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Using Computer Software to Solve Linear Programming Problems Graphically • SLM (Self Learning Material) for MBA

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Have you ever tried solving a linear programming problem by hand and found yourself drowning in graphs, equations, and calculations? You’re not alone! While understanding the manual graphical method is crucial for grasping the fundamentals, modern business problems often involve complex data sets that would take hours to solve manually. This is where computer software becomes your best friend, transforming what could be a tedious mathematical marathon into an efficient, accurate solution process.

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The digital revolution in linear programming

Linear programming has come a long way since its inception in the 1940s. What once required teams of mathematicians working with slide rules and graph paper can now be accomplished in minutes using specialized software. These digital tools don’t just speed up the process—they eliminate human error, handle multiple variables with ease, and provide visual representations that make complex problems understandable at a glance.

Think of it this way: if manual calculation is like walking to your destination, using computer software is like taking a high-speed train. Both will get you there, but one is significantly faster, more comfortable, and less prone to getting lost along the way.

Essential software tools for linear programming

The market offers several powerful software options, each with its own strengths and user interface. Let’s explore the most popular ones that have become staples in both academic and professional settings.

TORA (Totally Optimized Resource Allocation)

User-friendly interface: TORA stands out for its intuitive design that makes it perfect for beginners. The software presents a clean, straightforward interface where you can input your objective function and constraints without getting overwhelmed by technical jargon.

Educational focus: What makes TORA particularly valuable for students is its step-by-step solution display. It doesn’t just give you the final answer—it shows you how it arrived there, making it an excellent learning tool that bridges the gap between manual calculation and automated solving.

Graphical capabilities: The software excels at creating clear, colorful graphs that highlight the feasible region, constraint lines, and optimal points. This visual approach helps you understand the geometric interpretation of linear programming problems.

LINDO (Linear, Interactive, and Discrete Optimizer)

Professional-grade power: LINDO is the heavyweight champion of linear programming software. Used by Fortune 500 companies and major consulting firms, it can handle problems with thousands of variables and constraints without breaking a sweat.

Advanced features: Beyond basic linear programming, LINDO offers integer programming, nonlinear optimization, and stochastic programming capabilities. It’s like having a Swiss Army knife for optimization problems.

Scripting capabilities: For advanced users, LINDO allows you to write scripts and automate repetitive tasks, making it invaluable for large-scale business applications.

QM for Windows (Quantitative Methods)

Academic favorite: Widely used in business schools, QM for Windows offers a comprehensive suite of quantitative analysis tools, with linear programming being just one of its many modules.

Built-in templates: The software comes with pre-designed templates for common business problems like production planning, transportation, and assignment problems, allowing you to jump straight into problem-solving.

Report generation: QM for Windows automatically generates professional reports that include both numerical results and graphical representations, perfect for presentations and assignments.

Step-by-step guide to graphical solutions using software

Let’s walk through the process using a practical example. Imagine you’re helping a small bakery optimize their daily production of croissants and muffins to maximize profit.

Setting up your problem

Define your variables: Start by clearly identifying what you’re trying to optimize. In our bakery example, X₁ represents the number of croissants and X₂ represents the number of muffins to produce daily.

Input the objective function: Most software tools have a dedicated field for your objective function. You’ll enter something like “Maximize 3X₁ + 2X₂” where 3 and 2 represent the profit per unit.

Enter constraints: This is where software really shines. Instead of worrying about plotting each constraint line manually, you simply input the mathematical expressions. For example, if the bakery has limited oven time, you might enter “2X₁ + X₂ ≤ 100”.

Generating the graphical solution

Automatic plotting: Once you’ve entered all data, the software instantly generates a coordinate system with all constraint lines plotted. Each line is typically color-coded and labeled for easy identification.

Feasible region identification: The software automatically shades or highlights the feasible region—the area where all constraints are satisfied simultaneously. This visual representation makes it immediately clear which combinations of production levels are actually possible.

Optimal point location: Perhaps most impressively, the software identifies and marks the optimal solution point, often with a different color or symbol. It then provides the exact coordinates and the corresponding objective function value.

