# MGOC10H3 Lecture Notes - Lecture 2: Feasible Region, Sensitivity Analysis

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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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