A certain company has three plants with an excess in their monthly production capacity. All three can manufacture a new product that management wants to introduce to the market. The product can be made in three sizes: large, medium and small, which will give a net profit of $ 540, $ 490 and $ 420, respectively. The time it takes to manufacture each type of product is 2, 1.5 and 1 hour for large, medium and small products respectively. Each plant has capacity of 1300, 2000 and 1500 h. per month for the manufacture of this new product. The maximum estimated demands for each type of product are 1200, 1350 and 1600 products of each large, medium and small size respectively. As part of a market strategy, it is recommended that at least 50% of the medium product be produced, 70% of the small product and 60% of the large product with respect to its estimated demand. An important point is that the company makes its deliveries every month so that the production is stored during the previous period of being sent to its points of sale. Given the characteristics of the product this can not be stowed. In this sense it is known that each large-sized product occupies 7 m2 of space, 5m2 the medium and 3m2 the boy. There are 7,000, 5500 and 4000 m2 in each floor respectively. Additionally, the management does not want to overload one of the plants with respect to the others, for that reason it has managed to distribute the production equally. To achieve this, you can obtain the percentage of capacity assigned between the available capacity in hours in each sea floor, the same for all.
a) It is requested that the production commission that formulates a model to determine the units of each size must be carried out in each plant to enforce what is requested.
b) Determine the range of optimality for the utility of the medium product.
c) Determine the feasibility range for the minimum expected demands.