/ / 1Winston Chapter 7.6, Page 383, Number 2 (Transshipment)
Problem Statement: Sunco Oil produces oil at two wells. Well 1 can produce up to 150,000 barrels per day, and well 2 can produce up to 200,000 barrels per day. It is possible to ship oil directly from the wells to Sunco's customers in Los Angeles and New York. Alternatively, Sunco could transport oil to the ports of Mobile and Galveston and then ship it by tanker to New York or Los Angeles. Los Angeles requires 160,000 barrels per day, and New York requires 140,000 barrels per day. The costs of shipping 1,000 barrels between two points are shown in Table 59. Formulate a transshipment model (and equivalent transportation model) that could be used to minimize the transport costs in meeting the oil demands of Los Angeles and New York.
Create a balanced transportation tableau and find an initial basic feasible solution using the minimum cost method.
Table 59: Cost of shipping 1,000 barrels between two points.
ToFrom / Well 1 / Well 2 / Mobile / Galveston / N.Y. / L.A.
Well 1 / $0 / -- / $10 / $13 / $25 / $28
Well 2 / -- / $0 / $15 / $12 / $26 / $25
Mobile / -- / -- / $0 / $6 / $16 / $17
Galveston / -- / -- / $6 / $0 / $14 / $16
N.Y. / -- / -- / -- / -- / $0 / $15
L.A. / -- / -- / -- / -- / $15 / $0
The problem statement implies that wells 1 and 2 are supplies while Los Angeles, New York, Mobile and Galveston are warehouses or transshipment points. It may seem that Los Angeles and New York are demand points, but since they are capable of supplying each other with oil as seen only in the table, they are indeed transshipment points. A basic transportation graph is shown below to detail the possible routes of travel.
Transshipment Problem as Balanced Transportation Problem:
Supply
$10 / $13 / $25 / $28 / $0Well 1 / 150,000
$15 / $12 / $26 / $25 / $0
Well 2 / 200,000
$0 / $6 / $16 / $17 / $0 / (s + 0) =
Mobile / 350,000
$6 / $0 / $14 / $16 / $0 / (s + 0) =
Galveston / 350,000
+M / +M / $0 / $15 / $0 / (s + 0) =
N.Y. / 350,000
+M / +M / $15 / $0 / $0 / (s + 0) =
L.A. / 350,000
(s + 0) = / (s + 0) = / (s + 140,000)= / (s + 160,000)= / (√)
Demand / 350,000 / 350,000 / 490,000 / 510,000 / 50,000 / 1,750,000
Each supply point has a supply equal to its original supply, and each demand point has a demand equal to its original demand. Let s = total available supply of 350,000. Each transshipment point has a supply equal to (point’s original supply) + s and a demand equal to (point’s original demand) + s.
Since the total supply exceeds the demand by 50,000 a dummy demand is created to balance the problem.
Quant automatically includes a dummy demand to take care of the unbalanced supply and demand totals. The input information required by Quant in the transshipment program can be seen below. Since the barrel costs were given per 1,000 barrels, the barrel values were re-calculated so that the cost would be on a per-barrel basis.
Quant Transshipment Input:
Capacities of Sources Page 1
Well1: +150.0 Well2: +200.0
Capacities/Demands of Transshipment Points Page 1
Mobile: 0 Galves: 0 NY: -140.0 LA: -160.0
Cost/Profit Coefficients for 038302A Page 1
-- Minimization --
From To
Well1 Mobile:+10.00 Galves:+13.00 NY: +25.00 LA: +28.00
Well2 Mobile:+15.00 Galves:+12.00 NY: +26.00 LA: +25.00
Mobile Mobile: 0 Galves:+6.000 NY: +16.00 LA: +17.00
Galves Mobile:+6.000 Galves: 0 NY: +14.00 LA: +16.00
NY Mobile:M Galves:M NY: 0 LA: +15.00
LA Mobile:M Galves:M NY: +15.00 LA: 0
The supply and demand capacities of the Galveston and Mobile transshipment points are both zero because they are simply "handoff" points. The capacities of New York and Los Angeles are both negative values because they are also transshipment points, but have defined demands. Quant was able to minimize the costs of the oil shipments as seen in the minimized objective below.
Quant Summary of Results:
|------|
| Summary of Results for 038302A Page : 1 |
|------|
|From |To |Shipment|@ cost |Opp.Ct.|From |To |Shipment|@ cost |Opp.Ct. |
|------+------+------+------+------+------+------+------+------+------|
|Well1 |Mobile| 0 |+10.000| 0|Galves|Mobile| 0 |+6.0000|+8.0000 |
|Well1 |Galves| 0 |+13.000|+1.0000|Galves|Galves| 0 | 0| 0 |
|Well1 |NY |+140.00 |+25.000| 0|Galves|NY | 0 |+14.000|+1.0000 |
|Well1 |LA | 0 |+28.000|+3.0000|Galves|LA | 0 |+16.000|+3.0000 |
|Well1 |Dummy |+10.000 | 0| 0|Galves|Dummy | 0 | 0|+12.000 |
|Well2 |Mobile| 0 |+15.000|+5.0000|NY |Mobile| 0 |Infini.|Infini. |
|Well2 |Galves| 0 |+12.000| 0|NY |Galves| 0 |Infini.|Infini. |
|Well2 |NY | 0 |+26.000|+1.0000|NY |NY | 0 | 0| 0 |
|Well2 |LA |+160.00 |+25.000| 0|NY |LA | 0 |+15.000|+15.000 |
|Well2 |Dummy |+40.000 | 0| 0|NY |Dummy | 0 | 0|+25.000 |
|Mobile|Mobile| 0 | 0| 0|LA |Mobile| 0 |Infini.|Infini. |
|Mobile|Galves| 0 |+6.0000|+4.0000|LA |Galves| 0 |Infini.|Infini. |
|Mobile|NY | 0 |+16.000|+1.0000|LA |NY | 0 |+15.000|+15.000 |
|Mobile|LA | 0 |+17.000|+2.0000|LA |LA | 0 | 0| 0 |
|Mobile|Dummy | 0 | 0|+10.000|LA |Dummy | 0 | 0|+25.000 |
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| Minimized OBJ = 7500 Iteration = 4 Elapsed CPU second = .6015625 |
|------|
The results indicate that a minimum cost occurs when Well 1 supplies New York with 140,000 barrels at 2.5¢ per barrel for a cost of $3,500. Also, Well 2 will supply Los Angeles with 160,000 barrels at 2.5¢ per barrel for a cost of $4,000. It is important to note that the Mobile and Galveston transshipment points will not send or receive any barrels with this optimum plan. The total cost of shipping the 300,000 barrels to their destinations in Los Angeles and New York will be $7,500.