Operation and Planning Control
U7MEA37
Prepared by
Mr. Prabhu
Assistant Professor, Mechanical Department
VelTech Dr.RR & Dr.SR Technical University
1
Unit I
Linear programming
2
Introduction to Operations Research
• Operations research/management science
– Winston: “a scientific approach to decision making, which
seeks to determine how best to design and operate a
system, usually under conditions requiring the allocation
of scarce resources.”
– Kimball & Morse: “a scientific method of providing
executive departments with a quantitative basis for
decisions regarding the operations under their control.”
Introduction to Operations Research
• Provides rational basis for decision making
– Solves the type of complex problems that turn up in
the modern business environment
– Builds mathematical and computer models of
organizational systems composed of people, machines,
and procedures
– Uses analytical and numerical techniques to make
predictions and decisions based on these models
Introduction to Operations Research
• Draws upon
– engineering, management, mathematics
• Closely related to the "decision sciences"
– applied mathematics, computer science,
economics, industrial engineering and systems
engineering
Methodology of Operations Research*
The Seven Steps to a Good OR Analysis
Methodology of Operations Research*
The Seven Steps to a Good OR Analysis
•What are the
objectives?
•Is the proposed
problem too narrow?
•Is it too broad?
Methodology of Operations Research*
The Seven Steps to a Good OR Analysis
•What data should
be collected?
•How will data be
collected?
•How do different
components of the
system interact
with each other?
Methodology of Operations Research*
The Seven Steps to a Good OR Analysis
•What kind of model
should be used?
•Is the model accurate?
•Is the model too
complex?
Methodology of Operations Research*
The Seven Steps to a Good OR Analysis
•Do outputs match
current observations
for current inputs?
•Are outputs
reasonable?
•Could the model be
erroneous?
Methodology of Operations Research*
The Seven Steps to a Good OR Analysis
•What if there are
conflicting objectives?
•Inherently the most
difficult step.
•This is where software
tools will help us!
Methodology of Operations Research*
The Seven Steps to a Good OR Analysis
•Must communicate
results in layman’s
terms.
•System must be
user friendly!
Methodology of Operations Research*
The Seven Steps to a Good OR Analysis
•Users must be trained
on the new system.
•System must be
observed over time to
ensure it works
properly.
Linear Programming
14
Objectives
– Requirements for a linear programming model.
– Graphical representation of linear models.
– Linear programming results:
•
•
•
•
Unique optimal solution
Alternate optimal solutions
Unbounded models
Infeasible models
– Extreme point principle.
15
Objectives - continued
– Sensitivity analysis concepts:
•
•
•
•
•
•
Reduced costs
Range of optimality--LIGHTLY
Shadow prices
Range of feasibility--LIGHTLY
Complementary slackness
Added constraints / variables
– Computer solution of linear programming
models
• WINQSB
• EXCEL
• LINDO
16
3.1 Introduction to Linear
Programming
• A Linear Programming model seeks to
maximize or minimize a linear function,
subject to a set of linear constraints.
• The linear model consists of the following
components:
– A set of decision variables.
– An objective function.
– A set of constraints.
– SHOW FORMAT
17
• The Importance of Linear Programming
– Many real static problems lend themselves to linear
programming formulations.
– Many real problems can be approximated by linear
models.
– The output generated by linear programs provides
useful “what’s best” and “what-if” information.
18
Assumptions of Linear Programming
• The decision variables are continuous or divisible,
meaning that 3.333 eggs or 4.266 airplanes is an
acceptable solution
• The parameters are known with certainty
• The objective function and constraints exhibit
constant returns to scale (i.e., linearity)
• There are no interactions between decision variables
19
Methodology of Linear Programming
Determine and define the decision variables
Formulate an objective function
verbal characterization
Mathematical characterization
Formulate each constraint
20
MODEL FORMULATION
• Decisions variables:
– X1 = Production level of Space Rays (in dozens per
week).
– X2 = Production level of Zappers (in dozens per
week).
• Objective Function:
– Weekly profit, to be maximized
21
The Objective Function
Each dozen Space Rays realizes $8 in profit.
Total profit from Space Rays is 8X1.
Each dozen Zappers realizes $5 in profit.
Total profit from Zappers is 5X2.
The total profit contributions of both is
8X1 + 5X2
(The profit contributions are additive because of
the linearity assumption)
22
The Linear Programming Model
Max 8X1 + 5X2 (Weekly profit)
subject to
2X1 + 1X2 < = 1200 (Plastic)
3X1 + 4X2 < = 2400 (Production Time)
X1 + X2 < = 800
(Total production)
X1 - X2 < = 450
(Mix)
Xj> = 0, j = 1,2 (Nonnegativity)
23
3.4 The Set of Feasible Solutions
for Linear Programs
The set of all points that satisfy all the
constraints of the model is called a
FEASIBLE REGION
24
Using a graphical presentation
we can represent all the constraints,
the objective function, and the three
types of feasible points.
25
X2
1200
The plastic constraint:
The
Plastic constraint
2X1+X2<=1200
Total production constraint:
X1+X2<=800
Infeasible
600
Production
Feasible
Time
3X1+4X2<=2400
Production mix
constraint:
X1-X2<=450
600
800
Interior points.
Boundary
points.
• There are three
types
of feasible points
Extreme points.
X1
26
Linear Programming- Simplex method
• “...finding the maximum or minimum of linear
functions in which many variables are subject
to constraints.” (dictionary.com)
