Chapter 3.9, Problem 7P

Explanation of Solution

Formulation of linear problem

• The steps for defining a linear problem include:
  • Identifying the decision variables.
  • Writing the objective function.
  • Mentioning the constraints.
  • Explicitly stating the non-negativity restriction.
• A problem will be a linear problem if and only if the decision variables, objective function and constraints all are linear functions.
• If all the three conditions are satisfied, then it is a linear problem.
• Let $\text{C2}$ be the pounds of cream cheese produced, $\text{P2}$ be the pounds of cottage cheese produced, $\text{HF}$ be the pounds of high fat milk purchased, $\text{LF}$ be the pounds of low fat milk purchased, $\text{HFE}$ be the pounds of high fat milk put through evaporator, $\text{LFE}$ be the pounds of low fat milk put through evaporator, $\text{HF1}$ be the pounds of high fat milk used to make cream cheese, $\text{LF1}$ be the pounds of low fat milk used to make cream cheese, $\text{C1}$ be the cream input to make cream cheese, $\text{HF2}$ be the pounds of high fat milk used to make cottage cheese, $\text{LF2}$ be the pounds of low fat milk used to make cottage cheese and $\text{C2}$ be the cream input to make cottage cheese.
• Then the changed formulation is:

  $\text{Max z = 1}\text{.1P1 + }\text{.8P2 -}\text{.4HFE - }\text{.4LFE -}\text{.4LF - }\text{.8HF}$

  Subject to

  $\begin{array}{l}\text{P1>=1000, P2>=1000, P1<=1500, P2<=2000}\hfill \\ \text{P1 = HF1 + LF1 +C1}\hfill \\ \text{P2 = }\hfill \end{array}$