Question $3[25$ points$]$:

The manufacturing processes of tires is actually composed of three main steps: $(1)$ mixing, $(2)$ assembly, and $(3)$ curing. There are dedicated production lines for each step with a certain monthly capacity limit$.$ At the end of mixing and assembly steps, some semi$-$finished items are produced which can be also stored in the inventory for future use. The finished tires are produced by the end of curing step after which the tires are placed in the finished goods inventory to satisfy the customer orders.

The tire manufacturer ABC produces different types of tires where all types of tires are using the same production lines for each step. However, when different types of tires are produced at any production step in any month, a changeover is required per tire type. This changeover takes time and therefore consumes capacity of the production line. When production of a type of tire takes place in a month, a fixed cost per tire type is incurred. There exist unit production cost and unit inventory holding cost per tire type and per month. The goal of the production department is to satisfy the entire demand for each type of tire on time while respecting capacity constraints and minimizing costs of production and inventory holding.

Formulate the problem of finding the most cost efficient way of satisfying demand by this production plant. Define appropriate sets and parameters to represent data, and then present variables, constraints, and objective, including a word description for each of these model components.