• A linear program is a “problem that requires
the minimization of a linear form subject to
linear constraints...” (Dantzig vii)
Important Note
• Linear programming requires linear
inequalities
• In other words, first degree inequalities only!
Good: ax + by + cz < 3
Bad: ax2 + log2y > 7
Lets look at an example...
•
•
•
•
•
•
•
•
Farm that produces Apples (x) and Oranges (y)
Each crop needs land, fertilizer, and time.
6 acres of land: 3x + y < 6
6 tons of fertilizer: 2x + 3y < 6
8 hour work day: x + 5y < 8
Apples sell for twice as much as oranges
We want to maximize profit (z): 2x + y = z
We can't produce negative: x > 0, y > 0
Traditional Method
x = 1.71
y = .86
z = 4.29
• Graph the inequalities
• Look at the line we're trying to maximize.
Problems...
• More variables?
• Cannot eyeball the answer?
Simplex Method
• George B. Dantzig in 1951
• Need to convert equations
• Slack variables
Performing the Conversion
• -z
+ 2x + y = 0 (Objective Equation)
• s1 + x + 5y = 8
•
s2 + 2x + 3y = 6
•
s3 + 3x + y = 6
• Initial feasible solution
More definitions
•
•
•
•
•
Non-basic: x, y
Basic variables: s1, s2, s3, z
Current Solution: Set non-basic variables to 0
-z + 2x + y = 0 => z = 0
Valid, but not good!
Next step...
• Select a non-basic variable
– -z + 2x + 1y = 0
– x has the higher coefficient
• Select a basic variable
– s1 + 1x + 5y = 8 1/8
– s2 + 2x + 3y = 6 2/6
– s3 + 3x + y = 6
3/6
• 3/6 is the highest, use equation with s3
New set of equations
• Solve for x
– x = 2 - (1/3)s3 -(1/3)y
• Substitute in to other equations to get...
– -z – (2/3)s3 +(1/3)y = -4
– s1– (1/3)s3 + (14/3)y = 6
– s2 – (2/3)s3 +(7/3)y = 2
– x + (1/3)s3 +(1/3)y = 2
Redefine everything...
•
•
•
•
Update variables
Non-Basic: s3 and y
Basic: s1, s2, z, and x
Current Solution:
– -z – (2/3)s3 +(1/3)y = -4 => z = 4
– x + (1/3)s3 +(1/3)y = 2 => x = 2
–y=0
• Better, but not quite there.
Do it again!
• Repeat this process
• Stop repeating when the coefficients in the
objective equation are all negative.
Improvements
• Different kinds of inequalities
• Minimized instead of maximized
• L. G. Kachian algorithm proved polynomial
Artificial Variable Technique
(The Big-M Method)
Big-M Method of solving LPP
The Big-M method of handling instances with artificial
variables is the “commonsense approach”. Essentially, the
notion is to make the artificial variables, through their
coefficients in the objective function, so costly or unprofitable
that any feasible solution to the real problem would be
preferred....unless the original instance possessed no feasible
solutions at all. But this means that we need to assign, in the
objective function, coefficients to the artificial variables that are
either very small (maximization problem) or very large
(minimization problem); whatever this value,let us call it Big M.
In fact, this notion is an old trick in optimization in general; we
simply associate a penalty value with variables that we do not
want to be part of an ultimate solution(unless such an outcome
Is unavoidable).
Indeed, the penalty is so costly that unless any of the
respective variables' inclusion is warranted algorithmically,
such variables will never be part of any feasible solution.
This method removes artificial variables from the basis. Here,
we assign a large undesirable (unacceptable penalty) coefficients to
artificial variables from the objective function point of view. If the
objective function (Z) is to be minimized, then a very large positive
price (penalty, M) is assigned to each artificial variable and if Z is to
be minimized, then a very large negative price is to be assigned. The
penalty will be designated by +M for minimization problem and by –
M for a maximization problem and also M>0.
Example: Minimize Z= 600X1+500X2
subject to constraints,
2X1+ X2 >or= 80
X1+2X2 >or= 60 and X1,X2 >or= 0
Step1: Convert the LP problem into a system of linear
equations.
We do this by rewriting the constraint inequalities as equations by
subtracting new “surplus & artificial variables" and assigning them
zero & +M coefficientsrespectively in the objective function as
shown below.
So the Objective Function would be:
Z=600X1+500X2+0.S1+0.S2+MA1+MA2
subject to constraints,
2X1+ X2-S1+A1 = 80
X1+2X2-S2+A2 = 60
X1,X2,S1,S2,A1,A2 >or= 0
Step 2: Obtain a Basic Solution to the problem.
We do this by putting the decision variables X1=X2=S1=S2=0,
so that A1= 80 and A2=60.
These are the initial values of artificial variables.
Step 3: Form the Initial Tableau as shown.
Cj
Basic
Basic
CB Variab
Soln(XB)
le (B)
M
M
A1
A2
80
60
600
X1
500
X2
2
1
1
2
Zj 3M
3M
Cj - Zj 600-3M 500-3M
0
0
M
M
Min.Ratio
(XB/Pivotal
Col.)
S1
S2
A1
A2
-1
0
M
M
0
-1
M
M
1
0
M
0
0 80
1 60
M
0
It is clear from the tableau that X2 will enter and A2 will leave the
basis. Hence 2 is the key element in pivotal column. Now,the new
row operations are as follows:
R2(New) = R2(Old)/2
R1(New) = R1(Old) - 1*R2(New)
Cj
600
500
0
0
M
Basic
Basic
CB Variab
Soln(XB)
le (B)
X1
X2
S1
S2
A1
Min.Ratio
(XB/Pivota
l Col.)
M
500
3 2
1 2
0
1
-1
0
1 2
- 1/2
1
0
100/3
60
500
M
M/2-250
M
0
M
250-M/2
0
A1
X2
50
30
Zj 3M/2+250
Cj - Zj 350-3M/2
It is clear from the tableau that X1 will enter and A1 will leave the
basis. Hence 2 is the key element in pivotal column. Now,the new
row operations are as follows:
R1(New) = R1(Old)*2/3
R2(New) = R2(Old) – (1/2)*R1(New)
Cj
CB
600
500
Basic
Varia Basic
ble Soln(XB)
(B)
X1
X2
100/3
40/3
Zj
Cj - Zj
600
500
0
0
X1
X2
S1
S2
1
0
600
0
1
500
0
2 3
1 3
700 3
700 3
1 3
2 3
400 3
400 3
0
Min.
Ratio
(XB/P
ivotal
Col.)
Since all the values of (Cj-Zj) are either zero or positive and also
both the artificial variables have been removed, an optimum
solution has been arrived at with X1=100/3 , X2=40/3 and
Z=80,000/3.
Unit II
Dynamic Programming
Characteristics and Examples
48