Interpreting the results

Sensitivity analysis: Advanced software tools provide sensitivity analysis, showing how changes in constraints or objective function coefficients would affect the optimal solution. This is incredibly valuable for real-world decision-making.

Shadow prices: The software calculates shadow prices, indicating how much the objective function would improve if you could relax each constraint by one unit. For our bakery, this might show the value of adding one more hour of oven time.

Alternative solutions: In cases where multiple optimal solutions exist, quality software will identify and display all of them, giving you flexibility in implementation.

Real-world applications and benefits

Using computer software for linear programming isn’t just about convenience—it opens doors to solving problems that would be practically impossible to handle manually.

Handling complexity with ease

Multiple variables: While manual graphical methods are limited to two-variable problems, software can handle dozens or even hundreds of variables. A manufacturing company optimizing production across multiple product lines and facilities can benefit enormously from this capability.

Large constraint sets: Real business problems often involve numerous constraints—budget limitations, capacity restrictions, quality requirements, and regulatory compliance. Software handles these complex constraint systems effortlessly.

Dynamic problem solving: When market conditions change, you can quickly modify parameters and re-solve the problem. This agility is crucial in today’s fast-paced business environment.

Enhanced accuracy and reliability

Elimination of calculation errors: Manual calculations are prone to arithmetic mistakes, especially in complex problems. Software eliminates this risk entirely, ensuring that your decisions are based on accurate mathematical solutions.

Consistent results: Different people solving the same problem manually might arrive at slightly different answers due to rounding or graphing inaccuracies. Software provides consistent, repeatable results every time.

Validation capabilities: Most software includes built-in validation that checks for inconsistencies in your problem formulation, helping you identify errors before attempting to solve.

Time and cost efficiency

Rapid problem solving: What might take hours to solve manually can be completed in seconds using software. This time savings translates directly into cost savings and allows for more thorough analysis of alternatives.

Scenario analysis: The speed of computer solutions makes it practical to analyze multiple scenarios, helping you understand how different assumptions affect optimal decisions.

Professional presentation: Software-generated graphs and reports look professional and are easily incorporated into business presentations, making it easier to communicate your findings to stakeholders.

Choosing the right tool for your needs

Selecting the appropriate software depends on several factors including your experience level, problem complexity, and budget constraints.

For students and beginners: TORA or QM for Windows provide excellent starting points with their educational focus and user-friendly interfaces. They offer enough functionality to handle coursework while remaining approachable for newcomers.

For professionals: LINDO or similar enterprise-grade solutions offer the power and flexibility needed for complex business applications. While they require more investment in terms of both cost and learning time, they provide capabilities that can handle real-world optimization challenges.

For mixed environments: Many educational institutions and businesses use a combination of tools—simpler software for teaching and learning fundamental concepts, and more powerful tools for advanced applications and research.

What do you think? Have you experienced the frustration of solving complex linear programming problems manually, and how do you see software tools changing the way we approach optimization in business? What factors would be most important to you when choosing linear programming software for your specific needs?

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1 Operations Research-An Overview

  1. Introduction
  2. History of O.R.
  3. Approach, Techniques and Tools
  4. Phases and Processes of O.R. Study
  5. Typical Applications of O.R.
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  7. Models in Operations Research
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  3. Optimisation Models
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  6. Multiple, Unbounded Solution, and Infeasible Problems
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  1. Principle of Simplex Method
  2. Computational Aspect of Simplex Method
  3. Simplex Method with Several Decision Variables
  4. Two Phase and M-method
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  6. Sensitivity Analysis
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4 Transportation Problem

  1. Basic Feasible Solution of a Transportation Problem
  2. Modified Distribution Method
  3. Stepping Stone Method
  4. Unbalanced Transportation Problem
  5. Degenerate Transportation Problem
  6. Transhipment Problem
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  3. Graphical Method of Goal Programming
  4. The Simplex Method of Goal Programming
  5. Using Excel Solver to Solve Goal Programming Models
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