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Proceedings of the 1st International Technology, Education and Environment

Conference

(c) African Society for Scientific Research (ASSR)

Co-Published By: Human Resource Management Academic Research Society

MAXIMIZATION OF PROFIT IN MANUFACTURINGINDUSTRIES USING LINEAR PROGRAMMING TECHNIQUES: GEEPEE NIGERIA LIMITED

Fagoyinbo, I.S 1, Akinbo, R.Y 2, Ajibode, I.A 3

and Olaniran, Y.O.A

4

1, 2, 3 Department of Mathematics and Statistics Federal Polytechnic, Ilaro, Ogun State. Nigeria

4 Department of Marketing Federal Polytechnic, Ilaro, Ogun State. Nigeria

E-mail : [email protected] [email protected]@yahoo.com [email protected]

Abstract Any organization set up aims at maximization of profit from its

investment from minimum cost of objective function. This research work applies the concept of revised simplex method; an aspect of linear programming to solving industrial problems with the aim of maximizing profit. The industry GEEPEE

Nigeria Limited specialises in production of tanks of various types. Four different types of tank were sampled for study these are the Combo, Atlas, Ramboand Jumbo tanks of various sizes. Based on the analysis of the data collected it was observed that given the amount of materials available Polyethylene (Rubber)and Oxy—acetylene (Gas) used in the production of the different sizes of the

product, Combo tanks assures more objective value contribution and gives maximum profit at a given level of production capacity .

INTRODUCTIONLinear Programming is a subset of Mathematical

Programming that is concerned with efficient allocation of limitedresources to known activities with the objective of meeting a desiredgoal of maximization of profit or minimization of cost. In Statisticsand Mathematics, Linear Programming (LP) is a technique foroptimization of linear objective function, subject to linear equality and linear in equality constraint. Informally, linear programming determines the way to achieve the best outcome (such as maximumprofit or lowest cost) in a given mathematical model and given somelist of requirement as linear equation. Although there is a tendency

to think that linear programming which is a subset of operationsresearch has a recent development, but there is really nothing new about the idea of maximization of profit in any organization setting i.e. in a production company or manufacturing company. Forcenturies, highly skilled artisans have striven to formulate models thatcan assist manufacturing and production companies in maximizing their profit, that is why linear programming among other models inoperations research has determine the way to achieve the best

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Proceedings of the 1s

Conference

(c) African Society f

Co-Published By: Hu

outcome (i.e. maxand given some lis

Lineastudy. Most exsituation, but can

Some industriestransportation, emanufacturing coproved useful inrouting, schedulinand Tucker (199method developeoperations. Its orprocess of businefunds. This shortaenvironments. Altmodel is wide and

the elimination odecision-making (Bierman and Bon

Lineaproblems by dev distribution of limtechnological or pbe made (Andradin a canonical for

Max (Z) = X Subject to:

AX bFrom

variables (to bematrix of coeffici

objective functionconstraint whichobjective functionLinear Programmseveral reasons.operations researproblems. Certa

network flow prconsidered imporspecialised algoritprogramming haoptimization theimportance of cousual and most iproblem. When t

International Technology, Education and Environment

or Scientific Research (ASSR)

man Resource Management Academic Research Society

imization of profit) in a given mathematical mot of requirement represented as linear equations.

programming can be applied to various fieldstensively, it is used in business and econo

also be utilized in some engineering proble

that use linear programming models incluergy, telecommunications and productionpanies. To this extent, linear programming

modelling diverse types of problems in plannig assignment and design. David (1982), Neari ) noted operational research is a mathematito solve problems related to tactical and strate

igins show its application in the decision-makis analysis, mainly regarding the best use for sh

ge of funds is a characteristic of hyper-competitihough the practical application of a mathematicomplex, it will provide a set of results that ena

f a part of the subjectivism that exists in trocess as to the choice of action alternati

ini, 1973).Programming deals with special mathemati

eloping rules and relationships that aim at tited funds under the restrictions imposed by eitractical aspects when an attribution decision has, 1990). Generally, Linear Program can be writt

for profit maximization as:

the model above, x represent the vectoretermined) while c and b are vectors of knont. The expression to be maximized is called t

( in this case). The equation AX b is tspecifies a convex polytope over which t

is to be optimised.ing is a considerable field of optimization

any practical problems in operations problemsch can be expressed as linear programmiin special cases of linear programming such

blems and multi commodity flow problems aant enough to have generated much researchm for their solution. Historically, ideas from line inspired many of the central conceptsry such as Duality, Decomposition and t

vexity and its generalizations. Standard form is ttuitive form of describing a linear programmi

he problem involves “n” decision-making variab

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Conference


Co-Published By: Hu

and “m” restrictiothe form of eitfunction (Corrar aa maximization prlinear function to

Max(Z) =Problem constrain A11x1 + A12x2 + A

A21x1 + A22x2 + A

.

. .

