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CASE ON DISTRIBUTIONSYSTEM DESIGN
Presented By: Aakanksha Jain 211001
Amit Arora 211016
Farhan Aqeel 211045
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CASE DESCRIPTION
DARBY COMPANY
Manufactures and distributes meters used to measureelectric power consumption.
Started with a production plant in El Paso anddistribution center in Ft. Worth, Texas.
With expansion of business to north, new distributioncenter opened in Santa Fe, New Mexico.
With growth of business in the West Coast, newdistribution center opened in Las Vegas and a
production plant in San Bernardino, California.
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PRODUCTION PLANT COST OF PRODUCTION
El Paso 10.50
San Bernardino 10
PRODUCTION PLANT PRODUCTION CAPACITY
El Paso 30000San Bernardino 20000
Cost of production per meter at each production plant:
Quarterly production capacity at each production plant:
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El Paso
SanBernardino
2 PRODUCTIONPLANTS
Ft. Worth Santa Fe
Las Vegas
3 DISTRIBUTIONCENTERS Dallas
San Antonio
Wichita
Kansas City Denver
Salt Lake City
Phoenix
Los Angeles
San Diego
9 CUSTOMERZONES
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CU
RRENT
DI
STRIBUTION
SY
STEM
El Paso
Ft. Worth
Dallas
San
Antonio
Wichita
KansasCity
San
Bernardino
Santa Fe
Denver
Salt LakeCity
Phoenix
Las Vegas
LosAngeles
San Diego
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PART I
If the company does not change its current
distribution strategy, what will its manufacturingand distribution costs be for the following quarter?
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MODEL DESCRIPTION
To calculate the distribution cost from Production plants to the distribution centers
we use a transportation model with production plants as sources and distribution
centers as destinations.
The demand of the distribution centers is calculated by adding the individualdemands of the customer zones catered by them.
For Ft. Worth, demand = demand from Dallas+ San Antonio + Wichita +
Kansas City
=6300 + 4880 + 2130 + 1210
=14520
For Santa Fe, demand = demand from Denver + Salt lake city + Phoenix=6120 + 4830 + 2750
=13700
For Las Vegas, demand = demand from Los Angeles + San Diego
=8580 + 4460
=13040
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Applying Vogels Approximation Method (VAM)
FT.
WORTH
SANTA FE LAS
VEGAS
D1 SUPPLY
EL PASO3.214520 2.26740 4.2 08740 30000
15480
6740
(2.2)
(2.2)
SAN
BERNARDI
NO
3.96960 1.213040 0 200006960
(1.2)
(3.9)
DEMAND 14520 13700
6740
13040 8740 50000
() (1.7) (3) (0)
Initial Basic Feasible Solution:
FT. WORTH SANTA FE LAS VEGAS D1
EL PASO 14520 6740 -- 8740
SAN
BERNARDINO-- 6960 13040 --
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FT.
WORTH
SANTA FE LAS
VEGAS
D1 Ui
EL PASO -- -- 4.20 -- 0
SAN BERNARDINO -- -- 0 1.7
Vj 3.2 2.2 -0.5 0
Cost for unoccupied cells:
FT.
WORTH
SANTA FE LAS VEGAS D1
EL PASO -- -- 4.70 --
SAN BERNARDINO -- -- -1.70
Net Evaluation:
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Now to check for optimality, we again apply MODI method.
Cost for occupied cells:
FT.WORTH SANTA FE LASVEGAS D1 Ui
EL PASO 3.2 2.2 -- 0 0
SAN BERNARDINO -- -- 1.2 0 0
Vj 3.2 2.2 1.2 0
Cost for unoccupied cells:
FT.
WORTH
SANTA FE LAS
VEGAS
D1 Ui
EL PASO -- -- 4.20 -- 0
SAN BERNARDINO 3.9 -- -- 0
Vj 3.2 2.2 1.2 0
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As we are given fixed distribution pattern from distribution center to customer zones, to
calculate Distribution Cost from distribution centers to the customer zones we multiply
the per meter cost of transportation with the demand at each customer zone.=6300*0.3+4880*2.1+2130*3.1+1210*4.4+6120*2.7+4830*4.7+2750*3.4+858
0*2.1+4460*2.5
=1890+10248+6603+5324+16524+22701+9350+18018+11150
=$101808
Thus, Total distribution cost =92252+101808
=$194060
For manufacturing cost, we multiply the quantity supplied to each distribution center
from a production plant with the production cost at that production plant.
