The Heat$-$Aire Company has two plants that produce identical heat pump units. However,

production costs at the two differ due to the technology and labor used at each plant. The total

costs of production at the plants depend on the quantity produced, and are described as:

Total cost at plant $1$: $2$X$2$

$1-1$X$1+15$

Total cost at plant $2$: X$2$

$2+0\mathrm{.}3$X$2+10$

Note that X$1$ is the number of heat pumps produced at plant $1$ and X$2$ is the number of heat

pumps produced at plant$2.$ Neither plant can make more than $600$ heat pumps. Heat pumps

can be shipped from either plant to satisfy demand from four different customers. The unit

shipping costs and demands for each customer are summarized in the following table.

Customer $1$ Customer $2$ Customer $3$ Customer $4$

Plant $1$ $$23$ $$30$ $$32$ $$26$

Plant $2$ $$33$ $$27$ $$25$ $$24$

Demand $300250150400$

What is the optimal production and shipping plan if management wants to meet customer

demand at the lowest total cost?

a$.$ Formulate an NLP model for this problem.

b$.$ Implement your model in a spreadsheet and solve it$.$

c$.$ What is the optimal solution?