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LINEAR PROGRAMMING – DUAL : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 1 www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS LINEAR PROGRAMMING – DUAL AFTERSCHO? OL – DEVELOPING CHANGE MAKERS CENTRE FOR SOCIAL ENTREPRENEURSHIP PGPSE PROGRAMME – World’ Most Comprehensive programme in social entrepreneurship & spiritual entrepreneurship OPEN FOR ALL FREE FOR ALL

LINEAR PROGRAMMING – DUAL : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 2 www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS LINEAR PROGRAMMING – DUAL Dr. T.K. Jain. AFTERSCHO?OL Centre for social entrepreneurship Bikaner M: 9414430763 tkjainbkn@yahoo.co.in www.afterschool.tk, www.afterschoool.tk

STEPS… : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 3 STEPS… Put the primal in such a way that all requirements are put as constraints. If it is maximising problem, then convert it into less than or equal to type of inequality If it is a minimising problem, then convert it into a greater than or equal to (>=) inequality. If there is any wrong sign, convert it by multiplying it to (-1) If there is some equation, then convert it into 2 inequalities – one of less than type and one of greater than type.

Some fundamental rules… : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 4 Some fundamental rules… If primal is a minimisation problem, its dual will be a maximisation problem The number of variables in dual will be equal to number of constraints in the primal We convert the constraints into the coeffient of the dual objective and vice versa. If primal is a maximisation problem, with less than or equal to constraints, the dual will be a minimisation problem with greater than or equal to constraints (and vice versa also holds good).

Example… : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 5 Example… Minimise : Z = X1+X2+X3 Constraings : 2X1 – 2X2 <= 3 2X2 – X3 >= 5 X1 – 3X2 +4X3 = 5 X1>=0, X2>=0, X3 is unrestricted.

New problem (dual). … : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 6 New problem (dual). … Represent the constraints properly. Minimise : Z = X1+X2+X3 Constraings : 2X1 – 2X2 <= 3 (multiply it by (-1) to make it greater than) 2X2 – X3 >= 5 X1 – 3X2 +4X3 = 5 (convert it into two equations of inequality)

New solution… : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 7 New solution… Minimise : Z = X1+X2+X3 Constraings : -2X1 + 2X2 >= - 3 (multiply it by (-1) to make it greater than) 2X2 – X3 >= 5 X1 – 3X2 +4X3 <= 5(convert it) X1 – 3X2 +4X3 >= 5

…. : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 8 …. Minimise : Z = X1+X2+X3 Constraings : -2X1 + 2X2 >= - 3 2X2 – X3 >= 5 -X1 + 3X2 -4X3 >= - 5 X1 – 3X2 +4X3 >= 5

As X3 is unrestricted, we represent it as X4-X5 : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 9 As X3 is unrestricted, we represent it as X4-X5 Minimise : Z = X1+X2+X4-X5 Constraings : -2X1 + 2X2 >= - 3 2X2 – X4-X5 >= 5 -X1 + 3X2 -4X4 +4X5 >= - 5 X1 – 3X2 +4X3-4X5 >= 5

Matrix of Primal … : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 10 Matrix of Primal …

Now let us prepare dual : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 11 Now let us prepare dual Maximise : Z= - 3W1 +5W2-5W3+5W4 -2W1+0w2-1w3+1w4 <=1 2w1+2w2+3w3-3w4<=1 0w1 –w2 -4w3 +4w4 <= 1 0w1 +1w2 +4w3-4w4<=1

Now solve it as a normal problem. : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 12 Now solve it as a normal problem.

About AFTERSCHO?OL : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 13 About AFTERSCHO?OL PGPSE - World’s most comprehensive programme on social entrepreneurship – after class 12th Flexible – fast changing to meet the requirements Admission open throughout the year Complete support from beginning to the end – from idea generation to making the project viable.

Branches of AFTERSCHO?OL : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 14 Branches of AFTERSCHO?OL PGPSE programme is open all over the world as free online programme. Those who complete PSPSE have the freedom to start branches of AFTERSCHO?OL A few branches have already started - one such branch is at KOTA (Rajasthan).

Workshop on social entrepreneurship : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 15 Workshop on social entrepreneurship We conduct workshop on social entrepreneurship – all over India and out of India also - in school, college, club, association or any such place - just send us a call and we will come to conduct the workshop on social entrepreeurship. These workshops are great moments of learning, sharing, and commitments.

FREE ONLINE PROGRAMME : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 16 FREE ONLINE PROGRAMME AFTERSCHO?OL is absolutely free programme available online – any person can join it. The programme has four components : 1. case studies – writing and analysing – using latest tools of management 2. articles / reports writing & presentation of them in conferences / seminars 3. Study material / books / ebooks / audio / audio visual material to support the study 4. business plan preparation and presentations of those plans in conferences / seminars

100% placement / entrepreneurship : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 17 100% placement / entrepreneurship AFTERSCHO?OL has the record of 100% placement / entrepreneurship till date Be assured of a bright career – if you join AFTERSCHO?OL

Pursue professional courses along with PGPSE : 

www.afterschoool.tk AFTERSCHO?OL's MATERIAL FOR PGPSE PARTICIPANTS 18 Pursue professional courses along with PGPSE AFTERSCHO?OL permits you to pursue distance education based professional / vocational courses and gives you support for that also. Many students are doing CA / CS/ ICWA / CMA / FRM / CFP / CFA and other courses along with PGPSE. Come and join AFTERSCHO?OL

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