of 60
7/24/2019 lecture notes on transportation models management science
1/60
1
Transportation Problems
Transportation is considered as a specialcase of LP
Reasons?
it can be formulated using LP technique so isits solution
(to p!
7/24/2019 lecture notes on transportation models management science
2/60
"ere# $e attempt to firstl% define $hatare them and then stud%ing their solutionmethods& (to p'!
7/24/2019 lecture notes on transportation models management science
3/60
'
Transportation Problem
e ha)e seen a sample of transportationproblem on slide * in lecture
"ere# $e stud% its alternati)e solutionmethod
+onsider the follo$ing transportationtableau (to p,!
(to p-!
7/24/2019 lecture notes on transportation models management science
4/60
-
Re)ie$ of Transportation Problem
Warehouse supply of televisions sets: Retail store demand for television sets:
1- Cincinnati 300 A - New Yor 1!0
"- Atlanta "00 # - $allas "!0
3- %itts&ur'h "00 C - $etroit "00total (00 total )00
*o +tore,romWarehouse
A # C
1 1) 1. 11" 1/ 1" 13
3 13 1! 1(
LP formulation(to p.!
7/24/2019 lecture notes on transportation models management science
5/60
.
/ Transportation 0ample ( of '!2odel 3ummar% and +omputer 3olution $ith 0cel
inimie 2 1)41A5 1.41#5 1141C5 1/4"A5 1"4"#5 134"C5 1343A5 1!43#5 1(43C
su&6ect to
41A 5 41#5 41 300
4"A5 4"#5 4"C"00
43A5 43#5 43C"00
41A5 4"A5 43A 1!0
41#5 4"#5 43# "!0
41C5 4"C5 43C "00
4i60
Exhibit 4.15
(to p'!
7/24/2019 lecture notes on transportation models management science
6/60
,
Transportation Tableau
e 4no$ ho$ to formulate it using LP technique Refer to lecture note
"ere# $e stud% its solution b% firstl% attempting todetermine its initial tableau 5ust li4e the first simple tableau6
(to p7!
7/24/2019 lecture notes on transportation models management science
7/60
7
3olution toa transportation problem
8nitial tableau
9ptimal solution
8mportant :otes
Tutorials
(to p;!
(to p-'!
(to p'!
(to p.'!
7/24/2019 lecture notes on transportation models management science
8/60
;
initial tableau
Three different $a%s&
:orth$est corner method
The 2inimum cell cost method
7/24/2019 lecture notes on transportation models management science
9/60
*
:orth$est corner method
3teps&
1> assign largest possible allocation to the cellin the upper lefthand corner of the tableau
> Repeat step 1 until all allocations ha)e beenassigned
'> 3top> 8nitial tableau is obtained
0ample(to p1@!
7/24/2019 lecture notes on transportation models management science
10/60
1@
:ortheast corner
1.@
3tep 1
2a (1.@#@@!
3tep
1.@
.@
@
1.@
@
@
1.
(to p11!
7/24/2019 lecture notes on transportation models management science
11/60
11
8nitial tableau of : corner method
Repeat the abo)e steps# $e ha)e the follo$ingtableau>
3top> 3ince all allocated ha)e been assigned
0nsure that all columns and ro$s added up to its respecti)e totals>
(to p;!
7/24/2019 lecture notes on transportation models management science
12/60
1
The 2inimum cell cost method
"ere# $e use the follo$ing steps&
3teps&
3tep 1 Aind the cell that has the least cost
3tep & /ssign as much as allocation to this cell
3tep '& Bloc4 those cells that cannot be allocated
3tep -& Repeat abo)e steps until all allocation ha)ebeen assigned>
0ample& (to p1'!
7/24/2019 lecture notes on transportation models management science
13/60
1'
3tep 1 Aind the cell that has the least cost3tep & /ssign as much as allocation to this cell
3tep 1&
The min cost# so allocate as much resource as possible here
@@
3tep '
3tep &
(to p1-!
7/24/2019 lecture notes on transportation models management science
14/60
1-
3tep '& Bloc4 those cells that cannot be allocated3tep -& Repeat abo)e steps until all allocationha)e been assigned>
3tep '&
@@
@
7.
3econd iteration# step -
(to p1.!
7/24/2019 lecture notes on transportation models management science
15/60
1.
The initial solution
3top> The abo)e tableau is an initial tableaubecause all allocations ha)e been assigned
(to p;!
