# Linear Programming: Unlocking Optimization with Graphical and Simplex Methods

## TRANSPORTATION & ASSIGNMENT PROBLEM:

General structure of transportation problem, solution procedure for transportation problem, methods for finding initial solution, test for optimality. Maximization of transportation problem, unbalanced transportation problem, Assignment problem approach of the assignment model, solution methods of assignment problem, maximization in an assignment, unbalanced assignment problem, restriction on assignment

## Introduction

Linear programming is a powerful mathematical technique used to optimize the allocation of resources. It is widely used in various industries and fields, such as manufacturing, logistics, finance, and operations research. In this article, we will explore the formulation of linear programming models and discuss the graphical method and simplex method for solving linear programming problems. Additionally, we will delve into the transportation and assignment problems, which are specific applications of linear programming. We will examine the general structure of transportation problems, solution procedures, methods for finding initial solutions, tests for optimality, and approaches to solving assignment problems. Let's dive in and explore the fascinating world of linear programming and its applications.

## Formulation of Linear Programming Model

Linear programming models involve the optimization of a linear objective function subject to a set of linear constraints. The objective function represents the quantity that needs to be maximized or minimized, while the constraints define the limitations or restrictions on the decision variables. The decision variables represent the quantities that we can control or manipulate to achieve the desired objective.

To formulate a linear programming model, we follow these steps:

- Identify the decision variables: These variables represent the quantities we want to determine.
- Define the objective function: This function expresses the quantity we want to maximize or minimize in terms of the decision variables.
- Establish the constraints: These constraints specify the limitations or restrictions on the decision variables.
- Specify the non-negativity constraints: In many cases, the decision variables cannot take negative values.

Once the linear programming model is formulated, we can use various methods to solve it and obtain the optimal solution. Two commonly used methods are the graphical method and the simplex method.

## Graphical Method of Solving Linear Programming Problems

The graphical method is a graphical visualization technique used to solve linear programming problems with two decision variables. It involves plotting the constraints on a graph and identifying the feasible region, which represents the set of all feasible solutions. The optimal solution is then determined by finding the point within the feasible region that maximizes or minimizes the objective function.

To solve a linear programming problem using the graphical method, follow these steps:

- Plot the constraints: Each constraint is represented by a line or a boundary on the graph.
- Identify the feasible region: The feasible region is the intersection of all the constraint boundaries and represents the set of feasible solutions.
- Plot the objective function: The objective function is plotted on the graph, and its contours are drawn to represent the direction of optimization (maximization or minimization).
- Determine the optimal solution: The optimal solution is the point within the feasible region that maximizes or minimizes the objective function. It can be found by evaluating the objective function at the vertices of the feasible region or by using the corner-point method.

The graphical method provides an intuitive and visual approach to solving linear programming problems. However, it is limited to problems with only two decision variables and may not be practical for complex problems with a large number of constraints.

## Simplex Method (Maximization and Minimization)

The simplex method is a widely used algorithm for solving linear programming problems with multiple decision variables and constraints. It is an iterative procedure that systematically improves the solution until the optimal solution is reached.

The simplex method starts with an initial feasible solution and iteratively moves to adjacent feasible solutions that improve the objective function value. It continues this process until no further improvements can be made, indicating that the optimal solution has been reached.

The steps involved in the simplex method are as follows:

- Formulate the initial simplex tableau: The initial tableau is constructed using the coefficients of the objective function and constraints.
- Select the entering variable: The entering variable is the variable that can increase the objective function value the most. It is chosen based on the simplex tableau's coefficients and the objective function's optimization direction (maximization or minimization).
- Determine the leaving variable: The leaving variable is the variable that leaves the basis to make room for the entering variable. It is selected based on the minimum ratio test, which ensures that the solution remains feasible.
- Update the simplex tableau: The tableau is updated by performing row operations to pivot on the entering and leaving variables.
- Iterate until optimality: Steps 2 to 4 are repeated until the optimal solution is reached. The optimal solution is achieved when all the coefficients in the objective row of the tableau are non-negative for maximization problems (or non-positive for minimization problems).

The simplex method is a powerful algorithm that can handle linear programming problems with any number of decision variables and constraints. It provides an efficient way to find the optimal solution, even for large-scale problems. However, it may require a significant number of iterations in some cases.

