12.1 Optimizing Resources for Business Efficiency
Introduction to Linear Programming in Business
Linear Programming (LP) is a fundamental mathematical optimization technique used to determine the best possible outcome, such as maximizing profit or minimizing cost, subject to a set of linear constraints. These constraints typically represent limitations in resources such as time, labour, or capital.
LP is widely applied across various disciplines, including:
- Production planning and scheduling
- Transportation and logistics
- Financial portfolio optimization
- Healthcare resource allocation
- Network flow optimization
In the context of business decision-making, LP serves as a strategic tool for optimizing resource utilization and improving operational efficiency.
The Business Motivation: Profit Maximization
A central objective of most business enterprises is to maximize profit. Profit ([latex]\pi[/latex]) can be expressed as the difference between Total Revenue (TR) and Total Cost (TC):
[latex]\pi = TR − TC[/latex]
Expanding this further:
[latex]\pi= \text{Price} \times Q − (\text{Fixed Cost} + \text{Variable Cost}(Q))[/latex]
Where:
- Q is the quantity of goods or services produced,
- Fixed Costs (FC) are independent of output,
- Variable Costs (VC) vary with the level of output.
Thus, profit can be increased by:
- Maximizing total revenue, or
- Minimizing total cost
However, both strategies are constrained by the availability of resources. This necessitates the use of optimization techniques, such as linear programming, to make the most effective use of limited inputs.
Components of a Linear Programming Problem (LPP)
To apply LP in a business context, the real-world scenario must be translated into a mathematical model known as a Linear Programming Problem (LPP). The key components of an LPP are:
- Objective Function: A linear function representing the goal of the problem (e.g., maximize profit or minimize cost).
- Decision Variables: The unknowns to be determined (e.g., quantities of products to produce).
- Constraints: A set of linear inequalities or equations that represent resource limitations.
- Non-negativity Constraints: Decision variables are typically required to be non-negative, as negative quantities are not feasible in most business contexts.
Standard Form of an LPP
An LPP in standard form is written as:
Maximize (or Minimize):
[latex]Z = c_{1}x_{1} + c_{2}x_{2} + ... + c_{n}x_{n}[/latex]
Subject to:
[latex]\begin{align*}&a_{11}x_{1} + a_{12}x_{2} + ... + a_{1n}x_{n} \leq b_{1}\\ &a_{21}x_{1} + a_{22}x_{2} + ... + a_{2n}x_{n} \leq b_{2}\\ &\vdots\\ &a_{m1}x_{1} + a_{m2}x_{2} + ... + a_{mn}x_{n} \leq b_{m}\\ &x_{1}, x_{2}, ... x_{n} \geq 0 \end{align*}[/latex]
Methods for Solving LPPs
Once formulated, an LPP can be solved using one of the following methods:
- Graphical Method: Suitable for problems involving two decision variables. It provides a visual representation of the feasible region and optimal solution.
- Simplex Method: A systematic algebraic approach capable of solving problems with multiple variables and constraints.
Illustrative Example: Maximizing Bakery Profit
Scenario: Marietta owns a bakery that produces a variety of baked goods. For special orders, she bakes biscuits and cupcakes, earning a profit of $5 per biscuit and $4 per cupcake. Each biscuit requires 3 minutes in the mixer and 4 minutes in the oven, while each cupcake requires 5 minutes in the mixer and 1 minute in the oven. Due to other production commitments, she has only 78 minutes of mixer time and 36 minutes of oven time available for special orders.
Objective: Determine the number of biscuits and cupcakes Marietta should bake to maximize her profit.
Step 1: Define Decision Variables
Let:
- [latex]x_{1}[/latex] = number of biscuits
- [latex]x_{2}[/latex] = number of cupcakes
Step 2: Formulate the Objective Function
Maximize [latex]Z = 5x_{1} + 4x_{2}[/latex]
Step 3: Define the Constraints
Mixer time constraint:
[latex]3x_{1}+ 5x_{2} \leq 78[/latex]
Oven time constraint:
[latex]4x_{1}+ x_{2} \leq 36[/latex]
Non-negativity constraints:
[latex]x_{1}, x_{2} \geq 0[/latex]
This formulation represents a complete LPP. The next step would be to solve it using either the graphical method (since there are two variables) or the simplex method.
