OPTIMIZATION PRINCIPLE AND ITS APPLICATION IN OPTIMIZING LANDMARK UNIVERSITY BAKERY PRODUCTION USING LINEAR PROGRAMMING

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1 International Journal of Civil Engineering and Tehnology (IJCIET Volume 0, Issue 0, February 09, pp. 8-90, Artile ID: IJCIET_0_0_0 Available online at ISSN Print: and ISSN Online: IAEME Publiation Sopus Indeed OPTIMIZATION PRINCIPLE AND ITS APPLICATION IN OPTIMIZING LANDMARK UNIVERSITY BAKERY PRODUCTION USING LINEAR PROGRAMMING N. K Oladeo, A. Abolarinwa, S.O Salawu and A.F Lukman Physial Siene Department, Landmark University, Kwara State Nigeria H.I Bukari Bagabaga College of Eduation, Tamale, Ghana ABSTRACT This paper deals with the appliations of optimization priniple in optimizing profits of a prodution industry using linear programming to eamine the prodution ost and determine its optimal profit. Linear programming is an operation researh tehnique whih is widely used in finding solutions to managerial deision problems. However, many enterprises make more use of the trial-and-error method. As suh, firms have been finding it diffiult in alloating sare resoures in a manner that will ensure profit maimization and/or ost minimization. This paper makse use of seondary data olleted from the reords of the Landmark University Bakery on five types of bread produed in the firm whih inlude Family loaf, slied family bread, Choolate loaf, medium size bread, small size bread. A problem of this nature was identified as a linear programming problem, formulated in Mathematial terms and solved using AMPL software. The solution obtained revealed that Landmark bakery unit should onentrate muh more in prodution of,000 loaves of Family loaf and 0,7 loaves of Choolate bread while others type should be less produed sine their value is turning to zero in order to ahieve a maimum monthly profit of N,860,000. From the analysis, it was observed that Family loaf and the Choolate bread ontributed obetively to the profit. Hene, more of Family loaf and Choolate bread are needed to be produed and sold in order to maimize the profit. 8 editor@iaeme.om

2 N. K Oladeo, A. Abolarinwa, S.O Salawu, A.F Lukman and H.I Bukari Keywords: Linear, programming, prodution. Optimization, maimization, enterprises Cite this Artile: N. K Oladeo, A. Abolarinwa, S.O Salawu, A.F Lukman and H.I Bukari, Optimization Priniple and Its Appliation in Optimizing Landmark University Bakery Prodution Using Linear Programming, International Journal of Civil Engineering and Tehnology, 0(0, 09, pp INTRODUCTION The aim of every organization, ompany or firm is to make profit as that will guarantees its ontinuous eistene and produtivity. In this modern day, manufaturing industries at all levels are faed with the hallenges of produing goods of right quality, quantity and at right time and more espeially at minimum ost and maimum profit for their survival and growth. Thus, this demands an inrease in produtive effiieny of the industry. Linear programming (LP an be defined as a mathematial tehnique for determining the best alloation of a firm s limited resoures to ahieve optimum goal. It is also a mathematial tehnique used in operation researh or Management Sienes to solve speifi problems suh as alloation, transportation and assignment problems that permits a hoie or hoies between alternative ourses of ation. It is one of the most widely used optimization tehniques and perhaps the most effetive method. The term linear programming was oined by [] whih refers to problems in whih both the obetive funtion and onstraints are provided as the Simple method as published in []. [] Studied the optimal prodution ost of raw materials to its prodution output using Linear programming solver to solve and to optimize its monthly prodution output. Based on their result, the monthly optimal prodution output was.e-08. The ompany has to budget at least the optimal result to ahieve their monthly ost of prodution. The result helps the ompany to eliminate eess waste that inurs in their ost of prodution. Likewise, [] eamined optimization priniple and its appliation in solving the problem of over-alloation and under-alloation of the lassroom spae using Linear Programming in Landmark University where linear programming model was formulated based on the data obtained from the eamination and leture timetable ommittee on the lassroom failities, apaities and the number of students per programme in all the three ( Colleges to maimize the available lassroom spae and minimizes the ongestion and overrowding in a partiular leture room using AMPL software whih revealed that 6 out of lassrooms available with a seating apaity of 066 has always been used by the urrent student population of whih always auses overflow and ongestion in those onentrated lassroom while the remaining 6 lassrooms with the seating apaity of 80 were underutilized. Meanwhile it was revealed by the AMPL software if all these lassroom with seating apaity of were fully utilized, this indiated that an additional 0 i.e.(- students an be fully absorbed omfortably with the eisting lassrooms in both of the three ( olleges if the seating apaities are fully managed and maimized while the shool management will generate additional inome using the same lassroom faility with the eisting seating apaity. [] Foused on linear optimization for ahieving produt-mi optimization in terms of the produt identifiation and the right quantity in paint prodution for better profit and optimum firm performane in Nigeria. Their result showed that only two out of the five produts they 8 editor@iaeme.om

