Proceedings of the OAU Faculty of Technology Conference 2015
OPTIMIZATION OF PRODUCTION UNITS USING MATLAB PROGRAM: A CASE STUDY OF THE FEDERAL POLYTECHNIC, ADO-EKITI FOUNDRY SHOP S. O. Ejiko1, J. Adu2,* and P. O. Ajewole3 1
Department of Mechanical Engineering, The Federal Polytechnic, Ado-Ekiti, Nigeria. Department of Mechanical Engineering, Obafemi Awolowo University, Ile-Ife, Nigeria 3 Department of Agricultural & Bio-Environmental Engineering, The Federal Polytechnic, Ado-Ekiti, Nigeria. 2
*Email of Corresponding Author:
[email protected]
ABSTRACT This paper explores the possibility of maximizing the profit of a mechanical product manufacturing business by optimizing the production units using linear programming tool under varying constraints. A Visual Basic program was also developed to solve the linear programming problem. The foundry shop at the Mechanical Engineering Department of the Federal Polytechnic, Ado-Ekiti was used as a case study. The foundry produces three major components which include 5 litre pot, 3litre jug and grinding teeth plate of aluminium respectively, all by casting. Time for varying operations such as moulding, casting (pouring and solidifying) and finishing were collected for each product. A mathematical model equation for profit was developed based on the previous profit data of the foundry and subjected to constraint equations based on the time required to finish each of the three products denoted X, Y and Z. The profit function was analysed manually and by MATLAB program in order to obtain the quantity of each product that will give maximum profit. The result obtained revealed that 10 units, 4 units and 6 units of pot, jug and grinding teeth plate respectively will have to be produced per day in order to obtain a maximum profit of ₦1,880.00 in a day for both methods. This optimized profit margin reflects that the business is highly profitable when the resources and time available are efficiently utilized. Keywords: linear programming, optimization, production, units, MATLAB, profit resistance, fatigue strength, creep strength to mention but a few is used in metal casting. This is a liquid shaping process in which the liquid is made to conform to a desired geometry in a mould and then allowed to transform into a solid (Piwonka, 1986). The case of solidification as treated by Apelian (1986) which implies that the transport mechanism, kinetics of heat, fluid and mass flow during liquid to solid transformation of the alloy directly control the resultant cast structure which dictates the properties and performance of the cast components. At the end of simulation process, the numbers of component obtained will go a long way in optimizing the objectives of the company. This tool is not limited to production process only; it can also be applied to other social function in life provided the initial modelling conditions are met. Linear programming (L.P.) is a tool for solving optimization problems. It is a mathematical tool (model) for determining the optimal (maximum or minimum) value of a given function called the objective function, subject to a set of stated restrictions, or constraints, placed on the variable concerned (Croucher, 1980 and Stroud, 1996). Also, optimization can be referred to as the act of obtaining the best result under given circumstances. Again, it can be defined as a process of finding the conditions that give the maximum or minimum value of a given function (Beighter et al., 1979). Since invention is the mother of necessity, the simplex method was invented
INTRODUCTION In production engineering where the objectives of a firm are centred on the increment in profit and reduction in the cost of production, it becomes imperative to evaluate the equivalent units of production that favours the company's goal. Mathematical programming is a technique for solving certain kinds of problems (notably maximizing the profits and minimizing costs) subject to constraints on resources, capacities, supplies, demand and the like (Bender, 2000). The mathematical tools for attaining profitability index are of varying forms such as economic order quantity, break even method, and optimization techniques to mention but a few from several. The data obtained with respect to the formulated model will determine the technique to be utilized (Ikechukwu, 1989). The application of linear programming techniques into the production of Aluminium components through casting will go a long way in the expansion of the workshop activities (The Federal Polytechnic, Ado-Ekiti Foundry Workshop) knowing very much that Aluminium is one of the most abundant metals on earth (Abass, 2000; Ramsden, 1994). The basic processes such as mould preparation, pouring and solidification and finishing were considered against the time for varying products such as pot, jug and grinding teeth plate with respect to the available time per day. The case study of foundry shop largely depends on Aluminium alloy for casting which has properties such as compressive strength, corrosion
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external to the project (e.g. interest rate changes), and optimizing model will make the comparisons that management deems important; consider major risks and constraints on the projects and then select the best overall project or set of projects.. 3. Flexibility: The model should give valid results within the range of conditions the firm might experience. It should have the ability to be easily modified, or to be self-adjusting in response to changes in the firm's environment; for example, tax law change, new technological advancements alter risk levels and, above all, the organization's goals change. 4. Ease of use: the model should be reasonably convenient, neither should it take a long time to execute, but be easy to use and understand. It should not contain interpretation data that are hard to acquire, excessive personnel, or unavailable model's variables should also relate one to one with, those real world parameters the managers believe are significant to the project. Finally, it should be easy to stimulate the expected outcomes associated with investments in different project portfolios. . 5. Cost: Data-gathering and modelling costs should be low relative to the cost of production and must surely be less than the potential benefits of the project. All costs should be considered, including the costs of data management and of running the mode (Jack and Samuel, 1988). Ihueze and Okafor (2010) established an optimum forecast model to predict future production trends of 7UP Bottling Company in the presence of seasonal variation and trend components of the collected data. The data was a 60 months’ time series. He concluded that production increases by 0.002579KG/Month. Also, Salami (2014) solved the transportation problem of the distribution department of 7UP Bottling Company at Ilorin. He made use of three methods of solving transportation problem and tried to compare them. He concluded the results were the same. The scope of this research is to determine the optimum value for profitability for the particular Foundry Workshop under study. Based on the data available, linear programming tool is selected for determining the optimization value. The objective function is the profit while the constraints include the time for moulding, casting and finishing of three products (pot, jug and grinding teeth plate). Then, MATLAB was used to solve similar problem.
