Week 1 Assignment Problems W1_A1. An electronics firm is current manufacturing a product that has a variable cost of $0.60 per unit and a selling price of $1.10 per unit. Fixed costs are $15,500. Current sales volume is 32,000 units. The firm can substantially improve the product quality by adding a new piece of equipment that would increase the fixed cost by $8,000. Variable costs would increase to $0.70 per unit, but the volume is expected to increase to 50,000 units due to the higher quality of the product. a) b) c) d) Based on current equipment: What is profit when selling 32,000 units? Based on new equipment: What is profit when selling 50,000 units? Based on your profit calculations - should the company buy the new equipment? How many units would the company have to sell with the new equipment to generate a profit of $500? W1_A2. A start-up publishing company estimates that the fixed costs of its first major project will be $190,000, the variable cost will be $19 per book, and the selling price per book will be $34. a) How many books must be sold to break even? b) What is the total revenue at the break-even point? c) If the publishers take a total of $40,000 in salary - how many books must be sold to break even? d) What is the total revenue for the break-even volume when the publishers take a total of $40,000 in salary? 1 W1_A3. A distributor of prewashed shredded lettuce is opening a new plant and considering whether to use a mechanized process or a manual process to prepare the product. The manual process will have a fixed cost of $43,400 per month and a variable cost of $1.80 per bag. The mechanized process would have a fixed cost of $84,600 per month and a variable cost of $1.30 per bag. The company expects to sell each bag of shredded lettuce for $2.50 a) What is the break-even point for the manual process? b) What is the break-even point for the mechanized process? c) A point of indifference for two processes is quantity at which each process generates the same amount of profit. What is the point of indifference for the two processes? (Hint: If you have a break-even for each process - add another cell the calculates the difference for the profits between the two processes and have only one cell that represents quantity that be used to calculates costs/revenues for each process) d) Which process would you select if the quantity that is sold is greater than the quantity calculated as the point of indifference? e) Which process would you select in the quantity sold is less than the quantity calculated as the point of indifference? W1_A4. A small motor manufacturer assembles two types of motors, models A and B. The assembly process for each is similar in that both require a certain amount of wiring, drilling, and assembly. The table below shows the resource requirement along with the capacity for each department. Model A B Available Hours in Each Department Wiring (hrs./unit) 3 2 Resource Requirements Drilling (hrs./unit) 2 2.1 Assembly (hrs./unit) 3 0.5 240 210 180 Each model A sold yields a profit of $22 and each model B yields a profit of $15. Assuming that all motors that are assembled can be sold. Find the best combination of motors to yield the highest profit. Solve this two decision variable problem using the LP Graphing utility. a) b) c) d) What is the profit (value of the objective function) for the optimal solution? How many Model A's should be assembled? How many Model B's should be assembled? Is the assembly of 20 Model A's and 70 Model B's feasible (not asking if it is optimal). Does it fall in the feasible region? e) Is the assembly of 55 Model A's and 42 Model B's feasible (not asking if it is optimal). Does it fall in the feasible region? 2 W1_A5. A furniture cabinet maker produces two types of cabinets, Mission and Rustic, that house and hide LCD TVs. The resource requirements and profit for the two types of cabinets are shown below. Model Mission Rustic Resource Requirements and Profitability Materials ($/unit) Labor (hrs./unit) Profit ($/unit) 290 15 910 600 20 1400 The firm has a budget of $30,000 to spend on materials. The firm has 30 employees employed, so 1,200 labor hours are available for use. What is the best combination of furniture cabinets to be made? Solve this two decision variable problem using the LP Graphing utility. a) b) c) d) What is the profit (value of the objective function) for the optimal solution? How many Mission models should be produced? How many Rustic models should be produced? Is the production of 60 Mission models and 20 Rustic models feasible (not asking if it is optimal). Does it fall in the feasible region? e) Is the production of 10 Mission models and 40 Rustic models feasible (not asking if it is optimal). Does it fall in the feasible region? 3 W1_A6. The production department for an aluminum valve plant is scheduling its work for next month. Each valve must go through three separate machines during the fabrication process. After fabrication, each valve is inspected by a human being, who spends 15 minutes per valve. There are 535 inspection hours available for the month. The time required (in hours) by each machine to work on each valve is shown in the following table. Also shown are the minimum number of valves that must be produced for the month and the unit profit for each valve. DEPARTMENT V231 PRODUCTS V242 V784 DRILLING MILLING LATHE MINIMUM OF EACH PRODUCT TYPE NEEDED PROFIT ($/UNIT) 0.40 0.60 1.20 0.30 0.65 0.60 0.45 0.52 0.50 0.35 0.48 0.70 200 250 600 450 $16 $12 $13 $8 V906 CAPACITY OF EACH DEPARTMENT (hours) 700 890 1200 Formulate and solve the problem in Excel to determine the number of each product to manufacture that meets the requirements and maximizes profits. a) What is the maximum profit based on your optimal solution (the value of the objective function)? b) How many V231's should be manufactured based on your optimal solution? c) How many V242's should be manufactured based on your optimal solution? d) How many V784's should be manufactured based on your optimal solution? e) How many V906's should be manufactured based on your optimal solution? f) What is the total number of hours used in the drilling department based on your optimal solution? g) What is the total number of hours used in the milling department based on your optimal solution? h) What is the total number of hours used in the lathe department based on your optimal solution? i) What is the total number of hours used in the inspection department based on your optimal solution? 4 W1_A7. A snack company packages and sells three different canned party mixes that contain a total of 1 lb. of nuts. These three different products (Plain Nuts, Mixed Nuts, and Premium Mix) include a mix of four possible types of nuts (peanuts, cashews, almonds, and walnuts). The table below show the number of lbs. of each ingredient in each product type, the amount of ingredient available, and the revenue generated by selling each type of product. What should their production plan be to maximize their revenue? There is on additional piece of information that impacts their production plan and should be included in your formulation. Past demand indicates customers purchase at least twice as many cans of Plain Nuts as Premium Nuts. Your formulation should include a constraint that states that the number of cans of Plain Nuts produced should be at least two times the number of cans of Premium Nuts produced. Formulate and solve the problem in Excel to determine the number of each product to produce that meets the requirements and maximizes revenues. INGREDIENTS PEANUTS (lbs./can) CASHEWS (lbs./can) ALMONDS (lbs./can) WALNUTS (lbs./can) REVENUE ($/UNIT) PRODUCT PLAIN MIXED PREMIUM NUTS NUTS MIX 0.8 0.5 0.2 0.3 0.3 INGREDIENT AVAILABILITY (lbs.) 500 225 0.1 $2.25 0.3 100 0.1 $3.37 0.4 $6.49 80 a) What is the maximum revenue based on your optimal solution (the value of the objective function)? b) How many cans of Plain Nuts should be produced based on your optimal solution? c) How many cans of Mixed Nuts should be produced based on your optimal solution? d) How many cans of Premium Mix should be produced based on your optimal solution? e) After producing the number of cans of each product as suggested in your optimal solution, which of the ingredients has not been totally used by your production plan? 5