3 Optimization Priniple and Its Appliation in Optimizing Landmark University Bakery Prodution Using Linear Programming onsidered in their omputational eperiment are profitable. [6] Empirially eamined the impat of linear programming in entrepreneur deision making proess as an optimization tehnique for maimizing profit with the available resoures. Their work drew eamples from a fast food firm that enountered some hallenges in the prodution of meat pie, hiken pie and doughnut due to an inrement in the prie of raw materials. Their results showed that there should be disontinuity in the prodution of hiken pie and doughnut and that they should onentrate with prodution of meat pie. Using farm ativities, [7] developed a linear program that reflets hoies of seletion that is feasible given a set of fied farm onstraints and maimizing inome while ahieving other goals suh as food seurity. Their result obtained using linear programming is ompared with the traditional methods. Their results obtained using the linear programming model shows that they are more superior. [8] showed that the produt-mi problem an be used effiiently not only to determine the optimal operational points but also to provide information on how those optimal points ould be further inreased through hanging the onstraints of the optimization problem. Their results showed that this information ould be used to enhane prodution by informing epansion plans in whih management identifies and take advantage of the apaity of under-utilized onstraints and use them to epand the apaity of overutilized or limiting onstraints. Therefore, this paper deal with the appliation of linear programming (LP as an optimization priniple to optimize profit of manufaturing industries suh as Landmark University Bakery and determine the optimal solution for prodution and verify the output under normal operational environment using AMPL software.. MATHEMATICAL FORMULATION This paper investigates the overall quantity and quality ombination of the five produts produed by Landmark University Bakery and the alloation of resoures to the various produts through the reords kept by the manager of Landmark University Development Ventures (LMDV and the Bakery Unit manager relating to the different brands of bread produts produed by the firm, the tehnial oeffiients, the raw materials available and their relative pries is shown in Table below. We present Linear programm as a general standard form to display all properties required of a linear programming problem. This onsists of a linear obetive funtion f ( suh that real numbers,... n, then the funtion f of real variables,... n an be defined as: ( Other properties inlude a linear onstraint (whih is one that is either a linear equation or linear inequality and a non-negativity onstraint. These an be written in mathematial Notations as: (Linear onstraint ( (Non-negative onstraint ( Hene every linear program in the standard form an be generally presented as: Subet to ( 8 editor@iaeme.om

4 N. K Oladeo, A. Abolarinwa, S.O Salawu, A.F Lukman and H.I Bukari ( If satisfy all the onstraints of linear program, then the assignment of values to these variables are alled a feasible solution of the linear program... Linear programming We onsider a linear programming of the form: MaimizeF n C i X Subet to l n a i Bothl Where C X and u b. i... n X u.... n arethen obetive funtionoeffiients a i them linearinequalityonstrantsand l i maybe positiveornegative. and u and b are parametersin arelower and upperboundswith l u. (6.. Formulation of LP model Mathematial models were onstruted for the prodution of various type of bread produed by the LMU bakery unit. The obetive of the model was to minimize ost of produing a partiular produt after satisfying a set of onstraints. These onstraints were mainly those from nutrients requirements of the bread and the ingredients. The variables in the models were the ingredients while the ost of eah ingredients and the nutrient valued of eah ingredient was the parameter. The speified L.P model for the attainment of the obetive funtion is as follows: a a a a a ( b b b b b b b b ( editor@iaeme.om

