Quantitative Quantitative Analysis lecture 4
Date: 31/1/2017
Transportation Tableau Audio 134 05:00 I don't know where is the optimal solution, So, I will start with any correct solution , till i found the rer rerouting outing going to improve the situation, because I am not going to compare between them then didn't do any move . It will start with the minimum solution, minimum cost method , and then modify the distribution First:
Involve rst nding initial feasible solution using the minimum cost method.
Second: Make
Method
Third: !se
iteration to modify (Modied distribution
the Shadow "ost to solve the problem
#ransportation # ransportation $roblem $roblem % & product is produced at three plants and shipped to three warehouses. #he transportation costs per unit ( are are shown in the table below% Wareouse !lant
a
b
c
!lant "apacity
A
)
)
)*
#
+
+
*-
Prof. Maged Morcos
Quantitative Quantitative Analysis lecture 4
" Wareouse De$and
Date: 31/1/2017
/
0
*
*-
)-
+*
-
)*
a Show the #ransportation #ableau. 1see solution in attached sheets2 b 3evelop a model for minimi4ing transportation costs, solve the mode to determine the minimum5cost solution and the optimal solution.
0
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
(ini$i)ation Approac * "+,T Tableau . 5 $ar&s Wareouse To ro$
a
b
,uppl y
c
S"
s t
A
)
)
)*
n a l
#
+
+
*-
!
"
/
0
*
*-
De$an d
30
5
40
135 135
7hy )-, +*, - are di8erent9 :ecause it is the capacity of the warehouse or the demand in the area Supply ; demand it is a special case, the general case the supply is not ; 3emand
)
)
+
+
/
0
*
Te above nu$bers are te cost / unit o sendin ro$ eac !lant to eac Wareouse it is iven ro$ te operation (anaer %n&no'ns )Sending "ost of"ost of 6eceiving
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
we have 6 unknowns , each
one is divided for sending and receiving , this mean we can separate each cost <=ample % could be > - , >) , 0>0 , 0.* > .* 7e need to know that, when we know them, we can solve the problem for any route that we have to go to, because the plant that sends will be the same sending price, and the other one will receive according to the route that I sent to. #o solve it with the solver, it will be + unknowns, with + constraints and ? variables in the ob@ective function. In order to solve it using the solver, you need * formulas, (*eAuasions, * steps 7e say that the rules o o o's .
B6
"olu$ns
6
o's
o's
6
"olu$ns
(
6
(
1
3
6
3
1
1
6
1
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5
7e are going to search for the + unknowns. 1)
First Allocation
8 tout tat tis is 'ill be te best opti$u$ solution ro$ $y point o vie'9
8 'ill allocate te a$ount o units to any o te 'areouses accordin to $y &no'lede9
Prof. Maged Morcos
Quantitative Analysis lecture 4
*
Date: 31/1/2017
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
8 need 5 allocations otes:
. the supply of $lant & ; )* -
a
is A
-
-
b
,uppl y
"
)
)*
)
which better to send the )* to or
) or ) 7e choose the lowest number. I will take the )* from $lant A and will allocate them on 7arehouse ", because it can absorb - 5 )* ; * , 7ill take them from $lant C "C (the one who has the lowest price * Dor $lant C", *- 5 * ; *, I will allocate them on warehouse b because is te one 'o as te lo'est cost 2 ; so on Wareouse To ro$ s t
A
n a l
#
!
