TRANSPORTATION TRANSPORTATION AND ASSIGNMENT ASSIGNMEN T PROBLEM PROB LEM Transportation ansportation probe! " 1 . What do you understand by Tr
It is a special type of linear programming model in which th goods are shipped from various origins to different destinations. The objective is to find the best possible allocations of goods from various origins to different destinations such that the total transportation cost is minimum. #.
Write the !athe!ati$a %or! o% Transportation Probe!
m Min Z = ∑ i=1
n
∑
Cij ij subject to the constraints
j=1
n
∑ ij = ai i = 1!"! #####..! m j=1
m
∑ ij = b j j = 1!"! #####..! n $here i=1 ai = %umber of units available in iththe origin b j = %umber of units re&uired in j th destination Cij = transportation cost from iththe origin to jth destination &.
What is an unbaan$ed transportation probe! " 'o( to so)e it"
' transportation problem is said to to be unbalanced if the total total supply is not e&ual to the total demand ie m
n
∑ ai ≠ ∑ b j i=1
j=1
the unbalanced transportation problem is converted in to a balanced one by adding a dummy row or dummy column whichever is necessary. The unit transportation cost for the dummy row or column elements are assigned (ero. Then the problem is solved by the usual procedure. *. De%ine +i, -easibe soution +ii, Basi$ %easibe soution +iii, Nonde/enerate soution
)easible solution * ' set of non+negative values ij satisfies the constraint e&uation is called a feasible solution. ,asic feasible solution * ' basic feasible solution is said to be basic! if the number of positive allocations are m-n+1 . If the number of allocations are e&ual to m-n+1! it is called non+ degenerate basic feasible solution. 0. What do you understand by de/enera$y in a transportation probe!
If the number of occupied cells in a m n transportation problem! is less than m-n+1/ ! then the problem is said to be degenerate. 1. When does a TP ha)e a uni2ue soution "
$hile doing optimality test! if a empty cell evaluations ie ∆ij = Cij 0 ui - vj/ are positive! then the problem is said to be have an uni&ue solution. 3. What is the purpose o% MODI !ethod "
M2I method is the test procedure for optimality involves eamination of ea ch unoccupied cell to determine whether or not ma3ing an allocation in it reduce the total transportation cost and then repeating this procedure until lowest possible transportation cost is obtained. 4. List any three approa$hes used (ith TP %or deter!inin/ the startin/ sooution +or, the initia basi$ %easibe soution.
a. %orth 0 $est corner rule ii/ 4east cost entry method iii/ 5ogel6s approimations method. 5. 'o( (i you identi%y that a TP has /ot an aternate opti!a soution "
$hile doing optimality test! if any empty cell evaluation ie
∆ij = Cij 0 ui - vj/ = 7 then the problem is said to have an alternate optimal solution. 67. When do you say that the o$$upied $e is in independent position "
$hen it is not possible to draw a closed loop from the allocations! the occupied cell is in independent position . 66. What is !a8i!i9ation type transportation probe! and ho( these probe!s are so)ed"
The main objective of transportation problem is to minimi(e the transportation cost. In maimi(ation type problems! the objective is to maimi(e the profit or maimi(e the total sales.
To solve these problems! we have to convert all the cell entries by multiplying 01. Then the problem is solved by the usual method. 6#.
a.
Write do(n the basi$ steps in)o)ed in so)in/ a transportation probe!.
To find the initial basic feasible solution
b. To find an optimal solution by ma3ing successive improvements from the initial basic feasible solution. 6&.
State the ne$essary and su%%i$ient $ondition %or the e8isten$e o% a %easibe soution to a transportation probe!"
The necessary and sufficient condition for the eistence of feasible solution is a solution that satisfies all conditions of supply and demand. 6*. What is an assi/n!ent probe! " Gi)e t(o appi$ations "
It is a special type of transportation problem in which the number of jobs allocated for different machines or operators. The objective is to maimi(e the o verall profit or minimi(e the total cost for a given assignment schedule. a.