Overview
• What is dynamic programming?
• Examples
• Applications
49
What is Dynamic Programming?
• Design technique
– ‘optimization’ problems (sequence of related decisions)
– Programming does not mean ‘coding’ in this context, it
means ‘solve by making a chart’- or ‘using an array to save
intermediate steps”. Some books call this ‘memoization’
(see below)
– Similar to Divide and Conquer BUT subproblem solutions
are SAVED and NEVER recomputed
– Principal of optimality: the optimal solution to the
problem contains optimal solutions to the subproblems (Is
this true for EVERYTHING?)
50
Characteristics
• Optimal substructure
– Unweighted shortest path?
– Unweighted longest simple path?
• Overlapping Subproblems
– What happens in recursion (D&C) when this happens?
• Memoization (not a typo!)
– Saving solutions of subproblems (like we did in Fibonacci)
to avoid recomputation
51
Examples
• Matrix chain
• Longest common subsequence (called the LCS
problem”
52
Review of Technique
• You have already applied dynamic
programming and understand it why it may
result in a good algorithm
– Fibonacci
– Ackermann
– Combinations
53
Principal of Optimality
• Often called the “optimality condition”
• What it means in plain English: you apply the divide
and conquer technique so the subproblems are
SMALLER VERSIONS OF THE ORIGINAL PROBLEM: if
you solve “optimize” the answer to the small
problems , does that fact automatically mean that
the solution to the big problem is also
optimized????? If the answer is yes , then DP applies
to his problem
54
Example 1
• Assume your problem is to draw a straight line
between two points A and B. You solve this by
divide and conquer by drawing a line from A
to the midpoint and from the midpoint to B.
QUESTION: if you paste the two smaller lines
together will the RESULTING LINE FROM A TO
B BE THE SHORTEST DISTANCE FROM A to
B???
55
Example 2
• Say you want to buy Halloween candy – 100
candy bars. You do plan to do this by divide
and conquer – buying 10 sets of 10 bars. Is
this necessarily less expensive per bar than
just buying 2 packages of 50? Or perhaps 1
package of 100?
56
How To Apply DP
• There are TWO ways to apply dynamic
programming
– METHOD 1: solve the problem at hand recursively,
notice where the same subproblem is being ‘resolved’ and implement the algorithm as a TABLE
(example: fibonacci)
– METHOD 2: generate all feasible solutions to a
problem but prune (eliminate) the solutions that
cannot be optimal (example: shortest path)
57
Practice Is the Only Way to Learn This
Technique
• See class webpage for homework.
• Do problems in textbook not assigned but for
practice – even after term ends. It took me
two years after I took this course before I
could apply DP in the real world.
58
Dynamic programming
Design technique, like divide-and-conquer.
Example: Longest Common Subsequence (LCS)
• Given two sequences x[1 . . m] and y[1 . . n], find
a longest subsequence common to them both.
“a” not “the”
x: A B
C
B
D A B
BCBA =
LCS(x, y)
y: B
D C
A B
A
functional notation,
but not a function
59
Review: Dynamic programming
• DP is a method for solving certain kind of
problems
• DP can be applied when the solution of a
problem includes solutions to subproblems
• We need to find a recursive formula for the
solution
• We can recursively solve subproblems,
starting from the trivial case, and save their
solutions in memory
• In the end we’ll get the solution of the
whole problem
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Properties of a problem that can be
solved with dynamic programming
• Simple Subproblems
– We should be able to break the original problem
to smaller subproblems that have the same
structure
• Optimal Substructure of the problems
– The solution to the problem must be a
composition of subproblem solutions
• Subproblem Overlap
– Optimal subproblems to unrelated problems can
contain subproblems in common
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Review: Longest Common
Subsequence (LCS)
• Problem: how to find the longest pattern of
characters that is common to two text
strings X and Y
• Dynamic programming algorithm: solve
subproblems until we get the final solution
• Subproblem: first find the LCS of prefixes
of X and Y.
• this problem has optimal substructure: LCS
of two prefixes is always a part of LCS of
bigger strings
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Review: Longest Common
Subsequence (LCS) continued
• Define Xi, Yj to be prefixes of X and Y of
length i and j; m = |X|, n = |Y|
• We store the length of LCS(Xi, Yj) in c[i,j]
• Trivial cases: LCS(X0 , Yj ) and LCS(Xi,
Y0) is empty (so c[0,j] = c[i,0] = 0 )
• Recursive formula for c[i,j]:
if x[i]  y[ j ],
c[i  1, j  1]  1
c[i, j ]  
 max(c[i, j  1], c[i  1, j ]) otherwise
c[m,n] is the final solution
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Review: Longest Common
Subsequence (LCS)
• After we have filled the array c[ ], we can
use this data to find the characters that
constitute the Longest Common
Subsequence
• Algorithm runs in O(m*n), which is much
better than the brute-force algorithm: O(n
2m)
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0-1 Knapsack problem
• Given a knapsack with maximum capacity W,
and a set S consisting of n items
• Each item i has some weight wi and benefit
value bi (all wi , bi and W are integer values)
• Problem: How to pack the knapsack to achieve
maximum total value of packed items?
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0-1 Knapsack problem:
a picture
Weight
Items
This is a knapsack
Max weight: W = 20
W = 20
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Benefit value
wi
bi
2
3
3
4
5
5
8
9
10
4
66
0-1 Knapsack problem
• Problem, in other words, is to find
max bi subject to wi  W
iT


