CA5103 – Management Science
Analytical and Computer Solutions
of Transportation Problems
Learning Objectives:
At the end of this module, the students should be able to:
1. Identify the different methods used in finding the
analytical solution of transportation problems;
2.Solve transportation problems using the Northwest
Corner Rule, Least Cost Method and Modified
Distribution and;
2.Solve transportation and transshipment problems using
the Excel Solver.
Determination of a Starting
Feasible Solution (Uy et al.,
2013)
The general definition of the transportation model requires that:
Note: A starting basic feasible solution must include m + n -1 basic
variables (known as the Rim Requirement).
Analytical Solution of Transportation Problem
❑ Northwest Corner Method (NWC)
- provides a straightforward technique for obtaining the initial
solution (systematic, easily understandable method)
- the costs are not relevant in determining the initial solution
Listed below are the steps to find the initial basic feasible solution
(Uy et al., 2013) :
1. Starting with the northwest most (upper left hand) corner,
allocate the smaller amount of either the row supply or the
column demand thereby exhausting the supply and demand
requirements.
Analytical Solution of Transportation Problem
2.Subtract from the row supply and from the column demand the
amount allocated. If the column demand is now zero, move to
the cell next on the right; if the row supply is zero, move down
the cell in the next row. If both are zero, move first to the next
cell on the right, place 0, then down one cell.
3. Once a cell is identified ass per step 2, it becomes the new
northwest cell. Allocate an amount as in step 1.
4. Repeat the above steps 1 to 3 until all remaining supply and
demand is gone.
Analytical Solution of Transportation Problem
Example 1: Find the initial basic feasible solution using the Northwest
corner method. (Note: Consider the Acme Block Company
transportation problem below.)
Northwood
Westwood
Eastwood
To
Plant
From
Supply
40
24
30
40
Plant 1
40
30
40
42
Plant 2
80
Demand
25
45
10
80
Analytical Solution of Transportation Problem
Example 1 (using NWC):
To
Northwood Westwood
Eastwood
From
Plant 1
Plant
Supply
40
25
40
Plant 2
80
Demand
25
45
10
80
Step 1 : Begin in the upper
left hand corner of the table
by allocating the 25 units of
resources (smaller amount of
either row supply or column
demand) to exhaust the
requirements.
Analytical Solution of Transportation Problem
Example 1 (using NWC):
To
Northwood Westwood
Eastwood
From
Plant 1
25
Plant
Supply
40
15
40
Plant 2
80
Demand
25
45
10
80
Step 2 and Step 3 : Subtract
25 from the row supply in
Plant 1 (40) and allocate the
said amount of resources
(15) of Plant 1 to Westwood.
Note: The resources in row 1
are
fully
allocated
(exhausted).
Analytical Solution of Transportation Problem
Example 1 (using NWC):
To
Northwood Westwood
Eastwood
From
Plant 1
25
40
15
30
Plant 2
Plant
Supply
40
Step 4: Repeat steps 1 to 3
until all resources are
exhausted
and
all
requirements are satisfied.
10
80
Demand
25
45
10
80
Note that a solution becomes basic if the number of occupied cells, i.e. cells with allocations is m+n-1, where m represents the
number of rows and n, the number of columns (if m = 2 and n = 3 then m+n-1 = 4, thus the basic feasible solution must include 4
basic variables).
Analytical Solution of Transportation Problem
Example 1 (using NWC):
To
Northwood Westwood
Eastwood
From
24
25
Plant 1
30
30
15
40
30
Plant 2
Plant
Supply
40
40
42
40
10
80
Demand
25
45
10
80
Z = 24 (25) + 30 (15) + 40 (30) + 42 (10)
= 2670
The total cost in this solution is
obtained by multiplying the
cells allocation to the shipping
cost per unit.
Analytical Solution of Transportation Problem
Example 2: Find the initial basic feasible solution using the Northwest
corner method.
Transportation
Destination
Transportation
Government
W1
W2
W3
Mode
Regulations
Truck
12
6
5
3000
Railroad
20
11
9
3000
30
26
28
Airplane
Material
Requirements
3000
9000
4000
2500
2500
9000
Analytical Solution of Transportation Problem
Example 2 ( using NWC):
Transportation
Mode
W1
Truck
12
Railroad
20
Airplane
30
Material
Requirements
3000
1000
Destination
W2
6
11
26
2000
500
W3
Transportation
Government
Regulations
5
3000
9
3000
28
3000
2500
9000
4000
2500
2500
9000
Z = 12 (3000) + 20 (1000) + 11 (2000) + 26 (500) + 28 (2500)
= 161,000
Analytical Solution of Transportation Problem
❑ Least Cost (Minimum Cell) Method
- involves sequentially allocating the resources to the cells with the
minimum cost to obtain the initial solution
Here are the steps to find the initial basic feasible solution
(Uy et al., 2013) :
1. Select the cell with the lowest available cost. Allocate as much as
possible in view of the capacity of its row and the destination
requirement of its column.
