EM 602 FINAL EXAM NJIT Instructions: Complete all problems. You can use either Solver, Lindo or excel to resolve problems. Be sure to indicate all variables and explain all models. All work must be done individually. You must turn in the template with answers incorporated. The work should also be contained within the worksheets of the template or you can submit a word document that illustrates your work. In each worksheet or word document you must: Identify / define the variables that are used in your model List any assumptions Highlight the answers and include the answers on the front (answer sheet) of the template 1. (10 points) - A soda producer makes and sells two products, Classic Cola and Diet Cola. If this company spends x1 dollars on promotion of Classic Cola in a particular western city, it can sell 175 X1 0.75 12-packs of Classic Cola each week there. Moreover, if this company spends x2 dollars on promotion of Diet Cola in a particular western city, it can sell 150X2 0.75 12-packs of Diet Cola each week there. Each 12-pack of Classic Cola sells for $4.00 and costs $01.20 to produce and ship to customers in this western city. Each 12-pack of Diet Cola sells for $3.50 and costs $1.00 to produce and ship to customers in this western city. A total of $20,000 is available for promotion each week in this city. The soda producer seeks to maximize its weekly profit. Formulate and solve an appropriate optimization model to help this soda producer identify the best promotional strategies for its two products. 2. (20 points) Periodic preventive maintenance is carried out on aircraft engines, where an important component must be replaced. The number of aircraft scheduled for such maintenance over the next 6 months is estimated at 200,180, 300,198,230, and 290, respectively. All maintenance work is done in the month that parts are available, where a used component may be replaced with a new or an overhauled component. The overhauling of used components may be done in a local repair facility, where they will be ready for use at the beginning of next month, or they may be sent to a central repair shop where a delay of 2 months (as such maintenance can occur in the third month) is expected. The repair cost in the local shop is $120 per component. At the central facility, the cost is only $35 per component. An overhauled component used in a later month will incur an additional storage cost of $1.50 per unit per month. New components may be purchased at $200 each in month 1, with a 5% price increase every 2 months. A new component is available immediately (no delay). Formulate a model to the problem and solve to determine the optimal schedule for satisfying the demand for the component over the next 6 months. Also consider that the planning occurs within the first month. So all parts are ordered in the first month to satisfy maintenance requirements 1 3. (30 Pts) At the beginning of each day, a patient in a hospital is classified into one of three conditions; good, fair, or critical. At the beginning of the next day, the patient will either continue to be in the hospital and be in good, fair or critical condition or the patient will be discharged in one of three conditions: improved, unimproved or dead. The transition probabilities for these situations are as follows: Good Fair Critical Good .65 .50 .51 Good Fair Critical Improve d .06 .03 .01 Fair .20 .30 .25 Critical .05 .12 .20 Unimprove d .03 .02 .01 Dead .01 .03 .02 For example, a patient who begins the day in fair condition has a 12% chance of being in critical condition the next day and a 3% chance of being discharged the next day in improved condition. a. Consider a patient who enters the hospital in good condition. On average, how many days does this patient spend in the hospital? b. This morning, there were 500 patients in good condition, 300 in fair condition, and 200 patients in critical condition in the hospital. Tomorrow morning, the following admissions will be made: 50 patients in good condition, 40 in fair condition and 30 patients in critical condition. On the basis of this, predict tomorrow's hospital census (how many patients in good, critical and fair condition, total)? c. The hospitals daily admissions are as follows: 20 patients in good condition, 10 patients in fair condition and 10 patients in critical condition. On average, how many patients of each type would you expect to see in the hospital (think steady state conditions)? d. What fraction of patients who enter the hospital in good condition, will leave in improved condition? 2 4. Steco, a steel wholesaler, leases regional warehouses for its operation on a monthly basis. Steco currently has a list of three potential warehouses that it can lease. The cost per month to lease each warehouse is shown in table 1 below. There are four sales districts in Stecos operation. The monthly demand (in truck load requirements) is indicated in table 1 as well as the unit cost of sending a truck from a warehouse to a given district. Steco wants to know which warehouses to lease and how many trucks to send from each warehouse to each district that will minimize overall cost. Keep in mind that every warehouse does not have to be used. Formulate an LP program and solve this problem. (10 pts) Table 1 Cost per truck from warehouse to Sales District Sales Districts WAREHOUSE A B C 1 $170 $150 $100 2 $40 $195 $240 3 $70 $100 $140 4 $160 $10 $60 Monthly Demand (truck loads) 100 90 110 Monthly Capacity (number of truck loads) 200 250 300 Monthly Leasing Cost $7,750 $4,000 $5,500 60 5. Newcor's steel mill has received an order for 25 tons of steel. The steel must be 5% carbon and 5% molybdenum by weight. The steel is manufactured by combining three types of metals; steel ingots, scrap steel and alloys. Steel ingots can only be purchased in lot sizes of 1 at the specified weights given in the table below. The ingot can be cut to the desired amount needed in the process. The weight, cost per ton and chemical contents of each ingot is: INGOT WEIGHT 1 2 3 4 5 3 4 6 COST PER TON tons $350 tons $330 tons $310 tons $280 CARBON% 5 4 5 3 MOLYBDENUM% 3 3 4 4 Unlimited amounts of alloys and scrap steel can be purchased. Scrap steel cost $100 per ton and contains 3% carbon and 9% molybdenum. The cost per ton and chemical make up of the alloys is: ALLOY 1 2 3 COST PER TON $500 $450 $400 CARBON% 8 7 6 MOLYBDENUM% 6 7 8 Determine the appropriate mixture of the steel components to minimize the cost of fulfilling the requirements. 20 pts 3