Question

A local chemical processing plant is interested in producing b barrels of a special industrial sanitizing fluid by blending (mixing) n available ingredients. The quality of the output (industrial sanitizer) is measured by three quantitative indices. Let ai,j​ be the index value for each ingredient, where i=1,2,3 and j=1,2,…,n. Let also Li​ and Ui​ be, respectively, the minimum and maximum index values required in the blend, for i=1,2,3. Suppose also that the cost per barrel of ingredients be cj​, for j=1,2,…,n. The index of the blend in first index will be just a volume-weighted average of those chosen ingredients. However, the square of the second blend index will be the square of the volume-weighted average of those chosen ingredients. Finally, the logarithm of the third blend index will be the logarithm of the volume-weighted average of those chosen ingredients. The production manager wants to choose a blend that meets the minimum and maximum index requirements at minimum cost. Formulate the blending problem as a constrained non-linear program?

Answer 1

The blending problem at the chemical processing plant aimed at producing a sanitizing fluid can be formulated as a constrained non-linear program with the objective of minimizing the total cost while adhering to quality standards for the industrial sanitizer indices.

Step-by-step explanation:

To formulate the blending problem as a constrained non-linear program, we need to define decision variables, the objective function, and constraints. Let's define xj as the volume of ingredient j used in the blend. The objective function is to minimize the total cost, which can be represented as follows:

Minimize ∑j=1n cjxj

Subject to the following constraints:

1. For the first index: L1 ≤ ∑j=1n a1,jxj / b ≤ U1
2. For the second index (square of volume-weighted average): L22 ≤ [∑j=1n a2,jxj / b]2 ≤ U22
3. For the third index (logarithm of volume-weighted average): log(L3) ≤ log(∑j=1n a3,jxj / b) ≤ log(U3)
4. Volume constraint to produce b barrels: ∑j=1n xj = b
5. Non-negativity constraint: xj ≥ 0, for all j=1,2,...,n

This non-linear programming problem ensures that the combination of ingredients meets the quality standards within the specified index ranges for the cost to be minimum.