Here is the solution to the assignment problem using linear programming: - Let B = number of boxes of blocks, T = number of trucks - Profit function: P = 30B + 40T - Constraints: - Woodworking hours: B + 3T ≤ 120 - Painting hours: 2B + T ≤ 80 - Graphically, the feasible region is bounded by the two constraint lines in the first quadrant - The optimal solution occurs at the point where one constraint is binding - in this case, the painting hours constraint is binding at B = 40, T = 40 - At this point, profit is maximized at P = 30(40) + 40(40) = $3,

1. © Department of Mechanical Engineering
MS-301EngineeringManagement
Engineering
Management
Lecture By
Prof. Dr. Naseer Ahmed
Email: naseer@cecos.edu.pk
Department of Mechanical Engineering
CECOS University

2. © Department of Mechanical Engineering
MS-301EngineeringManagement
Management Science
Model and Their analysis
• A model is an abstraction or simplification of
reality, designed to include only the essential
features that determine the behaviour of a real
system
• For example, a three dimensional physical model
of a chemical processing plant might include
scale models of major equipment and large
diameter pipes, but it would not normally include
small piping or electrical wiring
• Most of the models of management science are
mathematical models.
• These can be as simple as the common equation
representing the financial operation of a company

3. © Department of Mechanical Engineering
MS-301EngineeringManagement
Management Science
Model and Their analysis
• Net income = revenue – expenses – taxes
• On the other hand, they may involve a very
complex set of equations
• As an example, the Urban Dynamics model was
created by Jay Forrester to simulate the growth
and decay of cities
• This model consisted of 154 equations
representing relationships between the factors
that he believed were essential: three economic
classes of workers (managerial/professional,
skilled, and unemployed), three corresponding
classes of housing, three types of industry (new,
mature and declining), taxation and land use

4. © Department of Mechanical Engineering
MS-301EngineeringManagement
Management Science
Model and Their analysis
• The value of these factors evolved through 250
simulated years to model the changing
characteristics of a city
• Even these 154 relationships still proved too
simplistic to provide any reliable guide to urban
development policies
• Management sciences uses a five step process
that begins in the real world, moves into the
model world to solve the problem, then returns to
the real world for implementation

5. © Department of Mechanical Engineering
MS-301EngineeringManagement
Management Science
Model and Their analysis
Real World Simulated (model) world
Formulate the problem (defining
objectives, variables and
constraints)
Construct a mathematical model
(a simplified yet realistic
representation of the system)
Test the model’s ability to predict
the present from the past, and
revise until you are satisfied
Derive a solution from the model
Apply the model’s solution to the
real system, document its
effectiveness, and revise further
as required

6. © Department of Mechanical Engineering
MS-301EngineeringManagement
Management Science
The Analyst and the Manager
• To be effective, the management science analyst
cannot just create models in an “ivory tower”
• The Problem-solving team must include
managers and others from the department or
systems being studied – to establish objectives,
explain system operations, review the model as it
develops from an operative perspective, and help
test the model
• The user who has been part of model
development, has developed some
understanding of it and confidence in it, and feels
a sense of “ownership” of it is most likely to use it
effectively

7. © Department of Mechanical Engineering
MS-301EngineeringManagement
Management Science
The Analyst and the Manager
• The manager is not likely to have a detailed knowledge
of management science techniques, nor the time for
model development
• Today’s manager should, however, understand the
nature of management science tools and the types of
management situations in which they might be useful
• Increasingly, management positions are being filled with
graduates of management (or engineering management)
programs that have included an introduction to the
fundamentals of management science and statistics
• Regrettably, all too few operations research or
management science programs require the introduction
to organization and behavioural theory that would help
close the manager/analyst gap from the opposite
direction

8. © Department of Mechanical Engineering
MS-301EngineeringManagement
Management Science
The Analyst and the Manager
• There is considerable discussion today of the effect
of computers and their applications (management
science, decision support systems, expert systems,
etc.) on managers and organizations
• Certainly, workers and managers whose jobs are so
routine that their decisions can be reduced to
mathematical equations have reason to worry about
being replaced by computers
• For most managers, however, modern methods
offer the chance to reduce the time one must spend
on more trial matters, freeing up time for the types
of work and decisions that only people can
accomplish

