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
- 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
- Determine the decision variables
- What are you trying to decide?
- What are their upper or lower bounds?
- 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)
- 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
A number of anomalies can occur in LPs:
- Alternate Optimal Solutions
- Redundant Constraints