6979(Print), ISSN (Online) Volume 4, Issue 1, January - April (2013), IAEME RESEARCH AND DEVELOPMENT (IJIERD) LINEAR PROGRAMMING IN MATLAB

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1 INTERNATIONAL International Journal of Industrial JOURNAL Engineering Research OF INDUSTRIAL and Development (IJIERD), ENGINEERING ISSN 0976 RESEARCH AND DEVELOPMENT (IJIERD) ISSN (Print) ISSN (Online) Volume 4, Issue 1, January - April (2013), pp IAEME: Journal Impact Factor (2013): (Calculated by GISI) IJIERD I A E M E LINEAR PROGRAMMING IN MATLAB Flt Lt Dinesh Kumar Gupta Department of Mechanical Engg, Lovely Professional University, Phagwara (Punjab) India ABSTRACT Since the late 1940s, linear programming models have been used for many different purposes. Airline companies apply these models to optimise their use of planes and staff. NASA has been using them for many years to optimize their use of limited resources. Oil companies use them to optimise their refinery operations. Small and medium-sized businesses use linear programming to solve a huge variety of problems, often involving resource allocation. In this paper, a typical product-mix problem in a manufacturing system producing two products (each product consists of two sub-assemblies) is solved for its optimal solution through the use of the latest versions of MATLAB having the command simlp, which is very much like linprog. As analysts, we try to find a good enough solution for the decision maker to make a final decision. Our attempt is to give the mathematical description of the productmix optimization problem and bring the problem into a form ready to call MATLAB s simlp command. The objective of this paper is to find the best product mix that maximizes profit. The graph obtained using MATLAB commands, give the shaded area enclosed by the constraints called the feasible region, which is the set of points satisfying all the constraints. To find the optimal solution we look at the lines of equal profit to find the corner of the feasible region which yield the highest profit. This corner can be found out at the farthest line of equal profit which still touches the feasible region. Keywords: Constraints, Decision variables, Duality, Feasible region, Optimization and Operating expenses 1. INTRODUCTION One of the most widespread commercial applications of scientific computation is linear programming. It is widely used for allocating resources to achieve an objective. Here, 19

2 in this paper the product-mix problem illustrated as a prototype example [1] refers to a manufacturing system which may occur in real life applications is being solved through MATLAB. The figure 1 represents a manufacturing system producing two products labeled P and Q. The rounded rectangles at the top of the figure indicate the revenue per unit and the maximum sales per week. For instance we can sell as many as 100 units of P for $90 per unit. The circles show the raw materials used, and the rectangles indicate the operations that the products must pass through in the manufacturing process. Each rectangle designates a machine used for the operation and the time required. For example product P consists of two subassemblies. RM stands for raw material.to manufacture the first subassembly, one unit of RM1 passes through machine A for 15 minutes. The output of machine A is moved to machine C where it is processed for 10 minutes. The second subassembly starts with RM2 processed in machine B for 15 minutes. The output is taken to machine C for 5 minutes of processing. The two subassemblies are joined with a purchased part in machine D. The result is a finished unit of P. Product Q is manufactured by a similar process as indicated in the figure1. P Q Fig 1 20