Am1x1 + Am2x2 +

Non negativity va

x1 , x2 , x3 , x4 0

Theleast four of theeffectively estimaattention or prodcourse of this stumake the best posof GeePee Nigerithe same vein, inbottlenecks may oin great demand

work is aimed atproduct that mustGeePee Nigeria LiMoreso, this reseNigeria Limited tprogramming usedecision for redumaterials etc. may

Linea

1940’s, motivatedproblems in war tiin the post war pelinear programmiregarded as Geor1947, and Johnthat same year.to the mathematic




ns, the model can be represented mathematically er maximization or minimization of the objnd Teophilo, 2003), (Salau, 1998). For instance,oblem: it consists of the following three namely:

e maximized:

+ + . . . . . +ts of the form:

13x3 + .........+ A1nxn ( ≤ or≥ ) b1

3x3 + .........+ A2nxn ( ≤ or≥ ) b2

. . . . . .

. .

Am3x3 + .........+ Amnxn ( ≤ or≥ ) bm

iables:

PURPOSE OF THE STUDY ain purpose of this study is to critically examineroducts produced in GeePee Nigeria Limited.e which of these products must be given mouced more in other to maximize profit. In tdy, linear programming technique will be usedsible use of the total available productive resourc

Limited (such as time, material, labours) etc.a production industry like GeePee Nigeria Limitccur. For example in the factory, a product may

hile others may lie idle. As such, this resear

highlightening such bottlenecks i.e. identifying tbe produced more in other to maximize profit

mited.rch work will assist the management of GeePmake valid decision with the technique of linin this research work so as to make an objecti

ction in wastage of resources like time, monbe avoided.

LITERATURE REVIEWProgramming was developed as a discipline in t

initially by the need to solve complex plannime operations. Its development accelerated rapiriods as many industries found its valuable usesng. The founders of the subject are generae B. Dantzig, who devised the simplex methodon Neumann, who establish the theory of dualhe noble price in economics was awarded in 19ian Leonid Kantorovich (USSR) and the econom

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Conference



Tjalling Koopmas (USA) for their contribution to the theory of optimal allocation of resources, in which linear programming played akey role. Many industries use linear programming as a standard tool,e.g. to allocate a finite set of resources in an optimal way. Exampleof important application areas include Airline crew scheduling,

shipping or telecommunication networks, oil refining and blending,stock and bond portfolio selection. The problem of solving a system of linear inequality also

dates back as far as Fourier Joseph (1768 – 1830) who was aMathematician, Physicist and Historian, after which the method of Fourier – Motzkin elimination is named. Linear programming arosea mathematical model developed during the Second World War toplan expenditure and returns in other to reduce cost to the army andincrease losses to the enemy. It was kept secret for years until 1947 whn many industries found its use in their daily planning. The linearprogramming problem was first shown to be solvable in polynomialtime by Leonid Khachiyan in 1979 but a large theo vertical and

practical breakthrough in the field came in 1984 when NarendraKarmarkar (1957 – 2006) introduced a new interior point method forsolving linear programming problems. A lot of applications wasdeveloped in Linear programming these includes: Lagrange in 1762solves tractable optimization problems with simple equality constraint. In 1820, Gauss solved linear system of equations by whatis now called Gaussian elimination method and in 1866, Whelhelm Jordan refined the method to finding least squared error as a measureof goodness-of-fit. Now it is referred to as Gausss- ordan method.Linear programming has proven to be an extremely powerful tool,both in modelling real-world problems and as a widely applicable

mathematical theory. However, many interesting optimizationproblems are non linear. The studies of such problems involve adiverse blend of linear Algebra, multivariate calculus, numericalanalysis and computing techniques.

The simplex method which is used to solve linearprogramming was developed by George B. Dantzig in 1947 as aproduct of his research work during World War II when he was working in the Pentagon with the Mil. Most linear programming problems are solved with this method. He extended his research work to solving problems of planning or scheduling dynamically overtime, particularly planning dynamically under uncertainty.

Concentrating on the development and application of specificoperations research techniques to determine the optimal choiceamong several courses of action, including the evaluation of specificnumerical values (if required), we need to construct (or formulate)mathematical model (Hiller et al, 1995) (ADAMS,1969), ( DANTZIG,1963) .

Conclusively, the development of linear programming has been ranked among the most important scientific advances of themid-20th century, and its assessment is generally accepted. Its impact




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since 1950 has behas saved thouscompanies.

METH

TheMethod for stanmethod the follo

Introductithe probleinto maxi

Find an in(identity m

B =

Considericonstraint,

A =

Compute t

- =

It sho If all -

solution.

If at least

-

Compute

if 0determine

Write dow revised si

Convert tthe colum

improve t




n extra ordinary. Today it is the standard tool tnd or million of dollars of many producti

ODOLOGY AND DATA ANALYSIS

ethod to be adopted is the Revised Simplard maximization problem. In order to use ting procedure is necessary:

n of slack or surplus variable if need be and brim into standard form after converting the proble

ization or minimization.

tial basic feasible solution with initial basic B =atrix) and form auxiliary matrix B such that:

and =

g the objective function as an additioform A and b such that:

and b =

he net evaluation:

. A

uld be noted that:0, the current basic solution is an optim

ne - 0, determine the most negative, s

orresponding to variables enters the basis.