2. Choose the next lowest-cost cell and make an allocation in view of
the remaining capacity and requirement of its row and column.
3. Repeat the process until all remaining supply and demand is
exhausted.
Analytical Solution of Transportation Problem
The Least Cost Method (LCM) yields not only an initial feasible
solution but also one that is close to optimal in small transportation
problems. This method is “heuristic” in nature.
Note: If there is a tie for a lowest-cost cell during any allocation, we
may choose any of these cells for allocation. If a single allocation
exhaust the capacity of a row and satisfies the requirement of a
column, we place a zero in one of the bordering cells (Uy et al., 2013).
Analytical Solution of Transportation Problem
Example 1 (using LCM):
Northwood
Westwood
Eastwood
Plant 1
24
30
40
Plant
Supply
40
Plant 2
30
40
42
40
To
From
Demand
25
25
45
10
80
80
Step 1: Select the cell with the least cost and allocate the shipment to
exhaust either the supply of plants or meet the demand requirements.
Note that the lowest cost is 24 (in cell Plant 1 to Northwood).
Analytical Solution of Transportation Problem
Example 1 (using LCM):
Northwood
Westwood
Eastwood
Plant 1
24
30
40
Plant
Supply
40
Plant 2
30
42
40
To
From
Demand
25
15
40
25
45
10
80
80
Step 2 : Choose the next lowest cost cell (with cost of 30) then make allocation of
15 units of resources meeting all the supplies in Plant 1.
Analytical Solution of Transportation Problem
Example 1 (using LCM):
Northwood
Westwood
Eastwood
Plant 1
24
30
40
Plant
Supply
40
Plant 2
30
42
40
To
From
Demand
25
15
40
25
30
45
10
10
80
80
Step 3 : Repeat the process until all remaining supply and demand is exhausted.
The total transportation cost: Z = 24 (25) + 30 (15) + 40 (30) + 42 (10) = 2670
Analytical Solution of Transportation Problem
Example 2: Find the initial basic feasible solution using the Least Cost
method.
Transportation
Mode
W1
Destination
W2
W3
Transportation
Government
Regulations
Truck
12
6
5
3000
Railroad
20
11
9
3000
Airplane
30
26
28
3000
Material
Requirements
9000
4000
2500
2500
9000
Analytical Solution of Transportation Problem
Example 2 ( using LCM):
Transportation
Mode
W1
Truck
12
Railroad
20
Airplane
30
Material
Requirements
Destination
W2
6
1000
3000
11
500
2000
26
W3
5
Transportation
Government
Regulations
3000
2500
9
3000
28
3000
9000
4000
2500
2500
9000
Z = 5 (2500) + 6 (500) + 11 (2000) + 20 (1000) + 30 (3000)
= 147,500
Analytical Solution of Transportation Problem
❑ Modified Distribution (MODI) Method (Uy et al., 2013)
- an evaluation procedure used to examine if it is more desirable
to move a shipment into one of the unused cells
- aims to determine whether a better schedule of shipments from
plants to warehouses can be developed
- used to compute improvement indices for each unused cell
without drawing all of the closed paths
Analytical Solution of Transportation Problem
The steps to find the initial basic feasible solution are as follows
(Uy et al., 2013) :
1. For each solution, compute the R and K values for the occupied or
used cells in the table using the formula:
Ri + Kj = Cij where R1 is always set to 0.
2. Calculate the improved indices for all empty or unused cells using:
Improvement index: Iij = Cij – (Ri + Kj)
3. Select the unused cell with the largest negative index.
Note: If all indices are equal to or greater than zero, the solution is
optimal.
Analytical Solution of Transportation Problem
4. Trace the close path for the unused cell having the largest negative
index.
5. Develop an improved solution.
6. Repeat steps 1 to 5 until an optimal solution has been found.
Analytical Solution of Transportation Problem
Example 1 (using MODI):
To
Northwood Westwood
Eastwood
24
40
From
25
Plant 1
30
30
15
40
30
Plant 2
Plant
Supply
40
42
40
10
80
Demand
25
45
10
80
Step 1 : Begin with the same
initial solution obtained using
NWC. To compute for R and K
values, consider the occupied
or used cells. Note that there
are four occupied cells (Plant
1 to Northwood, Plant 1 to
Westwood, Plant 2 to
Westwood and Plant 2 to
Eastwood).
Analytical Solution of Transportation Problem
Example 1 (using MODI):
To
Northwood Westwood
Eastwood
24
40
From
25
Plant 1
30
30
15
40
30
Plant 2
Plant
Supply
40
42
40
10
80
Demand
25
45
10
80
Step 1 : Begin with the same
initial solution obtained using
NWC. To compute for R and K
values, consider the occupied
or used cells. Note that there
are four occupied cells (Plant
1 to Northwood, Plant 1 to
Westwood, Plant 2 to
Westwood and Plant 2 to
Eastwood).
Analytical Solution of Transportation Problem
With four occupied cells (using Cij = Ri + Kj), the following are
obtained:
24 = R1 + K1; 30 = R1 + K2; 40 = R2 + K2 and; 42 = R2 + K2
Since R1 is assumed equal to zero, then
R2= 10; K1= 24; K2= 30; K3= 32;
Step 2 : After the row and column values are computed, the next step is
to evaluate each unused/ unoccupied cells by computing their
improvement indices using Iij = Cij – (Ri + Kj). Thus, I13 = C13 – (R1 + K3)
and I21 = C21 – (R2 + K1) are computed.
Note that there are two unused cells in the problem and their
improvement indices are: I13 = 8 and I21 = -4.
Analytical Solution of Transportation Problem
Step 3: Since the improvement index in the unused cell (row 2 to
column 1) is negative (I21 = -4), the solution in not yet optimal.
Step 4 and Step 5: Develop a new improved solution by tracing a
close path for the cell having the largest negative index (I21 = -4). This
is done by placing plus and minus signs at alternate corners of the
path, beginning with a plus sign at the unused cell (row 2 to column
1). The smallest number in a negative position in the close path
indicates the quantity that can be assigned to the unused cell being
entered in the solution. This quantity is added to all cells in the close
path with plus sign and subtracted from those cells with minus signs.
Analytical Solution of Transportation Problem
Example 1 (using MODI):
To
Northwood Westwood
Eastwood
24
40
40
42
40
From
Plant 1
-
30
Plant 2
25
30
15
+
40
+
30
-
Plant
Supply
10
80
Demand
25
45
S-smallest number in a
negative position in
the close path;
10
80
Ce cell having the
largest negative index
I21 = -4
Analytical Solution of Transportation Problem
Example 1 (using MODI):
To
Northwood Westwood
Eastwood
24
30
40
40
30
40
42
40
From
40
Plant 1
Plant 2
25
5
Plant
Supply
10
80
Demand
25
45
10
80
Note that the table shows
the results of adding and
subtracting the smallest
number (25) in a negative
position in the close path to
quantities with plus and
negative signs, respectively.
Step 6: Evaluate now the unused cells in this new solution and repeat
the process until all indices are equal to or greater than zero.
Analytical Solution of Transportation Problem
The evaluation of each used and unused cells of the second
solution is shown below:
With four occupied cells (using Cij = Ri + Kj), the following are
obtained:
30 = R1 + K2; 30 = R2 + K1; 40 = R2 + K2 and; 42 = R2 + K3
Since R1 is assumed equal to zero, then
R2= 10; K1= 20; K2= 30; K3= 32;
Now, evaluating the two unused cells in the problem by computing their
improvement indices using Iij = Cij – (Ri + Kj)
leads to I11 = 4 and I13= 8.
Since all improvement index values are either zero or positive, the
solution is optimal.