9. © Department of Mechanical Engineering
MS-301EngineeringManagement
Tools for Decision Making
Categories of Decision Making
• Decision making can be discussed conveniently
in three categories: decision making under
certainty(confidence), under risk, and under
uncertainty
• The payoff table, or decision matrix will help in
this discussion

10. © Department of Mechanical Engineering
MS-301EngineeringManagement
Tools for Decision Making
Categories of Decision Making
𝑆𝑡𝑎𝑡𝑒 𝑜𝑓 𝑁𝑎𝑡𝑢𝑟𝑒/𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦
𝑁1 𝑁2 … … 𝑁𝑗 … … 𝑁 𝑛
𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝑝1 𝑝2 … … 𝑝𝑗 … … 𝑝 𝑛
𝐴1 𝑂11 𝑂12 … … 𝑂1𝑗 … … 𝑂1𝑛
𝐴2 𝑂21 𝑂22 … … 𝑂2𝑗 … … 𝑂2𝑛
… … … … … … … … … … … … … …
𝐴𝑖 𝑂𝑖1 𝑂𝑖2 … … 𝑂𝑖𝑗 … … 𝑂𝑖𝑛
… … … … … … … … … … … … … …
𝐴 𝑚 𝑂 𝑚1 𝑂 𝑚2 … … 𝑂 𝑚𝑗 … … 𝑂 𝑚𝑛
Payoff Table

11. © Department of Mechanical Engineering
MS-301EngineeringManagement
Tools for Decision Making
Categories of Decision Making
• Our discussion will be made among some
number 𝑚 of alternatives, identified as
𝐴1, 𝐴2, … … . , 𝐴 𝑚
• There may be more that one future “State of
nature” 𝑁 (The model allows for 𝑛 different
futures)
• These future states of nature may not be equally
likely, but each state 𝑁𝑗 will have some (known or
unknown) probability of occurrence 𝑝𝑗. Since the
future must take on one of n values of 𝑁𝑗, the
sum of 𝑛 values of 𝑝𝑗 must be 1.0.

12. © Department of Mechanical Engineering
MS-301EngineeringManagement
Tools for Decision Making
Categories of Decision Making
• The outcome (or payoff, or benefit gained) will
depend on both the alternative chosen and the
future state of nature that occurs
• For example, if you choose alternative 𝐴𝑖 and the
state of nature 𝑁𝑗 takes place (as it will with
probability 𝑝𝑗), the payoff will be outcome 𝑂𝑖𝑗. A
full payoff table will contain 𝑚 times 𝑛 possible
outcomes.
• Let us consider what this model implies and the
analytical tools we might choose to use under
each of our three classes of decision making

13. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
• Decision making under certainty implies that we
are certain of the future state of nature (or we
assume that we are). In our model, this means
that the probability 𝑝1 of future 𝑁1 is 1.0, and all
other futures have zero probability
• The solution, naturally is to choose the alternative
𝐴𝑖 that gives us the most favourable outcome 𝑂𝑖𝑗.
Although this may seem like a trivial exercise,
there are many problems that are so complex
that sophisticated mathematical techniques are
needed to find the best solution

14. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
Linear Programming
• One common technique for decision making
under certainty is called linear programming
• In this method, a desired benefit (such as profit)
can be expressed as a mathematical function
(the value model or objective function) of several
variables
• The solution is the set of values for the
independent variables (decision variables) that
serves to maximize the benefits (or, in many
problems, to minimize the cost) subject to certain
limits (constraints)

15. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
Linear Programming: Example
• Consider a factory producing two products, product X
and Y
• The problem is this: if you can realize $10 profit per unit
of product X and $14 per unit of product Y, what is the
production level of x units of product X and y units of
product Y that maximizes the profit P? that is you seek
to
𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑃 = 10𝑥 + 14𝑦
• You can get a profit of
– $350 by selling 35 units of X or 25 units of Y
– $700 by selling 70 units of X or 50 units of Y
– $620 by selling 62 units of X or 44.3 units of Y; or (as
in the first two cases as well) any combination of X
and Y on the isoprofit line connecting these two
points

16. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
Linear Programming: Example

17. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
Linear Programming: Example
• Your production, and therefore your profit, is
subject to resource limitations, or constraints
• Assume in this example that you employ five
workers
– Three machinists
– Two assemblers
• And that each works only 40 hours a week
• Product X and/or Y can be produced by these
workers subject to the following constraints:
– Product X require three hours of machining and
one hour of assembly per unit
– Product Y require two hours of machining and two
hours of assembly per unit

18. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
Linear Programming: Example
• These constraints are expressed mathematically as
follows
3𝑥 + 2𝑦 ≤ 120 ℎ𝑜𝑢𝑟𝑠 𝑚𝑎𝑐ℎ𝑖𝑛𝑖𝑛𝑔 𝑡𝑖𝑚𝑒
𝑥 + 2𝑦 ≤ 80 ℎ𝑜𝑢𝑟𝑠 𝑎𝑠𝑠𝑒𝑚𝑏𝑙𝑦 𝑡𝑖𝑚𝑒
• Since there are only two products, these limitations
can be shown on a two-dimensional graph
• Since all relationships are linear, the solution to our
problem will fall at one of the corners
• To find the solution, begin at some feasible solution
(satisfying the given constraints) such as (x, y) = (0,
0), and proceed in the direction of “steepest ascent” of
the profit function (in this case, by increasing
production of Y at $ 14 profit per year) until some
constraint is reached

19. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
Linear Programming: Example
• Since assembly hours are limited to 80, no more than
80/2=40 units of Y can be made, earning 40X$14 =
$560 profit
• Then proceed along the steepest allowable ascent
from there (along the assembly constraint line) until
another constraint (machining hours) is reached
• At that point, (x, y) = (20, 30) and profit
𝑃 = 20 𝑋 $10 + 30 𝑋 $14 = $620
• Since there is no remaining edge along which profit
increases, this is the optimum solution

20. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
Linear Programming: Example

21. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
Linear Programming: Computer Solution
• About 50 years ago George Danzig of Stanford
University developed the simplex method, which
expresses the foregoing technique in a mathematical
algorithm that permits computer solution of linear
programming problems with many variables
(dimensions), not just the two (assembly and
machining) of this example
• Now linear programs in a few thousand variables and
constraints are viewed as “Small”
• Problems having tens or hundreds of thousands of
continuous variables and constraints are regularly
solved; tractable integer programs are necessarily
smaller, but are still commonly in the hundreds or
thousands of variables and constraints

22. © Department of Mechanical Engineering
MS-301EngineeringManagement
Decision making under certainty
Linear Programming: Computer Solution
• Another classic linear programming application is the oil
refinery problem, where profit is maximized over a set of
available crude oils, process equipment limitations,
product with different unit profits, and other constraints
• Other applications include assignment of employees with
differing aptitude to the jobs that need to be done to
maximize the overall use of skills; selecting the
quantities of items to be done to maximize the overall
use of skills; selecting the quantities of items to be
shipped from a number of warehouses to a variety of
customers while minimizing transportation cost; and
many more
• In each case there is one best answer, and the
challenge is to express the problem properly so that it
fits a known method of solution

23. © Department of Mechanical Engineering
MS-301EngineeringManagement
Assignment
• You operate a small wooden toy company
making two products: alphabet blocks and
wooden trucks. Your profit is $30 per box of
blocks and $40 per box of trucks. Producing a
box of blocks requires one hour of woodworking
and two hours of painting; producing a box of
trucks takes three hours of woodworking, but only
one hour of painting. You employ three
woodworkers and two painters, each working 40
hours a week. How many boxes of blocks (B) and
trucks (T) should you make each week to
maximize profit? Solve graphically as a linear
program and confirm analytically.