Question

Problem 1: Zippy motorcycle manufacturing produces two popular pocket bikes (miniature motorcycles with 49cc engines): the Razor and the Zoomer. In the coming week, the manufacturer wants to produce up to 700 bikes and wants to ensure the number of Razors produced does not exceed the number of Zoomers by more than 300. Each Razor produced and sold results in a profit of $70 while each Zoomer results in a profit of $A0. The bikes are identical mechanically and only differ in the appearance of the polymer-based trim around the fuel tank and seat. Each Razor's trim requires 2 pounds and 3 hours of production time while each Zoomer requires 1 pound of polymer and 4 hours of production time. Assume that 900 pounds of polymer and 2,400 labor hours are available for production of these items in the coming week. of polymer Find the best production strategy for Zippy Problem 2: T & L Manufacturing produces 3 unique products: X, Y, and Z. The office manager has recently been approached by upper management and asked to create an optimal production schedule for the coming month. The office manager knows the following about products X, Y, and Z. There are 2 steps to the production of each product: Plating and Heating. The plating process coats the product in a substance that will eventually harden and protect the outside of the product. Once the product has been plated it is heated until the outer shell has reached a temperature high enough to harden the material used in plating. Due to production stage, the manufacturer needs to produce at least 1500 units for each product. If made in house they cost $2.20, $1.50, and $1.20 Products X, Y, and Z need 12 minutes, 20 minutes, and 6 minutes each of plating. Under the current staffing situation there is currently 1500 hours of plating available for the coming month. Products X, Y and Z need 6, 12, and 3 minutes of heating. There are currently 1000 hours of heating available Due to limited resources all three of the products may either be built on site or bought from another manufacturer. It costs $3.00, $2.80, and $2.20 to buy these products from another manufacturer. The demand for products X, Y, and Z in the coming month is 3000, 5000, and 4000 Develop a plan to minimize the total costs for the coming month

Answer 1

Problem 1

Formulation

Let R and Z be the number of Razors and Zoomers to be produced respectively.

Max Z = total profit = 70 R + 40 Z
s.t.
R + Z <= 700 (max total demand)
R - Z <= 300 (Max R)
2R + Z <= 900 (polymer)
3R + 4Z <= 2400 (labor hours)
R, Z >= 0

Use graphical method to find an optimal solution as following:

1000 900 800 700 600 R + Z <= 700 240, 420 N 500 2R +Z<900 400 300 - Isoprofit line 200 100 0) 400 0 200 600 800

Note that the isoprofit line, moving away from the origin, cuts the feasiblity region, finally at (480, 420). So, the point at which the objective is maximized is (240, 420).

So, the optimal solution is:

R = 240
Z = 420

Max. Profit = 70*240 + 40*420 = $33,600