MGOC10H3 Lecture Notes - Feasible Region, Sensitivity Analysis
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MGTC74 Analysis for Decision Making
Chapter 2 – Solving LP Model – Graphical Method & Sensitivity Analysis of Constraints
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 = - ∞)
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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