7/24/2019 lecture notes on transportation models management science
16/60
1,
7/24/2019 lecture notes on transportation models management science
17/60
17
subtracting their t$o of theirleast cost
3tep 1
(;,!
(117!
(.-!
(,-! (;.! (111@!
(to p1;!
7/24/2019 lecture notes on transportation models management science
18/60
1;
3teps D '
"ighest penalt%cost
Step 2:
Step 3:this has the least cost
(to p1*!
7/24/2019 lecture notes on transportation models management science
19/60
1*
3tep -
(to p@!
7/24/2019 lecture notes on transportation models management science
20/60
@
3tep .3econd 8teration
(to p1!
7/24/2019 lecture notes on transportation models management science
21/60
1
'rd8teration of
7/24/2019 lecture notes on transportation models management science
22/60
8nitial tableau for
7/24/2019 lecture notes on transportation models management science
23/60
'
9ptimal solution?
8nitial solution from&
:ortheast cost# total cost EF.#*.
The min cost# total cost EF-#..@
7/24/2019 lecture notes on transportation models management science
24/60
-
3olution methods
e need a method# li4e the simplemethod# to chec4 and obtain the optimalsolution
T$o methods&
1> 3teppingstone method> 2odified distributed method (29G8!
(to p.!
(to p'@!
(to p7!
7/24/2019 lecture notes on transportation models management science
25/60
.
3teppingstone method
Let consider the follo$ing initial tableau from the 2in +ost algorithm
These are basic)ariables
There are:onbasic )ariables
Huestion& "o$ can $e introducea nonbasic )ariable into basic )ariable?
(to p,!
7/24/2019 lecture notes on transportation models management science
26/60
,
8ntroducea nonbasic )ariable into basic )ariables
"ere# $e can select an% nonbasic )ariable asan entr% and then using the I and steps toform a closed loop as follo$s&
let consider this nonbasic )ariable
Then $e ha)e
(to p7!
7/24/2019 lecture notes on transportation models management science
27/60
7
3tepping stone
I
I
The abo)e sa%ing that# $e add min )alue of all )e cells into cell that has I sign# and subtractsthe same )alue to the )e cellsThus# ma )e is min (@@#.! E .# and $e add . to cell /1 and /'# and subtract it from B1 and /'
(to p;!
7/24/2019 lecture notes on transportation models management science
28/60
;
3tepping stone
The abo)e tableaus gi)e min cost E .J, I 1@J1@ I 17.J11 17.J- I 1@@J . E F-..
e can repeat this process to all possible nonbasic cells in that abo)etableau until one has the min cost6 :9T a Kood solution method
(to p*!
7/24/2019 lecture notes on transportation models management science
29/60
*
Ketting optimal solution
8n such# $e introducing the netalgorithm called 2odified Gistribution(29G8!
(top-!
7/24/2019 lecture notes on transportation models management science
30/60
'@
2odified distributed method (29G8!
8t is a modified )ersion of stepping stone method
29G8 has t$o important elements&
1>8t determines if a tableau is the optimal one
>8t tells %ou $hich nonbasic )ariable should be firstl%considered as an entr% )ariable
'>8t ma4es use of steppingstone to get its ans$er ofnet iteration
"o$ it $or4s?(to p'1!
7/24/2019 lecture notes on transportation models management science
31/60
'1
Procedure (29G8!3tep @& let ui#) # ciL )ariables represent ro$s# columns# and cost in the
transportation tableau# respecti)el%
3tep 1& (a! form a set of equations that uses torepresent all basic )ariables
uiI ) E ciL
(b! sol)e )ariables b% assignone )ariable E @
3tep& (a! form a set of equations use torepresent nonbasic )ariable (orempt% cell! as such
ciL ui ) E 4iL
(b! sol)e )ariables b% using step 1b information
3tep '& 3elect the cell that has the most )e )alue in b3tep -& Mse steppingstone method to allocate resource to cell in step '3tep .& Repeat the abo)e steps until all cells in a has no negati)e
)alue0ample (to p-!(to p'!
7/24/2019 lecture notes on transportation models management science
32/60
'
29G8
+onsider to this initial tableau&
3tep @& let ui# ) # ci )ariables represent ro$s# columns# and cost in thetransportation tableau# respecti)el% (to p''!
7/24/2019 lecture notes on transportation models management science
33/60
''
3tep @
+'/
3tep 1& (a! form a set of equations that uses torepresent all basic )ariables
ui I ) E ci
(to p'-!
7/24/2019 lecture notes on transportation models management science
34/60
'-
ui I ) E ci
(b! sol)e )ariables b% assignone )ariable E @
(to p'.!
7/24/2019 lecture notes on transportation models management science
35/60
'.
3et one )ariable E @
Because $eadded annonbasic )ariable
3tep& (a D b!
(to p',!
7/24/2019 lecture notes on transportation models management science
36/60
',
3tep& (a D b!
:ote this ma% loo4 difficult and complicated# ho$e)er# $e can add these
7/24/2019 lecture notes on transportation models management science
37/60
'7
3tep& (a D b!# alternati)e
1
,@7
1 I
I.
3tep '& 3elect the cell that has the most )e )alue in b
(to p';!
7/24/2019 lecture notes on transportation models management science
38/60
';
3tep'
1
1 I
I.
3elect either one# (h%?!These cells mean# introduce it $ill reduce the min N to 1 cost unit
(to p'*!
7/24/2019 lecture notes on transportation models management science
39/60
'*
3tep -& Mse steppingstone method
I
I
Arom here $e ha)e O> (to p-@!
7/24/2019 lecture notes on transportation models management science
40/60
-@
3tep -& Mse steppingstone method
3tep .& $e repeat steps 1- again for the abo)e tableau# $e ha)e
(to p-1!
7/24/2019 lecture notes on transportation models management science
41/60
-1
3tep .
(to p-!
7/24/2019 lecture notes on transportation models management science
42/60
-
3tep . cont
/ll positi)e3T9P
(to p'1!
7/24/2019 lecture notes on transportation models management science
43/60
-'
8mportant :otes
hen start sol)ing a transportationproblem using algorithm# $e need toensure the follo$ing&
1> /lternati)e solution
> Total demand total suppl%
'> Gegenerac%
-> others
(to p-;!
(to p-.!
(to p.!
(to p7!
(to p--!
7/24/2019 lecture notes on transportation models management science
44/60
--
/lternati)e solution
hen on the follo$ing 4 E @
ciL ui ) E 4iL
h%?
(to p-'!
7/24/2019 lecture notes on transportation models management science
45/60
-.
Total demand total suppl%
:ote that# total demandE,.@# and total suppl% E ,@@
"o$ to sol)e it?
e need to add a dumm% ro$ and assign o cost to each cell as such >>
(to p-,!
7/24/2019 lecture notes on transportation models management science
46/60
-,
Gdss
0tra ro$# since Gemand Q suppl%
9ther eample OO (to p-7!
7/24/2019 lecture notes on transportation models management science
47/60
-7
Gdss
0tra column is added(to p-'!
7/24/2019 lecture notes on transportation models management science
48/60
-;
Gegenerac%
0ample O>> ie# basic )ariablesin the tableau
(to p-*!
7/24/2019 lecture notes on transportation models management science
49/60
-*
Gegenerac%
m ro$s I n column 1 E the number of cells $ith allocations' I ' 1 E .
8t satisfied>
8f failed? OO considering O>>
(fi)e basic )ariables# and abo)e has . as $ell6!
(to p.@!
7/24/2019 lecture notes on transportation models management science
50/60
.@
Gegenerac%
8f not matched# then $e select an nonbasic )ariable $ith least cheapestcost and considered it as a ne$ basic )ariable $ith assigned @ allocation to it
(note abo)e has onl% - basic )ariableonl%6!
(to p.1!
7/24/2019 lecture notes on transportation models management science
51/60
.1
Gegenerac%
/dded this:ote& $e pic4 this o)er others because it has the least cost for the 2in problem6
(to p-'!
7/24/2019 lecture notes on transportation models management science
52/60
.
others
1> hen one route cannot be used
/ssign a big 2 cost to its cell
(to p-'!
8f 1@ changed to+annot deli)eredThen $e assigned2 )alue here
7/24/2019 lecture notes on transportation models management science
53/60
.'
Tutorials
2odule B
1# .# ;# 1'# 1# '-
these questions are attached in the follo$ing
slides
7/24/2019 lecture notes on transportation models management science
54/60
.-
1
7/24/2019 lecture notes on transportation models management science
55/60
..
7/24/2019 lecture notes on transportation models management science
56/60
.,
7/24/2019 lecture notes on transportation models management science
57/60
.7
7/24/2019 lecture notes on transportation models management science
58/60
.;
7/24/2019 lecture notes on transportation models management science
59/60
.*
7/24/2019 lecture notes on transportation models management science
60/60