APPLICATION OF LINEAR PROGRAMMING PROBLEM IN MANUFACTURING OF LOCAL SOAP

Transcription

1 Volume, Issue, February ISSN -64 APPLICATION OF LINEAR PROGRAMMING PROBLEM IN MANUFACTURING OF LOCAL SOAP Igbinehi, E.M ', Oyebode Aminat Olaitana and Taofeek-Ibrahim Fatimoh Abidemi, Department of Mathematics and Statistics, School of Applied Science, The Federal Polytechnic, Offa. Kwara state,nigeria. Department of Marketing Kwara State Polytechnic, Ilorin,Kwara State, Nigeria Department of Computer Science School of Applied Science, The Federal Polytechnic Offa Nigeria. ABSTRACT This research Application of Linear Programming in a Local Soap Manufacturing Company was aimed to maximize the profit of the local soap Production. Linear Programming Problem was formulated based on the data collected from the company. From our findings and calculation, we observed that the company spends more on coloured soap and they gets more profit from white soap than coloured soap. We advise that the company should produce less coloured soap and produce more white soap of g and g to maximize thier profit. Keywords: Local, Soap, Coloured, Company, Profit..INTRODUCTION The complexity of today s business operations, the high cost of technology, materials and labour as well as competitive pressure and the shortened time frame in which many important decision must be made contribute to the difficultly of making effective decisions[,}. All this question are very difficult to answer because it depends on so many different economic, social and political factors and view point, very few business decisions are made which are not primarily based on quantitative measures of some nature[,4,]. It must be emphasized however that, timely and competent decision analysis should be an aid to the decision makers judgment, not a substitute for it[6]..algorithms FOR SIMPLE METHOD Step I: If the problem is of minimization, convert it to maximization problem by multiplying the objective function z by (-). Step II: See that all bi s, multiply it by (-) to make bi positive. Step III: Convert all the inequalities to equalities by addition of a slack variables artificial variables or by subtraction of surplus variables as the case may be. Step IV: Find the starting basic feasible solution. Step V: Construct the starting simplex table Step VI: Testing for the optimality of basic feasible solution by computing zj - cj if zj - cj >, the solution is optimal, otherwise, we proceed to the next step. Step VII: To improve on the basic feasible solution we find the IN-COMING VECTOR entering the basic matrix and the OUT-GOING VECTOR to be removed from the basic matrix. The reviable that corresponds to the most negative zj - cj is the IN-COMING VECTOR. while the variable that corresponds to the minimum ratio bi / aj for a particular j and aij > is the OUT-GOING VECTOR. Step VIII: the KEY ELEMENT or the pivot element is determined by considering the intersection between the arrows that correspond to both the in-coming and the out-going vectors. The key element is used to generate the next table in the next table, the pivot element will be replaced by zero. To calculate the new values for all other elements in the remaining rows of the pivot column we use the relation New row = former element in the old row ( intersectional element of the old row) x (corresponding element of replacing row). In this way we get the improved Step I: test this new basic feasible solution not optimal, repeat the process till optimal solution is obtained. Volume, Issue, February Page 6

2 Volume, Issue, February ISSN -64.METHODOLOGY LINEAR PROGRAMMING PROBLEM FORMULATION Optimize z = cxi + C x C Cn x n (i) St aii xi + a ainn * bi ai xi + a ann * b ai xi + a ann * b (ii) ami xi + am amnn * bm i,, n > (iii) Where * means = <> and (m<n). 4.DATA COLLECTION Interview method was used to collected data in local soap manufacturing company which manufactures three different types of local soap, g of white soap, g of white soap and g of coloured soap. They uses litres of water of soak litres of the chemical and kgs of caustic soda. There are litres of water, 6 litres of chemical and kg of caustic soda are available to be used. It takes hrs to produce g of white soap, hrs to produce g of white soap and hrs to produce g of coloured soap. It takes a total of hrs for all the three types of soap to be strong and ready for use. They spent N, to produce g of white soap, N, to produce g of white soap and N, to produce g of coloured soap and the budget N, to produce soap every week. They sells three of g of white soap at the rate of N, she also sell three of g of white soap at the rate of N and sells three of g of coloured soap at the rate of N. After the production and the selling, she have the total of N, on g of white soap, she also have the total of N, on g of white soap and she have the total of N, on g of coloured soap. Let represent g of white soap Let represent g of white soap Let represent g of coloured soap No. of soap produced from each different type For = = 98 She produces 98 pieces of g of white soap For = = 9 She produces 9 pieces of g of white soap For = = 9 She produces 9 pieces of g of coloured soap The total no of soap produced =8 Resource Constraint = Time Constraint + + Bound Constraint + + Objective Function (Profit) Profit = selling price cost price (cost of production) For = N - N = N, For = N - N = N, For = N - N = N Max Z = N Volume, Issue, February Page

3 Volume, Issue, February ISSN -64 = Max Z= + + S.t Convert to Standard Form Max Z = OS + OS +OS S.t S + OS +OS = OS + S +OS = OS + OS +S = TABLE B V S S S C B S S S B MR 46 46/ / / 46 ( - j BV CB S S - TABLE S S S B MR / - -/ - 4/ / / / /.. 68 (- j -4 - / - / Volume, Issue, February Page 8

4 Volume, Issue, February ISSN -64 TABLE BV CB S S S B MR S - 4 -/ - / /6 -/ 488/ 68/ -68/ / -68/ 488/ -44 9/ (- j 44-9/ TABLE 4 BV CB S S S B / -/ -/ -/ /6 / -/ 44/ 9/ -/ / -8/ 6/ (- j - 8/ -6/.CONCLUSION Max Z = + + = [44] + [9] - [] = = = 498 = 66. Max Z = N6,6 From our findings and calculation, we observed that the company spends more on coloured soap due to the palm oil used to produced coloured soap and gets more profit from white soap than coloured soap. We advise that the company should produce less coloured soap and produce more white soap of g and g to maximize her profit. Volume, Issue, February Page 9

5 Volume, Issue, February ISSN -64 REFERENCES [] Pal Inc (99): Operation Research Techniques Published by Phenum Press New York. [] Gerdness Inc. (99): Quantitative Approaches to Management 8 th edition [] William I. S. & Irwin.S. (99): Introduction to Management Science nd Edition. [4] Vanderbei, R.J 996.gramming: Foundations and Extensions, Kluwer Academic Publishers, Boston, 996. [] Wright S.J. (99): Primal-Dual Interior-Point Methods, Society for Industrial and Applied Mathematics, Philadelphia. [6] Jalaluddin Abdullah, Optimization by the Fixed-Pont Method, Version..[] ( Volume, Issue, February Page