This Case Study deals with production-mix problem in Red Brand Canners, medium-sized company that cans and distributes a variety of fruit and vegetable products. This particular problem deals with three tomato-products:
1. Whole tomato cans
2. Tomato juice
3. Tomato paste cans
Production objective is to define such product mix, which will bring maximum total profit, subject to various production constraints. General objective function can be defined as

(1)

Since three different products are discussed here, this function can be also expressed as

(1a)

where - total revenue of whole tomato, tomato juice and tomato paste respectively - total production cost of whole tomato, tomato juice and tomato paste respectively

Total revenue and total cost of each of the products can be calculated with respect to units of product produced (sold), unit price and unit cost of product. For whole tomato this can be expressed as follows:

(2) (2a)

where - units of the whole tomato sold - price of the unit of whole tomato - cost per unit of whole tomato

Similarly, total revenue and total cost may be calculated for tomato juice

(3) (3a)

and for tomato paste

(4) (4a)

where - units of the tomato juice and tomato paste sold, respectively - unit price of the tomato juice and tomato paste, respectively - cost per unit tomato juice and tomato paste, respectively

Therefore, equation for total profit can now be written as follows:

(5)

MODEL FORMULATION

Decision Variables

First step in formulating the problem is to define decision variables. If we take a look at the objective function expressed in equation (5) we can say: OK, we need to know how much we will produce whole tomato, how much tomato juice and how much tomato paste,