O.S. Balogun, E.T. Jolayemi, T.J. Akingbade, H.G. Muazu / International Journal of      Engineering Research and Applicati...
O.S. Balogun, E.T. Jolayemi, T.J. Akingbade, H.G. Muazu / International Journal of          Engineering Research and Appli...
O.S. Balogun, E.T. Jolayemi, T.J. Akingbade, H.G. Muazu / International Journal of              Engineering Research and A...
O.S. Balogun, E.T. Jolayemi, T.J. Akingbade, H.G. Muazu / International Journal of             Engineering Research and Ap...
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Ln2520042007

  1. 1. O.S. Balogun, E.T. Jolayemi, T.J. Akingbade, H.G. Muazu / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.2004-2007 Use Of Linear Programming For Optimal Production In A Production Line In Coca –Cola Bottling Company, Ilorin *O.S. Balogun, ** E.T. Jolayemi, ***T.J. Akingbade and *H.G. Muazu*Department of Statistics and Operations Research, Modibbo Adama University of Technology, P.M.B. 2076, Yola, Adamawa State, Nigeria. **Department of Statistics, University of Ilorin, Ilorin, Kwara State, Nigeria. ***Department of Mathematical Sciences, Kogi State University, Anyigba, Kogi State, Nigeria.Abstract Many companies were and are still respective customers who should also not violateestablished to derive financial profit. In this National Agency for Food and Drug Administrationregard the main aim of such establishments is to Control (NAFDAC) and Standard Organization ofmaximize (optimize) profit. This research is on Nigeria (SON). The problem then is on how tousing Linear programming Technique to derive utilize limited resources to the best advantage, tothe maximum profit from production of soft maximize profit and at the sometime selecting thedrink for Nigeria Bottling Company Nigeria, products to be produced out of the number ofIlorin plant. Linear Programming of the products considered for production that willoperations of the company was formulated and maximize profit.optimum results derived using Software that The research is aimed at deciding howemployed Simplex method. The result shows that limited resources,raw materials of Nigeria Bottlingtwo particular items should be produced even Company (Coca-cola), Ilorin plant , Kwara Statewhen the company should satisfy demands of the would be allocated to obtain the maximumother - not - so profitable items in the contribution to profit. It is also aimed at determiningsurrounding of the plants. the products that contribute to such profit. The scope of the research is to use LinearKEYWORDS: Optimization, Linear Programming on some of the soft drinks producedprogramming, Objective function, Constraints. by Nigeria Bottling Company, Ilorin plant. The data on which this is based are quantity of raw materialsIntroduction available in stock, cost and selling prices and Company managers are often faced with therefore the profit of crate of each product. Thedecisions relating to the use of limited resources. profit constitutes the objective function while rawThese resources may include men, materials and materials available in stock are used as constraints.money. In other sector, there are insufficient If demands which must be met are to be available,resources available to do as many things as such can be included in the constraints. The data ismanagement would wish. The problem is based on secondary data collected in the year 2007 at thehow to decide on which resources would be Nigeria Bottling Company, Ilorin plant, Kwaraallocated to obtain the best result, which may relate State.to profit or cost or both. Linear Programming is The Simplex method, also called Simpleheavily used in Micro-Economics and Company technique or Simplex Algorithm, was invented byManagement such as Planning, Production, George Dantzig, an American Mathematician, inTransportation, Technology and other issues. 1947. It is the basic workhorse for solving LinearAlthough the modern management issues are error Programming Problems up till today. There havechanging, most companies would like to maximize been many refinements to the method, especially toprofits or minimize cost with limited resources. take advantage of computer implementations, butTherefore, many issues can be characterized as the essentials elements are still the same as theyLinear Programming Problems (Sivarethinamohan, were when the method was introduced (Chinneck,2008). 