A new premium whiskey will be produced by blending up to 5 different distilling products, and the quality of the results will be measured by 3 performance indices. The following table shows the value of each index for the 5 ingredients, along with lower and upper limits for the index of the blend and costs per unit volume.

index ingredient 1 2 3 4 5 1 12.6 15.8 17.2 10.1 11.7 2 31.4 30.2 29.6 40.4 28.9 3 115 202 184 143 169 Cost 125 154 116 189 132 index Blend Low High 1 12 16 2 31 36 3 121 164 The index of the blend in the first two cases will be just a volume-weighted sum of those of the chosen ingredients. However, the third index is logarithmic (i.e., the natural logarithm of the blend value will be the logarithm of the volume-weighted ingredient sum). 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.