It is used in production environment in which the number of jobs are assigned to number of wor3ers or machines in such a way that the total time to complete all the jobs will be minimum.
b. It is used in traveling salesman. 60. What do you !ean by an unbaan$ed AP"
8ince the assignment is one to one basis the problem have a s&uare matri. If the given problem is not a s&uare matri ie the number of rows and columns are not same then it is called unbalanced assignment problem. To ma3e it a balanced assignment add a dummy row or dummy column and then convert it into a balanced one. 'ssign (ero cost values for the dummy row or column and solve it by usual assignment method. 61. State the di%%eren$e bet(een TP : AP
'ssignment 'llocations are made one to one
Transportation More than one allocation is possible in each
basis.Therefore only one occupied cell
row and each column . :ence it neet not be
will be 9resent in each row and each
a s&uare matri.
column. :ence the table will be a s&uare matri. It will always provide degeneracy
It will not provide degeneracy
The supply at any row and demand at
The supply and demand may have any
any column will be e&ual to 1
positive &uantity.
63. 'o( do you $on)ert the !a8i!i9ation probe! in to a !ini!i9ation one "
To solve the maimi(ation problem in to minimi(ation assignment problem! first convert the given maimi(ation matri in to an e&uivalent minimi(ation matri form by multiplying 01 in all the cost elements. Then the problem is a maimi(ation one and can be solved by the usual assignment method. 64. Gi)e the inear pro/ra!!in/ %or! o% A.P.
bjective is to minimi(e the total cost involve.
Min Z
n
n
=∑
∑
i=1
Cij ij subject to the constraints
j=1
11 -1" - 1; - #### - 1n = 1
"1 -"" - ";- #### - "n
= 1
;1 -;" - ;; - #### - ;n
= 1
##############
##############
############## ##############.
n1 - n" - n; - ####. - nn
=
1
65. What is the na!e o% the !ethod used in /ettin/ the opti!u! assi/n!ent "
:ungarian method. #7. What is the indi$ation o% an aternate soution in an assi/n!ent probe! "
If the final cost matri contains more than the re&uired number of (ero for assignment at independent position then it indicated that the problem has an alternate optimal solution.
#6. What do you inderstand by resti$ted assi/n!ents " E8pain ho( shoud one o)er$o!e it "
In assignment problems! it is assured that the performance of all the machines and operators are same. :ence any machine can be assigned to any job. ,ut in practical cases! a machine cannot do all the operations of a job and operator cannot do all 3inds of tas3s. Therefore a high processing time is assigned to the impossible cell M or ∞/ and then it will be solved by the usual assignment method. In the final assignment the restricted cell will not be present. ##. Write t(o theore!s that are used %or so)in/ assi/n!ent probe!s
Theorem 1 * The optimum assignment sschedule remains unaltered if we add or subtract a constant to < from all the elements of the row or column of the assignment cost matri. Theorem " * If for an assignment problem all Cij ≥ 7! then an assignment schedule ij/ which satisfies ∑ Cij ij = 7 ! must be optimal. #&. Write the !athe!ati$a %or!uation o% an assi/n!ent probe!.
n
n
Min Z = ∑
∑
i=1
Cij ij subject to the constraints
j=1
n
∑
ij
=1
ij
=1
i=1
n
∑ j=1
ij
= {1! if i th job is assigned to j th operator
{7! otherwise Cij = Cost of assigning the n th job to m th machine. #*. What is tra)ein/ saes!an probe! and (hat are its ob;e$ti)es "
In this model a salesman has to visit n cities. :e has to start from a particular city! visit each city once and then return to his starting point. The main objective of a salesman is to select the best se&uence in which he visited all cities in order to minimi(e the total distance traveled or minimi(e the total time. #0. Why assi/n!ent probe! (i a(ays pro)ide de/enera$y "
In assignment problem! the allocation is one to o ne basis therefore! the number of occupied cells in each row and each column will be eactly e&ual to 1. :ence assignment problem will always provide degeneracy. #1. Why a transportation te$hni2ue or the si!pe8 !ethod $annot used to so)e an assi/n!ent probe!