2000; Gupta and Hira, 2006). A linear programming model can be The Simplex method is a Pivot Algorithmformulated and solutions derived to determine the that transverses the through Feasible Basic Solutionsbest course of action within the constraint that while Objective Function is improving. The Simplexexists. The model consists of the objective function method is, in practice, one of the most efficientand certain constraints. For example, the objective algorithms but it is theoretically a finite algorithmof Nigeria Bottling Company (Coca-cola) is to only for non-degenerate problems (Feiring, 1986).produce quality products needed by its customers, To derive solutions from the LP formulated usingsubject to the amount of resources (raw materials) the Simplex method, the objective function and theavailable to produce the products needed by their constraints must be standardized. 2004 | P a g e
  2. 2. O.S. Balogun, E.T. Jolayemi, T.J. Akingbade, H.G. Muazu / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.2004-2007The characteristics of the standard form are: Maximize Z r Cj j(i) All the constraints are expressed in the form of Xequations except the non-negativity constraints j 1 rwhich remain inequalities( 0). a X(ii) The right-hand-side of each constraint Subject to ij j Si bi, i 1,2,3,....,m j 1equation is non-negative. Xj 0, j 1,2,3,....,n(iii) All the decision variables are non-negative.(iv) The Objective function is of maximization or Si 0, i 1,2,3,....,mminimization type. Before attempting to obtain the Any vector X satisfying the constraints of the Linearsolution of the linear programming problem, it must Programming Problems is calledbe expressed in the standard form is then Feasible Solution of the problem (Fogiel, 1996; Schulze, 1998; Chinneck, 2000).expressed in the “the table form” or “matrix form”as given below: r Algorithm to solve linear programming problem:Maximize Z Cj X j (i)See that all bi s are positive, if a constraint has j 1 negative bi multiply it by 1 to make bi positive.Subject to r (ii) Convert all the inequalities by the addition of bi,(bi 0), i 1,2,3,...,m slack or by subtraction of surplus variable as the a ijX j j 1 case may be. (iii)Find the starting Basic Feasible Solution.X j 0, j 1,2,3,...,mIn standard form (Canonical form), it is (iv) Construct the Simplex table as follows: Basic Ej X1 X2 X3.... Xn y1 y2... ym Xb Variable y1 0 a11 a12 a13... a1m 1 0 0 b1 y2 0 a21 a22 a23... a2m 0 1 0 b2 y3 0 a31 a32 a33... a3m 0 0 1 bm Zj Z1 Z2 Z3 Z4 0 0 0 Z Z0 Ej E1 E2 E3 E4 0 0 0 . . . . . . . . bj Zj Ej X X2 X3... Xn y1 y y 1 m 2... bi(v) Testing for optimality of Basic Feasible minimum ratio aij for a particular j, andSolution by computing Z j Ej.If aij 0, i 1,2,3,...,m is the OUTGOING Z j Ej 0, the solution is optimal; otherwise, VECTOR.we proceed to the next step. (vii)The key element or the pivot element is(vi) To improve on the Basic Feasible Solution, we determined by considering the intersection betweenfind the basic matrix. The variable that corresponds the arrows that corresponds to both incoming and outgoing vectors. The key element is used toto the most negative of Z j Ej is the INCOMING generate the next table. In the next table, pivotVECTOR while the variable that corresponds to the element is replaced by UNITY, while all other 2005 | P a g e