by George Dantzig in 1947 (Nelder and MeadT, 1995). The method is simple and very powerful algorithm in solving real world problems relating to linear programming. The L.P. is a tool for project evaluation and selection which we adopt for decision aiding model (Richard et al., 1974). There are various project selection model for deriving varying solutions such as that of economic-batch/lot size, payback period and regression analysis (Adejuyigbe, 2002). There are criteria for selecting an appropriate model. We live in the midst of what has been called the "knowledge explosion”. We frequently hear such comments as "90 percent of all we know about physics has been discovered since Albert Einstein published his original work on special relativity"; and "80 percent of what we know about the human body has been discovered in the past fifty years”. In addition, evidence is cited to show that knowledge is growing exponentially. Such statements emphasize the importance of the management of change. To survive, firms must develop strategies for reassessing the use of their resources. Every allocation of resources is an investment in the future. Due to the complex nature of most strategies, many of these investments are in projects. The proper choice of investment projects is crucial to the long-run survival of every firm. On daily basis, we witness the results of both good and bad investment choices. On the front pages of our newspapers, we read about the success or failure of past decisions made by Ashland Oil regarding the maintenance of its fuel Storage tanks, by IBM concerning the timing of introducing its PS/2 line of computers, and by General Motors about building and marketing the Cadillac Allante. But can such important choices be made sensibly? Once made, do they ever change and if so how? These questions reflect the need for effective selection models. Within the limits of their capabilities, such models can be used to increase profits, to select investments for limited capital resources, or to improve the competitive position of the organization. They can be used for ongoing evaluation as well as initial selection, and thus are a key to the allocation of the organization's scarce resources. When a firm chooses a project selection model, the following criteria, based on Souder (1973) are most important. 1. Realism: The model should reflect the reality of the manager's decision situation, including the multiple objectives of both the firm and its managers. Without a common measurement system, direct comparison of different projects is impossible. For example, Project A may strengthen a firm's market share by extending facilities and Project B might improve its competitive position by strengthening staff. Other things being equal, which is better? The model shows the realities of the firm's limitations on facilities, capital, personal etc. The model should also include factors for risk both the technical risks of performance, cost, and time and the market risk of customer rejection. 2. Capability: The model should be sophisticated enough to deal with multiple time periods, simulate various situations both internal and
METHODOLOGY The Foundry Shop of the Federal Polytechnic, Ado-Ekiti was selected as the case study. The shop produces three major components which include 5 litre pot, 3 litre jug and grinding teeth plate of aluminium by casting. Data were collected during the workshop operation. Time of varying operations were determined by the use of diamond stop watch through observation and monitoring of the product processes which includes moulding, casting and finishing. The data gathered, as shown in Table 1.0, gives the average
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moulding time of 40 minutes, 30 minutes and 20 minutes; casting time of 20, 10, and 40 minutes; and finishing time of 10, 20 and 10 minutes for the 5 litre pot, 3 litre jug and grinding teeth plate of aluminium respectively. The data were tabulated and analysed by linear programming method to determine the optimum units of production and profit which when adopted by the workshop will go a long way to increase the financial capability of the shop. Thus, bring about the expansion of the Foundry business.
