/ / 1Winston Chapter 7.6, Page 383, Number 3 (Transshipment)
Problem Statement: In Problem 2, assume that before being shipped to Los Angeles or New York, all oil produced at the wells must be refined at either Galveston or Mobile. To refine 1,000 barrels of oil costs $12 at Mobile and $10 at Galveston. Assuming that both Mobile and Galveston have infinite refinery capacity, formulate a transshipment and balanced transportation model to minimize the daily cost of transporting and refining the oil requirements of Los Angeles and New York.
The transportation tableau remains the same as the previous problem except that certain costs have risen and the possibilities of shipping directly from either of the wells to Los Angeles or New York have disappeared. A basic transportation graph is shown below to detail the possible routes of travel.
There are two ways to incorporate the added refining costs into the transportation tableau.
The first way would be to add $12 to every shipment that leaves Mobile and add $10 to every shipment that leaves Galveston, as shown below. Additionally, New York and Los Angeles are now blocked from receiving directly from Well 1 or 2 by using +M for those costs. The new shipment costs are incorporated into the tableau below (Transportation Tableau 1).
Transportation Tableau 1:
$10 + 12 / $13 + 10 / +M / +M / $0
Well 1 / 150,000
$15 + 12 / $12 +10 / +M / +M / $0
Well 2 / 200,000
$0 / $6 / $16 / $17 / $0
Mobile / 350,000
$6 / $0 / $14 / $16 / $0
Galveston / 350,000
+M / +M / $0 / $15 / $0
N.Y. / 350,000
+M / +M / $15 / $0 / $0
L.A. / 350,000
Demand / 350,000 / 350,000 / 490,000 / 510,000 / 50,000 / 1,750,000
For curious minds, the results from Quant using this first method are shown below:
Quant 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 038303A Page 1
-- Minimization --
From To
Well1 Mobile:+22.00 Galves:+23.00 NY: M LA: M
Well2 Mobile:+27.00 Galves:+22.00 NY: M LA: M
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
Quant Output:
|------|
| Summary of Results for 038303A Page : 1 |
|------|
|From |To |Shipment|@ cost |Opp.Ct.|From |To |Shipment|@ cost |Opp.Ct. |
|------+------+------+------+------+------+------+------+------+------|
|Well1 |Mobile|+100.00 |+22.000| 0|Galves|Mobile| 0 |+6.0000|+7.0000 |
|Well1 |Galves| 0 |+23.000| 0|Galves|Galves| 0 | 0| 0 |
|Well1 |NY | 0 |Infini.|Infini.|Galves|NY |+140.00 |+14.000| 0 |
|Well1 |LA | 0 |Infini.|Infini.|Galves|LA |+60.000 |+16.000| 0 |
|Well1 |Dummy |+50.000 | 0| 0|Galves|Dummy | 0 | 0|+23.000 |
|Well2 |Mobile| 0 |+27.000|+6.0000|NY |Mobile| 0 |Infini.|Infini. |
|Well2 |Galves|+200.00 |+22.000| 0|NY |Galves| 0 |Infini.|Infini. |
|Well2 |NY | 0 |Infini.|Infini.|NY |NY | 0 | 0| 0 |
|Well2 |LA | 0 |Infini.|Infini.|NY |LA | 0 |+15.000|+13.000 |
|Well2 |Dummy | 0 | 0|+1.0000|NY |Dummy | 0 | 0|+37.000 |
|Mobile|Mobile| 0 | 0| 0|LA |Mobile| 0 |Infini.|Infini. |
|Mobile|Galves| 0 |+6.0000|+5.0000|LA |Galves| 0 |Infini.|Infini. |
|Mobile|NY | 0 |+16.000|+1.0000|LA |NY | 0 |+15.000|+17.000 |
|Mobile|LA |+100.00 |+17.000| 0|LA |LA | 0 | 0| 0 |
|Mobile|Dummy | 0 | 0|+22.000|LA |Dummy | 0 | 0|+39.000 |
|------|
| Minimized OBJ = 11220 Iteration = 2 Elapsed CPU second = .3896484 |
|------|
The second way would be to add $12 to every shipment that enters Mobile and add $10 to every shipment that enters Galveston, as shown below. Again, New York and Los Angeles are blocked from receiving directly from Well 1 or 2 by using +M for those costs. The new shipment costs for are incorporated into the tableau below (Transportation Tableau 2).
Transportation Tableau 2:
Mobile / Galveston / N.Y. / L.A. / Dummy / Supply$10 / $13 / +M / +M / $0
Well 1 / 150,000
$15 / $12 / +M / +M / $0
Well 2 / 200,000
$0 / $6 / $16 + 12 / $17 + 12 / $0
Mobile / 350,000
$6 / $0 / $14 + 10 / $16 + 10 / $0
Galveston / 350,000
+M / +M / $0 / $15 / $0
N.Y. / 350,000
+M / +M / $15 / $0 / $0
L.A. / 350,000
Demand / 350,000 / 350,000 / 490,000 / 510,000 / 50,000 / 1,750,000
For even more curious minds, the results from Quant using this second method are shown below:
Quant 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 038303B Page 1
-- Minimization --
From To
Well1 Mobile:+10.00 Galves:+13.00 NY: M LA: M
Well2 Mobile:+15.00 Galves:+12.00 NY: M LA: M
Mobile Mobile: 0 Galves:+6.000 NY: +28.00 LA: +29.00
Galves Mobile:+6.000 Galves: 0 NY: +24.00 LA: +26.00
NY Mobile:M Galves:M NY: 0 LA: +15.00
LA Mobile:M Galves:M NY: +15.00 LA: 0
Quant Output:
|------|
| Summary of Results for 038303B Page : 1 |
|------|
|From |To |Shipment|@ cost |Opp.Ct.|From |To |Shipment|@ cost |Opp.Ct. |
|------+------+------+------+------+------+------+------+------+------|
|Well1 |Mobile|+100.00 |+10.000| 0|Galves|Mobile| 0 |+6.0000|+9.0000 |
|Well1 |Galves| 0 |+13.000| 0|Galves|Galves| 0 | 0| 0 |
|Well1 |NY | 0 |Infini.|Infini.|Galves|NY |+140.00 |+24.000| 0 |
|Well1 |LA | 0 |Infini.|Infini.|Galves|LA |+60.000 |+26.000| 0 |
|Well1 |Dummy |+50.000 | 0| 0|Galves|Dummy | 0 | 0|+13.000 |
|Well2 |Mobile| 0 |+15.000|+6.0000|NY |Mobile| 0 |Infini.|Infini. |
|Well2 |Galves|+200.00 |+12.000| 0|NY |Galves| 0 |Infini.|Infini. |
|Well2 |NY | 0 |Infini.|Infini.|NY |NY | 0 | 0| 0 |
|Well2 |LA | 0 |Infini.|Infini.|NY |LA | 0 |+15.000|+13.000 |
|Well2 |Dummy | 0 | 0|+1.0000|NY |Dummy | 0 | 0|+37.000 |
|Mobile|Mobile| 0 | 0| 0|LA |Mobile| 0 |Infini.|Infini. |
|Mobile|Galves| 0 |+6.0000|+3.0000|LA |Galves| 0 |Infini.|Infini. |
|Mobile|NY | 0 |+28.000|+1.0000|LA |NY | 0 |+15.000|+17.000 |
|Mobile|LA |+100.00 |+29.000| 0|LA |LA | 0 | 0| 0 |
|Mobile|Dummy | 0 | 0|+10.000|LA |Dummy | 0 | 0|+39.000 |
|------|
| Minimized OBJ = 11220 Iteration = 2 Elapsed CPU second = .5498047 |
|------|
Indeed, the refining costs can be incorporated into the problem by adding them to the shipping costs either going to or coming from Galveston and Mobile to obtain the same solution and objective function.