Thus, manufacturing cost = 14520*10.50+13700*10.50+13040*10
= $426710
Therefore, Total Cost = 194060+426710
=$620770
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PART IISuppose that the company is willing to consider droppingthe distribution center limitations; that is, customers
could be served by any of the distribution centers forwhich costs are available. Can costs be reduced? By howmuch?
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Initial Basic Feasible Solution:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO-- -- -- -- -- -- -- -- -- -- 11180 18820 --
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- -- -- -- -- -- 20000
FT.
WORT
H
6300 4880 -- -- -- -- -- -- -- -- 38820 -- --
SANTA
FE-- -- 2130 1210 6120 -- -- -- 620 8740 -- 31180 --
LAS
VEGAS-- -- -- -- -- 4830 2750 8580 3840 -- -- -- 30000
Check for non-degeneracy:
m+n-1 = 5+13-1=17 which is equal to the number of basic variables.
Thus, solution is non-degenerate.
Now, to check for optimality, we apply MODI method.
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Cost for occupied cells:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
Ui
EL
PASO-- -- -- --- -- -- -- -- -- -- 3.2 2.2 -- 2.4
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- -- -- -- -- -- 1.2 1.2
FT.
WORT
H
0.3 2.1 -- -- -- -- -- -- -- -- 0 -- -- -0.8
SANTA
FE-- -- 4.5 6.0 2.7 -- -- -- 2.7 0 -- 0 -- 0.2
LAS
VEGAS-- -- -- -- -- 3.3 2.4 2.1 2.5 -- -- -- 0 0
Vj 1.1 2.9 4.3 5.8 2.5 3.3 2.4 2.1 2.5 -0.2 0.8 -0.2 0
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DALLAS SANANTO
NIO
WICHITA KANSAS
CITY
DENVER SALTLAKE
CITY
PHOENIX LOSANGEL
ES
SANDIEG
O
D1 FT.WORT
H
SANTAFE LASVEGAS Ui
EL
PASO -- -- 4.2 2.4
SAN
BERN
ARDIN
O
3.9 -- 1.2
FT.WORT
H
-- -- 3.1 4.4 6.0 0 -- -0.8
SANTA
FE5.2 5.4 -- -- -- 4.7 3.4 3.3 -- -- -- 0.2
LAS
VEGAS 5.4 -- -- -- -- 0 -- 0
Vj 1.1 2.9 4.3 5.8 2.5 3.3 2.4 2.1 2.5 -0.2 0.8 -0.2 0
Cost for unoccupied cells:
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Net Evaluation:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO -- -- 1.8
SAN
BERN
ARDIN
O
2.9 --
FT.WORT
H
-- -- -0.4 -0.6 4.3 1 --
SANTA
FE3.9 2.3 -- -- -- 1.2 0.8 1 -- -- --
LAS
VEGAS 2.9 -- -- -- -- 0.2 --
As all Cij is not equal to or greater than zero, thus solution is not optimal. To obtainan optimal solution, we insert in the solution and form a loop.
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DALL
AS
SAN
ANTONIO
WICH
ITA
KANS
ASCITY
DENV
ER
SALT
LAKECITY
PHOE
NIX
LOS
ANGELES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
EL
PASO-- -- -- -- -- -- -- -- -- -- 11180
+
18820
-
--
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- -- -- -- -- -- 20000
FT.WORT
H
6300 4880 -- -- -- -- -- -- -- 38820-
-- --
SANTA
FE-- -- 2130 1210
-
6120 -- -- -- 620 8740 -- 31180
+
--
LAS
VEGAS-- -- -- -- -- 4830 2750 8580 3840 -- -- -- 30000
Here, =1210
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New Basic Feasible Solution:
DALLAS
SANANTO
NIO
WICHITA
KANSAS
CITY
DENVER
SALTLAKE
CITY
PHOENIX
LOSANGEL
ES
SANDIEG
O
D1 FT.WORT
H
SANTAFE
LASVEGAS
EL
PASO-- -- -- -- -- -- -- -- -- -- 12390 17610 --
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- -- -- -- -- -- 20000
FT.