"
De$an +
a
b
,up ply
c
S" 4 30
+
3 20
/ 45 0 30
5
35
5 40
3
35
+
*0
*
*135
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
d
135
Total Cost - T9"9 . 35 < 3 6 30 < 6 20 < 7 6 45 < 2 6 5 < 5 . = 540 How
to
proof
that
$!"
is
the
#ptiu
Solution%%%
7e have to e=amine the Shadow "ost of sending and receiving - If this shadow cost will gives the same cost of the route that I go through, this mean that it will be the same 55555, - If it improves , I have to change the units to the improved case - If it increase the cost, this will be the optimum solution - #o get of the e=pensive routes - #o proof that my route is optimum, I have to proof that other routes are costly and wrong - In this case the problem stops. - #hat 7hy I have to reroute, if I didnEt get it from the rst time. -
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
First Step : &sed 'outs : (the empty slots in
the table (1) Ac *
A
+
c
() -a *
-
+
a
(.) -/ *
-
+
/
() C/ *
C
+
/
(!) Cc *
C
+
c
Shadow ,ado' "ost o eceivin Shadow Cos "ost oft of
(1) Ac *
A
+
c
() -a *
-
+
a
(.) -/ *
-
+
/
() C/ *
C
+
/
(!) Cc *
C
+
c
%n&no'n 8 ave to >nd te$
Is this shadow, I have to nd the unknowns - Drom nding them , I try new routes - #he shadow cost ; the 555555 routes -
There is no iproveent (it will /e #0)
- #he /
shadow cost will be negative Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
have to reove soe of the 2uantities, and put it into the ne3ative si3n which ean that there is another iproveent solution
- #he shadow cost will be positive - will increase the total cost /4 ore than $!" - This 'oute will /e e5cluded - This ean that reach the optiu solution (1) Ac *
A
+
c
*
.
() -a *
-
+
a
*
6
(.) -/ *
-
+
/
*
() C/ *
C
+
/
*
(!) Cc *
C
+
c
*
!
- #hese
are + unknown in * problems - Fet anyone ; A. - Fet #he Shadow cost of * . - #herefore % ( A + c "
+
c
*
.
c. 3
(*
-
C
+
.
*
!
". 2
-
Search and complete%
-
(+
C
+
/
*
b. 0
?
0
()
-
+
/
*
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
#. 7
(0
-
-
+
a
*
6
a.*1-
- #hese
are the shadow "ost that I will use them in the second step% - #hese are unknowns that I can get them from the rst iteration. -
,"
A
0
#
7
"
2
a
b
"
1
0
3
s t n a l !
De$and
-
30
+ /
30
) 20 45 5
35
0
,uppl y
) +
5 40
*
)* **135 135
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
Second Step :&nused 'outs : (the empty
slots in the table (1) Aa () A/ (.) -c () Ca
- #he
cost of &a should be ; the shadow cost 5 GGH JKL Shadow "ost NKOP QNRTRU VWXN YZ[\ ] ^N The cost of Aa should /e shadow cost
*1-9 .
Aa
*A6 a
4
*
the
-
* 0 6 ?1@ - .
65
_R[U `R[ R\ R[ OW * will increase the *- to be **. - I will not go for this
-
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
Wareouse To ro$
,"
A
0
#
7
"
2
a
b
"
1
0
3
s t n a l !
De$and
30
) 20
+
45
/ 30
35
0
5
,uppl y
) +
5
*
40
)* **135 135
75planation of Shadow cost allocation: - 7hen & sends he take nothing - : takes when he send - " takes 0 when he send - a gives 5 when it receives - b gives nothing when it receives - c gives ) when it receives To ro$ ," A
-
#
c
1
0
3
0
3 ) 5 )
- > (5 - > - ) - > ) ;) To ro
0
a
,"
a
#
c
1
0
3
,uppl y
)*
,uppl y
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
$ #
-
7
> (5 ; + > - ; > ) +
3 0- + 0
To
-
ro$
,"
"
2
a
#
c
1
0
3
/ 45 0
0 > (5 / 0 > - ;0 0 > ) ;*
+
*-
,uppl y
*-
5 *
Bnly the used routes are ; to their shadow cost j the unused routes are to their shadow cost If they are ; so the problem nished , If they are higher , that means that my solution is reached - If they are less, I have to reroute, which means I didnEt reach the optimum solution -
The cost of 8888 should /e shadow cost *1-9 .
Aa 4
*2-9 .
Ab 3
*3-9 .
#c * # 6 c *763 - .
