimum demand for F1 and F2 is 200 and 400 bbl/day, respectively. Develop an LP model to determine the optimal production mix for F1 and F2,
- A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 780, 900, and 720, respectively. The time requirements (in hours per unit) and the profit per unit for each product are as follows. ProductA ProductB ProductC Dept. I 2 1 2 Dept. II 3 1 2 Dept. II 2 2 1 Profit $18 $12 $15 If management decides that the number of units of Product B manufactured must equal or exceed the number of units of products A and C manufactured, how many units of each product should the company produce to maximize its profit? product A product B product C What is the maximum profit (in dollars)?Kane Manufacturing has a division that produces two models of fireplace grates, x units of model A and y units of model B. To produce each model A grate requires 3 lb of cast iron and 6 min of labor. To produce each model B grate requires 4 lb of cast iron and 3 min of labor. The profit for each model A grate is $4.00, and the profit for each model B grate is $3.50. Also, 1000 lb of cast iron and 20 labor-hours are available for the production of fireplace grates per day.Because of a backlog of orders for model A grates, Kane's manager had decided to produce at least 150 of these grates a day. Operating under this additional constraint, how many grates of each model should Kane produce to maximize profit? (x,y)= What is the optimal profit?