3 The rectangle at the upper left indicates that one machine of each type is available. Each machine operates for minutes per week. OE stands for operating expenses. For this case the operating expenses, not including the raw material cost is $6000. This amount is expended regardless of amounts of P and Q produced. Our problem is to find out the product mix that maximizes profit using MATLAB software. 2. FORMULATION OF LINEAR PROGRAMMING MODEL Let X= Number of P products produced per week Y= Number of Q products produced per week Max Z = 45X + 60Y (Weekly revenue) subject to 15X + 10Y (Machine A) 15X + 30Y (Machine B) 15X + 5Y (Machine C) 10X + 5Y (Machine D) X 100 (Product P) Y 50 (Product Q) X 0,Y 0 (Nonnegativity) 3. SOLUTION USING MATLAB Since there are only two unknowns involved, we may graph all constraints. The MATLAB commands [2] given are as follows: x=0:160 % range of graph y1=(-15.*x)./10; % 15x+10y machine A area(x,y1) y2=max((-15.*x)./30,0); % 15x+30y machineb y3=max((-15.*x)./5,0); % 15x+5y machinec y4=max((-10.*x)/5,0); % 10x+5y machine D y5=max((100-1.*x)./0,0); % 1x+0y 100 market constraint P product y6=max((50-0.*x)./1,0); % 0x+1y 50 market constraint Q product ytop=min([y1;y2;y3;y4;y5;y6]); % array of minima area(x,ytop); % filled area plot The shaded area enclosed by the constraints is called the feasible region, which is the set of points satisfying all the constraints. If this region is empty, then the problem is said to be infeasible, and it has no solution. The lines of equal profit Z are given by Z = 45x + 60y. If we fix Z to, say 100, then all points (x; y) which satisfy 45x + 60y yield the same profit 100. hold on; [u v]=meshgrid(0:160,0:160); contour(u,v,45*u+60*v); hold off; To find the optimal solution, we look at the lines of equal profit to find the corner of the feasible region which yields the highest profit. This corner can be found at the farthest line of equal profit which still touches the feasible region as shown in Fig 2. 21

4 3.1 THE COMMAND simlp Fig 2 The command simlp from the optimization toolbox implements the simplex algorithm to solve a linear programming problem in the form min f * x x subject to A * x b where f is any vector and the matrix A and vector b define the linear constraints. So our original problem is translated into the format max 45x + 60y x,y subject to min 45x 60y x,y subject to 15x +10y x+ 30y x + 5y 15 5 x 10x + 5y 10 5 x y 100 y x y

5 Observe the switching of signs to turn the max into a min and to deal with the constraints. Duality in linear programming is a very important concept, more than just a matter of formatting. The economical interpretation of duality can be simplified into the saying that minimizing the cost of production is equivalent to maximizing the profit. Now we are ready to solve the problem. First we set up the vectors and matrices: f=[-45-60] A=[15 10; 15 30;15 5; 10 5; 1 0; 0 1; -1 0; 0-1] b=[; ; ; ; 100; 50; 0; 0] The optimization toolbox has the command simlp : simlp(f,a,b) -f*ans % optimize % compute profit f = A = b = ans = ans =

6 4. CONCLUSION The objective of this research work in finding the maximum units of products for product mix problem is achieved. The final result as derived using MATLAB are as follows: The values obtained are X = 100 units and Y= 30 units which mean that for the product mix to optimize profits the manufacturer should produce 100 units of P products per week and 30 units of Q products per week. The net profit would be equal to maximum revenue, Z minus total operating expenses, OE which comes out to be $300 per week. The extensions of this research work would be to identify bottlenecks and sensitivity analysis using Fuzzy Algorithm which allows us to determine how sensitive the optimal solution is to changes in data values. This includes analyzing changes in: i) An Objective Function Coefficient (OFC) ii) A Right Hand Side (RHS) value of a constraint. REFERENCES [1] Operations Research Models and Methods by Paul A Jensen [2] B.R Hunt, R.L Lipsman and J.M. Rosenberg. A Guide to MATLAB, for beginners and experienced users, Cambridge University Press, 2001 [3] Varun Shukla, Anand Handa and Arun Agarwal, Spectrum Analysis by using MATLAB Script International journal of Electronics and Communication Engineering & Technology (IJECET), Volume 3, Issue 1, 2012, pp , ISSN Print: , ISSN Online: [4] Gopichand Allaka, Prasad Raju Kalidindi, Koteswara Rao S, Manibabu Daadi and Abhay Patnala, Design of Solid Shafts using MATLAB International Journal of Mechanical Engineering & Technology (IJMET), Volume 3, Issue 3, 2012, pp , ISSN Print: , ISSN Online: