Learning Outcomes

• Mahasiswa dapat menghitung solusi model transportasi dengan menggunakan program komputer..

Outline Materi:

• Masalah Transportasi• Pembuatan program komputer• Contoh & Penyelesaian..

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.

• The network representation for a transportation problem with two sources and three destinations is given on the next slide.

Transportation Problem

• Network Representation 11

22

33

11

22

cc11

11cc1212

cc1313

cc2121 cc2222

cc2323

dd11

dd22

dd33

ss11

s2

SOURCESSOURCES DESTINATIONSDESTINATIONS

Transportation Problem

• LP FormulationThe LP formulation in terms of the amounts

shipped from the origins to the destinations, xij , can be written as:

Min cijxij

i j

s.t. xij < si for each origin i j

xij = dj for each destination j

i

xij > 0 for all i and j

Transportation Problem• LP Formulation Special Cases

The following special-case modifications to the linear programming formulation can be made:– Minimum shipping guarantee from i to j:

xij > Lij

– Maximum route capacity from i to j:

xij < Lij

– Unacceptable route:

Remove the corresponding decision variable.

Example: BBCExample: BBC

Building Brick Company (BBC) has Building Brick Company (BBC) has orders for 80 tons of bricks at three orders for 80 tons of bricks at three suburban locations as follows: Northwood suburban locations as follows: Northwood -- 25 tons, Westwood -- 45 tons, and -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. BBC has two plants, Eastwood -- 10 tons. BBC has two plants, each of which can produce 50 tons per each of which can produce 50 tons per week. Delivery cost per ton from each week. Delivery cost per ton from each plant to each suburban location is shown plant to each suburban location is shown on the next slide.on the next slide.

How should end of week shipments be How should end of week shipments be made to fill the above orders?made to fill the above orders?

Example: BBCExample: BBC

Delivery Cost Per TonDelivery Cost Per Ton

NorthwoodNorthwood WestwoodWestwood EastwoodEastwood

Plant 1 24 Plant 1 24 30 30 4040

Plant 2 Plant 2 30 40 30 40 4242

Example: BBCExample: BBC

Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem Data

A B C D E F G H

1

2 Constraint X11 X12 X13 X21 X22 X23 RHS

3 #1 1 1 1 50

4 #2 1 1 1 50

5 #3 1 1 25

6 #4 1 1 45

7 #5 1 1 10

8 Obj.Coefficients 24 30 40 30 40 42 30

LHS Coefficients

Example: BBCExample: BBC

Partial Spreadsheet Showing Optimal SolutionPartial Spreadsheet Showing Optimal Solution

A B C D E F G

10 X11 X12 X13 X21 X22 X23

11 Dec.Var.Values 5 45 0 20 0 10

12 Minimized Total Shipping Cost 2490

13

14 LHS RHS

15 50 <= 50

16 30 <= 50

17 25 = 25

18 45 = 45

19 10 = 10E.Dem.

W.Dem.

N.Dem.

Constraints

P1.Cap.

P2.Cap.

Optimal SolutionOptimal Solution

FromFrom ToTo AmountAmount CostCost

Plant 1 Northwood 5 Plant 1 Northwood 5 120 120

Plant 1 Westwood 45 Plant 1 Westwood 45 1,3501,350

Plant 2 Northwood 20 Plant 2 Northwood 20 600 600

Plant 2 Eastwood 10 Plant 2 Eastwood 10 420 420

Total Cost = Total Cost = $2,490$2,490

Example: BBCExample: BBC

Example: BBCExample: BBC

Partial Sensitivity Report (first half)Partial Sensitivity Report (first half)

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$C$12 X11 5 0 24 4 4$D$12 X12 45 0 30 4 1E+30$E$12 X13 0 4 40 1E+30 4$F$12 X21 20 0 30 4 4$G$12 X22 0 4 40 1E+30 4$H$12 X23 10.000 0.000 42 4 1E+30

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$C$12 X11 5 0 24 4 4$D$12 X12 45 0 30 4 1E+30$E$12 X13 0 4 40 1E+30 4$F$12 X21 20 0 30 4 4$G$12 X22 0 4 40 1E+30 4$H$12 X23 10.000 0.000 42 4 1E+30

Example: BBCExample: BBC

Partial Sensitivity Report (second half)Partial Sensitivity Report (second half)

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$E$17 P2.Cap 30.0 0.0 50 1E+30 20$E$18 N.Dem 25.0 30.0 25 20 20$E$19 W.Dem 45.0 36.0 45 5 20$E$20 E.Dem 10.0 42.0 10 20 10$E$16 P1.Cap 50.0 -6.0 50 20 5

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$E$17 P2.Cap 30.0 0.0 50 1E+30 20$E$18 N.Dem 25.0 30.0 25 20 20$E$19 W.Dem 45.0 36.0 45 5 20$E$20 E.Dem 10.0 42.0 10 20 10$E$16 P1.Cap 50.0 -6.0 50 20 5