The transportation techni&ue or simple method canno t be used to solve the assignment problem because of degeneracy .
PARTB 6. Obtain the initia soution %or the %oo(in/ TP usin/ NWAM e c r u o 8
1 " ; ? 2emand
' " ; @ 1 >
2estination , > ; ? B
C ? 1 > " 1A
8upply @ A > 1? ;?
#. So)e the TP (here the $e entries denote the unit transportation $osts. e c r u o 8
9 F e E c r u o 2emand 8
2estination ' , C 2 8upply &. So)e the %oo(in/ TP. @ ? " B "7 A ; @ > ;7 2estination @ ? B @7 1 " ; Capacity 17 ?7 "7 ;7 177 1 " " ; 17 " ? 1 " 1@ ; 1 ; 1 ?7 2emand "7 1@ ;7
*. -ind the !ini!u! transportation $ost. y r o t c a )
)1 )" ); 2emand
21 1 >7 ?7 @
$arehouse 2" 2; ;7 @7 ;7 ?7 A >7 A >
2? 17 B7 ;7 1?
8upply > 1A
0. -ind the opti!a soution by usin/ >AM. y r o t c a )
1 " ; ? Ee&uirement
' > B B ?
, 1" ; @ A ?
$arehouse C 2 B > > 11 11 " B "
D @ ; " ?
1. So)e the TP. e c r u o 8
1 " ; 2emand
' 11 "1 A ;7
2estination , C "7 > 1B "7 1" A "@ ;@
2 A 1" ?7
8upply @7 ?7 >7
) 17 @ 11 17 "
'vailable @ B "
3. So)e the %oo(in/ TP to !a8i!i9e the pro%it. e c r u o 8
2estination ' 1 ?7 " ?? ; ;A 2emand ?7
, "@ ;@ ;A "7
C "" ;7 "A B7
2 ;; ;7 ;7 ;7
8upply 177 ;7 >7
4. So)e the %oo(in/ unbaan$ed TP. M 1 E )
1 " ; 2emand
1 @ B ; >@
T " 1 ? " "7
; > B @ @7
8upply 17 A7 1@
5.
81 8" 8; 8? 21 2"
7 17 1@ 1A 1@ 17
8" B 7 "7 "@ "7 "@
8; "? B 7 17 B7 "@
8? > 1" A 7 1@ ";
21 "? @ ?@ ;7 7 ?
2" 17 "7 > B 17 7
66. A %ir! ha)in/ # sour$es S6 : S# (ishes to ship its produ$ts to # destinations D6 : D#. The nu!ber o% units a)aiabe at S6 : S# are 0 : #0 resp. and the produ$t de!anded at D6 : D# are #7 : 67 units respe$ti)ey. The %ir! instead o% shippin/ dire$ty de$ides to in)esti/ate the possibiity o% transship!ent. The unit transportation $osts +in rupees, are /i)en in the %oo(in/ tabe. -ind the opti!a shippin/ s$hedue.
8ource
2estination
'vailable
81 7 " ; ? +
8ource
81 8" 2estination 21 2" 2emand
8" " 7 " ? +
21 ; " 7 1 "7
6#. So)e the AP?
' 1 ? A
, ? > @ >
C B 17 11 A
2 ; > @
' 17 1@ ;@ 1>
, "@ ;7 "7 "@
C 1@ @ 1" "?
2 "7 1@ "? "7
I II III I5 6&. So)e the AP?
I II III I5
6*. So)e the Assi/n!ent probe!? s 3 s a T
I II III I5 5
' 1 " @ ; 1
Men , C ; " ? ; B ; 1 ? @ B
2 A 1 ? " @
D A @ B " ?
