A chemical manufacturer needs to produce 1250 barrels of a special industrial cleaning fluid by blending 5 available ingredients. The quality of the result is measured by 3 quantitative indices. The following table show the index values for each ingredient, along with the minimum and maximum required in the blend and the cost per barrel of ingredients.

index ingredient 1 2 3 4 5 1 50.4 45.2 33.1 29.9 44.9 2 13.9 19.2 18.6 25.5 10.9 3 89.2 75.4 99.8 84.3 68.8 Cost 531 339 128 414 307 Blend index Minimum Maximum 1 33 43 2 17 20 3 81 99 The index of the blend in the first case will be just a volume-weighted average of those of the chosen ingredients. However, the square of the second blend index will be the square of the volume-weighted ingredient average, and the logarithm of the third blend index will be the logarithm of the volume-

weighted ingredient average. Production planners want to choose a blend that meets upper and lower index requirements at minimum cost.

(a) Formulate this blending problem as a constrained NLP.

(b) Use class optimization software to start from a feasible solution and compute at least a local optimum.