linear programming problem ( graphical Analysis)(LPP) operation research

3. project
Within 15 days Sales target within
a month
Within a budget of
INR 25000
:
Maximum score Maximum sales
in a month Minimize the cost
TIME TIME
Pre Decided
Budget

4. Such type of problems are called
• Maximum Profit
• Minimum Cost
• Minimum use of Resources
There can be many day to day problems which need to be
solved using:

5. Linear Programming (LP) is a versatile technique for assigning a fixed amount of
Resources in such a way that some objectives are optimized and other defined
conditions are also satisfied.
The Objectives may be cost minimization or inversely profit maximization
The Technique Of Linear Programming was developed by GEORGE B.
DANTZIG while he was working with US Air force during World War 2.
primarily it was developed for solving military problems.
But now it is being used for solving wide range of problems relating to oil
refining , energy planning, pollution control education and almost in all
functional areas of Management-production,finance,marketing,personnel.

6. Linear programming and Optimization are used in various industries.
The manufacturing and service industry uses linear programming on a
regular basis. In this section, we are going to look at the various
applications of Linear programming.
1. Manufacturing industries use linear programming for analyzing
their supply chain operations. Their motive is to maximize efficiency
with minimum operation cost. As per the recommendations from
the linear programming model, the manufacturer can reconfigure
their storage layout, adjust their workforce and reduce the
bottlenecks.
2. Linear programming is also used in organized retail for shelf
space optimization. Since the number of products in the market
has increased in leaps and bounds, it is important to understand
what does the customer want. Optimization is aggressively used in
stores like Walmart, Hypercity, Reliance, Big Bazaar, etc.

7. 3. Optimization is also used for optimizing Delivery Routes. This is an
extension of the popular traveling salesman problem. The service
industry uses optimization for finding the best route for multiple
salesmen traveling to multiple cities. With the help of clustering and
greedy algorithm, the delivery routes are decided by companies like
FedEx, Amazon, etc. The objective is to minimize the operation cost
and time.
4. Optimizations are also used in Machine Learning. Supervised
Learning works on the fundamental of linear programming. A system is
trained to fit on a mathematical model of a function from the labeled
input data that can predict values from an unknown test data.

8. 1. A decision amongst alternative courses of action is required.
2. The decision is represented in the model by decision variables.
3. The problem encompasses a goal, expressed as an objective function, that the
decision maker wants to achieve.
4. Restrictions (represented by constraints) exist that limit the extent of achievement
of the objective.
5. The objective and constraints must be definable by linear mathematical
functional relationships.
1. Proportionality - The rate of change (slope) of the objective function and constraint
equations is constant.
2. Additivity - Terms in the objective function and constraint equations must be
additive.
3. Divisibility -Decision variables can take on any fractional value and are therefore
continuous as opposed to integer in nature.
4.Certainty - Values of all the model parameters are assumed to be known with
certainty (non-probabilistic).

9. ADVANTAGES OF LINEAR PROGRAMMING
Following are certain advantages of linear programming:
1.Linear programming helps in attaining the optimum use of productive
resources. It also indicates how a decision-maker can employ his productive
factors effectively by selecting and distributing (allocating) these resources.
2.Linear programming techniques improve the quality of decisions. The
decision-making approach of the user of this technique becomes more
objective and less subjective.
3.linear programming techniques provide possible and practical solutions since
there might be other constraints operating outside the problem which must be
taken into account. Just because we can produce so many units docs not
mean that they can be sold. Thus, necessary modification of its mathematical
solution is required for the sake of convenience to the decision-maker.
4.Highlighting of bottlenecks in the production processes is the most significant
advantage of this technique. For example, when a bottleneck occurs, some
machines cannot meet demand while other remains idle for some of the time.
5.Linear programming also helps in re-evaluation of a basic plan for changing
conditions. If conditions change when the plan is partly carried out, they can
be determined so as to adjust the remainder of the plan for best results.

