The Blending (minimization) Problem

Blending problems involve the determination of the best blend of available ingredients to form a certain quantity of a product under strict specifications. The best blend means the least-cost blend of inputs required to meet a designated level of output or given specifications. Blending problems are especially important in the process industries such as petroleum, chemicals, and food, and in fields where a certain level of service is desired at minimum cost. The decision maker must determine the ingredients to use and in what quantities.

In this respect, the blending problem is similar to the product-mix problem. However, the objectives usually differ. In the product-mix problem, the profit derived from selling the products made from the given amount of resources is to be maximized. In blending, the cost of the ingredients is to be minimized, while adhering to certain specifications and using certain ingredients (resources). In the blending problem, an attempt is made to use as few resources as possible to provide a given product (or service) level. Therefore, a blending problem is also considered a problem of allocating resources in the best manner.

Example:

In preparing sungold paint, it is required that the paint have a brilliance rating of at least 300 degrees and a hue level of at least 250 degrees. Brilliance and hue levels are determined by two ingredients: Alpha and Beta. Both Alpha and Beta contribute equally to the brilliance rating, one ounce (dry weight) of either producing one degree of brilliance in one drum of paint. However, the hue is controlled entirely by the amount of Alpha, one ounce of it producing three degrees of hue in one drum of paint. The cost of Alpha is 45 cents Per ounce, and the cost of Beta is 12 cents Per ounce. The problem is minimize the cost of the resources, the problem is to find the quantity of Alpha and Beta to be included in the preparation of each drum of paint.

Answer Report