WORT
H
6300 4880 -- 1210 -- -- -- -- -- -- 37610 -- --
SANTA
FE-- -- 2130 -- 6120 -- -- -- 620 8740 -- 32390 --
LAS
VEGAS-- -- -- -- -- 4830 2750 8580 3840 -- -- -- 30000
Now, to check for optimality, we again apply MODI method.
Check for non-degeneracy:
m+n-1 = 5+13-1=17 which is equal to the number of basic variables.
Thus, solution is non-degenerate.
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DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
Ui
EL
PASO -- -- 4.2 2.4
SAN
BERN
ARDIN
O
3.9 -- 1.2
FT.WORT
H
-- -- 3.1 -- 6.0 0 -- -0.8
SANTA
FE5.2 5.4 -- 6.0 -- 4.7 3.4 3.3 -- -- -- 0.2
LAS
VEGAS 5.4 -- -- -- -- 0 -- 0
Vj 1.1 2.9 4.3 5.2 2.5 3.3 2.4 2.1 2.5 -0.2 0.8 -0.2 0
Cost for unoccupied cells:
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Net Evaluation:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO -- -- 1.8
SAN
BERN
ARDIN
O
4.1 --
FT.WORT
H
-- -- -0.4 -- 4.3 1 --
SANTA
FE3.9 2.3 -- 0.6 -- 1.2 0.8 1 -- -- --
LAS
VEGAS 2.9 -- -- -- -- 0.2 --
As all Cij is not greater than or equal to zero, thus solution is not optimal. To obtainan optimal solution, we again insert in the solution and form a loop.
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DALL
AS
SAN
ANTONIO
WICH
ITA
KANS
ASCITY
DENV
ER
SALT
LAKECITY
PHOE
NIX
LOS
ANGELES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
EL
PASO-- -- -- -- -- -- -- -- -- -- 12390
+
17610
-
--
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- -- -- -- -- -- 20000
FT.WORT
H
6300 4880 1210 -- -- -- -- -- -- 37610-
-- --
SANTA
FE-- -- 2130
-
6120 -- -- -- 620 8740 -- 32390
+
--
LAS
VEGAS-- -- -- -- -- 4830 2750 8580 3840 -- -- -- 30000
Here, =2130
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New Basic Feasible Solution:
DALLAS SANANTO
NIO
WICHITA KANSAS
CITY
DENVER SALTLAKE
CITY
PHOENIX LOSANGEL
ES
SANDIEG
O
D1 FT.WORT
H
SANTAFE LASVEGAS
EL
PASO-- -- -- -- -- -- -- -- -- -- 14520 15480 --
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- -- -- -- -- -- 20000
FT.
WORT
H
6300 4880 2130 1210 -- -- -- -- -- -- 35480 -- --
SANTA
FE-- -- -- -- 6120 -- -- -- 620 8740 -- 34520 --
LAS
VEGAS-- -- -- -- -- 4830 2750 8580 3840 -- -- -- 30000
Now, to check for optimality, we again apply MODI method.
Check for non-degeneracy:
m+n-1 = 5+13-1=17 which is equal to the number of basic variables.
Thus, solution is non-degenerate.
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Cost for occupied cells:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
Ui
EL
PASO-- -- -- --- -- -- -- -- -- -- 3.2 2.2 -- 2.4
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- -- -- -- -- -- 1.2 1.2
FT.
WORT
H
0.3 2.1 3.1 4.4 -- -- -- -- -- -- 0 -- -- -0.8
SANTA
FE-- -- -- -- 2.7 -- -- -- 2.7 0 -- 0 -- 0.2
LAS
VEGAS-- -- -- -- -- 3.3 2.4 2.1 2.5 -- -- -- 0 0
Vj 1.1 2.9 3.9 5.2 2.5 3.3 2.4 2.1 2.5 -0.2 0.8 -0.2 0
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Net Evaluation:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO -- -- 1.8
SAN
BERN
ARDIN
O
2.9 --
FT.WORT
H
-- -- -- -- 4.3 1 --
SANTA
FE3.9 2.3 0.4 0.6 -- 1.2 0.8 1 -- -- --
LAS
VEGAS 2.9 -- -- -- -- 0.2 --
As all Cij is greater than or equal to zero, thus solution is optimal.
