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PART – 1 SOLVE THE FOLLOWING MATHEMATICAL MODEL Solve this problem using graphical linear programming and answer the questions that follow. Use simultaneous equations to determine the optimal values of the decision variables. Maximize Z = 6X1 + 10X2 Subject to Material 3X1 + 6X2 ≤ 48 kg Labour 4X1 + 8X2 ≤ 80 hr X1, X2 ≥ 0 PART – 2 WRITE THE LINEAR PROGRAMMING MODEL FOR EACH PROBLEM. 1. An RTW manufacturer produces two types of men’s T-shirt; with collar and without collar labeled style A and style B respectively. Each unit of style A needs one man-hour; while each unit of style B requires 2 man-hours. No more than 16 units of each style A and style B can be sold each day. Manpower capacity is limited to 24 hours a day. The contribution to profit of each unit of style A and style B is BD4 and BD3 respectively. Develop linear programming model for this problem? 2. A research team which is composed of a senior researcher, an assistant researcher and a secretary who receive BD280, BD230 and BD180 per month respectively must finish a particular project within a period of 100 days. However, it is stipulated that the senior researcher and the assistant researcher render at least 12 days and 25 days respectively, and the secretary be on the job no more than 70 days. Set up an LP MODEL that will optimize the cost of the project. PART – 3 SOLVING THE LINEAR PROGRAMMING APPLICATIONS (EASY) PROBLEM#1 Cycle Trends is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from aluminum and steel alloys. The anticipated unit profits are $10 for the Deluxe and $15 for the Professional. The number of pounds of each alloy needed per frame is summarized on the next slide. A supplier delivers 100 pounds of the aluminum alloy and 80 pounds of the steel alloy weekly. How many Deluxe and Professional frames should Cycle Trends produce each week? Pounds of each alloy needed per frame are Aluminum Alloy Steel Alloy Deluxe 2 3 Professional 4 2 PROBLEM#2 An appliance manufacturer produces two models of microwave ovens: H and W. Both models Require fabrication and assembly work; each H uses four hours of fabrication and two hours of assembly, and each W uses two hours of fabrication and six hours of assembly. There are 600 fabrication hours available this week and 480 hours of assembly. Each H contributes $40 to profits, and each W contributes $30 to profits. What quantities of H and W will maximize profits? ABI201 Management Science-1 Abdulaziz Al-Saadi PROBLEM#3 GM operates a plant that assembles and finishes cars and trucks. It takes 5 man-days to assemble a truck and 2 man-days to assemble a car. It takes 3 man-days to finish each type of vehicle. Because of man-power limitations, assembly can take no more than 180 man-days per week and finishing no more than 135 man-days per week. If the profit on each truck is $300 and $200 on each car, how many of each should they produce to maximize profit? PROBLEM#4 XYZ Furniture Co. produce two type of table, Classic and Professional, The scarcity material is high quality Red Wood which is limited to 1500 ft2. Classic chair required 3 ft2 and Professional required 5 ft2. XYZ Co. obligate to Supply a new establishment by 180 professional Chairs this week. If the Classic Chair and professional Chair contribute a profit of BD 25 and BD 40 Respectively. What is the best mix that can produce to gain maximum profit? Solve this problem using Graphical Method? PROBLEM#5 The Outdoor Furniture Corporation manufacturer's two products, Professional and picnic tables, for use in offices and parks. The firm has two main resources; its carpenters (labor force) and a supply of red-wood for use in the furniture. During the next production cycle, 1200 hours of labor are available under union agreement. The firm also has a stock of 3500 feet of good quality red-wood. Each professional that outdoor furniture produces requires 5 labor hours and 10 feet of red-wood; each picnic table take 5 labor hours and 35 feet of red- wood. Completed Professional will yield a profit of $9 each, and tables will result in a profit of $ 20. (A) What is the equation of the objective function? (B) List all the inequality constrains for this problem? (C) Solve the above Linear Programming Model using graphical method and identify how many Professional and tables should Outdoor Furniture produce to obtain the largest possible profit? PROBLEM#6 A craftsman named Sunil Babar builds two kinds of birdhouses, one for wrens and a second for bluebirds. Each wren birdhouse takes four hours of labor and four units of lumber. Each bluebird house requires two hours of labor and twelve units of lumber. The craftsman has available 60 hours of labor and 120 units of lumber. Wren houses yield a profit of $6 each and bluebird houses yield a profit of $15 each. Write out the objective and constraints and Solve graphically to find the maximum profit. PROBLEM#6 A furniture manufacturing company manufactures dining room tables and chairs. A table requires 1 labor-hour for assembling and 1 labor-hour for finishing. A chair requires 2 labor- hours for assembling and 1 labor-hour for finishing. The maximum labor-hours available per day for assembly and finishing are 8 and 10, respectively. The profit made on a table is BD 60 and the profit on a chair is BD 20. How many tables and chairs should be produced to maximize their profit? ABI201 Management Science-1 Abdulaziz Al-Saadi PROBLEM#7 The production planner for a private label soft drink maker is planning the production of two soft drinks: Root light (R) and Sassafras Soda (S). Two resources are constrained: production time (T), of which she has at most 720 minute per day; and carbonated water (W), of which she can get at most 1500 gallons per day. A case of Root Light requires 2 minutes of time and 5 gallons of water to produce, while a case of sassafras soda requires 3 minutes of time and 5 gallons of water. Further more Private label soft drink maker has an obligation to supply grocer store 80 case per day. Profits for the Root Light are $6.00 per case, and profits for the Sassafras soda are $4.00 per case. Find the best mix that lead to maximum profit. PROBLEM#8 A small candy shop is preparing for the holiday season. The owner must decide how many bags of deluxe mix and how many bags of standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3 kg raisins and 1/3 kg peanuts, and the standard mix has 1/2 kg raisins and 1/2 kg peanuts per bag. The shop has 90 kg of raisins and 60 kg of peanuts to work with. Peanuts cost $.60 per kg and raisins cost $1.50 per kg. The deluxe mix will sell for $2.90 per kg, and the standard mix will sell for $2.55 per kg. The owner estimates that no more than 110 bags of one type can be sold. If the goal is to maximize profits, how many bags of each type should be prepared? What is the expected profit? PART – 3 CHALLENGED LINEAR PROGRAMMING APPLICATION (HARD) CHALLENGED PROBLEM # 1 The National Credit Union has $250,000 available to invest in a 12-month commitment. The money can be placed in Treasury notes yielding an 8% return or in municipal bonds at an average rate of return of 9%. Credit union regulations require diversification to the extent that at least 50% of the investment be placed in Treasury notes. Because of defaults in such municipalities as Cleveland and New York, it is decided that no more than 40% of the investment be placed in bonds. How much should the National Credit Union invest in each security so as to maximize its return on investment? CHALLENGED PROBLEM # 2 A banker has funds available to invest. She can purchase a Type A bond yielding a 5% return on the amount invested and she can purchase a Type B bond yielding a 10% return on the amount invested. Her client insists that she invest a t least twice as much in A bonds as in B founds. How much should be invested in each type of bond to maximize the client’s return if not more that $17000 is to be invested in B bonds and at least $5000 must be invested in A bonds? ABI201 Management Science-1 Abdulaziz Al-Saadi