Linear programming manufacturing application

1. PROJECT REPORT
MCOM
QUANTITATIVE TECNIQUES FOR BUSINESS
SUBMITTED BY:
MUNEEB
SUBMITTED TO:

2. MANUFACTURING
. APPLICATION

3. Table Of Content
1: Introduction 02
2: Classic applications of Linear
programming: 03
3: Uses of Linear Programming: 04
4: Advantages of Linear Programming: 04
5: Application of Linear Programming: 05
6: Basic requirements of a Linear
Programming model: 05
7: Graphical method for solving a Linear
Programming problem: 06
8: Limitations of Linear Programming
model: 06
9: Manufacturing Application: 08
10: Production Scheduling: 09
11: Problem: 09
12: Solution: 10
13: Conclusion: 14

4. ABSTRACT
Linear programmingisthe process of taking variouslinear inequalities
relating to somesituation, and findingthe "best" valueobtainable under those
conditions. Objectives of businessdecisions frequently involvemaximizing
profitor minimizingcosts. Linear programminguseslinear algebraic
relationships to representa firm’sdecisions, given a businessobjective, and
resourceconstraints. Linear programming(LP)is a widely used mathematical
technique designed to help operationsmanagersplan and make the decisions
necessary to allocate resources. In business, it is often desirable to find the
production levelsthat will producethe maximum profitor the minimum cost.
The production processcan often be described with a set of linear inequalities
called constraints. The profit or cost function to be maximized or minimized is
called the objective function. The processof findingthe optimal levels with the
system of linear inequalities is called linear programming(as opposed to non-
linear programming).

5. INTRODUCTION:
Linear programmingis not a programminglanguagelike C++, Java, or Visual
Basic. Linear programmingcan be defined as:
"A method to allocate scarce resourcesto competing activities in an optimal
manner when the problem can be expressed usinga linear objective function
and linear inequality constraints."
A linear program consists of a set of variables, a linear objective function
indicating the contribution of each variable to the desired outcome, and a set
of linear constraints describing the limits on the valuesof the variables. The
answer"to a linear program is a set of valuesfor the problem variables that
results in the best largest or smallest valueof the objective function and yet is
consistent with all the constraints. Formulation is the process of translating a
real-world problem into a linear program. Once a problem has been
formulated asa linear program, a computer program can be used to solve the
problem. In this regard, solvinga linear program is relatively easy. The
hardest part about applyinglinear programmingisformulatingthe problem
and interpretingthe solution.
Decision Variables:
The variables in a linear program are a set of quantities that need to be
determined in order to solve the problem; i.e., the problem is solved when the
best valuesof the variables have been identified. The variables are sometimes
called decision variables because the problem is to decidewhat value each
variable should take. Typically, the variables representthe amountof a
resourceto useor the level of some activity. Frequently, definingthe variables
of the problem is oneof the hardest and/or most crucial steps in formulatinga
problem as a linear program. Sometimes creative variable definition can be
used to dramatically reducethe size of the problem or makean otherwise
non-linear problem linear.
Objective Function:
The objective of a linear programmingproblem willbe to maximizeor to
minimizesomenumericalvalue. This valuemay be the expected net present
valueof a project or a forest property; or it may be the cost of a project; it
could also be the amountof wood produced, theexpected number of visitor-
daysat a park, the number of endangered speciesthat will be saved, or the
amountof a particular typeof habitat to be maintained. Linear programming

6. is an extremely generaltechnique, and its applications are limited mainly by
our imaginations and our ingenuity.
The objective function indicates how mucheach variable contributes to the
valueto be optimized in the problem.
Constraints:
These are mathematical expressionsthat combine the variables to express
limits on the possible solutions. For example, they may express the idea that
the number of workersavailable to operate a particular machine is limited, or
that only a certain amountof steel is available per day.
Classic applications of Linear programming:
1. Manufacturing:
Productchoice Several alternative outputswith differentinputrequirements
Scarce inputsMaximizeprofit.
2. Agriculture:
Feed choice Several possible feed ingredientswith differentnutritional
content NutritionalrequirementsMinimizecosts.
3. The Transportation Problem:
Several depotswith variousamountsof inventory Severalcustomersto whom
shipmentsmust be madeMinimizecost of servingcustomers.
4. Scheduling:
Many possiblepersonnelshifts Staffing requirementsat varioustimes
Restrictions on shift timing and length Minimizecost of meeting staffing
requirements.
5. Finance:
Several typesof financialinstrumentsavailable Cash flow requirementsover
time Minimizecost.

