Exam 1 Fall 2007

IOE 202, Fall 2007 1 IOE 202 Exam I 11/20/2007 Problem 1 /25 Problem 2 /25 Problem 3 /25 Problem 4 /25 Total /100 Name: This is a closed book exam; you are allowed to use two pages of your own notes and calculators (calculators should be used for arithmetics only; not for graphing and other advanced tasks). Please write your solutions to each problem in this exam book. Be clear and show your work. Your grade will depend in part on the completeness and clarity of your answers. Do not use the exam book as scrap paper. There are four equally weighted problems in the exam. You are obligated to comply with the Honor Code of the College of Engineering. You are not allowed to receive or give aid on this examination; in particular, you are not allowed to discuss this exam with anyone who will be taking it at a later date. After you have completed the examination, please write the following Honor Pledge: \I have neither given nor received aid on this examination," and sign your name below. Instructors are not required to grade tests in which the signed Honor Pledge does not appear. Honor Pledge: Signed: IOE 202, Fall 2007 2 1. A camera store sells an average of 100 cameras per month. The cost of holding a camera in inventory for a year is 30% of the price the camera shop pays for the camera. There is a xed cost of $120 each time the camera store places an order with its supplier. The price the store is charged per camera depends on the number of cameras ordered, as speci ed in the table below. No. of cameras Price per ordered camera 1-40 $90 41-100 $70 More than 100 $55 (a) Identify all the parameters of the appropriate EOQ model (and their units) (b) Each time the camera store places an order, how many cameras should it order? How frequently should these orders be placed? Show your work. (If you get fractional values, do not round them to integers, but report them with two digits after the decimal point, i.e., 765.4321 should be reported as 765.43 and 123.4567 should be reported as 123.46.) IOE 202, Fall 2007 3 Additional space for problem 1 IOE 202, Fall 2007 4 2. Rupert Murdoch, the media mogul, has just told you of his plan to invest $1.2 million in one or more of the following four major entertainment companies (Artisan, Disney, Fox, Lions Gate). As his investment advisor, you are to determine which companies Mr. Murdoch should invest in and how many shares of those companies he needs to buy in order to maximize his total estimated annual return. His nancial research team has provided you with the information you need to make the decision (of course, in this problem, all the number provided and ctional and overly simpli ed). Entertainment companies Stock Price per share Est. Annual return per share ARTISAN $14 $0.98 DISNEY $25 $1.42 FOX $24.5 $2.11 LIONS GATE $23.8 $0.39 Shares can be purchased in fractional amounts. Besides this data, Mr. Murdoch also gave you some additional guidelines that you need to follow: The investment should not contain more than 100,000 shares of any one company. Total number of Artisan and Disney shares bought has to be greater than 5,000 You can only invest in at most 3 companies You can invest in either Disney or Fox (or neither), but not both You can invest in Lions Gate only if you invest in Artisan Formulate a linear programming model (with integer variables, if necessary) to maximize Mr. Murdoch?s estimated return on investment. Clearly de ne your decision variables and identify the meaning of the objective function and the constraints. IOE 202, Fall 2007 5 Additional space for problem 2 IOE 202, Fall 2007 6 3. Coalco produces coal at three mines and ships it to four customers. The cost per ton of producing coal, the ash and sulfur content of the coal (by weight), and the production capacity in tons is given in Table 1. The number of tons of coal demanded by each customer are given in Table 2; these demands must be met exactly. The cost (in dollars) of shipping a ton of coal from a mine to each customer is given in Table 3. Production cost Capacity Ash content Sulfur content Mine 1 $50 per ton 120 tons 8% 5% Mine 2 $55 per ton 100 tons 6% 4% Mine 3 $62 per ton 140 tons 4% 3% Table 1: Information on coal production at each mine Customer 1 Customer 2 Customer 3 Customer 4 80 70 60 90 Table 2: Customer demands (tons) Customer 1 Customer 2 Customer 3 Customer 4 Mine 1 4 6 8 12 Mine 2 9 6 7 11 Mine 3 8 12 3 5 Table 3: Shipping costs from mines to customers ($ per ton) It is required that the total amount of coal shipped by the company (i.e., total shipped from all mines to all customers) contain at most 5% ash and at most 4% sulfur. Formulate a linear programming model that minimizes the cost of meeting customer demand. Clearly de ne your decision variables and identify the meaning of the objective function and the constraints. IOE 202, Fall 2007 7 Additional space for problem 3 IOE 202, Fall 2007 8 4. A computer parts manufacturer produces two types of monitors | 20 inch and 24 inch. There are two production lines, one for each type of monitor. The small monitor line has a daily capacity of 700 units per day. The large monitor line has a daily capacity of 500 units per day. In department A, the LCD planes are produced for both monitor lines. The production of a small panel requires 1 hour of labor, and a large panel requires 2 hours of labor. Total daily labor availability in department A is 1,200 hours. In department B, the monitors are inspected. Each small monitor requires 0.3 hours of labor for inspection, while each large monitor requires 0.2 hours of labor for inspection. A total of 240 hours of labor are available in department B. The small monitor nets an earning contribution of $40 per unit; the large monitor nets an earnings contribution of $30 per unit. (a) Formulate a linear programming model to nd a daily production schedule that maxi- mizes the net earnings of the company, using as your variables S | the number of small monitors to produce each day, in hundreds L | the number of large monitors to produce each day, in hundreds. Clearly identify the meaning of the objective function and the constraints. (Make sure the units of the left and right hand sides of all constraints match.) IOE 202, Fall 2007 9 (b) On the graph paper on the next page, identify the optimal solution to the LP in (a) graphically. Clearly label each component of your graph. Which two constraints are tight at this solution? (c) What is the optimal production plan and how much pro t will it bring? (Hint: you can identify the optimal values of S and L by \reading" them o your graph. To make sure your numbers are right, plug them into the left hand sides of the tight constraints you identi ed in (b), and make sure you indeed satisfy them as equalities.) (d) Consider the labor constraint for the department A. Suppose the constraint changed to \S + 2L 13," i.e., we had an additional 100 labor hours in department A. What are the new optimal values of the variables? How much additional net earnings did the extra labor in department A bring in? IOE 202, Fall 2007 10 Graph paper for problem 4 6 - L S