CPF SOLUTIONSCorner-Point Feasible (CPF) Solution is a Feasible

Solution that Lies at a Corner of the Feasible Region

Optimal Solutions & CPF Solutions For an LP with Feasible Solutions & Bounded

Feasible Region, it Must Possess CPF Solutions & At Least One Optimal Solution.

The Best CPF Solution Must be an Optimal Solution. If Exactly One Optimal Solution Exists, it Must be a

CPF Solution. If Multiple Optimal Solutions Exist, At Least Two

Must be CPF Solutions

STANDARD FORM OF LPTwo-Variable Problems May be Solved

Graphically

Real-Life Problems Usually Have More Than Two Variables. LP Problems May Have Both Equality & Inequality Constraints. Variables May be Required to be Non-Negative & May be ‘urs’ (Unrestricted in Sign)

The Simplex Method (Algorithm): Convert LP Problem into an Equivalent Problem with All Constraints as Equations & All Variables as Non-Negative

SLACK VARIABLESFor Inequality Constraints with ‘≤,’ Introduce a Slack Variable (si for ith Constraint) i.e. Amount of Resource Unutilized in ith Constraint:

Max z = 4 x1 + 3 x2 Subject Tox1 + x2 ≤ 40 (Constraint-1->Material in Sq. Yards)2 x1 + x2 ≤ 60 (Constraint-2->Labour in Hours)x1, x2 ≥ 0

Introducing Slack Variables s1 & s2:x1 + x2 + s1 = 40 (Standardized or Augmented Constraint-1)2 x1 + x2 + s2 = 60 (Standardized or Augmented Constraint-2)

In Standard (or Augmented) Form,(x1, x2) Satisfies ith Constraint if & only if si ≥ 0 i.e. Resource Usage is Within Availability Limits

STANDARD FORM OF LPMax z = 4 x1 + 3 x2

Subject Tox1 + x2 + s1 = 402 x1 + x2 + s2 = 60 x1, x2, s1, s2 ≥ 0

EXCESS (SURPLUS) VARIABLESFor Inequality Constraints with ‘≥,’ Introduce an Excess (Surplus) Variable (ei for ith Constraint) i.e. Amount by Which ith Constraint is Over-Satisfied: Min z = 50 x1 + 20 x2 + 30 x3 + 80 x4 Subject To400 x1 + 200 x2 + 150 x3 + 500 x4 ≥ 500 (Constraint-1->Calories)3 x1 + 2 x2 ≥ 6 (Constraint-2->Chocolate)2 x1 + 2 x2 + 4 x3 + 4 x4 ≥ 10 (Constraint-3->Sugar)2 x1 + 4 x2 + x3 + 5 x4 ≥ 8 (Constraint-4->Fat)x1, x2, x3, x4 ≥ 0

Introducing Excess Variables e1, e2, e3 & e4:3 x1 + 2 x2 – e2 = 6 (Standardized Constraint-2)

In Standard Form,(x1, x2) Satisfies ith Constraint if & only if ei ≥ 0 i.e. Minimum Requirement is Exceeded

STANDARD FORM OF LPMin z = 50 x1 + 20 x2 + 30 x3 + 80 x4

Subject To400 x1 + 200 x2 + 150 x3 + 500 x4 – e1 = 5003 x1 + 2 x2 – e2 = 62 x1 + 2 x2 + 4 x3 + 4 x4 – e3= 102 x1 + 4 x2 + x3 + 5 x4 – e4 = 8xi, ei ≥ 0(i = 1, 2, 3, 4)

SIMPLEX METHODAdjacent CPF Solutions: An LP with ‘n’

Decision Variables, Two CPF Solutions are Adjacent if ‘n – 1’ Constraint Boundaries are Common

Optimality Test: For Any LP with At Least One Optimal Solution, a CPF Solution with No Adjacent CPF Solution(s) that is Better Must be an Optimal Solution

EXAMPLE: Woodcarving Inc.Max z = 3 x1 + 2 x2

Subject Tox1 + x2 ≤ 80 (Constraint-1)2 x1 + x2 ≤ 100 (Constraint-2)x1 ≤ 40 (Constraint-3)x1 ≥ 0 (Constraint-4)x2 ≥ 0 (Constraint-5)

