As we have explained in last article ( Introduction to Optimization), Constrained is similar to unconstrained optimization in some points like below:

• Requires a prescriptive model.
• Uses an objective function.
• Solution is an extreme value.

And different mainly in:

• Multiple decision variables.
• Constraints on resources.

We will study business situation through simple example and explain to to optimize the solution step by step. Our motivation problem is called Banner Chemicals

Situation

• Banner Chemicals manufactures specialty chemicals. One of their products comes in two grades, high and supreme. The capacity at the plant is 110 barrels per week.
• The high and supreme grade products use the same basic raw materials but require different ratios of additives. The high grade requires 3 gallons of additive A and 1 gallon of additive B per barrel while the supreme grade requires 2 gallons of additive A and 3 gallons of additive B per barrel.
• The supply of both of these additives is quite limited. Each week, this product line is allocated only 300 gallons of additive A per week and 280 gallons of additive B.
• A barrel of the high grade has a profit margin of \$80 per barrel while the supreme grade has a profit margin of \$200 per barrel.

Question: How many barrels of High and Supreme grade should Banner Chemicals produce each week?

Problem Formulation Process

1. Determine the decision variables
• What are you trying to decide?
• What are their upper or lower bounds?
1. Formulate the objective function
• What are we trying to minimize or maximize?
• Must include the decision variables and the form of the function determines approach (linear for LP)
1. Formulate each constraint
• What is my feasible region?  What are my limits?
• Must include the decision variables and will almost always be linear functions
Step 1.  Determine Decision Variables
• XH = Number of High grade barrels to produce per week
• XS = Number of Supreme grade barrels to produce per week
• Bounds XH ≥ 0     XS ≥ 0 Step 2. Formulate Objective Function
• Profit = 80XH + 200XS
• Maximize z(XH, XS) = 80XH + 200XS Step 3. Formulate Constraints

Plant Capacity is 110 barrels    : XH + XS ≤ 110

 3XH + 2XS ≤ 300XH + 3XS ≤ 280 High Supreme Available Additive A 3 gal 2 gal 300 gal Additive B 1 gal 3 gal 280 gal

Anomalies

A number of anomalies can occur in LPs:

• Alternate Optimal Solutions
• Redundant Constraints
• Infeasibility