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DI
STRIBU
TION
SY
SYTEM
El Paso
Ft. Worth
Dallas
SanAntonio
Wichita
KansasCity
Santa Fe
Denver
San Diego
SanBernardino
Las Vegas
Salt LakeCity
Phoenix
LosAngeles
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To calculate Distribution cost from production plant to the distribution centers
we add the corresponding contributed costs from each production plant to a
distribution center. We also subtract the cost of shipping 8740 units as they are
being shipped to a dummy destination. Thus, the cost of distribution from ElPaso to Santa Fe includes the cost of these extra units which needs to be
subtracted.
=14520*3.2+15480*2.2+20000*1.2-8740*2.2
=46464+34056+24000-19228
=$85292
Distribution cost from distribution centers to the customer zone is calculated by
adding the corresponding contributed costs from each distribution center to a
customer zone.
=6300*0.3+4880*2.1+2130*3.1+1210*4.4+6120*2.7+
4830*3.3+2750*2.4+8580*2.1+620*2.7+3840*2.5
=1890+10248+6603+5324+16524+15939+6600+18018+1674+9600
=$92420
Thus, Total distribution cost=85292+92420
=$177712
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Manufacturing cost = 14520*10.50+15480*10.50+20000*10-8740*10.5
=152460+162540+200000-91770
=$423230
Thus, Total cost = 177712+423230
=$600942
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CHANGES IN COST
In case of a distribution plan without limitations:
The total cost would be reduced to $600,942, which is a savings of $19,828
or 3.2% The distribution cost is reduced to $177,712 i.e. a savings of $16,888.
The manufacturing cost is reduced to $423230 i.e. a savings of $3480.
This new plan allows the company to produce more of its meters at its more
cost efficient San Bernardino plantin fact, under this plan it produces at
capacity, which is 20,000 units.
This plan allows more than one center to supply to a specific zone and the
solution shows that this is the case for the San Diego customer zone.
This plan allows a different center to service customers if it is cheaper.
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PART IIIThe company wants to explore the possibility of satisfying some of the
customer demand directly from the production plants. In particular, the
shipping cost is $0.30 per unit from San Bernardino to Los Angeles and$0.70 from San Bernardino to San Diego. The cost of direct shipments
from EI Paso to San Antonio is $3.50 per unit. Can distribution costs be
further reduced by considering these direct plant-customer shipments?
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PRODUCTION
PLANT
CUSTOMER ZONE COST
El Paso San Antonio $3.50
San Bernardino Los Angeles $0.30
San Bernardino San Diego $0.70
Distribution costs for some direct plant-customer shipments:
With an option of direct plant-customer shipments we replace the cells
corresponding to these direct shipments with the costs given in place of
from the earlier table.
The supply of the sources remains the same as production capacity
remains the same.
The demand of the destinations is also the same.
The demand and supply of the transient nodes i.e. the distribution
centers is the total supply available.
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DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
SUPPL
Y
EL
PASO 3.50 3.2 2.2 4.2 30000
SAN
BERN
ARDIN
O
0.30 0.70 3.9 1.2 20000
FT.WORT
H
0.3 2.1 3.1 4.4 6.0 0 0 50000
SANTA
FE5.2 5.4 4.5 6.0 2.7 4.7 3.4 3.3 2.7 0 0 50000
LAS
VEGAS 5.4 3.3 2.4 2.1 2.5 0 0 50000
DEMA
ND6300 4880 2130 1210 6120 4830 2750 8580 4460 8740 50000 50000 50000
Modified Transshipment table is as follows:
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D SA W K DE SLC P LA SD D1 FT. SANT LAS SUPPL
Y
EL 3.5048
80
3.26300 2.29460 4.29360 30000
23700
18820
9460
(1)(1.
