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# Linear programming

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• ### Linear programming

1. 1. Linear programming<br />SIMPLEX METHOD<br />
2. 2. SIMPLEX METHOD IS AN OPERATIONAL RESEARCH OR MATHEMATICAL TECHNIQUE BY WHICH THE LIMITED RESOURCES OF A FIRM ARE DISTRIBUTED TWO OR MORE COMPETITIVE PROGRAMMES IN SUCH A WAY THAT THE OPTIMAL SOLUTION IS OBTAINED.<br />THE OPTIMAL SOLUTION OBTAINED BY SIMPLEX METHOD IS SAID TO BE THE BEST SOLUTION OF THE PROBLEM.<br />THE SIMPLEX METHOD PROVIDES THE TECHNIQUE OF SOLVING LPP OF ANY MAGNITUDE WHERE THERE ARE TWO OR MORE DECISION VARIABLES.<br />THE SIMPLEX METHOD IS BASED ON THE ASSUMPTION THAT THE SOLUTION OF THE PROBLEM EXIST AT THE CORNER POINTS OF THE FEASIBLE REGION OF BASIC FEASIBLE SOLUTION. <br />MEANING<br />
3. 3. OBJECTIVE FUNCTION: OBJECTIVE IS REPRESENTED BY Z IF PROBLEM IS BASED ON PROFIT THAN MAXIMIZE Z AND ON COST THAN MINIMIZE Z.<br />NON NEGETIVE INSTRUCTION (NNR): VARIABLES CANNOT BE LESS THAN ZERO.<br />CONSTRAINTS: IT MEANS RESTRICTIONS IN ACHIEVING THE OBJECTIVES.<br />FEASIBLE SOLUT ION: THE BEST POSSIBLE SOLUTION WHICH FULFILLS ALL THE ASSUMPTIONS AND THE RESTRICTIONS OF THE PROBLEMS<br />TERMINOLOGY<br />
4. 4. KEY COLUMN: THE COLUMN WITH MAXIMUM POSITIVE NER IS CONSIDERED AS KEY COLUMN.<br />KEY ROW: THE ROW WITH MINIMUM NON NEGETIVE RATIO IS CONSIDERED AS KEY ROW.<br />KEY NUMBER: THE COEFFICIENT AT THE INTERSECTION POINT OF KEY ROW AND KEY COLUMN IS CONSIDERED AS KEY NUMBER.<br />NET EVALUATION ROW (NER): IT IS ALSO KNOWN AS INDEX ROW WHICH REPRESENTS THE NET CONTRIBUTION OF A UNIT OF EACH OF VARIABLES IF ADDED TO THE PRODUCT MIX.<br />
5. 5. FORMULATE LPP.<br />CONVERT INEQUALITIES INTO EQUALITIES.<br />In case of less than equals to constraints slack variable is added in the equation. it represents unused capacity and it is denoted by S1, S2, S3 and so on….<br />In case of greater than equal to constraint surplus variable is deducted from the equation. It represent over used capacity and it is denoted by S1,S2,S3 and so on…….And artificial variable is added in the equation. It represents a very large quantity. And it is denoted by A1,A2,A3 and so on ….<br />In case of equal to constraint artificial variable Is added to make the equation mathematically correct.<br /><ul><li>Profit contribution of slack and surplus variable is considered as ‘0’ and profit contribution of artificial variable is considered as ‘-M’.</li></ul>STEPS FOR SIMPLEX METHOD<br />
6. 6. REFORMULATE LPP.<br />PREPARE FIRST SIMPLEX TABLE.<br /><ul><li>In this basis variable column first priority is given to artificial variables. Than to slack or surplus variables and at last to name variables.</li></ul>DETERMINATION OF KEY COLUMN , KEY ROW AND KEY NUMBER.<br /><ul><li>In case of artificial variables the NER with maximum positive M is considered as key column.
7. 7. When an artificial variables is eliminated from basis variable column it is also eliminated from the main table. </li></li></ul><li>RATIO = QUANTITY/CORRESPONDING COEFFECIENT IN KEY COLUMN.<br />IN NEW TABLE BASIS VARIABLES OF KEY ROW IS REPLACED WITH THE VARIABLE OF KEY COLUMN.<br />NEW VALUE OF KEY ROW = OLD VALUE/KEY NUMBER.<br />NEW VALUE OF REMAINING ROW = OLD VALUE(FROM OLD TABLE) – CORRESPONDING NEW VALUE OF KEY ROW(FROM NEW TABLE) *CORRESPONDING COEFFICIENT IN KEY COLUMN(FROM OLD TABLE).<br />FORMULAS<br />
8. 8. M<br />QUESTION: MAX Z : 2X1+ 3X2 + 4X3<br />SUBJECT TO:<br />3X1 + X2 + 6X3 ≤ 600<br />2X1 + 4X2 + 2X3 ≥ 480 <br />2X1 + 3X2 + 3X3 = 540<br />NNR : X1,X2,X3 ≥ 0<br />ANS: <br />MAX Z : 2X1 + 3X2 + 4X3 + OS1 +OS2 – MA1 – MA2<br />SUBJECT TO:<br />3X1 + X2 + 6X3 + S1 = 600<br />2X1 + 4X2 + 2X3 – S2 + A1 = 480<br />2X1 + 3X2 + 3X3 + A2 = 540<br />NNR : X1,X2,X3,S1,S2,A1,A2 ≥ 0 <br />
9. 9. FIRST SIMPLEX TABLE<br />
10. 10. SECOND SIMPLEX TABLE<br />
11. 11. THIRD SIMPLEX TABLE<br />
12. 12. FOURTH SIMPLEX TABLE<br />
13. 13. Since all the entries in the cj-zj row are either zero or negative , the optimum solution is given by: x1=0, x2=96, x3=84 with maximum z= Rs.624 <br />SOLUTION<br />