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iT
The problem is called a “0-1” problem,
because each item must be entirely
accepted or rejected.
Just another version of this problem is the
“Fractional Knapsack Problem”, where we
can take fractions of items.
67
0-1 Knapsack problem: brute-force
approach
Let’s first solve this problem with a
straightforward algorithm
• Since there are n items, there are 2n possible
combinations of items.
• We go through all combinations and find
the one with the most total value and with
total weight less or equal to W
• Running time will be O(2n)
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0-1 Knapsack problem: brute-force
approach
• Can we do better?
• Yes, with an algorithm based on dynamic
programming
• We need to carefully identify the
subproblems
Let’s try this:
If items are labeled 1..n, then a subproblem
would be to find an optimal solution for
Sk = {items labeled 1, 2, .. k}
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Defining a Subproblem
If items are labeled 1..n, then a subproblem
would be to find an optimal solution for Sk
= {items labeled 1, 2, .. k}
• This is a valid subproblem definition.
• The question is: can we describe the final
solution (Sn ) in terms of subproblems (Sk)?
• Unfortunately, we can’t do that. Explanation
follows….
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Defining a Subproblem
w1 =2 w2
b1 =3 =4
b2 =5
w3 =5
b3 =8
w3 =5
b3 =8
w4 =9
b4 =10
For S5:
Total weight: 20
total benefit: 26
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wi
bi
1
2
3
2
3
4
3
4
5
4
5
8
5
9
10
Item
#
?
Max weight: W = 20
For S4:
Total weight: 14;
total benefit: 20
w1 =2 w2
b1 =3 =4
b2 =5
Weight Benefit
w4 =3
b4 =4
S4
S5
Solution for S4 is
not part of the
solution for S5!!!
71
Defining a Subproblem (continued)
• As we have seen, the solution for S4 is not
part of the solution for S5
• So our definition of a subproblem is flawed
and we need another one!
• Let’s add another parameter: w, which will
represent the exact weight for each subset
of items
• The subproblem then will be to compute
B[k,w]
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Recursive Formula for subproblems