Table 1: Data gathered from the production process
SXitre Pot 3 Litre Jug Grinding Teeth Plate Available Time per day
Table 3: Optimisation Table Basis X
z 2 4 1 -10
W1 1 0 0 0
W2 0 1 0 0
W3 0 0 1 0
Casting (Mins) 20 10 40 480
Finishi ng (Mins) 10 20 10 240
Profit Per component (N) 80 120 100
Problem Solving Procedure In solving this problem, each equation above was divided through by 10 and slack variables (W1, W2 and W 3) were introduced to the constraints. Then, this becomes: 4 x + 3y + 2z + W1 = 64 2x + y + 4z +W2=48 x + 2y + z+ W 3 = 24 P-8x-12y-10z=0
Formulation of Mathematical Model Decision variables x, y and z are stated as follows: X = Number of 5 litre Pot produced in a day Y = Number of 3 litre Jug produced in a day Z = Number of Grinding Teeth Plate produced in a day The target is to maximize the company profit. Considering Table 1, the objective function is now: P=80x+ 120y+ 100z Subject to: 40x + 30y + 20z =640 (moulding time) 20x+10y + 40z =480 (casting time) 10x + 20y +10z =240 (finishing time) These are the constraints. With x, y, z ≥ 0 constants
Table 2: Simplex Tableau Basis X Y W1 4 3 W2 2 1 W3 1 2 P -8 -12
Mould ing (Mins) 40 30 20 640
Referring to Table 2, the simplex tableau is constructed by starting the basic solution. Recall that there are now three constraints and six variables, x, y, z, W1, W2, W3 . If we start by letting x, y and z to be zero, then we have the temporary solutions: W1 = 64, W2 = 48 and W3 = 24. The columns with the slack variable form a unit matrix. The objective function is moved across the equal sign so as to obtain a function that corresponds to the simplex tableau.
B 64 48 24 0
Check 74 56 29 -30
Y
z
W1
W2
W3
B
Check
W1 W2 W3 P W1 W2
4 2 1 -8 5/2 3/2
3 1 2 -12 0 0
2 4 1 -10 ½ 7/2
1 0 0 0 1 0
0 1 0 0 0 1
0 0 1 0 -3/2 -1/2
64 48 24 0 28 36
74 56 29 -30 61/2 83/2
Y
½
(1)
½
0
0
½
12
29/2
P W1 Z Y P
-2 16/7 3/7 2/7 -2/7
0 0 0 l 0
—4 0 (1) 0 0
0 1 0 0 0
0 -1/7 2/7 -1/7 8/7
6 -10/7 -1/7 4/7 38/7
144 160/7 72/7 48/7 1296/7
144 172/7 83/7 60/7 1340/7
X Z Y P
1 0 0 0
0 0 l 0
0 1 0 0
7/16 -3/16 -2/16 2/16
-1/16 5/16 -2/16 18/16
-10/16 2/16 ¾ 84/16
10 6 4 188
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172/16 116/16 88/16 3112/16
Procedure for Analysing The procedure for solving the linear program includes: (i) The column containing the most negative entry in the index row was selected and enclosed. This is referred to as the key column. (ii) Referring to Table 2, the values in the “B” column were divided by the positive entry in the key column. The smallest ratio determines the key row. The number at the intersection of the key column and the key row becomes the required pivot. (iii) The pivot was made unity by dividing all the values in its row by itself. (iv) In order to generate a new set of numbers for other rows, a number was picked on the table and traced vertically as well as horizontally so as to obtain its corresponding entries on the key row and key column. Subtracted the product of these entries from this number. The result was used to replace the old number on the tableau. This was repeated until all the numbers on the tableau were used. (v) The process i to iv was applied continuously until the x, y, z column form a unity matrix and the values on the “B” column became the corresponding optimized value. Upon completing the manual computation, MATLAB program was written to solve the same problem. The program for optimizing the linear equations is clearly described in appendix 1.
CONCLUSION The result clearly reflects the optimized value of the product which a layman can easily understand and apply to maximize his profit. The application of linear optimization tool will go a long way in evaluating the economic viability of the enterprise. Hence, deduce the number of components to be produced, the number of workers to be employed and the capacity of equipment to be purchased.