= . ,also the solution is unbound

and if at least one 0, consider the athe leaving variable.

n the result obtained from steps 2 and 5 above iplex table.

e leaving element to unity and all other elementk to zero and

e current basic feasible solution.

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Go to stepfeasible sunbounde

For the purpGEEPEE Ni

Table 1: There aFollows

NAME OF TANK

LITR

COMBO 46000

ATLAS 32000

RAMBO 23000

JUMBO 16000

Source: Field Survey

Table 2: The DatNAME OF

TANK

MAT

POL(kg)

COMBO 14

ATLAS 12

RAMBO 12

JUMBO 10

MATERIALS AVAILABLE

97

Source: Field Survey

The Linear Progra

Max (Z) = 60Subject to:

+ 12 +

+ 4 + 3

+ 15 +

By introducing thhave:

Max (Z) = 60Subject to:

+ 12 +

+ 4 + 3+ 15 +

For non negativit

The augmented m

A =




4 and repeat the procedure until an optimum baolution is obtained or an indication of

solution.

ose of this research work the bye producteria Limited were used.e Four Products Produced in the Company,

S GALLON HEIGHT(mm)

CODE

10000 2600 2XWT

7000 2900 2XWT

5100 2600 WT23

3500 1900 WT160C

October, 2010.

Collected were as FollowsRIAL

ETHYLENE

TIME

(Min)

OXYACETYLENE

(kg)

PROFIT

(DOLLA

6 20 60

4 15 45

3 15 30

3 12 27

42 100

ctober, 2010.

mming model is formulated as:

+ 45 + 30 + 27

12 + 10 97

+ 3 42

15 + 12 100

for all non-negativity conditioslack variable in the objective functions above

+ 45 + 30 + 27 + 0 +0 0

12 + 10 97

+ 3 4215 + 12 100condition:

,atrix is obtained as:

ican

of

as

R)

.e




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B =

C =

=

=

=

A =

B=

B =

To obtain the net

- =

==[ -60 -45 -30 -

In the net evaluaenters the bases s

= . b

=

=

Table 3: To DraX

Z

97421000




]

evaluation ( - we have:

. A

]7 0 0 0 ]

ion of - is the highest negative, thenthat:

the Revised Simplex Tableau we have:Ratio

14620-60

6.929750




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1st IterationConverting the lezero, we have:

( )

To obtain the nex

- = [0 0 3

- = [ 0 0

Since -evaluation, the cur

= . b

=

=

Finally = 5;

The Tanks produced,subjected to statis was observed thafive units of CoDollars, it will giv

Dollars. To this extent ancarried out that ththe production of allotted time atta2,600mm, containa profit margin ostood at 5 within t




ading element to unity and all other, element

this becomes

=

net valuation - we have:

1]

1 5 9 0 0 3 ]

0, i.e. there is non-negative value in therent feasible solution is optimum. Hence;

= 300

CONCLUSIONata collected from the industry on four typesnamely Combo, Atlas, Rambo and Jumboical analysis using the Revised Simplex Method.

if GEEPEE NIGERIA LIMITED can produbo Tanks with an objective Coefficient of Sian objective value contribution of Three Hundr

based on the model formulated, and the analy e amount of polyethylene and oxyacetylene useddifferent sizes of plastic material produced with tched to each product; Combo Tanks (of heig ing 4,600 litres i.e. 10,200 gallons of liquid) assuf three hundred dollars if the quantity produche specified period of time.

to

et

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cety

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isin

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RECOMMENDATIION After it has been established in the course of this

research work that among other products, Combo Tanks assuresmore profit in GEEPEE NIGERIA LIMITED. We thereby

recommend that: The management of GEEPEEE NIGERIALIMITED should give more attention to the production of theproduct (Combo) than other products because it seems to give thehighest profit.In the same vein, critical examination should be carried out on otherproducts on their contribution to the growth or success of thecompany, if their profit margin is very low, then their production canbe ignored. Finally, any product having an adverse effect orcontributing losses to the profit margin of the company can bestopped and the company invest more on the production of Comboto generate more profit.

REFERENCES Adams W. J. (1969): Element of Linear Programming . VanNostrand Reinhold Publishing Company International. Andrade, E. L. (1990): Introdução à Pesquisa Operacional. LTC,Rio deJaneiro.Bierman Jr., Bonini H., & Charles P. (1973): Quantitative Analysis for Business Decisions. 4th edition, Richard D. Irwin,Illinois.Corrar, Teophilo L., Carlos Renato (2003): Pesquisa Operacionalpara Decisão em Contabilidade e Administração. Editora Atlas, Rio

de Janeiro.Dantzig G. B. (1963) : Linear Programming and Extension,Priceton University Press.David Charles. H. (1982): Introduction Concept and Application of Operation ResearchHiller, F.S., G.J. Lieberman and G. Liebeman (1995):Introduction to Operations Research, New York: McGraw-Hill.

Nearing E.D and Tucker A.W. (1993): Linear Programming andRelated Problems. Academic Press Boston.

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Manufacturing Industries - 1