Recursive formula for subproblems:
B[k  1, w]
if wk  w

B[k , w]  
max{B[k  1, w], B[k  1, w  wk ]  bk } else
• It means, that the best subset of Sk that has
total weight w is one of the two:
1) the best subset of Sk-1 that has total weight
w, or
2) the best subset of Sk-1 that has total weight
w-wk plus the item k
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Recursive Formula
B[k  1, w]
if wk  w

B[k , w]  
max{B[k  1, w], B[k  1, w  wk ]  bk } else
• The best subset of Sk that has the total
weight w, either contains item k or not.
• First case: wk>w. Item k can’t be part of the
solution, since if it was, the total weight
would be > w, which is unacceptable
• Second case: wk <=w. Then the item k can
be in the solution, and we choose the case
with greater value
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0-1 Knapsack Algorithm
for w = 0 to W
B[0,w] = 0
for i = 0 to n
B[i,0] = 0
for w = 0 to W
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Running time
for w = 0 to W
O(W)
B[0,w] = 0
for i = 0 to n
Repeat n times
B[i,0] = 0
for w = 0 to W
O(W)
< the rest of the code >
What is the running time of this algorithm?
O(n*W)
Remember that the brute-force algorithm
takes O(2n)
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Example
Let’s run our algorithm on the
following data:
n = 4 (# of elements)
W = 5 (max weight)
Elements (weight, benefit):
(2,3), (3,4), (4,5), (5,6)
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Example (2)
i
W
0
0
0
1
2
3
4
0
0
0
0
5
0
1
2
3
4
for w = 0 to W
B[0,w] = 0
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Example (3)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
5
0
W
for i = 0 to n
B[i,0] = 0
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Example (4)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
5
0
W
i=1
bi=3
wi=2
w=1
w-wi =-1
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Example (5)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
5
0
W
i=1
bi=3
wi=2
w=2
w-wi =0
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Example (6)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
5
0
W
i=1
bi=3
wi=2
w=3
w-wi=1
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Example (7)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
5
0
W
i=1
bi=3
wi=2
w=4
w-wi=2
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Example (8)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
5
0
3
W
i=1
bi=3
wi=2
w=5
w-wi=2
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Example (9)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
5
0
3
W
i=2
bi=4
wi=3
w=1
w-wi=-2
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Example (10)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
3
5
0
3
W
i=2
bi=4
wi=3
w=2
w-wi=-1
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Example (11)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
3
4
5
0
3
W
i=2
bi=4
wi=3
w=3
w-wi=0
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Example (12)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
3
4
4
5
0
3
W
i=2
bi=4
wi=3
w=4
w-wi=1
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
4/9/2015
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Example (13)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
3
4
4
5
0
3
7
W
i=2
bi=4
wi=3
w=5
w-wi=2
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
4/9/2015
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Example (14)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
3
4
4
0
3
4
5
0
3
7
W
i=3
bi=5
wi=4
w=1..3
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
4/9/2015
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Example (15)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
3
4
4
0
3
4
5
5
0
3
7
W
i=3
bi=5
wi=4
w=4
w- wi=0
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
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Example (15)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
3
4
4
0
3
4
5
5
0
3
7
7
W
i=3
bi=5
wi=4
w=5
w- wi=1
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
4/9/2015
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Example (16)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
3
4
4
0
3
4
5
0
3
4
5
5
0
3
7
7
W
i=3
bi=5
wi=4
w=1..4
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
4/9/2015
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Example (17)
i
0
1
2
3
4
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
3
3
3
0
3
4
4
0
3
4
5
0
3
4
5
5
0
3
7
7
7
W
i=3
bi=5
wi=4
w=5
Items:
1: (2,3)
2: (3,4)
3: (4,5)
4: (5,6)
if wi <= w // item i can be part of the solution
if bi + B[i-1,w-wi] > B[i-1,w]
B[i,w] = bi + B[i-1,w- wi]
else
B[i,w] = B[i-1,w]
else B[i,w] = B[i-1,w] // wi > w
4/9/2015
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Comments
• This algorithm only finds the max possible
value that can be carried in the knapsack
• To know the items that make this maximum
value, an addition to this algorithm is
necessary
• Please see LCS algorithm from the previous
lecture for the example how to extract this
data from the table we built
4/9/2015
95
Conclusion
• Dynamic programming is a useful technique
of solving certain kind of problems
• When the solution can be recursively
described in terms of partial solutions, we
can store these partial solutions and re-use
them as necessary
• Running time (Dynamic Programming
algorithm vs. naïve algorithm):
– LCS: O(m*n) vs. O(n * 2m)
– 0-1 Knapsack problem: O(W*n) vs. O(2n)
4/9/2015
96
Unit III
Network Models
Chapter Outline
12.1 Introduction
12.2 Minimal-Spanning Tree Technique
12.3 Maximal-Flow Technique
12.4 Shortest-Route technique
Learning Objectives
Students will be able to
– Connect all points of a network while minimizing
total distance using the minimal-spanning tree
technique.
– Determine the maximum flow through a network
using the maximal-flow technique.
– Find the shortest path through a network using
the shortest-route technique.
– Understand the important role of software in
solving network problems.
Minimal-Spanning Tree Technique
• Determines the path through the network
that connects all the points while minimizing
total distance.
Minimal-Spanning Tree
Steps
1. Select any node in the network.
2. Connect this node to the nearest node that minimizes the
total distance.
3. Considering all of the nodes that are now connected, find
and connect the nearest node that is not connected.
4. Repeat the third step until all nodes are connected.
5. If there is a tie in the third step and two or more nodes that
are not connected are equally near, arbitrarily select one and
continue. A tie suggests that there might be more than one
optimal solution.
Minimal-Spanning Tree
Lauderdale Construction
2
3
1
3
5
3
2
5
2
8
3
6
2
4
7
7
3
5
4
6
1
Minimal-Spanning Tree
Iterations 1&2
2
3
1
3
5
3
2
5
Second Iteration
First Iteration
2
8
3
6
2
4
7
7
3
5
4
6
1
2
3
1
3
5
5
3
2
2
8
3
6
2
4
7
7
3
5
4
6
1
Minimal-Spanning Tree
Iterations 3&4
2
3
1
3
5
3
2
5
Third Iteration
2
8
3
6
2
4
7
7
3
5
4
6
1
2
3
1
3
5
5
3
2
Fourth Iteration
2
8
3
6
2
4
7
7
3
5
4
6
1
Minimal-Spanning Tree
Iterations 4&5
2
3
1
3
5
5
3
2
2
8
3
6
2
4
Fourth Iteration
7
7
3
5
4
6
1
2
3
1
3
5
5
3
2
Fifth Iteration
2
8
3
6
2
4
7
7
3
5
4
6
1
Minimal-Spanning Tree
Iterations 6&7
2
3
1
3
5
3
2
5
Sixth iteration
2
8
3
6
2
4
7
7
3
5
4
6
1
Seventh & final
iteration
Minimum Distance: 16
2
3
1
3
5
5
3
2
2
8
3
6
2
4
7
7
3
5
4
6
1
