Optimization

Transportation
Planning

Example#1: Transportation, Problems
Can be formulated as linear
programs and solved by general
purpose linear programming codes.
However, there are many computer packages, which
contain separate computer codes for these models
which take advantage of their network structure.

Transportation Problem
The transportation problem seeks to
minimize the total shipping costs of
transporting goods from m origins
(each with a supply si ) to n destinations
(each with a demand dj ), when the unit
shipping cost from an origin, i, to a
destination, j, is cij.

N-
DELHI
S-DELHI
E-DELHI
W-DELHI
Destinations
Sources
FARIDABAD
GURGAON
GAHZIABAD
S1=250
S2=200
S3= 250
D1=100
D2=150
D3=250
D4=100
17
19
16
17
13
19
16
15
15
17
17
16
CASE STUDY#1

TP: a Linear Programming Model
•The structure of the model is:
Minimize Total Shipping Cost
ST
[Amount shipped from a source] <= [Supply at that
source]
[Amount received at a destination]=[Demand at
that Destination]
• Decision variables
Xij = the number of cases shipped from plant i to warehouse j.
where: i=1 (FARIDABAD), 2 (GURGAON), 3 (GHAZIABAD)
j=1 (N-DELHI), 2 (S-DELHI), 3 (E-DELHI), 4(W-DELHI)

N-Delhi
S-Delhi
E-Delhi
W-Delhi
D1=100
D2=150
D3=250
D4=100
The supply constraints
Faridabad
S1=250
X11
X12
X13
X14
Supply from FARIDABAD X11+X12+X13+X14 = 250
Gurgaon
S2=200
X21
X22
X23
X24
Supply from GURGAON X21+X22+X23+X24 = 200
Ghaziabad
S3= 250
X31
X32
X33
X34
Supply from Ghaziabad X31+X32+X33+X34 = 250
CASE STUDY

The complete
mathematical model
Minimize 13X11+16X12+19X13+ 17X14 +17X21+19X22+16X23+15X24+
15X31+17X32+17X33+16X34
ST
Supply constrraints:
X11+ X12+ X13+ X14 250
X21+ X22+ X23+ X24 200
X31+ X32+ X33+ X34 250
Demand constraints:
X11+ X21+ X31 100
X12+ X22+ X32 150
X13+ X23+ X33 250
X14+ X24+ X34 100
All Xij are nonnegative
≤
≤
≤
=
=
=
=
Total shipment out of a supply node
cannot exceed the supply at the node.
Total shipment received at a destination
node, must equal the demand at that node.
CASE STUDY

The complete
mathematical model
Minimize 13X11+16X12+19X13+ 17X14 +17X21+19X22+16X23+15X24+
15X31+17X32+17X33+16X34
ST
Supply constrraints:
X11+ X12+ X13+ X14 250
X21+ X22+ X23+ X24 200
X31+ X32+ X33+ X34 250
Demand constraints:
X11+ X21+ X31 100
X12+ X22+ X32 150
X13+ X23+ X33 250
X14+ X24+ X34 100
All Xij are nonnegative
≤
≤
≤
=
=
=
=
Total shipment out of a supply node
cannot exceed the supply at the node.
Total shipment received at a destination
node, must equal the demand at that node.
CASE STUDY#6

– Reduced costs
• The unit shipment cost between
Gurgaon and N-Delhi must be reduced
by at least Rs.5, before it would become
economically feasible to utilize it
• If this route is used, the total cost will
increase by Rs. 5 for each case shipped
between Gurgaon and N-Delhi .

– Allowable Increase/Decrease
• This is the range of optimality.
• The unit shipment cost between Gurgaon
and N-Delhi may increase up to any level
or decrease up to Rs. 5 with no change in
the current optimal transportation plan.

Another solution:
Yet optimal

Network Planning

Solution Techniques
Mathematical optimization techniques:
1. Exact algorithms: find optimal solutions
2. Heuristics: find “good” solutions, not
necessarily optimal
Simulation models: provide a mechanism to
evaluate specified design alternatives created by
the designer.

Example
Single product
Two plants p1 and p2
• Plant p2 has an annual capacity of 60,000 units.
The two plants have the same production costs.
There are two warehouses w1 and w2 with
identical warehouse handling costs.
There are three markets areas c1,c2 and c3 with
demands of 50,000, 100,000 and 50,000,
respectively.

Unit Distribution Costs
Facility
warehouse
p1 p2 c1 c2 c3
w1 0 4 3 4 5
w2 5 2 2 1 2

CASE : NETWORK PROBLEM
Network Representation
ARNOLD
WASH
BURN
ZROX
HEWES
200,000200,000
60,000060,0000
50,00050,000
100,000100,000
50,00050,000
00
55
44
22
33
44
55
22 11
22
P1P1
P2P2
C2C2
C1C1
W1W1
W2W2
C3C3

Heuristic #1:
Choose the Cheapest Warehouse to Source
Demand
D = 50,000
D = 100,000
D = 50,000
Cap = 60,000
$5 x 140,000
$2 x 60,000
$2 x 50,000
$1 x 100,000
$2 x 50,000
Total Costs = $1,120,000

