i don't understaond the transportation problem in example 11of ncert book.plz elaboratly solve it ??

Asked by sweety | 23rd Oct, 2015, 09:14: PM

Expert Answer:

H o w space m a n y space u n i t s space s h o u l d space b e space t r a n s p o r t e d space f r o m e a c h space f a c t o r y space t o space e a c h space d e p o t space i n space o r d e r space t h a t space t h e t r a n s p o r t a t i o n space cos t space i s space m i n i m u m. space W h a t space w i l l space b e space t h e space m i n i m u m space t r a s p o r t a t i o n space cos t ?
 
S o l u t i o n colon L e t space apostrophe x apostrophe space b e space t h e space n u m b e r space o f space u n i t s space t r a n s p o r t e d space f r o m space t h e space f a c t o r y space a t space P space t o space t h e space d e p o t space a t space A. A n d L e t space apostrophe y apostrophe space b e space t h e space n u m b e r space o f space u n i t s space t r a n s p o r t e d space f r o m space t h e space f a c t o r y space a t space P space t o space t h e space d e p o t space a t space B. T h e space p r o d u c t i o n space c a p a c i t y space o f space P space i s space 8 space u n i t s. T h u s comma space P space c a n space s e n d space 8 minus x minus y space u n i t s space t o space t h e space d e p o t space a t space C. S i n c e space 8 minus x minus y greater or equal than 0 comma space w e space h a v e comma space x plus y less or equal than 8... left parenthesis 1 right parenthesis  N o w space c o n s i d e r space t h e space f a c t o r y space Q. S i n c e space P space i s space s e n d i n g space apostrophe x apostrophe space u n i t s space t o space d e p o t space A space a n d space t h e space r e q u i r e m e n t space o f space A space i s space 5 space u n i t s. T h u s comma space Q space w i l l space s e n d space 5 minus x space u n i t s space t o space d e p o t space A. S i n c e space 5 minus x greater or equal than 0 comma space w e space h a v e comma space x less or equal than 5... left parenthesis 2 right parenthesis  S i m i l a r l y comma space P space i s space s e n d i n g space apostrophe y apostrophe space u n i t s space t o space d e p o t space a t space B space a n d space t h e space r e q u i r e m e n t space o f space B space i s space 5 space u n i t s comma a n d space h e n c e comma space Q space w i l l space s e n d space 5 minus y space u n i t s space t o space d e p o t space B. S i n c e space 5 minus y greater or equal than 0 comma space w e space h a v e comma space y less or equal than 5... left parenthesis 3 right parenthesis space  T h e space p r o d u c t i o n space c a p a c i t y space o f space t h e space f a c t o r y space Q space i s space 6 space u n i t s. H e n c e space Q space c a n space s e n d space open square brackets 6 minus open parentheses 5 minus x plus 5 minus y close parentheses close square brackets equals x plus y minus 4 space u n i t s space t o space t h e space d e p o t space C. S i n c e space x plus y minus 4 greater or equal than 0 comma space w e space h a v e comma space x plus y greater or equal than 4.... left parenthesis 4 right parenthesis  T r a n s p o r t a t i o n space cos t space f r o m space P space t o space A space i s space 160 cross times x. space T r a n s p o r t a t i o n space c o s t space f r o m space P space t o space B space i s space 100 cross times y. T r a n s p o r t a t i o n space c o s t space f r o m space P space t o space C space i s space 150 cross times open parentheses 8 minus x minus y close parentheses space.  T r a n s p o r t a t i o n space c o s t space f r o m space Q space t o space A space i s space 100 cross times open parentheses 5 minus x close parentheses. space T r a n s p o r t a t i o n space c o s t space f r o m space P space t o space B space i s space 120 cross times open parentheses 5 minus y close parentheses. T r a n s p o r t a t i o n space c o s t space f r o m space P space t o space C space i s space 100 cross times open parentheses x plus y minus 4 close parentheses space.  T h u s comma space t h e space t o t a l space t r a n s p o r t a t i o n space cos t space i s equals 160 x plus 100 y plus 150 open parentheses 8 minus x minus y close parentheses plus 100 open parentheses 5 minus x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space plus 120 open parentheses 5 minus y close parentheses plus 100 open parentheses x plus y minus 4 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 10 open parentheses x minus 7 y plus 190 close parentheses O u r space a i m space i s space t o space m i n i m i z e space t h e space t r a n s p o r t a t i o n space cos t. H e n c e space t h e space p r o b l e m space r e d u c e s space t o space M i n i m i z e space Z equals 10 open parentheses x minus 7 y plus 190 close parentheses s u b j e c t space t o space t h e space c o n s t r a i n t s space x greater or equal than 0 comma space y greater or equal than 0 space x plus y less or equal than 8... left parenthesis 1 right parenthesis x less or equal than 5 space space space space space.... left parenthesis 2 right parenthesis y less or equal than 5 space space space space space space... left parenthesis 3 right parenthesis x plus y greater or equal than 4.... left parenthesis 4 right parenthesis  G r a p h space t h e space a b o v e space e q u a t i o n s space a s space f o l o w s colon

Answered by Vimala Ramamurthy | 24th Oct, 2015, 02:02: PM