The Maximal-Flow Technique
1. Pick any path (streets from west to east) with
some flow.
2. Increase the flow (number of cars) as much as
possible.
3. Adjust the flow capacity numbers on the path
(streets).
4. Repeat the above steps until an increase in
flow is no longer possible.
Maximal-Flow
Road Network for Waukesha
2
1 2
1
2
3
West
Point
1
6
1
2
0
0 1 1
10
1
4
6
0
3
3 2
5
1
East
Point
Maximal-Flow
Road Network for Waukesha
Add 2
2
1 2
1
Subtract 2
3
West
Point
1
2
2
6
1
0
0 1 1
10
1
4
6
0
3
3 2
5
1
East
Point
Road Network for Waukesha
First Iteration
Add 1
0
3 2
1
4
1
West
Point
1
6
1
2
0
0 1 1
10
1
4
Subtract 1
6
0
5
3
1
3 2
12-110
East
Point
Road Network for Waukesha
Second Iteration
0
4 2
0
4
0
West
Point
6
2
1 2
0
0 2 0
10
1
0
4
Subtract 2
3
3 2
6
1
Add 2
5
East
Point
Road Network for Waukesha
Third Iteration
0
4 2
0
West
Point
4
0
1
6
2
2
2
0 2 0
8
1
East
Point
4
Path
4
2
3
3 0
5
3
1-2-6
1-2-4-6
1-3-5-6
Total
Flow (Cars
Per Hour)
200
100
200
500
The Shortest-Route Technique
• 1. Find the nearest node to the origin (plant). Put the
distance in a box by the node.
– In some cases, several paths will have to be checked to
find the nearest node.
• 2. Repeat this process until you have gone through the entire
network. The last distance at the ending node will be the
distance of the shortest route. You should note that the
distances placed in the boxes by each node are the shortest
route to this node. These distances are used as intermediate
results in finding the next nearest node.
Shortest-Route Problem
Ray Design, Inc.
Roads from Ray’s Plant to the
Warehouse
4
2
200
100
100
Plant
1
50
100
150
200
6
100
3
40
5
Warehouse
Ray Design, Inc.
First Iteration
100
4
2
200
100
100
Plant
1
50
100
150
200
6
100
3
40
5
Warehouse
Ray Design, Inc.
Second Iteration
100
4
2
200
100
100
Plant
1
50
100
150
200
6
100
3
150
40
5
Warehouse
Ray Design, Inc.
Third Iteration
100
4
2
200
100
100
Plant
1
50
100
150
200
6
100
3
150
40
5
190
Warehouse
Ray Design, Inc.
Fourth Iteration
100
4
2
200
100
100
290
Plant
1
50
100
150
200
6
100
3
150
40
5
190
Warehouse
Project Management - CPM/PERT
Project Scheduling and Control Techniques
Gantt Chart
Critical Path Method (CPM)
Program Evaluation and Review Technique (PERT)
120
History of CPM/PERT
• Critical Path Method (CPM)
– E I Du Pont de Nemours & Co. (1957) for construction of new
chemical plant and maintenance shut-down
– Deterministic task times
– Activity-on-node network construction
– Repetitive nature of jobs
• Project Evaluation and Review Technique (PERT)
–
–
–
–
U S Navy (1958) for the POLARIS missile program
Multiple task time estimates (probabilistic nature)
Activity-on-arrow network construction
Non-repetitive jobs (R & D work)
121
Project Network
• Network analysis is the general name given to certain specific
techniques which can be used for the planning, management and
control of projects
• Use of nodes and arrows
Arrows
 An arrow leads from tail to head directionally
– Indicate ACTIVITY, a time consuming effort that is required to perform a
part of the work.
Nodes
 A node is represented by a circle
- Indicate EVENT, a point in time where one or more activities start and/or
finish.
• Activity
– A task or a certain amount of work required in the project
– Requires time to complete
– Represented by an arrow
• Dummy Activity
– Indicates only precedence relationships
– Does not require any time of effort
122
CPM calculation
• Path
– A connected sequence of activities leading from
the starting event to the ending event
• Critical Path
– The longest path (time); determines the project
duration
• Critical Activities
– All of the activities that make up the critical path
123
Forward Pass
• Earliest Start Time (ES)
– earliest time an activity can start
– ES = maximum EF of immediate predecessors
• Earliest finish time (EF)
– earliest time an activity can finish
– earliest start time plus activity time
EF= ES + t
Backward Pass
Latest Start Time (LS)
Latest time an activity can start without delaying critical path
time
LS= LF - t
Latest finish time (LF)
latest time an activity can be completed without delaying
critical path time
LS = minimum LS of immediate predecessors
124
CPM analysis
• Draw the CPM network
• Analyze the paths through the network
• Determine the float for each activity
– Compute the activity’s float
float = LS - ES = LF - EF
– Float is the maximum amount of time that this activity can
be delay in its completion before it becomes a critical
activity, i.e., delays completion of the project
• Find the critical path is that the sequence of activities and
events where there is no “slack” i.e.. Zero slack
– Longest path through a network
• Find the project duration is minimum project completion time
125
Consider below table summarizing the details of a project
involving 10 activities
Activity
Immediate precedence duration
a
6
b
8
c
5
d
b
13
e
c
9
f
a
15
g
a
17
h
f
9
i
g
6
j
d,e
12
Construct the CPM network. Determine the critical path and project
completion time .Also compute total float and free floats for the126non-
CPM Example:
• CPM Network
f, 15
h, 9
g, 17
a, 6
i, 6
b, 8
d, 13
j, 12
c, 5
e, 9
127
CPM Example
• ES and EF Times
f, 15
h, 9
g, 17
a, 6
0 6
i, 6
b, 8
0 8
d, 13
j, 12
c, 5
0 5
e, 9
128
CPM Example
• ES and EF Times
f, 15
6 21
h, 9
g, 17
a, 6
0 6
6 23
i, 6
b, 8
0 8
c, 5
0 5
d, 13
j, 12
8 21
e, 9
5 14
129
CPM Example
• ES and EF Times
f, 15
6 21
g, 17
a, 6
0 6
6 23
i, 6
23 29
h, 9
21 30
b, 8
0 8
c, 5
0 5
d, 13
8 21
e, 9
5 14
j, 12
21 33
Project’s EF = 33
130
CPM Example
• LS and LF Times
a, 6
0 6
b, 8
0 8
c, 5
0 5
f, 15
6 21
g, 17
6 23
d, 13
8 21
i, 6
23 29
27 33
h, 9
21 30
24 33
j, 12
21 33
21 33
e, 9
5 14
131
CPM Example
• LS and LF Times
a, 6
0 6
4 10
b, 8
0 8
0 8
c, 5
0 5
7 12
f, 15
6 21
18 24
g, 17
6 23
10 27
d, 13
8 21
8 21
e, 9
5 14
12 21
i, 6
23 29
27 33
h, 9
21 30
24 33
j, 12
21 33
21 33
132
CPM Example
• Float
a, 6
3 0 6
3 9
b, 8
0 0 8
0 8
c, 5
7 0 5
7 12
f, 15
3 6 21
9 24
g, 17
4 6 23
10 27
d, 13
0 8 21
8 21
h, 9
3 21 30
24 33
i, 6
4 23 29
27 33
j, 12
0 21 33
21 33
e, 9
7 5 14
12 21
133
CPM Example
• Critical Path
f, 15
h, 9
g, 17
a, 6
i, 6
b, 8
d, 13
j, 12
c, 5
e, 9
134
PERT
• PERT is based on the assumption that an activity’s duration
follows a probability distribution instead of being a single value
• Three time estimates are required to compute the parameters
of an activity’s duration distribution:
– pessimistic time (tp ) - the time the activity would take if
things did not go well
– most likely time (tm ) - the consensus best estimate of the
activity’s duration
– optimistic time (to ) - the time the activity would take if
things did go well
Mean (expected time):
te =
tp + 4 tm + to
6
2
Variance: Vt = =
2
darla/smbs/vit
tp - to
6
135
PERT analysis
• Draw the network.
• Analyze the paths through the network and find the critical path.
• The length of the critical path is the mean of the project duration
probability distribution which is assumed to be normal
• The standard deviation of the project duration probability
distribution is computed by adding the variances of the critical
activities (all of the activities that make up the critical path) and
taking the square root of that sum
• Probability computations can now be made using the normal
distribution table.
136
Probability computation
Determine probability that project is completed within specified time
x-
Z=