REFERENCES Abass, A.O., Introductory organic and inorganic chemistry, 2nd Edition. OGFAT Publications, Ibadan, 2000. Adejuyigbe, S.B., Production management, 1st Edition. Topfun Publisher, Akure, 2002. Apelian, D., Cast structure of alloys. In: Encyclopedia of materials science and engineering, Vol. 1: Bever, M.B. ed. Pergamon Press Ltd., Uk, 1986. Beighter, C. S., Philip, D.T and Wilde D.J., Foundation of optimization, 2nd Ed. Prentice Hall, Englewood, 1979. Bender, E., An introduction to mathematical modeling, Dover Publications, NY, 2000. Croucher, J. S., Operation research: A first course. 1 st Ed. Pergamon Press Pty Ltd., Australia, 1980. Ihueze, C. C. and Okafor, E.C., Multivariate time series analysis for optimum production forecast: A case study of 7up Soft Drink Company in Nigeria. African Research Review. 8 (3a): 276-305, 2010. Ikechukwu, J. M., Economic optimization of a refinery using linear programming. Unpublished MSc. Project, Department of Mechanical Engineering, University of Benin, 1989. Jack, R. M. and Samuel, J. M., Project management a managerial approach, 2nd Ed. John Wiley and Sons Publisher, New York, 1988. Nelder, J.A. and MeadT, R., A simplex method for function minimization. Computer Journal, Vol. 7(4), 308-313, 1965. Piwonka, Y.S., Casting of metals. In: Encyclopedia of materials science and engineering, Vol. 1: Bever, M.B. ed. Pergamon Press Ltd., Uk, 1986. Ramsden, E.N., A- level chemistry, 3rd Ed. Stanley Thornes Publishers Ltd., 1994. Richard, A.J., William, T.N. and Roger, C.V., Production and operation management: A system concept. Houghton Mifflin, Boston, 1974. Salami, A.O., Application of transportation linear programming algorithms to cost reduction in Nigeria soft drinks industry. World academy of science, engineering and technology, 8 (2): 416-422, 2014. Souder, W.E., Utility and perceived acceptability of R&D project selection models. Management Science Journal, 19(12): 1384-1394, 1973. Stroud, K. A., Further Engineering Mathematics, 2nd Edition. ELBS Publisher, China, 1994.
RESULTS AND DISCUSSION The results of computation show the optimized quantity of product to be produced in order to achieve the maximum profit. The products quantity as estimated includes 10 units of Pot (X) per day, 4 units of jug (Y) per day and 6 units of Teeth plate (Z) per day with a maximum profit of ₦1,880.00. The manually calculated results correspond perfectly with the MATLAB program. Considering the fact that an average of 22 days can be utilized for components production on monthly basis. Where the total products are sold, a monthly profit of ₦41,360.00 shows that the business is highly profitable and suitable for small scale entrepreneur. The linear optimization gives the optimum units and profit of the products for efficient utilization of the available resources and time. This result also shows that greater attention should be given to production of pot because of its volume and profit generated that is almost half of the entire profit. Without going through a rigor of manual computation, the program written can be used to solve similar problem. This is because the result generated from both the manual and the program are the same. This can be verified by comparing the result presented on Table3 and appendix II. Also, optimization tool box under MATLAB was used to confirm the results as shown in appendix iii and iv.
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[row] = MRT(A(1:m-1,n),t);%get the row of the pivot
APPENDIX I: MATLAB Program function [X, Pmax] = simplex2(P, A, b) %P is the vector [a1,a2,a3...] of the coefficients of the profit function % z = a1*x1+a2*x2+a3*x3+... to be maximised % % If z is to be minimised, P = -(a1,a2,a3...) should be used instead
fprintf('pivot row-> %g pivot column-> %g \n',row,col) %divide the pivot row to make the pivot element 1 A(row,:)= A(row,:)/A(row,col); % eliminate the other elements on the pivot column by row operations for i = 1:m if i ~= row A(i,:)= A(i,:)-A(i,col)*A(row,:); end end [mi,col] = Br(A(m,1:n-1));
%A is the matrix of the left hand side to the constraint equations in the %form A eps t = A(1:m-1,col); %elements of the pivot column singled out
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m = d(j); else m = []; j = []; end
APPENDIX III: Command line interphase for Linprog function
APPENDIX II: Result From the MATLAB Program >> [a,b] = simplex2([80 120 100], [40 30 20;20 10 40;10 20 10], [640 480 240]) Initial tableau 40 30 20 1 0 0 640 20 10 40 0 1 0 480 10 20 10 0 0 1 240 -80 -120 -100 0 0 0 0 pivot row-> 1 pivot column-> 1 Tableau 1 1 0.75 0.5 0.025 0 0 16 0 -5 30 -0.5 1 0 160 0 12.5 5 -0.25 0 1 80 0 -60 -60 2 0 0 1280 pivot row-> 3 pivot column-> 2 Tableau 2 1 0 0.2 0.04 0 -0.06 11.2 0 0 32 -0.6 1 0.4 192 0 1 0.4 -0.02 0 0.08 6.4 0 0 -36 0.8 0 4.8 1664 pivot row-> 2 pivot column-> 3 Tableau 3 1 0 0 0.04375 -0.00625 0.0625 10 0 0 1 -0.01875 0.03125 0.0125 6 0 1 0 -0.0125 -0.0125 0.075 4 0 0 0 0.125 1.125 5.25 1880 a= 10 4 6 b= 1880
APPENDIX IV: MATLAB optimization tool interphase
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