# Linear Programming

26/03/2020

The technique of linear programming was formulated by a Russian mathematician L.V. Kantorovich. But the present version of simplex method was developed by Geoge B. Dentzig in 1947. Linear programming (LP) is an important technique of operations research developed for optimum utilization of resources.

It is an important optimization (maximization or minimization) technique used in decision making is business and everyday life for obtaining the maximum or minimum values as required of a linear expression to satisfying certain number of given linear restrictions.

#### Common terminologies used in Linear Programming

• Decision Variables: The decision variables are the variables which will decide my output. They represent my ultimate solution. To solve any problem, we first need to identify the decision variables. For the above example, the total number of units for A and B denoted by X & Y respectively are my decision variables.
• Objective Function: It is defined as the objective of making decisions. In the above example, the company wishes to increase the total profit represented by Z. So, profit is my objective function.
• Constraints: The constraints are the restrictions or limitations on the decision variables. They usually limit the value of the decision variables. In the above example, the limit on the availability of resources Milk and Choco are my constraints.
• Non-negativity restriction: For all linear programs, the decision variables should always take non-negative values. Which means the values for decision variables should be greater than or equal to 0.

### Characteristics:

(a) Objective function:

There must be clearly defined objec­tive which can be stated in quantitative way. In business problems the objective is generally profit maximization or cost minimization.

(b) Constraints:

All constraints (limitations) regarding resources should be fully spelt out in mathematical form.

(c) Non-negativity:

The value of variables must be zero or positive and not negative. For example, in the case of production, the manager can decide about any particular product number in positive or minimum zero, not the negative.

(d) Linearity:

The relationships between variables must be linear. Linear means proportional relationship between two ‘or more variable, i.e., the degree of variables should be maximum one.

(e) Finiteness:

The number of inputs and outputs need to be finite. In the case of infinite factors, to compute feasible solution is not possible.

#### Advantages of Linear Programming

1. LP makes logical thinking and provides better insight into business problems.
2. Manager can select the best solution with the help of LP by evaluating the cost and profit of various alternatives.
3. LP provides an information base for optimum alloca­tion of scarce resources.
4. LP assists in making adjustments according to changing conditions.
5. LP helps in solving multi-dimensional problems.

#### Assumptions:

(i) There are a number of constraints or restrictions- expressible in quantitative terms.

(ii) The prices of input and output both are constant.

(iii) The relationship between objective function and constraints are linear.

(iv) The objective function is to be optimized i.e., profit maximization or cost minimization.

### Application Area

In business, executives need to make a plethora of decisions. From which company will provide and stock the break room’s vending machines to which costs get cut during a lean time, all require careful thought and planning. It’s essential that every decision, no matter what its perceived importance may be, be made with the best intentions and the company’s best interests at heart.

The Health of the Business Depends on It: Making snap business decisions can make or break a business. Changing business practices on a whim or because an owner is in a bad mood can have irreversible consequences. All business decisions should be carefully considered, and should preferably have input from several others. The business owner has the final say, but other voices should be brought to the table so that all sides of an issue can be considered.

It Takes a Team: A business is only as good as its weakest member, so putting together a strong management group is paramount to the business’ ability to make good decisions, which leads to the success of the business. All members of management need to be on the same page, as to what constitutes the mission and objectives of the business. Once everybody knows the mission, deciding how to meet its objectives becomes easier. If you value a person enough to put her on the team, then you should value that person’s opinion, even if it disagrees with yours.

Consult Someone With More Expertise: If you don’t feel secure enough in your abilities or in your team’s abilities to make a solid decision, there is nothing wrong with getting help from someone outside of the company. For example, a graphics design firm might consult with an attorney to look over a new contract, or an attorney might hire an accountant to assist with the financial statements for the law firm. No one is an expert at everything, and good decision making includes knowing when to seek additional assistance.

Don’t Be Afraid to Reverse: Sometimes, a decision made needs to be revisited. A business move that was ideal in January might not be as effective in July. Evaluating previous business decisions and choosing to modify or completely reverse that decision is acceptable. Although long-term, permanent decisions are typically the goal, sometimes, you have to think short-term and then revisit the issue at a later date.

Gray Areas: Not every decision is black and white. In some cases, you have to look at the gray area, as well. This is where having a strong management team or additional expertise comes into play. These people can help a business owner see all sides of an issue, which improves the chances of making a keen business decision that’s in the company’s best interest.