where  = tp = project mean time
 = project standard mean time
x = (proposed ) specified time
137
Normal Distribution of Project Time
Probability
Z
 = tp
x
Time
138
PERT Example
Immed. Optimistic Most Likely Pessimistic
Activity Predec. Time (Hr.) Time (Hr.) Time (Hr.)
A
-4
6
8
B
-1
4.5
5
C
A
3
3
3
D
A
4
5
6
E
A
0.5
1
1.5
F
B,C
3
4
5
G
B,C
1
1.5
5
H
E,F
5
6
7
I
E,F
2
5
8
J
D,H
2.5
2.75
4.5
K
G,I
3
5
7
139
PERT Example
PERT Network
D
A
E
H
J
C
B
I
F
K
G
140
PERT Example
Activity
A
B
C
D
E
F
G
H
I
J
K
Expected Time
6
4
3
5
1
4
2
6
5
3
5
Variance
4/9
4/9
0
1/9
1/36
1/9
4/9
1/9
1
1/9
4/9
141
PERT Example
Activity ES
A
B
C
D
E
0
0
6
6
6
F
G
H
I
J
K
9
9
13
13
19
18
EF
6
4
9
11
7
13
11
19
18
22
23
LS
LF
0
5
6
15
12
9
16
14
13
20
18
Slack
6
9
9
20
13
13
18
20
18
23
23
0 *critical
5
0*
9
6
0*
7
1
0*
1
0*
142
PERT Example
Vpath = VA + VC + VF + VI + VK
= 4/9 + 0 + 1/9 + 1 + 4/9
= 2
path = 1.414
z = (24 - 23)/(24-23)/1.414 = .71
From the Standard Normal Distribution table:
P(z < .71) = .5 + .2612 = .7612
143
PROJECT COST
Cost consideration in project
• Project managers may have the option or requirement to crash
the project, or accelerate the completion of the project.
• This is accomplished by reducing the length of the critical path(s).
• The length of the critical path is reduced by reducing the
duration of the activities on the critical path.
• If each activity requires the expenditure of an amount of money
to reduce its duration by one unit of time, then the project
manager selects the least cost critical activity, reduces it by one
time unit, and traces that change through the remainder of the
network.
• As a result of a reduction in an activity’s time, a new critical path
may be created.
• When there is more than one critical path, each of the critical
paths must be reduced.
• If the length of the project needs to be reduced further, the
process is repeated.
145
Project Crashing
• Crashing
– reducing project time by expending additional resources
• Crash time
– an amount of time an activity is reduced
• Crash cost
– cost of reducing activity time
• Goal
– reduce project duration at minimum cost
146
Time-Cost Relationship
 Crashing costs increase as project duration decreases
 Indirect costs increase as project duration increases
 Reduce project length as long as crashing costs are less than
indirect costs
Time-Cost Tradeoff
Min total cost =
optimal project
time
Total project cost
Indirect
cost
Direct cost
time
147
Benefits of CPM/PERT
•
•
•
•
•
Useful at many stages of project management
Mathematically simple
Give critical path and slack time
Provide project documentation
Useful in monitoring costs
CPM/PERT can answer the following important
questions:
•How long will the entire project take to be completed? What are the
risks involved?
•Which are the critical activities or tasks in the project which could
delay the entire project if they were not completed on time?
•Is the project on schedule, behind schedule or ahead of schedule?
•If the project has to be finished earlier than planned, what is the best
way to do this at the least cost?
148
Limitations to CPM/PERT
•
•
•
•
•
•
Clearly defined, independent and stable activities
Specified precedence relationships
Over emphasis on critical paths
Deterministic CPM model
Activity time estimates are subjective and depend on judgment
PERT assumes a beta distribution for these time estimates, but
the actual distribution may be different
• PERT consistently underestimates the expected project
completion time due to alternate paths becoming critical
To overcome the limitation, Monte Carlo simulations can be
performed on the network to eliminate the optimistic bias
149
PERT vs
CPM
CPM
PERT
CPM uses activity oriented network.
PERT uses event oriented Network.
Durations of activity may be estimated
with a fair degree of accuracy.
Estimate of time for activities are not so
accurate and definite.
It is used extensively in construction
projects.
It is used mostly in research and
development projects, particularly
projects of non-repetitive nature.
Deterministic concept is used.
Probabilistic model concept is used.
CPM can control both time and cost
when planning.
PERT is basically a tool for planning.
In CPM, cost optimization is given prime
importance. The time for the completion
of the project depends upon cost
optimization. The cost is not directly
proportioned to time. Thus, cost is the
In PERT, it is assumed that cost varies
directly with time. Attention is therefore
given to minimize the time so that
minimum cost results. Thus in PERT, time
150
is the controlling factor.
Unit IV
Inventory Management
The objective of inventory
management is to strike a balance
between inventory investment and
customer service
Inventory control
It means stocking adequate number and kind of
stores, so that the materials are available
whenever required and wherever required.
Scientific inventory control results in optimal
balance
What is inventory?
Inventory is the raw materials, component
parts, work-in-process, or finished products
that are held at a location in the supply chain.
Input
Material
Management
department
Inventory
(money)
Goods in stores
Work-in-progress
Finished products
Equipment etc.
Basic inventory model
Output
Production
department
Zero Inventory?
• Reducing amounts of raw materials and purchased
parts and subassemblies by having suppliers deliver
them directly.
• Reducing the amount of works-in process by using
just-in-time production.
• Reducing the amount of finished goods by shipping
to markets as soon as possible.
Importance of Inventory
 One of the most expensive assets of
many companies representing as much
as 50% of total invested capital
 Operations managers must balance
inventory investment and customer
service
FUNCTIONS OF
INVENTORY
• To meet anticipated demand.
• To smoothen production requirements.
• To decouple operations.
INVENTORY
SUPPLY
PROCESS
PRODUCTS
DEMAND
PRODUCTS
DEMAND
DEMAND
PROCESS
Functions Of Inventory
(Cont’d)
• To protect against stock-outs.
• To take advantage of order cycles.
• To help hedge against price increases.
• To permit operations.
• To take advantage of quantity discounts.
Types of Inventory
 Raw material