  3. 3. O.S. Balogun, E.T. Jolayemi, T.J. Akingbade, H.G. Muazu / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.2004-2007 elements of the pivot column are replaced by zero. (viii) Test of this new Basic Feasible Solution for To calculate the new values for all other elements in optimality as (6) it is not optimal; repeat the the remaining rows of that first column, we use the relation. process till optimal solution is obtained. This was New row = Former element in old rows implemented by the software Management Scientist (intersection element in the old row) Version 6.0. (Corresponding element of replacing row). Table 1: Quantity of raw materials available in stock Rawmaterials Quantityavailable Concentrates 4332(units) Sugar 467012(kg) Water(H2O) 16376630(litres) Table 2:(iv)oxide(C2O) 8796(Vol.perpressure) Carbon Quantity of raw materials needed to produce a crate of each product listedFlavors Concentrate Sugar Water Cabon(iv)oxideCoke35cl 0.00359 0.54 6.822 0.0135FantaOrange35cl 0.00419 1.12 7.552 0.007FantaOrange35cl 0.00217 0.803 4.824 0.005FantaLemon35cl 0.0042 1.044 7.671 0.0126Schweppes 0.00359 0.86 6.539 0.0125Sprite50cl 0.00359 0.73 7.602 0.0133FantaTonic35cl 0.00359 0.63 6.12 0.0133Krestsoda35cl 0.0036 0.89 7.055 0.0146Coke50cl 0.00438 0.23 7.508 0.0156 Table 3: Average Cost and Selling of a crate of each product Product Average Average Profit(₦) Cost Selling price(₦) price(₦) Coke35cl 358.51 690 331.49 FantaOrange35cl 370.91 690 319.09 FantaOrange50cl 489.98 790 300.02 FantaLemon35cl 368.67 690 321.33 Schweppes 341.85 690 348.15 Sprite50cl 486.04 790 303.96 Model Formulation FantaTonic35cl 690 367.91 Maximize Z 322.09 Krestsoda35cl 266.72 690 423.28 Coke50cl 489.89 790 300.11 2006 | P a g e
  4. 4. O.S. Balogun, E.T. Jolayemi, T.J. Akingbade, H.G. Muazu / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.2004-2007 Z 331.49X1 319.09X2 300.02X3 321.33X4 348.15X5 303.96X6 367.91X7 323.28X8 300.11X9 Subjectto: 0.00359X1 0.00419X2 0.0021X3 0.0042X4 0.00359X5 0.00359X6 0.00359X7 0.0036X8 0.00438X9 4332 0.54X1 1.12X2 0.803X3 1.044X4 0.86X5 0.73X6 0.63X7 0.89X8 0.23X9 467012 6.822X1 7.552X2 4.824X3 7.671X4 6.539X5 7.602X6 6.12X7 7.055X8 7.508X9 16376630 0.0135X1 0.007X2 0.005X3 0.0126X4 0.0125X5 0.0133X6 0.0133X7 0.0149X8 0.0156X9 8796 Xi 0, fori 1,2,3,...,9 Now, introducing the slack variable to convert inequalities to equations, it gives: Z 331.49X1 319.09X2 300.02X3 321.33X4 348.15X5 303.96X6 367.91X7 323.28X8 300.11X9 Subjectto: 0.00359X1 0.00419X2 0.00217X3 0.0042X4 0.00359X5 0.00359X6 0.00359X7 0.0036X8 0.00438X9 X10 4332 0.54X1 1.12X2 0.803X3 1.044X4 0.86X5 0.73X6 0.63X7 0.89X8 0.23X9 X11 467012 6.822X1 7.522X2 4.834X3 7.671X4 6.539X5 7.602X6 6.12X7 7.055X8 7.058X9 X12 16376630 0.00135X1 0.007X2 0.005X3 0.0126X4 0.0125X5 0.0133X6 0.0133X7 0.0149X8 0.0156X9 X13 8796 Xi 0, fori 1,2,3,...,13 Where, X7= Fanta Tonic 35cl X1 = Coke 35cl X8 = Krest soda 35cl X2 = Fanta Orange 35cl X9 = Coke 50cl X3 = Fanta Orange 50cl X4 = Fanta Lemon 35cl Analysis and Result X5 = Schweppes Optimal Solution X6 = Sprite 50cl Objective function value = 263,497,283 Variables Value X1 0.00000 X2 0.00000 X3 462547.21890 X4 0.00000 X5 0.00000 X6 0.00000 The X7 Management Scientist Version 6.0 gives: 0.00000Z 263,497,283 X3 462,547 and X9 0.00000 X8 415,593 References:Interpretation of Result Chinneck, .J.W. Practical Optimization, A Gentle X9 Based on the data collected the optimum 415593.00000 introduction; 2000. results derived from the model indicate that two www.sce.carleton.ca/faculty/chinneck/po.html products should be produced 50cl Coke and 50cl Feiring, B.R. Linear Programming: An Fanta Orange. Their production quantities should be Introduction; Sage Publications Inc., USA, 1986. 462,547 and 415,593 crates respectively. This will Fogiel, .M. Operations Research Problems Solver; produce a maximum profit of ₦263,497,283. Research and Education Association; Piscataway, New Jersey, 1996. Gupta, .P.K.and Hira, .D.S. Operations Research; Conclusion Rajendra Ravindra Printers Ltd., New Delhi, Based on the analysis carried out in this 2006. research and the result shown, Nigeria Bottling Schulze, .M.A. Linear Programming for Company, Ilorin plant should produce Fanta Orange Optimization; Perspective Scientific Instruments 50cl and 35cl,Coke 50cl and 35cl, Fanta lemon 35cl, Inc., Sprite 50cl,Schweppes,Krest soda 35cl but more of USA, 1998. Coke 50cl and Fanta Orange 50cl in order to satisfy Sivarethinamohan, .R. Operations Research; Tata their customers. Also, more of Coke 50cl and Fanta McGraw Hill Publishing Co.Ltd., New Orange 50cl should be produced in order to attain Delhi, 2008. maximum profit because they contribute mostly to the profit earned. 2007 | P a g e

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