EXAMPLE: Woodcarving Inc.100

80

50 8040

x1

x2

(2)

(1)

(3)

(5)

(4)

O

FEASIBLE

REGION

SIMPLEX METHOD CONCEPTSConcept-1: Simplex Method Focuses Only on CPF

Solutions i.e., Reduces Infinite Number of Feasible Solutions to Finite Number of Solutions to Examine

Concept-2: Simplex Method is an Iterative Algorithm: Initialize -> Optimality Test -> If Yes, Stop ; If No, Find Better CPF Solution & Repeat Iteration

Concept-3: When Possible, Simplex Method Usually Chooses the Origin (All Decision Variables = 0) as the Initial CPF Solution (Unless Origin is Infeasible)

SIMPLEX METHOD CONCEPTSConcept-4: Given a CPF Solution, Easier Computationally

to Analyze Adjacent CPF Solutions (Compared to Any Other CPF Solution) i.e. Always Follows Edges of the Feasible Region

Concept-5: Simplex Method Identifies Rate of Improvement in ‘z’ by Analyzing Available Edges. Chooses Edge with Largest Rate of improvement in ‘z’

Concept-6: Optimality Test Focuses on Positive Rate of Improvement in ‘z’ Along Adjacent Edges of Current CPF Solution. If No Adjacent Edge Gives a Positive Rate of Improvement in ‘z,’ Current CPF Solution is Optimal!

BASIC PROPERTIES OF SOLUTIONS

Standardized Solution: A Solution to Original Variables (Decision Variables) Standardized by Corresponding Slack (or Excess) Variables

Basic Solution: A Corner-Point Solution (May be Feasible or Infeasible) to Original Variables (Decision Variables) Standardized by Corresponding Slack (or Excess) Variables

Basic Feasible (BV) Solution: A Corner-Point Feasible (CPF) Solution to Original Variables (Decision Variables) Standardized by Corresponding Slack (or Excess) Variables

BASIC / NONBASIC VARIABLESTotal Number of Variables in LP ‘n’Number of Functional Constraints ‘m’For All Basic Solutions:1.Each Variable is Either Basic (BV) or Nonbasic (NBV) 2.Number of Basic Variables = Number of Functional

Constraints (i.e. Equations). 3.Number of Nonbasic Variables = Total Number of

Variables – Number of Functional Constraints4.Nonbasic Variables = 05.Values of Basic Variables Obtained as Solution to System

of Equations (Functional Constraints in Standardized Form)

6.If Basic Variables ≥ 0, Basic Solution is a BF Solution

EXAMPLE: ORIGINAL LPMax z = 60 x1 + 30 x2 + 20 x3

Subject To 8 x1 + 6 x2 + x3 ≤ 48 4 x1 + 2 x2 + 1.5 x3 ≤ 20 2 x1 + 1.5 x2 + 0.5 x3 ≤ 8x2 ≤ 5 x1, x2, x3 ≥ 0

EXAMPLE: STANDARD FORMMax z = 60 x1 + 30 x2 + 20 x3

Subject To 8 x1 + 6 x2 + x3 + s1 = 48 4 x1 + 2 x2 + 1.5 x3 + s2 = 20 2 x1 + 1.5 x2 + 0.5 x3 + s3= 8x2 + s4 = 5 x1, x2, x3, s1, s2, s3, s4 ≥ 0

EXAMPLE: INITIAL BF SOLUTION (x1, x2, x3 =0)

ROW BASIC VARIABLE 0 z – 60 x1 – 30 x2 – 20 x3 = 0 z = 0 1 8 x1 + 6 x2 + x3 + s1 = 48 s1 = 48 2 4 x1 + 2 x2 + 1.5 x3 + s2 = 20 s2

= 20 3 2 x1 + 1.5 x2 + 0.5 x3 + s3 = 8

s3 = 8 4 x2 + s4 = 5 s4

= 5

BV = s1, s2, s3, s4 (With Coefficient of ‘1’ in Associated Row) NBV = x1, x2, x3 = 0