7. Uses of Linear Programming:
There are many uses of L.P. It is notpossible to list them all here. However L.P
is very usefulto find outthe following:
1: Optimum productmixto maximizethe profit.
2: Optimum schedule of ordersto minimizethe total cost.
3: Optimum media-mixto get maximum advertisementeffect.
4: Optimum schedule of suppliesfrom warehousesto minimizetransportation
costs.
5: Optimum line balancing to have minimum idlingtime.
6: Optimum allocation of capital to obtain maximum R.O.I
7: Optimum allocation of jobs between machines for maximum utilization
of machines.
8: Optimum assignments of jobs between workersto have maximum labor
productivity.
9: Optimum staffing in hotels, police stations and hospitals to maximize
efficiency.
10: Optimum number of crew in buses and trains to have minimum operating
costs.
11: Optimum facilities in telephone exchange to have minimum break downs.
Advantages of Linear Programming:
1: Providethe best allocation of available resources.
2: Meet overall objectives of the management.
3: Assist managementto take proper decisions.
4: Provideclarity of thought and better appreciation of problem.
5: Improveobjectivity of assessment of the situation.
6: Put across our view points moresuccessfully by logical argument
supported by scientific methods.

8. Application of Linear Programming:
The primary reason for usinglinear programmingmethodology isto ensure
that limited resourcesare utilized to the fullestextent without any waste and
that utilization is madein such a way that the outcomes are expected to be the
best possible. Some of the examples of linear programmingare:
a) A production manager planningto producevariousproductswiththe given
resourcesof raw materials, man-hours, and machine-timefor each product
mustdeterminehow many productsand quantitiesof each productto
produceso as to maximizethe total profit.
b) An investor has a limited capital to invest in a number of securities such as
stocks and bonds. He can use linear programmingapproachto establish a
portfolio of stocks and bondsso as to maximizereturn at a given level of risk.
c) A marketing manager has at his disposala budget for advertisementin
such mediaas newspapers, magazines, radio and television. The manager
would like to determinethe extent of mediamix which would maximizethe
advertisingeffectiveness.
d) A Farm has inventoriesof a number of items stored in warehouseslocated
indifferentpartsof the country that are intended to servevariousmarkets.
Within the constraintsof the demand for the productsand location of
markets, the company would liketo determinewhich warehouseshould ship
which productand how muchof it to each market so that the total cost of
shipmentis minimized.
Basic requirements of a Linear Programming model:
1: The system under consideration can be described in terms of a series of
activities and outcomes. These activities (variables) mustbe competingwith
other variables for limited resourcesand relationships amongthese variables
mustbe linear and the variables must be quantifiable.

9. 2: The outcomesof all activities are known with certainty.
3: A well defined objective function exist which can be used to evaluate
differentoutcomes. The objective function should be expressed as a linear
function of the decision variables. The purposeisto optimizethe objective
function which may be maximization of profitsor minimization of costs, and
so on.
4: The resourceswhich are to be allocated amongvariousactivities mustbe
finite and limited.
5: There mustnot be just a single course of action but r a number of feasible
coursesof action open to the decision maker, one of which would givethe
best result.
Graphical method for solving a Linear Programming problem:
If the linear programmingproblem istwo variable problems, it can be solved
graphically. The steps required for solvinga linear programmingproblem by
graphic method are:
1: Formulate the problem into a linear programmingproblem.
2: Each inequality in the constraints may be written as equality.
3: Draw straight lines correspondingto the equations obtained in step 2. So
there will be as many straight lines as there are questions.
4: Identify the feasible region. Feasible region is the area which satisfies all
constraints simultaneously.
5: The permissible region or feasible region is a many sided figure. The corner
pointsof the figureare to be located and their co-ordinates are to be
measured.
Limitations of Linear Programming model:
1: There is no guaranteethat linear programmingwillgive integer valued
equations. For instance, solution may resultin producing8.291cars. In such a
situation, the manager will examine the possibility of producing8 as well as 9
cars and will take a decision which ensureshigher profitssubject to given

10. constraints. Thus, rounding can give reasonably good solutionsin many cases
but in somesituations we will get only a poor answer even by rounding. Then,
integer programmingtechniques alonecanhandlesuchcases
2: Under linear programmingapproach, uncertainty isnot allowed. The linear
programmingmodeloperatesonly when values for costs, constraints etc. are
known butin real life such factors may be unknown.
3: The assumption of linearity is another formidablelimitation of linear
programming. The objective functionsand the constraintfunctionsin the
Linear programming modelareall linear. Weare thus dealingwith a system
that has constant returnsto scale. In many situations, the input-outputrate
for an activity varies with the activity level. The constraints in real life
concerningbusinessand industrial problemsare not linearly related to the
variables, in most economicsituations, sooner or later, the law of diminishing
marginal returnsbegins to operate.
4: Linear programmingwillfail to give a solution if management has
conflicting multiplegoals. In Linear programming model, thereis only one
goal which is expressed in the objective function.
Eg. Maximizingthe value of the profitfunction or minimizinghe cost function,
one should resort to Goal programmingin situationsinvolvingmultiplegoals.
All these limitations of linear programmingindicateonly onething- that linear
programmingcannotbe madeuse of in all businessproblems. Linear
programmingis not a panacea for all managementand industrialproblems.
Butfor those problemswhere it can be applied, the linear programmingis
considered a very usefuland powerfultool.

