Introduction to Linear Programming (LP) – Linear programming is a tool for solving
optimization models.
3.1 What Is a Linear Programming Problem?
Decision Variables – In any linear programming model, the decision variables should
completely describe the decisions to be made.
Objective Function – In any linear programming model, the decision maker wants to
maximize or minimize some function of the decision variables.
Objective Function Coefficient – The coefficient of a variable in the objective function.
In many problems, the objective function coefficient for each variable is simply the
contribution of the variable to the company’s profit.
Constraints – it is IMPERATIVE that all terms in a constraint have the same units.
Technological Coefficients – the coefficients of the decision variables in the
constraints.
Right-hand side (RHS) – the number to the right on a constraint e.g. <= 100 or >= 40
Unrestricted in Sign (URS) - a variable that can assume positive, zero, and negative
values.
The Proportionality and Additivity Assumptions
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The fact that the objective function for an LP must be a linear function of the
decision variables has two implications.
1. The contribution of the objective function from each decision variable is
proportional to the value of the decision variable. For example, the contribution to
the objective function from making four soldiers (4 x 3 = $3) (see 3.1 LP example
on iPad) is exactly four times the contribution to the objective function from
making one soldier ($3).
2. The contribution to the objective function for any variable is independent of the
values of the other decision variables. E.g. no matter what the value of y, the
manufacture of x soldiers will always contribute 3x dollars to the objective
function.
- Analogously, the fact that each LP constraint must be a linear inequality or linear
equality has two implications.
1. The contribution of each variable to the left-hand side of each constraint is
proportional to the value of the variable. E.g. it takes 3x the hours to make 3x the
trains.
2. The contribution of each variable to the left-hand side of each constraint is
independent of the values of the variable. For example, no matter what the value
of x, the manufacture of y trains uses y finishing hours and y carpentry hours.
The first implication given in each list is called the Proportionality Assumption of
Linear Programming.
The second implication given in each list is called the Additivity Assumption of Linear
Programming.
The Divisibility Assumption – requires that each decision variable be allowed to
assume fractional values. Is not satisfied in the Giapetto problem.
The Certainty Assumption – is that each parameter (objective function coefficient, rhs,
and technological coefficient) is known with certainty.
Feasible Region – all set of points that satisfies all the LP’s constraints and sign
restrictions
Optimal Solution – a point in the feasible region with the largest objective function
value. Similarly, for a minimization problem, an optimal solution is a point in the feasible
region with the smallest objective function value.