Purchased but not processed
 Work-in-process

Undergone some change but not completed

A function of cycle time for a product
 Maintenance/repair/operating (MRO)

Necessary to keep machinery and processes productive
 Finished goods

Completed product awaiting shipment
The Material Flow Cycle
Cycle time
95%
Input
Wait for
inspection
Wait to
be moved
Move Wait in queue Setup
time
for operator
time
5%
Run
time
Output
Safety stock =
(safety factor z)(std deviation in LT demand)
Service level
Probability
of stock-out
Safety
Stock
Read z from Normal table for a given service level
Average Inventory =
(Order Qty)/2 + Safety Stock
Inventory
Level
Order
Quantity
EOQ/2
Safety Stock (SS)
Lead Time
Place
order
Receive
order
Time
Average
Inventory
Managing Inventory
1. How inventory items can be classified
2. How accurate inventory records can be
maintained
Inventory Models for
Independent Demand
Need to determine when and how
much to order
1. Basic economic order quantity
2. Production order quantity
3. Quantity discount model
Economic order of quantity
EOQ = Average Monthly Consumption X Lead Time [in
months] + Buffer Stock – Stock on hand
ECONOMIC ORDER OF
QUANTITY(EOQ)
PURCHASING
COST
CARRYING
COST
•Re-order level: stock level at which fresh order
is placed.
•Average consumption per day x lead time +
buffer stock
•Lead time: Duration time between placing an
order & receipt of material
•Ideal – 2 to 6 weeks.
Basic EOQ Model
Important assumptions
1. Demand is known, constant, and independent
2. Lead time is known and constant
3. Receipt of inventory is instantaneous and
complete
4. Quantity discounts are not possible
5. Only variable costs are setup and holding
6. Stockouts can be completely avoided
An EOQ Example
Determine optimal number of needles to order
D = 1,000 units
S = $10 per order
H = $.50 per unit per year
Q* =
2DS
H
Q* =
2(1,000)(10)
=
0.50
40,000 = 200 units
An EOQ Example
Determine optimal number of needles to order
D = 1,000 units
Q* = 200 units
S = $10 per order
H = $.50 per unit per year
Expected
Demand
D
number of = N =
=
Order
quantity
Q*
orders
1,000
N=
= 5 orders per year
200
An EOQ Example
Determine optimal number of needles to order
D = 1,000 units
Q* = 200 units
S = $10 per order
N = 5 orders per year
H = $.50 per unit per year
Number of working
Expected time
days per year
between orders = T =
N
T=
250
= 50 days between orders
5
Classification of Materials for
Inventory Control
Classification
Criteria
A-B-C
Annual value of consumption of the items
V-E-D
Critical nature of the components with respect to
products.
H-M-L
Unit price of material
F-S-N
Issue from stores
S-D-E
Purchasing problems in regard to availability
S-O-S
Seasonality
G-O-L-F
X-Y-Z
Channel for procuring the material
Inventory value of items stored
Relevant Inventory Costs
 Holding costs - the costs of holding
or “carrying” inventory over time
 Ordering costs - the costs of
placing an order and receiving
goods
 Setup costs - cost to prepare a
machine or process for
manufacturing an order
Ordering Costs
•
•
•
•
•
Stationary
Clerical and processing, salaries/rentals
Postage
Processing of bills
Staff work in expedition /receiving/
inspection and documentation
Holding/Carrying Costs
•
•
•
•
•
•
•
•
Storage space (rent/depreciation)
Property tax on warehousing
Insurance
Deterioration/Obsolescence
Material handling and maintenance, equipment
Stock taking, security and documentation
Capital blocked (interest/opportunity cost)
Quality control
Stock out Costs
• Loss of business/ profit/ market/ advise
• Additional expenditure due to urgency of
purchases
a) telegraph / telephone charges
b) purchase at premium
c) air transport charges
• Loss of labor hours
Balancing Carrying against
Ordering Costs
Higher
Annual Cost ($)
Minimum
Total Annual
Stocking Costs
Lower
Total Annual
Stocking Costs
Annual
Carrying Costs
Annual
Ordering Costs
Smaller
EOQ
Larger
Order Quantity
16
Unit V Queuing Theory
Outlines
Introduction
Single server model
Multi server model
Introduction
• Involves the mathematical study of queues or waiting line.
• The formulation of queues occur whenever the demand for a
service exceeds the capacity to provide that service.
• Decisions regarding the amount of capacity to provide must
be made frequently in industry and elsewhere.
• Queuing theory provides a means for decision makers to
study and analyze characteristics of the service facility for
making better decisions.
Basic structure of queuing model
• Customers requiring service are generated over time by an
input source.
• These customers enter the queuing system and join a queue.
• At certain times, a member of the queue is selected for
service by some rule know as the service disciple.
• The required service is then performed for the customer by
the service mechanism, after which the customer leaves the
queuing system
The basic queuing process
Queuing
system
Input
source
Customer
s
Queue
Service
mechanis
m
Served
Customer
s
Characteristics of queuing models
•
•
•
•
•
•
Input or arrival (interarrival) distribution
Output or departure (service) distribution
Service channels
Service discipline
Maximum number of customers allowed in the system
Calling source
Kendall and Lee’s Notation
Kendall and Lee introduced a useful notation representing the 6
basic characteristics of a queuing model.
Notation: a/b/c/d/e/f
where
a = arrival (or interarrival) distribution
b = departure (or service time) distribution
c = number of parallel service channels in the system
d = service disciple
e = maximum number allowed in the system (service + waiting)
f = calling source
Conventional Symbols for a, b
M = Poisson arrival or departure distribution (or equivalently
exponential distribution or service times distribution)
D = Deterministic interarrival or service times
Ek = Erlangian or gamma interarrival or service time distribution
with parameter k
GI = General independent distribution of arrivals (or interarrival
times)
G = General distribution of departures (or service times)
Conventional Symbols for d
•
•
•
•
FCFS = First come, first served
LCFS = Last come, first served
SIRO = Service in random order
GD = General service disciple
Transient and Steady States
Transient state
• The system is in this state when its operating characteristics
vary with time.
• Occurs at the early stages of the system’s operation where its
behavior is dependent on the initial conditions.
Steady state
• The system is in this state when the behavior of the system
becomes independent of time.
• Most attention in queuing theory analysis has been directed
to the steady state results.
Queuing Model Symbols
n = Number of customers in the system
s = Number of servers
pn(t) = Transient state probabilities of exactly n customers in the
system at time t
pn = Steady state probabilities of exactly n customers in the
system
λ = Mean arrival rate (number of customers arriving per unit
time)
μ = Mean service rate per busy server (number of customers
served per unit time)
Queuing Model Symbols (Cont’d)
ρ = λ/μ = Traffic intensity
W = Expected waiting time per customer in the system
Wq = Expected waiting time per customer in the queue
L = Expected number of customers in the system
Lq = Expected number of customers in the queue
Relationship Between L and W
If λn is a constant λ for all n, it can be shown that
L = λW
Lq = λ Wq
If λn are not constant then λ can be replaced in the above
equations by λbar,the average arrival rate over the long run.
If μn is a constant μ for all n, then
W = Wq + 1/μ
Relationship Between L and W (cont’d)
These relationships are important because:
• They enable all four of the fundamental quantities L, W, Lq
and Wq to be determined as long as one of them is found
analytically.
• The expected queue lengths are much easier to find than that
of expected waiting times when solving a queuing model from
basic principles.
Single server queuing models
• M/M/1/FCFS/∞/∞ Model
when the mean arrival rate λn and mean service μn are all constant we
have
n


cn       n , for n  1,2,...



T herefore
p n   n p0
, for n  1,2,...
where
1
p0 
1
 1 
  n
1

  1  









1  
1    n   n  0 
 n 1 
T hus
pn  1    n
, for n  1,2,...
Single server queuing models (cont’d)
Consequently
n
d

L   n(1   )  n  (1   )  
n 0
n  0 d


d   n
d  1 


 (1   ) 
     (1   ) 
d  n 0 
d  1   


1 


1 
Single server queuing models
(cont’d)
Similarly
2
Lq   (n  1) pn  L  1  p0 
 (   )
n 1

T heexpectedwaiting timeis
1
L
W 
  
Lq


Wq 
     
Multi server queuing models
• M/M/s/FCFS/∞/∞ Model
When the mean arrival rate λn and mean service μn,
are all constant, we have the following rate diagram
Multi server queuing models (cont’d)
In this case we have
n
n  1,2,  , s

n!
cn   n
 
n  s, s  1, 
 s! s n  s
T herefore
n
p0 n  1,2, , s where p0 

n
!

pn  
 n
  p n  s, s  1, 
 s! s n  s 0
1

n
 n!

s
s!1   s 
Multi server queuing models
(cont’d)
It follows t hat
 s 1
Lq 
( s  1)!( s   )
L  Lq  
Wq 
Lq

W  Wq 
1

2
p0
Thank You
198