šÆ What is an Objective Function?
Definition: An objective function is a mathematical expression that represents the goal of a linear programming problem. It defines what we want to maximize or minimize.
The Simplex Method is an algorithm that finds the optimal solution to linear programming problems by evaluating the objective function at the corner points of the feasible region.
š§ Basic Structure of an Objective Function
The general form of an objective function is:
Z = cāxā + cāxā + cāxā + ... + cāxā
Where:
- Z = Objective function value
- cā, cā, cā, ..., cā = Coefficients (profit, cost, etc.)
- xā, xā, xā, ..., xā = Decision variables
š Steps to Build an Objective Function
- Identify the Goal - Determine if you want to maximize or minimize
- Define Decision Variables - What are you deciding about?
- Identify Coefficients - What is the contribution of each variable?
- Write the Expression - Combine variables and coefficients
- Include Constraints - Define limitations (non-negativity, resource limits)
- Verify Linearity - Ensure all relationships are linear
š” Real-World Examples
Example 1: Profit Maximization
Problem: A factory produces chairs and tables.
- Chair profit: $50
- Table profit: $80
- Goal: Maximize profit
Decision Variables:
xā = number of chairs
xā = number of tables
Objective Function:
Maximize Z = 50xā + 80xā
Example 2: Cost Minimization
Problem: A company needs vitamins from two food sources.
- Food A cost: $2 per unit
- Food B cost: $3 per unit
- Goal: Minimize total cost
Decision Variables:
xā = units of Food A
xā = units of Food B
Objective Function:
Minimize Z = 2xā + 3xā
Example 3: Production Mix
Problem: Manufacturing company with 3 products.
- Product A revenue: $150
- Product B revenue: $200
- Product C revenue: $120
- Goal: Maximize revenue
Objective Function:
Maximize Z = 150xā + 200xā + 120xā
š Common Constraints in Simplex Method
| Constraint Type |
Example |
Interpretation |
| Resource Limit |
2xā + 3xā ā¤ 100 |
Cannot exceed available resources |
| Minimum Requirement |
xā + xā ā„ 50 |
Must meet minimum demand |
| Equality |
xā + xā = 75 |
Must be exactly satisfied |
| Non-negativity |
xā ā„ 0, xā ā„ 0 |
Cannot produce negative units |
āļø Maximize vs Minimize
Maximization
Goal: Increase value
Examples:
- Profit
- Revenue
- Production
- Efficiency
Maximize Z = cāxā + cāxā
Minimization
Goal: Decrease value
Examples:
Minimize Z = cāxā + cāxā
š Complete Simplex Problem Example
Problem Statement:
A bakery produces bagels and donuts. Each bagel requires 2 hours of labor and 3 units of flour. Each donut requires 1 hour of labor and 2 units of flour. Available resources: 80 hours of labor, 120 units of flour. Bagel profit: $5, Donut profit: $3.
Solution:
1. Decision Variables:
xā = number of bagels
xā = number of donuts
2. Objective Function:
Maximize Z = 5xā + 3xā
3. Constraints:
2xā + 1xā ā¤ 80 (labor constraint)
3xā + 2xā ā¤ 120 (flour constraint)
xā ā„ 0, xā ā„ 0 (non-negativity)
š” Best Practices & Tips
- Check Linearity: All terms must be linear (no xĀ² or xy)
- Consistent Units: Ensure coefficients have same units
- Clear Variables: Document what each variable represents
- Verify Feasibility: Ensure constraints allow for a solution
- Sensitivity Analysis: Test how changes affect the solution
- Round Appropriately: Consider practical constraints (whole units)
ā
Conclusion
Building an objective function is the first critical step in solving linear programming problems with the Simplex Method. By following these steps and understanding the structure, you can effectively model real-world optimization problems.
Key Takeaway: The objective function is simply a weighted sum of decision variables, where weights represent the contribution of each variable to your goal.
š See More: Simplex & Linear Programming