*4-9
"a
)
*
*A6 a * 0 6 ?1@ - . *A6 b* 0 6 0 -.
*"6
a
the
65 63
4
Prof. Maged Morcos
Quantitative Analysis lecture 4
.
Date: 31/1/2017
* 2 6 ?1@ -
.
67
Total Cost . 35 < 3 6 30 < 6 20 < 7 6 45 < 2 6 5 < 5 - T9"9 . = 540 - T9"9 . = 540 6 * 4 - < 5 . = 520
means, I have the chance to reducing my total cost by ( 5 number of units go there,
- Audio 5:00 * 4 -
-
If I will send unit it will be reduced *-
Wareouse To
s t n a l
ro$
,"
A
0
#
7
b
c
1
0
3
30
+
!
"
2
,uppl y
a
/
) 35 15 20 9 50 45 0 9
5 5 9
) +
*
35
50
50
Prof. Maged Morcos
Quantitative Analysis lecture 4
De$a nd
-
5 5 -
Date: 31/1/2017
30
5
40
135 135
7e will go to reroute, from the same problem ( Stepping Stone I have to make sure that %( )*, *-, *- ()-, +*, - are the same because these are constraints. I have to make the colored numbers inside the table * , because we have (* eAuations :c ( need 5 , H q P^ RX "c, GGGH GGx zGGX\ GG{|T RGG\ KGG}~ •GG\ €KGG‚T GG}ƒ •GG constraints + …GG†TN YGG‡ ˆNPq KLGG‰RU )- •GG\ RZ‚}GG„ WGGH * }GG„q P^ Š|‹[O R\ ŒŽ .^ˆR|\ *- …†T * … RZ{qN 0- •\ RZ†q P^ I should proof that the *0- is the correct one . I have to show the Shadow cost again .
First Step :&sed 'outs : (the lled slots in the
table (1) Ac *
A
+
c
*
.
() -a *
-
+
a
*
6
(.) -/ *
-
+
/
*
() -c *
-
+
c
*
6
(!) C/ *
C
+
/
*
-
Fet anyone ; -
-
Fet #he Shadow cost of
-
*
#herefore % ( A
+
A. 0 c
*
. Prof. Maged Morcos
Quantitative Analysis lecture 4
"
Date: 31/1/2017
+
c
*
. c. 3
(
-
-
+
c
*
6 #.
()
-
-
+
/
*
b. 4
(0
-
-
+
a
*
6 a. 3
(*
-
3
C
+
/
*
".*2-
-
Second Step :&nused 'outs : (the empty
slots in the table -
#ake the above number in the unused slots The cost of 8888 should /e shadow cost
*A6 a * 06 3 -
*
*1-9 .
Aa 4
*2-9 .
Ab 3
*A6 b* 0 6 4 -.
*3-9 .
"a
* "6a* 263-
.
"c 5
* "6 c* 263 -
.
.
the
61 1
67
*4-9 . +
64
Prof. Maged Morcos
Quantitative Analysis lecture 4
-
Date: 31/1/2017
&ll of them are > only one is negative #his mean that the cell which has the negative value * Ab - has to have a number inside it 7e will make a re5allocation #he is in a wrong route, I have to move it. #he )* can go in &:, or the * , or the * #he best number to move is the * , 7e have to send it up
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
Wareouse To ro$ A s t n a l
#
!
" De$a nd
,"
,uppl y
a
b
c
3
4
3
20 0 15 ) 35 ) 9 20 15 3 30 5 + + 9 9 2
/ 30
50 5
0
9
*
40
35
50
50 135 135
Total Cost
- T9"9 . = 520 6 * 1 - < 15 . = 505 How to proof that $ !"! , is the #ptiu Solution%%%
- ‘ow to do it in the e=am the correct one, - ’ou have to do some trials of values, before
starting. - ’ou block ) blocks, - #hen you move 0 moves - you will nd one ; * , the other is ?* , the third ; * - #hen , you will begin with ?* - and do the calculation - if it didn't come, I will need the iteration /
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
- ’ou are shortening the distance of beginning - In breif ( make * or ) assumption - and begin with the best - #his will reduce the reroute - !ntil another problem comes, then repeat it
So the whole company improve
?
