CHAPTER 7 OVERVIEW OF OPTIMIZATION 237 EXERCISE 7.3 Product Mix In product mix problems, the objective of optimization is to find a most profitable allocation of a set of limited resources over a set of desired products or activities. The BOATS model discussed in the tutorial earlier in this chapter is an example of this problem type. Industrial LP software can now solve such models involving tens of thousands of products and resources. MANUFACTURING ATHLETIC SHOES A firm manufactures three types of athletic shoes: Basketball with a $10 per pair profit, Running with a $9 per pair profit, and Tennis with a $7.50 per pair profit. The resources consumed per pair of shoes and the quantities available are shown in the following table. Resources Canvas Labor hours Machine hours Rubber Basketball Running Tennis Available 12,000 2 222 21,000 19,500 16,500 a. What production quantities maximize profit? b. Given optimal production, what is the economic value to the firm of each of the resources ? Blending Blending problems are an important class of linear programs They occur in the production of gasoline and other petroleum products, chemicals, paints, alloys, Fertilizers, and processed foods. Unlike the product mix problem in which a single set of raw materials could be carbined into many different types of products, here a single type of product can be produced from many different combinations of raw materials In blending problems, the objective is generally to minimize cost per unit of final product, and the constraints and to enforce correct proportions of ingredients In the following example, vanous constituents are blended to forms a metal alloy The chemical requirements are that the alloy must contain at least 9% of clement A and between 6,5% and 11% of element B