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Linear programming model problems have several properties and assumptions in common. The seven basic properties of linear programming are: (1) one objective function, meaning problems seek to maximize or minimize an objective. (2) One or more constraints, meaning constraints limit the degree which the objective can be obtained. (3) Alternative courses of action, meaning there must be more than one course of action offered as optional paths. (4) Objective function and constraints are linear – proportionally and divisibility, meaning that each variable is equally multiple and divisible in relation to the quantity. (5) Certainty, meaning the number in the objective and constraints are known with certainty and do not change during the period being studied. (6) Divisibility, meaning solutions are divisible and may take any fractional value. (7) Nonnegative variables, meaning negative values of physical quantities are impossible.
To demonstration linear programming in an example form, the book used the Flair Furniture Company to explain the concept. The objective for Flair is to determine the best possible combination of tables and chairs to manufacture in order to reach maximum profits (maximize profit), the constraints are (1) carpentry time cannot exceed 240 hours per week (2) painting time cannot exceed 100 hours per week. The variables for this model are the quantity of tables and chairs, with the constants being the 4 hours per table and 3 hours per chair manufacturing time, and 2 hours per table and 1 hour per chair painting time. The formulas that are created from these are:
4T + 3C ≤ 240 (carpentry constraint)
2T + 1C ≤ 100 (painting constraint)
T ≥ 0 (first nonnegativity constraint)
C ≥ 0 (second nonnegativity constraint)
There are no real alternative courses of action, other than changing the variables to maximize the profit. Each of the manufacturing hours and painting hours variables are proportional and divisible to the quantity of production and the constants per product for each item have certainty. As you solve the objective by manipulating the variables, they results may be fractional but cannot be negative solutions. The key steps that need to be considered while formulating a linear programming problem are: (1) completely understand the managerial problem being faced (2) Identify the objective and the constraints (3) Define the decision variables (4) Use the decision variables to write mathematical expressions for the objective function and the constraints.
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