3)(2)(
2.2)
SA
N 0.3085
80
0.704460 3.9 1.26960 20000
11420
6960
(0.4)(
0.4)(0.
4)(0.5
)(2.7)
FT. 0.36300 2.1 3.1 4.4 6.0 0 043700 50000
43700
(0)
SA
NT5.2 5.4 4.521
30
6.012
10
2.7612
0
4.7 3.4 3.3 2.7 0 040540 50000
48790
46660
40540
(0)(0)
LA
S 5.4 3.348
30
2.4275
0
2.1 2.5 08740 033680 50000
41260
3851033680
(0)(0)
DE
MA
ND
6300 4880 2130 1210 6120 4830 2750 8580 4460 874
0
50000
6300
50000
9460
50000
16320
9360
(4.9) (1.4)
(1.9)
()
(1.4)
()
(1.6)
()
(2.7)
(2.7)
(1.4)
(1.4)
()
(1)
(1)
()
(1.8)
(1.8)
(1.8)
()
(1.8)
(1.8)
(1.8)
()
(0)
(0)
()
(3.2)
()
(2.2)
(2.2)
(1.7)
(1.7)(2.2)
(1.2)
(1.2)
(1.2)
(3)(4.2)
Applying Vogels Approximation Method (VAM)
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DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 6300 9460 9360
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.
WORT
H
6300 -- -- -- -- -- -- -- -- -- 43700 -- --
SANTA
FE-- -- 2130 1210 6120 -- -- -- -- -- -- 40540 --
LAS
VEGAS-- -- -- -- -- 4830 2750 -- -- 8740 -- -- 33680
Initial Basic Feasible Solution:
Check for non-degeneracy:
m+n-1 = 5+13-1=17 which is equal to the number of basic variables.
Thus, solution is non-degenerate.
Now, to check for optimality, we apply MODI method.
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Cost for occupied cells:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
Ui
EL
PASO-- 3.5 -- --- -- -- -- -- -- -- 3.2 2.2 4.2 4.2
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 0.3 0.7 -- -- -- 1.2 1.2
FT.
WORT
H
0.3 -- -- -- -- -- -- -- -- -- 0 -- -- 1
SANTA
FE-- -- 4.5 6.0 2.7 -- -- -- -- -- -- 0 -- 2
LAS
VEGAS-- -- -- -- -- 3.3 2.4 -- -- 0 -- -- 0 0
Vj -0.7 -0.7 2.5 4 0.7 3.3 2.4 -0.9 -0.5 0 -1 -2 0
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DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
Ui
EL
PASO -- -- -- -- 4.2
SAN
BERN
ARDIN
O
-- -- 3.9 -- 1.2
FT.WORT
H
-- 2.1 3.1 4.4 6.0 0 -- 1
SANTA
FE5.2 5.4 -- -- -- 4.7 3.4 3.3 2.7 0 -- 2
LAS
VEGAS 5.4 -- -- 2.1 2.5 -- -- 0
Vj -0.7 -0.7 2.5 4 0.7 3.3 2.4 -0.9 -0.5 0 -1 -2 0
Cost for unoccupied cells:
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Net Evaluation:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO -- -- -- --
SAN
BERN
ARDIN
O
-- -- 4.7 --
FT.WORT
H
-- 1.8 -0.4 -0.6 4.3 -1 --
SANTA
FE3.9 4.1 -- -- -- -0.6 -1 2.2 1.2 -2 --
LAS
VEGAS 4.7 -- -- 3.0 3.0 -- --
As all Cij is not greater than or equal to zero, thus solution is not optimal. To obtainan optimal solution, we insert in the solution and form a loop.
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DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOEN
IX
LOS
ANGEL
ES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 6300 9460+
9360-
SAN
BERNA
RDINO
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.