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
(aBi$i)ation Approac * !+8T -
-
Solve the problem with the prot approach. I didnEt calculate the cost , I calculate the prot for sending from one factory to the warehouse ( “ow I target the big numbers See which cell I have to send the ma=imum amount in. 7ill use the cell of ( / , to put the ma=imum number ; 30 because the capacity *- )- ; 07ill allocate them in the ( * cell because it is bigger than (0 Wareouse To ro$
,"
A
0
#
7
"
2
a
b
"
1
0
3
s t n a l !
De$and
+ 30 30
35 30
/
) 0
5
,uppl y
) 20 20 40
+ *
)* **135 135
7ill make the iteration, and all the unused should be negative
- -
0-
Prof. Maged Morcos
Quantitative Analysis lecture 4
-
Date: 31/1/2017
If all the unused is negative, it means that I reached the best prot - If there is a plus , it means that this will improve the prot - It is the contrary of the minimi4ation - #his is not the optimal solution , with a prot of * - I can bring a better solution Wareouse To ro$ A
,"
a
b
"
1
0
3
0
s t n a l
#
7
"
2
+
!
De$and
30 30
15 30 ) 9 50 30 9
/
0 5
,uppl y
20
)
)*
20 + 9
*-
20
*-
40
*
135 135
Solve it at hoe : T9 * !
T9 * ! 0
Prof. Maged Morcos
Quantitative Analysis lecture 4
00
Date: 31/1/2017
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
,upply C De$and Wareouse To ro$ s t n a l !
a
b
,uppl y
"
,"
A
)
)
35
#
+
+
50
"
/
0
*
0
De$a nd
30
5
40
145 135
7e found that the company can produce +- in ". - #his transportation tableau cannot be solved !nless the supply is ; 3emand - ‘ow the manager (6“3 deptcan maneuver the modeling in order to solve the problem - Maneuver means play around the tableau till the supply is ; 3emand. - 7hich Market is this% :ooming or 6ecession Market9 - Is a recession market because the supply is ” the 3emand - Meaning that #he capacity of the warehouse is not enough - 7hich mean one of these factories will carry (store inventories.
-
0)
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
‘e produce them, but, he canEt distribute them - 7e have to solve the problem - 7here to put the e=tra - 7e have to rent 7arehouse CdC -
Wareouse To ro $
a
s t
#
5
) 30 )
,uppl y
-
)*
1 10 + 0
-
*-
0
-
+-
30 +
" De$and
!
0
d
c
,"
A
n a l
b
/ 30
0 5
* 40
10
145
145
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
De$and C ,upply Wareouse To ro$ s t n a l !
a
b
,uppl y
"
,"
A
)
)
35
#
+
+
50
"
/
0
*
50
De$a nd
30
5
- If the demand ” Supply - this Market is a :ooming - #hat means one of the
50
135 145
warehouses will not
take the - units - I am not going to distribute this - units in its area - 7here to send it9 - I have to add a new plant , to allocate the e=tra - in order to have 3emand ; Supply
0*
Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
Wareouse To ro$ s t n a l
A
,uppl y
"
,"
A
)
)
35
#
+
+
50
"
/
0
*
50
D
-
-
-
10
!
De$a nd
-
b
30
5
50
145 145
"
,uppl y
7ill &llocate Wareouse To ro$
A ,"
A s t
15
"
!
+ 50
- #his 0+
)
20
)
35
+
50
*
50
-
10
30 /
0
20
D De$a nd
#
n a l
b
30
5
10 50
145 145
mean that this guy will take only Prof. Maged Morcos
Quantitative Analysis lecture 4
Date: 31/1/2017
:ecause the Market is booming, I 7ill build a new $lant - I get the correct transportation cost - Solve the problem for Ma=imum $rot -
0
Prof. Maged Morcos