Heuristic #2:
Choose the warehouse where the total delivery costs
to and from the warehouse are the lowest
[Consider inbound and outbound distribution costs]
D = 50,000
D = 100,000
D = 50,000
Cap = 60,000
$4
$5
$2
$3
$4
$5
$2
$1
$2
$0
P1 to WH1 $3
P1 to WH2 $7
P2 to WH1 $7
P2 to WH 2 $4
P1 to WH1 $4
P1 to WH2 $6
P2 to WH1 $8
P2 to WH 2 $3
P1 to WH1 $5
P1 to WH2 $7
P2 to WH1 $9
P2 to WH 2 $4
Market #1 is served by WH1, Markets 2 and 3
are served by WH2

D = 50,000
D = 100,000
D = 50,000
Cap = 60,000
Cap = 200,000
$5 x 90,000
$2 x 60,000
$3 x 50,000
$1 x 100,000
$2 x 50,000
$0 x 50,000
P1 to WH1 $3
P1 to WH2 $7
P2 to WH1 $7
P2 to WH 2 $4
P1 to WH1 $4
P1 to WH2 $6
P2 to WH1 $8
P2 to WH 2 $3
P1 to WH1 $5
P1 to WH2 $7
P2 to WH1 $9
P2 to WH 2 $4
Total Cost = $920,000
Heuristic #2:
Choose the warehouse where the total delivery
costs to and from the warehouse are the lowest
[Consider inbound and outbound distribution
costs]

The Optimization Model
The problem described earlier can be framed as the
following linear programming problem.
Let
x(p1,w1), x(p1,w2), x(p2,w1) and x(p2,w2) be the flows from
the plants to the warehouses.
x(w1,c1), x(w1,c2), x(w1,c3) be the flows from the
warehouse w1 to customer zones c1, c2 and c3.
x(w2,c1), x(w2,c2), x(w2,c3) be the flows from warehouse
w2 to customer zones c1, c2 and c3

The problem we want to solve is:
min 0x(p1,w1) + 5x(p1,w2) + 4x(p2,w1)
+ 2x(p2,w2) + 3x(w1,c1) + 4x(w1,c2)
+ 5x(w1,c3) + 2x(w2,c1) + 2x(w2,c3)
subject to the following constraints:
x(p2,w1) + x(p2,w2) ≤ 60000
x(p1,w1) + x(p2,w1) = x(w1,c1) + x(w1,c2) + x(w1,c3)
x(w1,c1) + x(w2,c1) = 50000
x(w1,c2) + x(w2,c2) = 100000
x(w1,c3) + x(w2,c3) = 50000
all flows greater than or equal to zero.

CASE : NETWORK DESIGN
ARNOLD
WASH
BURN
ZROX
HEWES
200,000200,000
60,00060,000
50,00050,000
100,000100,000
50,00050,000
0 (A)0 (A)
5 (B)5 (B)
4 (C)4 (C)
2 (D)2 (D)
3 (E)3 (E)
4 (F)4 (F)
5 (G)5 (G)
2 (H)2 (H)
1 (I)1 (I)
2 (J)2 (J)
P1P1
P2P2
C2C2
C1C1
W1W1
W2W2
C3C3

The problem we want to solve is:
min 0x(p1,w1) + 5x(p1,w2) + 4x(p2,w1)
+ 2x(p2,w2) + 3x(w1,c1) + 4x(w1,c2)
+ 5x(w1,c3) + 2x(w2,c1) + 2x(w2,c3)
subject to the following constraints:
x(p2,w1) + x(p2,w2) ≤ 60000
x(w1,c1) + x(w2,c1) = 50000
x(w1,c2) + x(w2,c2) = 100000
x(w1,c3) + x(w2,c3) = 50000
all flows greater than or equal to zero.

Optimal Solution
Facility
warehouse
p1 p2 c1 c2 c3
w1 140,000 0 50,000 40,000 50,000
w2 0 60,000 0 60,000 0
Total cost for the optimal strategy is $740,000

CASE : NETWORK PROBLEM
ARNOLD
WASH
BURN
ZROX
HEWES
200,000200,000
60,00060,000
50,00050,000
100,000100,000
50,00050,000
00
55
44
22
33
44
55
22 11
22
P1P1
P2P2
C2C2
C1C1
W1W1
W2W2
C3C3
140,000140,000
60,00060,000
50,000
50,000
40,,000
40,,000
50,000
50,000
60,000
60,000

New Supply Chain Strategy
OBJECTIVES:
• Reduce inventory and financial risks
• Provide customers with competitive response times.
ACHIEVE THE FOLLOWING:
• Determining the optimal location of inventory across the various
stages
• Calculating the optimal quantity of safety stock for each component at
each stage
Hybrid strategy of Push and Pull
• Push Stages produce to stock where the company keeps safety stock
• Pull stages keep no stock at all.
Challenge:
• Identify the location where the strategy switched from Push-based to
Pull-based
• Identify the Push-Pull boundary
Benefits:
• For same lead times, safety stock reduced by 40 to 60%
• Company could cut lead times to customers by 50% and still reduce
safety stocks by 30%

Three Different Product Categories
High variability - low volume products
Low variability - high volume products, and
Low variability - low volume products.

Supply Chain Strategy Different for
the Different Categories
High variability low volume products
• Position them mainly at the primary warehouses
demand from many retail outlets can be aggregated
reducing inventory costs.
Low variability high volume products
• Position close to the retail outlets at the secondary
warehouses
• Ship fully loaded tracks as close as possible to the
customers reducing transportation costs.
Low variability low volume products
• Require more analysis since other characteristics are
important, such as profit margins, etc.

Modeling Supply Chain and Network Planning

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