## General Structure of Transportation Problem

The transportation problem is a specific application of linear programming that deals with the optimal allocation of goods from sources to destinations. It is commonly encountered in logistics and supply chain management, where goods need to be transported efficiently and cost-effectively.

The general structure of a transportation problem can be defined as follows:

- Sources: The sources represent the locations where the goods are produced or available.
- Destinations: The destinations represent the locations where the goods need to be delivered.
- Supply: The supply represents the capacity or availability of goods at each source.
- Demand: The demand represents the requirement or demand for goods at each destination.
- Costs: The costs represent the transportation costs or distances between sources and destinations.

The objective of the transportation problem is to minimize the total transportation cost while satisfying the supply and demand constraints.

## Solution Procedure for Transportation Problem

The transportation problem can be solved using various solution procedures, such as the northwest corner method, the least cost method, and Vogel's approximation method. These methods help find the initial feasible solution, which is then improved iteratively to reach the optimal solution.

The solution procedure for the transportation problem typically involves the following steps:

- Finding the initial feasible solution: This step involves allocating the available supply to the demand in a way that satisfies the supply and demand constraints. Methods such as the northwest corner technique, the least cost approach, and Vogel's approximation method can be used to find the initial viable option.
- Improving the solution: Once the initial feasible solution is obtained, it can be improved iteratively by applying optimization techniques such as the stepping stone method or the modified distribution method. These methods identify and exchange units of supply and demand to reduce the total transportation cost.
- Testing for optimality: After each improvement iteration, the solution is tested for optimality to determine whether further improvements can be made. Various optimality tests, such as the reduced cost method or the u-v method, can be applied to check if the solution is optimal or if additional iterations are required.
- Reaching the optimal solution: The solution is iteratively improved and tested for optimality until no further improvements can be made. At this point, the optimal solution has been reached, and the transportation problem is solved.

The solution procedure for the transportation problem is systematic and ensures that the optimal solution is obtained while satisfying the supply and demand constraints.

## Methods for Finding Initial Solution

Finding the initial feasible solution for the transportation problem is a crucial step in solving the problem optimally. Several methods can be employed to obtain the initial solution:

- Northwest Corner Method: This method starts allocating units of supply from the northwest corner (top-left corner) of the transportation table and proceeds sequentially along the rows and columns.
- Least Cost Method: The least cost method selects the cell with the lowest transportation cost and assigns the maximum possible units to that cell. This process continues until the supply and demand constraints are satisfied.
- Vogel's Approximation Method: Vogel's approximation method considers the differences between the two lowest transportation costs in each row and column. The cell with the largest difference is assigned units until either the supply or demand is exhausted.

These methods provide initial feasible solutions, which can then be improved iteratively to reach the optimal solution.

## Test for Optimality

To determine whether a solution to the transportation problem is optimal or if further improvements can be made, various optimality tests can be applied. Two commonly used tests are the reduced cost method and the u-v method.

The reduced cost method involves calculating the reduced costs for each unused cell in the transportation table. The reduced cost represents the amount by which the transportation cost should change for a unit increase in the allocated quantity. If all the reduced costs are non-negative, the current solution is optimal.

The u-v method, also known as the stepping stone method, involves identifying a closed loop of allocations in the transportation table. The closed loop is formed by following a path from an unoccupied cell to an occupied cell, and then to another unoccupied cell, and so on. The optimality test is based on the existence of a closed loop with zero net change in the total allocation.

By applying these optimality tests, it is possible to determine whether the current solution is optimal or if further improvements are required.

## Maximization of Transportation Problem

In some cases, the transportation problem may involve maximizing an objective function rather than minimizing it. The objective of the problem becomes to maximize the total allocation subject to the supply and demand constraints.

To solve a maximization transportation problem, the following steps can be followed:

- Convert maximization to minimization: To convert the maximization problem into a minimization problem, take the negative of the objective function coefficients.
- Follow the solution procedure: Apply the same solution procedure discussed earlier for the minimization transportation problem, including finding the initial feasible solution, improving the solution iteratively, and testing for optimality.
- Interpret the results: Since it is a maximization problem, the optimal solution obtained will represent the maximum allocation that satisfies the supply and demand constraints.