11. Manufacturing Application:
Manufacturingproblems: In these problems, we determinethe number of
unitsof differentproductswhich should be produced and sold by a firm when
each productrequiresa fixed manpower, machinehours, labor hour per unit
of product, warehousespaceper unit of the outputetc., in order to make
maximum profit.
Production Scheduling:
Setting a low-cost production scheduleover a period of weeks or months is a
difficultand importantmanagementproblem in most plants.
The production manager hasto consider many factors: labor capacity,
inventory and storage costs, space limitations, productdemand, and labor
relations. Becausemost companiesproducemorethan one product, the
schedulingprocess is often quite complex.
Basically, the problem resembles the productmix modelfor each period in the
future. The objective is either to maximizeprofit or to minimizethe total cost
(production plusinventory)of carryingoutthe task.
Production schedulingis amenable to solution by LP because it is a problem
that mustbe solved on a regular basis. When the objective function and
constraints for a firm are established, the inputscan easily be changed each
Manufacturing
Application
Production
Scheduling
Production
MIx

12. month to providean updated schedule.
Production Mix:
A fertile field for the useof LP is in planningfor the optimal mix of productsto
manufacture. A company mustmeeta myriad of constraints, ranging from
financial concernsto sales demand to material contracts to union labor
demands. Itsprimary goal is to generate the largest profitpossible.
Problem:
The Outdoor FurnitureCorporation manufacturestwo products, benches and
picnic tables, for use in yardsand parks. The firm has two main resources: its
carpenters(labor force)and a supply of redwood for usein the furniture.
Duringthe next production cycle, 1,200 hoursof labor are available under a
union agreement. The firm also has a stock of 3,500 feetof good quality
redwood. Eachbench that Outdoor Furnitureproducesrequires4 labor hours
and 10 feet of redwood: each picnictable takes 6 labor hours and 35 feet of
redwood. Completed bencheswill yield a profitof $9 each, and tables will
result in a profitof $20 each. How many benches and tables should Outdoors
Furnitureproduceto obtain the largest possible profit? Use graphical linear
programmingapproach.

13. Solution:
Step 1:
Translate the real life problem in to the linear programmingmodel.
Labor (Hrs) Material
(Redwood)
Profit
Benches 04 10 09
Picnic Table 06 35 20
1200 3500
Step 2:
Introduce objective function and all the constraints along Non-
negative condition upon decision variables.
Objective function:
Our objective is to maximize the profit so profit function is
£ = 20x + 9y
Constraints:
1: The labor constraint:
6x + 4y ≤ 1200
If,
x = 0 , y = 300 and y = 0 , x = 200
2: The material constraint:
35x + 10y ≤ 3500
If,
x = 0 , y = 350 and y = 0 , x = 100

14. Step 3:
After that we plot the given constraints into a graphical
presentation.
Graph: 1
Red Line = The labor constraints
Green Line = The material constraints
Yellow Line = Optimal Point

15. Step 4:
We investigate the feasible region.
Graph: 2
Red Line = The labor constraints
Green Line = The material constraints
Yellow Line = Optimal Point
Black Lines = Feasible Region

16. Step: 5
with the help of feasible region, we are able to find the points on the
corner that can optimize our solution.
Profit Function:
£ = 20x + 9y
1: where x = 200 , y = 300
Put,
= 20 (0) + 9 (300 ) = 2700
2: where x = 34 , y = 278
Put,
= 20 (34) + 9 (278) = 3182
3: where x = 100 , y = 350
Put
= 20 (100) + 9 (0) = 2000
SO, know we know that the point where x= 34 and y = 278 is the
point where companies profit is optimal.

17. Conclusion:
From this project we came to a conclusion that 'Linear
programming' is like a vast ocean where many methods, advantages,
uses, requirements etc. can be seen. Linear programming can be
done in any sectors where there is less waste and more profit. By
this, the production of anything is possible through the new
methods of L.P. As we had collected many data about Linear
programming, we came to know more about this, their uses,
advantages and requirements. Also, there are many different ways
to find out the most suitable L.P. Also, we formulate an example for
linear programming problem and done using the two methods
simplex method and dual problem. And came to a conclusion that
L.P is not just a technique but a planning the process of determining
a particular plan of action from amongst several alternatives. Even
there are limitations; L.P is a good technique, especially in
the business sectors.