? 1B B 1"
@ "7 1@ 1B
60. So)e the Assi/n!ent probe!? s b o G
' , C
Machine 1 " 11 1> > 1; 1B
; A 1" 1@
2" ? ? 1 7 17
@ "@ + +
2 D
"1 1?
"? 17
1> 1"
"A 11
"B 1@
61. A $o!pany is %a$ed (ith the probe! o% assi/nin/ * !a$hines to di%%erent ;obs +one !a$hine to one ;ob ony,. The pro%its are esti!ated as %oo(s. b o G
' ; > ; B @ @
1 " ; ? @ B
Machine , B 1 A ? " >
C " ? @ ; ? B
2 B ? A > ; ?
8olve the problem to maimi(e the profit. 63. Deter!ine the opti!u! assi/n!ent s$hedue %or the %oo(in/ assi/n!ent Probe!. The $ost !atri8 is /i)en beo(. b o G
Machine 1 " ; ? @ B ' 11 1> A 1B "7 1@ , > 1" B 1@ 1; C 1; 1B 1@ 1" 1B A 2 "1 "? 1> "A " 1@ D 1? 17 1" 11 1@ B If the job C cannot be assigned to machine B! will the optimum solution changeH 64. A $o!pany has %our !a$hines to do three ;obs. Ea$h ;ob $an be assi/ned to one and ony one !a$hine. The $ost o% ea$h ;ob on ea$h !a$hine is /i)en in the %oo(in/ tabe.
1 1A A 17
' , C
" "? 1; 1@
; "A 1> 1
? ;" 1 ""
$hat are job assignments which will minimi(e the cost. 65. Write the a/orith! %or 'un/arian !ethod. #7. There are %our !a$hines in a !a$hine shop. On a parti$uar day the shop /ot Orders %or e8e$utin/ %i)e ;obs +A= B= <= D : E,. the e8pe$ted pro%it %or ea$h ;ob on ea$h ;ob on !a$hine is as %oo(s?
1
"
;
?
' ;" ?1 @> 1A , ?A @? B" ;? C "7 ;1 A1 @> 2 >1 ?; ?1 ?> D @" " @1 @7 )ind the optimal assignment of job to machines to maimi(e the profit. $hich job should be rejected. #6. A !ar@etin/ !ana/er has %i)e saes!en (or@in/ under his $ontro to be assi/ned to %i)e saes territories. Ta@in/ into a$$ount the saes potentia o% the territories and the $apabiities o% the saes!an= the !ar@etin/ !ana/er esti!ated the saes per !onth+in thousands o% Rs., %or ea$h $o!bination and is presented as beo(. -ind the assi/n!ent o% saes!en to saes Territories to !a8i!i9e the saes )aue per !onth.
8alesmen 1 " ; ? @
' ;" ?7 ?1 "" "
8ales Territories , C 2 ;A ?7 "A "? "A "1 "> ;; ;7 ;A ?1 ;B ;; ?7 ;@
D ?7 ;B ;> ;B ;
##. So)e the %oo(in/ assi/n!ent probe! usin/ 'un/erian !ethod. The !atri8 entries are pro$essin/ ti!es in hours.
Gob 1 " ; ? @
1 "7 ? "; 1> 1B
perator " ; "" ;@ "B "? 1? 1> 1@ 1B 1 "1
? "" "? 1 1A 1
@ 1A > 1 1@ "@
#&. -i)e (a/ons are a)aiabe at %i)e stations 6= #= &= * and 0. These are re2uired at %i)e stations I= II= III= I> and >. the !iea/es bet(een )arious stations are /i)en by
)rom 1
I 17
II @
To III
I5 1A
5 11
" ; ? @
1; ; 1A 11
" " B
B ? 1" 1?
1" ? 1> 1
1? @ 1@ 17
'o( shoud the (a/ons be transported so as to !ini!i9e the tota !iea/e $o)ered"