WORT
H
6300 -- -- -- -- -- -- -- -- -- 43700 -- --
SANTA
FE-- -- 2130 1210 6120 -- -- -- -- -- 40540-
--
LAS
VEGAS-- -- -- -- -- 4830 2750 -- -- 8740-
-- -- 33680
+
Here = 8740
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New Basic Feasible Solution:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 6300 18200 620
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.WORT
H
6300 -- -- -- -- -- -- -- -- -- 43700 -- --
SANTA
FE-- -- 2130 1210 6120 -- -- -- -- 8740 -- 31800 --
LAS
VEGAS-- -- -- -- -- 4830 2750 -- -- -- -- -- 42430
Now, to check for optimality, we again apply MODI method.
Check for non-degeneracy:
m+n-1 = 5+13-1=17 which is equal to the number of basic variables.Thus, solution is non-degenerate.
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Cost for occupied cells:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
Ui
EL
PASO-- 3.5 -- --- -- -- -- -- -- -- 3.2 2.2 4.2 4.2
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 0.3 0.7 -- -- -- 1.2 1.2
FT.
WORT
H
0.3 -- -- -- -- -- -- -- -- -- 0 -- -- 1
SANTA
FE-- -- 4.5 6.0 2.7 -- -- -- -- 0 -- 0 -- 2
LAS
VEGAS-- -- -- -- -- 3.3 2.4 -- -- -- -- -- 0 0
Vj -0.7 -0.7 2.5 4 0.7 3.3 2.4 -0.9 -0.5 -2 -1 -2 0
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DALL
AS
SAN
ANTONIO
WICH
ITA
KANS
ASCITY
DENV
ER
SALT
LAKECITY
PHOE
NIX
LOS
ANGELES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
Ui
EL
PASO -- -- -- -- 4.2
SAN
BERN
ARDIN
O
-- -- 3.9 -- 1.2
FT.WORT
H
-- 2.1 3.1 4.4 6.0 0 -- 1
SANTA
FE5.2 5.4 -- -- -- 4.7 3.4 3.3 2.7 -- -- 2
LAS
VEGAS 5.4 -- -- 2.1 2.5 0 -- 0
Vj -0.7 -0.7 2.5 4 0.7 3.3 2.4 -0.9 -0.5 -2 -1 -2 0
Cost for unoccupied cells:
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Net Evaluation:
DALL
AS
SAN
ANTONIO
WICH
ITA
KANS
ASCITY
DENV
ER
SALT
LAKECITY
PHOE
NIX
LOS
ANGELES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
EL
PASO -- -- -- --
SAN
BERN
ARDIN
O
-- -- 4.7 --
FT.WORT
H
-- 1.8 -0.4 -0.6 4.3 1 --
SANTA
FE3.9 4.1 -- -- -- -0.6 -1 2.2 1.2 -- --
LAS
VEGAS 4.7 -- -- 3.0 3.0 2 --
As all Cij is not greater than or equal to zero, thus solution is not optimal. To obtainan optimal solution, we insert in the solution and form a loop.
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DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOEN
IX
LOS
ANGEL
ES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 6300 18200
+
620-
SAN
BERNA
RDINO
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.
WORTH
6300 -- -- -- -- -- -- -- -- -- 43700 -- --
SANTA
FE-- -- 2130 1210 6120 -- -- -- 8740 -- 31800-
--
LAS
VEGAS-- -- -- -- -- 4830 2750-
-- -- -- -- -- 42420
+
Here = 620
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New Basic Feasible Solution:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 6300 18820 --
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.WORT
H
6300 -- -- -- -- -- -- -- -- -- 43700 -- --
SANTA
FE-- -- 2130 1210 6120 -- 620 -- -- 8740 -- 31180 --
LAS
VEGAS-- -- -- -- -- 4830 2130 -- -- -- -- -- 43050
Now, to check for optimality, we again apply MODI method.
Check for non-degeneracy:
m+n-1 = 5+13-1=17 which is equal to the number of basic variables.Thus, solution is non-degenerate.
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Cost for occupied cells:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
Ui
EL
PASO-- 3.5 -- --- -- -- -- -- -- -- 3.2 2.2 -- 3.2
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 0.3 0.7 -- -- -- 1.2 1.2
FT.