10. 1. Linear programming model does not take into consideration the
effect of time uncertainty. Thus, the LP model should be defined in
such a way that any change due to internal as well as external
factors can be incorporated.
2. Parameters appearing in the model are assumed to be constant
but in real-life situations, they are frequently neither known nor
constant.
3. Parameters like human behaviour, weather conditions, stress of
employees, demotivated employee can’t be taken into account
which can adversely effect any organisation.
4. Only one single objective is dealt with while in real life situations,
problems come with multi-objectives.
5. Sometimes large-scale problems can be solved with linear
programming techniques even when assistance of computer is
available. For it, the main problem can be fragmented into several
small problems and solving each one separately.

11. Linear optimization models are among the most successful application of
operational research. In fact, they rank highest in economic impact. Suppose, we
consider the operations of a major integrated oil company. The key decisions in
this type of industry related to processes of:
1.Exploring for oil deposits.
2.Producing crude oil.
3.Shipping crude to various refineries.
4.Cracking the crude into several blending stock.
5.Combining the stocks into several types of petroleum
products.
6.Shipping the manufacture products from the refineries to
marketing areas.

12. understanding the given problem
Convert to a Linear Programming
problem
Two Aspect:
 Formulation Part
 Solution Part

13. A Company manufactures and sells 2 types of products ‘A’ and
‘B’. The cost of production of each unit of ‘A’ and ‘B’ is RS.200
and RS.150 respectively. Each unit of ‘A’ yield a profit of 20 and
each unit of ‘B‘ yields a profit of 15 on selling.
Company estimates the monthly demand of ‘A’ and ‘B’ to be
maximum value of 500 units in All. The production budget for
the month is 50000.How many units should be company
manufactures in order to earn maximum profit from its monthly
sales of ‘A’ and ‘B’?
Maximum profit ?????
Constraints Production Budget

14. products
Cost of
production per
unit
Profit per unit Total demand
A 200 20
500 units
B 150 15
Production
budget
50000 _ _
Tabular form
Now, express the problem in mathematical form.

16. Step 1. Formulate the LPP problems and develop objective
function along with all the constraints function.
Step 2. Graph the feasible region and find the corner
points. The coordinates of the corner points can be
obtained by either inspection or by solving the two
equations of the lines intersecting at that point.
Step 3. Make a table listing the value of the objective
function at each corner point.
Step 4. Determine the optimal solution from the table in step
3. If the problem is of maximization (minimization) type, the
solution corresponding to the largest (smallest) value of the
objective function is the optimal solution of the LPP.

17. A furniture company produces inexpensive tables and chairs. The production process for
each is similar in that both require a certain number of hours of carpentry work and a
certain number of labour hours in the painting department. Each table takes 4 hours of
carpentry and 2 hours in the painting department. Each chair requires 3 hours of carpentry
and 1 hour in the painting department. During the current production period, 240 hours of
carpentry time are available and 100 hours in painting is available. Each table sold yields a
profit of E7; each chair produced is sold for a E5 profit. Find the best combination of tables
and chairs to manufacture in order to reach the maximum profit.
Example 2

18. The decision variables can be defined as X = number of tables to
be produced & Y = number of chairs to be produced.
Now linear programming (LP) problem can be formulated in
terms of X and Y and Profit (P).
maximize P = 7X + 5Y (Objective function) subject to 4X + 3Y ≤ 240
(hours of carpentry constraint) 2X + Y ≤ 100 (hours of painting
constraint) X ≥ 0, Y ≥ 0 (Non-negativity constraint).
Therefore the mathematical formulation of the LPP is:
Maximize: P = 7X + 5Y
Subject to: 4X + 3Y ≤ 240
2X + Y ≤ 100 X
≥ 0 , Y ≥ 0 To find the optimal solution to this LP using the
graphical method we first identify the region of feasible solutions
and the corner points of the of the feasible region.
The graph for this example is plotted in the next slide

19. In this example the corner points are (0,0),
(50,0), (30,40) and (0,80). Testing these
corner points on P = 7X + 5Y gives
Because the point (30,40) produces the
highest profit we conclude that producing 30
tables and 40 chairs will yield a maximum
profit of E410.
The graphical method is one of the easiest
way to solve a small LP problem. However
this is useful only when the decision variables
are not more than two.