5 Optimization Priniple and Its Appliation in Optimizing Landmark University Bakery Prodution Using Linear Programming. DATA COLLECTION AND ANALYSIS OF RESULTS Table below presents five different types of breads produed by LMU bakery, their prodution ost, selling prie and profit. Table : shows basi eight (8 raw materials used for the prodution of bread at Landmark University Bakery, The ombinations of the quantities of these eight basi raw materials (raw material mi for bread prodution per loaf (in grams, and the maimum quantity of eah raw material held in stok for monthly prodution is also aptured in the table. This information is used to determine the prodution ost (in terms of raw materials per loaf of bread produed by the bakery. Table Shows types of Bread, Cost and selling prie with the profits Name of Produt Prodution ost per loaf(n Selling prie per loaf (N Profit (N Family loaf ( Family loaf slie ( Choolate bread ( Medium size loaf ( Small size loaf ( Soure: Landmark bakery 08 Table shows the raw material Mi used for Bread Prodution per Baking Raw Materials Type of Bread and their Raw Material Mi X X X X X Total Quantity Per month in (grams (appro. Flour Yeast Milk 0 Egg Water Flavour Butter Sugar Soure: Landmark Bakery reords editor@iaeme.om

6 N. K Oladeo, A. Abolarinwa, S.O Salawu, A.F Lukman and H.I Bukari.. FORMULATION OF LINEAR PROGRAMMING Both the obetive funtion and the onstraints values were inserted into the linear programming model as shown below Maimize P :80 60 Subet to : Flour : 00 0 Yeast : 0 Milk : 0 Egg :00 00 Water : Flavour: 0 0 Butter : Sugar: (,... n non negativity Formation of Slak Variables In order to represent the above LP model in anonial form, si slak variables w i ( i,...6 were introdued into the model. This hanged the inequalities signs in the onstraint aspet of the model to equality signs. A slak variable will aount for the unused quantity of raw material (if any at end of the prodution. As a result, the above LP model yields: Maimize P : Subet to : Flour : 00 0 Yeast : 0 Milk : 0 Egg :00 00 Water : Flavour : 0 0 Butter : Sugar : (,... n non negativity w 0 w w w w w 6 w 7 w We analyse this program by Simple method proposed by George Danzig (97 and published in Danzig(96 whih have been found to be more effiient and onvenient for omputer software implementation (AMPL program whih is a present day appliation used for solving Mathematial equations editor@iaeme.om

7 Optimization Priniple and Its Appliation in Optimizing Landmark University Bakery Prodution Using Linear Programming.. Program written in AMPL to generate the Results var >= 0; #family size bread var >= 0; #slied family size bread var >= 0; #hoolate bread var >= 0; #medium var >= 0; #00 naira Maimize z: 80* + 60* + 70* + 0* + 0*; s.t. M: 00* + 0* + 0* + 0* + 00* <= ; #flour s.t. M: 0* + * + 0* + * + 0* <= ; #yeast s.t. M: 0* + * + * + * + * <= ; #milk s.t. M: 00* + 00* + 80* + 7* + 0* <= ; #egg s.t. M: 0* + 0* + * + 0* + 0* <= 00000; #sugar s.t. M6: 80* + 80* + 80* + 0* + 0* <= ; #water s.t. M7: 0* + 0* + 0* + 0* + 0* <= 00000; #flavour s.t. M8: * + * + * + * + * <= ; #butter reset; model bs.mod; solve; display,,,,, z; ampl: inlude bs.run; MINOS.: optimal solution found. iterations, obetive = 000 = 0 = 07. = 0 = 0 z = ANALYSIS OF RESULTS GENERATED BY THE AMPL Results from the analysis arried out on the Linear Programming model using Simple method through AMPL software estimated the value of the obetive funtion to be N The ontributions of the five deision variables into the obetive funtion are 000, 0, 07, 0 and 0 respetively. This simply shows that only and variables ontributed meaningfully to improve the value of the obetive funtion of the Linear Programming model with 000 and 07 respetively. From the results of the Linear Programming model, it is therefore desirable and profitable for Landmark University Bakery unit to onentrate muh more on the prodution of (family size bread and (hoolate bread prodution. By this, total sales of about 000 loaves of and 07 loaves of would be sold by the LMU Bakery per month. This would feth the Bakery an optimal profit of about N, 860,000 per month based on the osts of raw materials and the apaity of the oven only. CONCLUSION In this paper, we have suessfully eamines various type, quantities and the ost of Landmark University Bakery prodution. We determine its optimal solution using the 89 editor@iaeme.om