Maximization transportation problems can be effectively solved using the same techniques and methods as minimization problems, with slight modifications in the objective function coefficients.

## Unbalanced Transportation Problem

In some situations, the total supply does not equal the total demand in the transportation problem. This condition is referred to as an unbalanced transportation problem. Unbalanced problems arise when the available supply does not match the required demand, or vice versa.

To solve an unbalanced transportation problem, we can introduce a dummy source or dummy destination to balance the supply and demand. The dummy source represents a fictitious supplier with a supply equal to the unmet demand, while the dummy destination represents a fictitious destination with a demand equal to the excess supply.

By introducing these dummy entities, the unbalanced transportation problem can be transformed into a balanced transportation problem, which can then be solved using the methods discussed earlier.

## Assignment Problem Approach of the Assignment Model

The assignment problem is another specific application of linear programming that deals with the optimal assignment of resources to tasks. It is often encountered in workforce management, project planning, and scheduling, where tasks need to be assigned to individuals or machines in the most efficient manner.

The assignment problem can be represented as a special case of the transportation problem, where the sources and destinations are one-to-one correspondences. Each source represents a resource, each destination represents a task, and the transportation cost represents the assignment cost or efficiency.

The objective of the assignment problem is to minimize the total assignment cost while ensuring that each task is assigned to exactly one resource and each resource is assigned to exactly one task.

## Solution Methods of Assignment Problem

Several solution methods can be employed to solve the assignment problem and find the optimal assignment. Two commonly used methods are the Hungarian method and the branch-and-bound method.

The Hungarian method is an efficient algorithm that solves the assignment problem by iteratively finding augmenting paths in a weighted bipartite graph. It achieves the optimal assignment by performing a series of row and column operations on a cost matrix.

The branch-and-bound method is a more general approach that can handle larger instances of the assignment problem. It systematically explores the assignment space by branching on different assignments and pruning infeasible or suboptimal branches. The branch-and-bound method guarantees the optimal solution but may require more computational resources.

These solution methods ensure that the assignment problem is solved optimally, providing the most efficient assignment of resources to tasks.

## Maximization in an Assignment

In some cases, the assignment problem may involve maximizing an objective function rather than minimizing it. The objective becomes to maximize the total assignment benefit or efficiency while ensuring that each task is assigned to exactly one resource and each resource is assigned to exactly one task.

To solve a maximization assignment problem, the same solution methods discussed earlier can be applied, with slight modifications. The objective function coefficients are adjusted accordingly to represent the benefits or efficiencies, and the assignment is determined based on maximizing the total objective function value.

## Unbalanced Assignment Problem

Similar to the transportation problem, the assignment problem can also be unbalanced when the number of resources does not match the number of tasks. Unbalanced assignment problems arise when there are more resources than tasks or more tasks than resources.

To solve an unbalanced assignment problem, we can introduce dummy resources or dummy tasks to balance the assignment. The dummy resources represent fictitious resources with zero assignment benefits or efficiencies, while the dummy tasks represent fictitious tasks that need to be assigned to the excess resources.

By introducing these dummy entities, the unbalanced assignment problem can be transformed into a balanced assignment problem, which can then be solved using the methods discussed earlier.

## Restriction on Assignment

In some assignment problems, additional restrictions or constraints may be imposed on the assignments. These restrictions can include constraints on the availability of resources, the compatibility between resources and tasks, or the precedence relationships among tasks.

To handle restrictions on assignment, the assignment problem can be extended to include additional variables and constraints in the linear programming formulation. These variables and constraints represent the restrictions and help ensure that the assignments satisfy the specified conditions.

The inclusion of restrictions on assignment enhances the applicability of the assignment problem in various real-world scenarios, where specific constraints need to be considered during the assignment process.

## Conclusion

In conclusion, linear programming is a valuable tool for optimizing resource allocation and solving complex decision-making problems. The formulation of linear programming models, along with the graphical method and simplex method, provides effective means to find optimal solutions. Additionally, the transportation problem and assignment problem, which are specific applications of linear programming, address challenges related to efficient transportation and task assignment. By applying the solution procedures and methods discussed, organizations can enhance their operational efficiency and make informed decisions. Understanding the concepts and techniques of linear programming opens up a world of possibilities for improving processes, maximizing resources, and achieving better outcomes.