Linear Programming Problems (LPP)

Linear Programming, also knows as linear optimization is considered as a process which brings best possible solution to the mathematical model with the help of certain linear relationships. Linear programming deals with the constrained problems of increasing profits, decreasing costs, reasonable use of resources and so on, thus solving problems through the principles of linear inequality, referred as Linear Programming Problems (LPP).

This process finds its applications across a range of fields such as business, marketing, economics, commerce, industry and military etc.

Forms of Linear Programming Problems

There are two forms of Linear Programming Problems which are given below;

1. Canonical Form which uses “Boolean Algebra” to present “Boolean outputs” of digital circuits.
2. Standard Form is the simplified version of canonical form of linear programming problem.

Types of Linear Programming Problems (LPP)

There are different types of Linear Programming Problems (LPP) which are discussed with their constraints and objective functions such as:

• Manufacturing Problems
• Transportation Problems
• Optimal Assignment Problems
• Diet Problems

Manufacturing Problems

This type of linear programming problems deals with the problems related to manufactured products such as maximizing the net profits or increasing the production rate. This may be a function of availability of the workspace, the number of employees, the number of machine hours, utilization of packing materials, the requirement of raw materials and so on. This program is used in industrial sector as a forecast or predictor of company’s capital rise with time. Here, constraints could be factors such as employee working hours, the cost of packing materials etc. whereas the objective function could be the production rate.

Transportation Problems

The type of linear programming problem deals with the study of routes for the effective transportation, ways of efficient transportation of the products from several sources to the various markets in minimal cost of transportation. This type of problem is important to resolve for large organizations which comprise of multiple units of production and have large customer base. Here, the constraints could be the patterns of supply-demand whereas the objective function could be the const of transportation.

Optimal Assignment Problems

The optimal assignment problems are the type of linear programming problems which are directly related to a firm or company’s task completion in which selective number of employees are allocated for the completion of a specific assignment within the due period of time. This process works through division of labor in which each individuals perform one job within the given task. These kinds of problems are usually observed large corporations during event planning and management. The constraints in this problem could be number of individuals working, number of hours by each worker etc. whereas, the objective function could be the total number of completed tasks in a due time.

Diet Problems

This type of linear programming problem consists of the diet problems such as increasing the specific food intake which are high in specific nutrients which is helpful in devising and applying a specific diet plan. The purpose of diet problems is to find such group of foods, contributory in meeting daily nutritional requirements in minimum budget. Here, the constraints could be the factors such as nutritional needs to be met e.g., optimal level of calorie, sugar or cholesterol in diet. Whereas, the objective function could be the cost of food consumption.

Formulation of Linear Programming Problems

The formulation of LLP is similar to the mathematical modeling, which are given below.

1. Identification Of The Decision Variable

This step is involved in the identification of the decision variable which acts as a predictor of the performance of the objective function; which has to be optimized by all means. The decision variables could be denoted by X, Y, ZZ etc.

2. Create an Objective function

After the identification of the decision variable, devise an algebraic expression for the given condition which presents the quantity, aimed to maximize.

3. Identification Of The Constraints

This step simply involves the selection and writeup of the linear equalities with the help of given data.

4. Graphical Representation

The step involves the use of constraints to plot the graph in order to solve the linear programming problems.

5. Construct the Feasible Region

This is the section of the graph which aims to satisfy all the constrained problems by showing a “valid solution for the objective function” at any point withing the region.

6. Selecting the Optimal Point

The last step focuses on choosing the optimal point for which the given linear function needs to have optimized values i.e., minimum or maximum.

Optimal Solution

The optimal solution for the linear programming problems hides in the optimization of the linear function value i.e., either to maximize or minimize (e.g., xx and yy variables) within the conditions of linear inequalities via specific constraints. The solution region, also known as viable region of linear programming problem is the one common to all the constraints under study. Within/on the boundary of the feasible region, lies the feasible solutions to the constraints, represented by points. The region which is not feasible to the case is referred as “infeasible region”.

Limitations of Linear Programming Problems

• Linear programming problems (LLP) can be used only when all the given conditions of the problem are quantifiable.
• These are usable only when all the constraints and objective functions discussed in the case are used when presented via linear inequalities.