8 N. K Oladeo, A. Abolarinwa, S.O Salawu, A.F Lukman and H.I Bukari seondary data olleted from the reords of the Landmark University Bakery on five types of bread produed in the firm through a linear programming problem formulated as a Mathematial terms using AMPL software. The solution revealed that the bakery manager should onentrate muh more in prodution of loaves of Family loaf and loaves of Choolate bread while others type should be less produed sine their value is gradually turning to zero in order to ahieve a maimum monthly profit of N,860,000. From the analysis, it was also revealed that Family loaf and the Choolate bread ontributed obetively to the highest and optimal profit. Hene, more of Family loaf and Choolate bread are needed to be produed and sold in order to maimize the profit. REFERENCES [] George Danzig. The Dantzig simple method for linear programming. IEEEplore Vol. Issue (97. [] Danzig Linear Programming and Etension. Prineton University Press (96.ISBN [] Ezeliora and Obiafudo.Optimization of prodution ost using Linear Programming solver. Journal of Sientifi and Engineering (0.pp. - [] Oladeo N.K, Abolarinwa A, Salawu S.O, Bamiro M.O, Lukman A.F and Bukari H.I. Appliation of Optimization Priniples in Classroom Alloation using Linear Programming. International Journal of Mehanial Engineering and Tehnology, Vol.0 Issue 0 (09 pp [] Adebiyi, S.O., Amole, B.B., and Soile, I.O Linear optimization tehniques for produt- Mi of paints prodution in Nigeria. AUDCE. Vol.0. No., (0: pp [6] Ibitoye, O., Atoyebi, K.O., Genevieve, K., and Kadiri, K., Entrepreneur Deision making proess and appliation of linear programming tehnique. European Journal of Business, Eonomis and Aountany Vol., No.,. (0 pp.- [7] Feli, M., Judith, M., Jonathan, M., and Munashe, S., Modelling a small farm livelihood system using linear programming in Bindura, Zimbabwe. Researh Journal of Management Sienes, ( (0 pp. 0-. [8] Vakilifard, H., Esmalifalak, H., and Behzadpoor, M., Profit Optimization and Post Optimality Analysis using Linear Programming. World Journal of Soial Sienes Vol., No., (0: pp.7-7 [9] Izaz, U.K., Norkhairul, H.B.., and Imran, A.J., Optimal Prodution Planning for ICI Pakistan using Linear Programming and Sensitivity Analysis. International Journal of Business and Soial Siene, Vol., No., (0: pp.06-. [0] Junaid, A.A., and Mukhtar, H.S., Development of Optimal Cutting Plan using Linear Programming Tools and MATLAB Algorithm. International Journal of Innovation, Management and Tehnology, Vol., No., (00: pp.8-9. [] Raeiyan, K., Neati, F.K., Haati, R., Safari, H.R., and Alizadeh, E., Using Linear Programming in Solving the Problem of Servies Company s Costs.Singaporean Journal of Business Eonomis and Management Studies, Vol., No.0, (0: pp [] Veselovska, L., A Linear Programming Model of Integrating Fleibility Measures into Prodution Proesses with Cost Minimization. Journal of Small Business and Entrepreneurship Development, Vol., No., (0: pp [] Anieting, A.E., Ezugwu, V.O., and Ologun, S. Appliation of Linear Programming Tehnique in the Determination of Optimum Prodution Capaity. IOSR Journal of Mathematis (IOSR-JM, (6, (0: editor@iaeme.om