WORT
H
0.3 -- -- -- -- -- -- -- -- -- 0 -- -- 0
SANTA
FE-- -- 4.5 6.0 2.7 -- 3.4 -- -- 0 -- 0 -- 1
LAS
VEGAS-- -- -- -- -- 3.3 2.4 -- -- -- -- -- 0 0
Vj 0.3 0.3 3.5 5 1.7 3.3 2.4 -0.9 -0.5 -1 0 -1 0
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Net Evaluation:
DALL
AS
SAN
ANTONIO
WICH
ITA
KANS
ASCITY
DENV
ER
SALT
LAKECITY
PHOE
NIX
LOS
ANGELES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
EL
PASO -- -- -- 1
SAN
BERN
ARDIN
O
-- -- 3.7 --
FT.WORT
H
-- 1.8 -0.4 -0.6 4.3 1 --
SANTA
FE3.9 4.1 -- -- -- 0.4 -- 3.2 2.2 -- --
LAS
VEGAS 4.7 -- -- 3.0 3.0 1 --
As all Cij is not greater than or equal to zero, thus solution is not optimal. To obtainan optimal solution, we insert in the solution and form a loop.
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DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 6300+
18820
-
--
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.WORT
H
6300 -- -- -- -- -- -- -- -- 43700-
-- --
SANTA
FE-- -- 2130 1210
-
6120 -- 620 -- -- 8740 -- 31180
+
--
LAS
VEGAS-- -- -- -- -- 4830 2130 -- -- -- -- -- 43050
Here =1210
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New Basic Feasible Solution:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 7510 17610 --
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.WORT
H
6300 -- -- 1210 -- -- -- -- -- -- 42490 -- --
SANTA
FE-- -- 2130 -- 6120 -- 620 -- -- 8740 -- 32390 --
LAS
VEGAS-- -- -- -- -- 4830 2130 -- -- -- -- -- 43050
Now, to check for optimality, we again apply MODI method.
Check for non-degeneracy:
m+n-1 = 5+13-1=17 which is equal to the number of basic variables.Thus, solution is non-degenerate.
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DALL
AS
SAN
ANTONIO
WICH
ITA
KANS
ASCITY
DENV
ER
SALT
LAKECITY
PHOE
NIX
LOS
ANGELES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
Ui
EL
PASO -- -- -- 4.2 3.2
SAN
BERN
ARDIN
O
-- -- 3.9 -- 1.2
FT.WORT
H
-- 2.1 3.1 -- 6.0 0 -- 0
SANTA
FE5.2 5.4 -- 6 -- 4.7 -- 3.3 2.7 -- -- 1
LAS
VEGAS 5.4 -- -- 2.1 2.5 0 -- 0
Vj 0.3 0.3 3.5 4.4 1.7 3.3 2.4 -0.9 -0.5 -1 0 -1 0
Cost for unoccupied cells:
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Net Evaluation:
DALL
AS
SAN
ANTONIO
WICH
ITA
KANS
ASCITY
DENV
ER
SALT
LAKECITY
PHOE
NIX
LOS
ANGELES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
EL
PASO -- -- -- 1
SAN
BERN
ARDIN
O
-- -- 3.7 --
FT.WORT
H
-- 1.8 -0.4 -- 4.3 1 --
SANTA
FE3.9 4.1 -- 0.6 -- 0.4 -- 3.2 2.2 -- --
LAS
VEGAS 4.7 -- -- 3.0 3.0 1 --
As all Cij is not greater than or equal to zero, thus solution is not optimal. To obtainan optimal solution, we insert in the solution and form a loop.
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DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 7510+
17610
-
--
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.WORT
H
6300 -- 1210 -- -- -- -- -- -- 42490-
-- --
SANTA
FE-- -- 2130
-
-- 6120 -- 620 -- -- 8740 -- 32390
+
--
LAS
VEGAS-- -- -- -- -- 4830 2130 -- -- -- -- -- 43050
Here =2130
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Cost for occupied cells:
DALL
AS
SAN
ANTO
NIO
WICH
ITA
KANS
AS
CITY
DENV
ER
SALT
LAKE
CITY
PHOE
NIX
LOS
ANGEL
ES
SAN
DIEG
O
D1 FT.
