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Lecture 2

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Vinh QuanLecture

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**preview**shows half of the first page. to view the full**2 pages of the document.**MGOC10 Analysis for Decision Making

Lecture 02

Chapter 2 – Solving LP Model – Graphical Method & Sensitivity Analysis of Constraints

LP terminology

Example – Giapetto’s problem

MAX Z = 3X1 + 2X2

S.T. 2X1 + X2 100 (finishing)

X1 + X2 80 (carpentry)

X1 40 (soldier demand)

X1 0 , X2 0

Any specification of values for variables is called a solution eg. X1 = 60, X2 = 10.

Feasible solution is a solution that satisfies all constraints and sign restrictions

eg. [X1 = 40, X2 = 20] or [X1 = 30.5, X2 = 20.001] or …

The feasible region is set of all feasible solutions.

Given many feasible solutions, goal is to find the best or optimal solution.

For max problem, optimal (best) solution is one which maximizes value of objective.

For min problem, it is one which minimizes objective.

Solutions which are intersection of 2 or more constraints are called corner feasible solutions (at corner

of feasible region). The optimal solution for an LP must be a corner feasible solution.

LPs can have one optimal solution, multiple optimal solutions, or problem can be infeasible (no

feasible solution) or unbounded ( Z = ∞ or Z = - ∞)

Slack

Given optimal solution, can compute the slack for any “≤” constraints. Slack = right hand side – left hand side.

Optimal solution for Giapetto, X1* = 20, X2* = 60

For X1 + X2 80 hrs, Slack = 80 – (X1* + X2*) = 80 – (20 + 60) = 0.

For X1 40 units, Slack = 40 – X1* = 40 – 20 = 20.

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