WORT
H
SANTA
FE
LAS
VEGAS
Ui
EL
PASO-- 3.5 -- --- -- -- -- -- -- -- 3.2 2.2 -- 3.2
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 0.3 0.7 -- -- -- 1.2 1.2
FT.
WORT
H
0.3 -- 3.1 4.4 -- -- -- -- -- -- 0 -- -- 0
SANTA
FE-- -- -- -- 2.7 -- 3.4 -- -- 0 -- 0 -- 1
LAS
VEGAS-- -- -- -- -- 3.3 2.4 -- -- -- -- -- 0 0
Vj 0.3 0.3 3.1 4.4 1.7 3.3 2.4 -0.9 -0.5 -1 0 -1 0
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Optimal Table:
DALL
AS
SAN
ANTONIO
WICH
ITA
KANS
ASCITY
DENV
ER
SALT
LAKECITY
PHOE
NIX
LOS
ANGELES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 9640 15480 --
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.WORT
H
6300 -- 2130 1210 -- -- -- -- -- -- 40360 -- --
SANTA
FE-- -- -- -- 6120 -- 620 -- -- 8740 -- 34520 --
LAS
VEGAS-- -- -- -- -- 4830 2130 -- -- -- -- -- 43050
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DISTRIBU
TION
SY
SYTEM
El Paso
Ft. Worth
Dallas
Wichita
KansasCity
Santa Fe
Denver
PhoenixSan
Antonio
SanBernardino
Las Vegas
Salt LakeCityLos
Angeles
San Diego
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Distribution cost from production plant to the distribution centers
=9640*3.2+15480*2.2+6960*1.2-8740*2.2
=30848+34056+8352-19228
=$54028
Distribution cost from distribution centers and production plants to the customer
zone =6300*0.3+4880*3.5+2130*3.1+1210*4.4+6120*2.7
+4830*3.3+620*3.4+2130*2.4+8580*0.3+4460*0.7
=1890+17080+6603+5324+16524+15939+2108+5112+2574+3122
=$76276
Thus, Total distribution cost=54028+76276
=$130304
Manufacturing cost = 9640*10.5+15480*10.5+6960*10+4880*10.5+
8580*10+4460*10-8740*10.5=101220+162540+69600+51240+85800+ 44600-
91770
=$423230
Thus, Total cost = $553534
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CHANGES IN COST
Darby Companys total cost will be reduced to $553,534which is a savings
of $47,408, or 7.9% from the limitation-free distribution system and a savings
of $67,236, or 10.8% from the original distribution system. The manufacturing costs are the same as those in Part II, but it is the cost of
shipping in this system that reduces the total cost.
Darby should definitely adopt this plan of allowing customer zones to be
supplied by numerous distribution centers as well as shipping directly the
mentioned customer zones.
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PART IVOver the next five years, Darby is anticipating
moderate growth (5000 meters) to the North and
West. Would you recommend that they consider
plant expansion at this time?
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DALL
AS
SAN
ANTONIO
WICH
ITA
KANS
ASCITY
DENV
ER
SALT
LAKECITY
PHOE
NIX
LOS
ANGELES
SAN
DIEGO
D1 FT.
WORTH
SANTA
FE
LAS
VEGAS
EL
PASO-- 4880 -- -- -- -- -- -- -- -- 9640 15480 --
SAN
BERN
ARDIN
O
-- -- -- -- -- -- -- 8580 4460 -- -- -- 6960
FT.WORT
H
6300 -- 2130 1210 -- -- -- -- -- -- 40360 -- --
SANTA
FE-- -- -- -- 6120 -- 620 -- -- 8740 -- 34520 --
LAS
VEGAS-- -- -- -- -- 4830 2130 -- -- -- -- -- 43050
Optimal Solution from Part III:
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El Paso plant is producing 8740 units below their capacity, which could be used to
satisfy increased demand. Thus, if the demand is focused in North there is no need
to go for plant expansion at this time.
San Bernardino plant is producing at full capacity i.e. 20000 meters.
The plant should only be expanded if the majority of the demand comes from the
West where it is cheaper to produce and ship from the San Bernardino plant.
The cost of expanding a plant should of course be weighed against the cost ofshipping to determine whether it is worth it.
RECOMMENDATIONS
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