Analytical Solution of Transportation Problem
Example 1 (using MODI):
To
Northwood Westwood
Eastwood
24
30
40
40
30
40
42
40
From
40
Plant 1
Plant 2
25
5
Plant
Supply
10
80
Demand
25
45
10
80
Thus, the minimum cost is
Z = 30 (40) + 30 (25) + 40 (5)
+ 42 (10)
= 2570
Analytical Solution of Transportation Problem
Example 2:Use MODI method to determine the optimal allocation of the
given transportation.(Note: Consider the initial solution
obtained in LCM)
Transportation
Mode
W1
Destination
W2
W3
Transportation
Government
Regulations
Truck
12
6
5
3000
Railroad
20
11
9
3000
Airplane
30
26
28
3000
Material
Requirements
9000
4000
2500
2500
9000
Analytical Solution of Transportation Problem
Example 2 (using MODI):
Transportation
Mode
W1
Destination
W2
Truck
12
Railroad
20
11
Airplane
30
26
Material
Requirements
1000
3000
6
2000
500
W3
Transportation
Government
Regulations
5
3000
9
3000
2500
3000
28
9000
4000
2500
2500
9000
Z = 12 (1000) + 6 (2000) + 11 (500) + 9(2500) + 30 (3000)
= 142,000
Excel Solution of Transportation Problem
The steps in solving transportation problem (Example 1) using Excel Solver
is presented below (Anderson et al., 2019):
Data entry: Enter the data for the
transportation costs, the origin supplies,
and the destination demand in the top
portion of the worksheet. A linear
programming model is developed in the
bottom part of the worksheet. Note the four
key elements in the worksheet model: DV,
≤
≤
OF, LHS and RHS constraints. The data
and descriptive labels are contained in
A1:E5, transportation costs in B3:D4,
origin supplies in E3:E4 and destination
demands in B5: D5
Excel Solution of Transportation Problem
Formulation: Listed below are the key
elements required by the Excel Solver.
Decision Variables (DV) – Cells B15:D16
are reserved for the DV while the other
decision variables equal zero indicate
that nothing will be shipped over the
corresponding routes.
Objective Function (OF) – The formula
SUMPRODUCT(B3:D4,B15:D16) has
been placed into cell A11 to compute the
cost of the solution.
≤
Excel Solution of Transportation Problem
Left- Hand Sides (LHS) – Cells E15:16 contain
the LHS for the supply constraints while cells
B17:D17 contain the LHS for the demand
constraints.
Cell E15 = SUM(B15:D15) (Copy to E16)
Cell B17 = SUM(B15:B16) (Copy to C17:D17)
Right-Hand Sides (RHS) – Cells G15:16
contain the RHS for the supply constraints
while cells B19:D19 contain the RHS for the
demand constraints.
Cell G15 = E3 (Copy to G16)
Cell B19 = B5 (Copy to C19:D19)
Excel Solution of Transportation Problem
Select Solver from the Analysis Group in
the Data Ribbon. When the Solver
Parameters dialog box appears, enter the
proper values for the constraints and the
objective function, select Simplex LP,
and click the checkbox for Make
Unconstrained Variables Nonnegative. Then click Solve.
≤
≤
Computer Solution of Transportation Problem
The excel solution displays that
the optimal values are shown to
be x12= 40, x21= 25, x22= 5, and
x23= 10. Also, the minimum cost
is 2570.
≤
≤
Excel Solution of Transshipment Problem
The steps in solving transshipment problem (Example 1 on Northside and Southside
Facilities) using Excel Solver is listed below (Anderson et al., 2019). Note that the
worksheet is organized into sections: an arc section and a node section.
Formulation: The arc section uses
cells A4:C14. Each arc is identified
in cells A5:A14. The costs are
identified in cells in B5:B14, and
cells C5:C14 are reserved for the
values of the decision variables (the
amount shipped over the arcs). The
node section uses cells F5:K13.
Each of the nodes is identified in
cells F7:F13.
Excel Solution of Transshipment Problem
The following formulas are entered into cells
G7:H13 to represent the flow out and the flow in
for each node:
Units shipped in:
Cell G9 = C5 + C7
Cell G10 = C6 + C8
Cell G11 = C9 + C12
Cell G12 = C10 + C13
Units
shipped
Cell G13
= C11out:
+ C14
Cell H7 = SUM(C5:C6)
Cell H8 = SUM(C7:C8)
Cell H9 = SUM(C9:C11)
Excel Solution of Transshipment Problem
The net shipments in cells I7:I13
are the flows out minus the flows
in for each node. For supply
nodes, the flow out will exceed the
flow in, resulting in positive net
shipments. For nodes, the flow out
will be less than the flow in,
resulting
in
negative
net
shipments. The “net” supply
appears in cells K7:K13. Note that
the net supply is negative for
demand nodes.
Excel Solution of Transshipment Problem
Decision Variables (DV) – Cells
C5:C14 are reserved for the DV.
Objective Function (OF) – The formula
=SUMPRODUCT(B5:B14,C5:C14) is
placed into cell G18 to show the total
cost associated with the solution.
Excel Solution of Transshipment Problem
Left-Hand Sides (LHS) – The LHS of
the constraints represent the net
shipments for each node. Cells
I7:I13 are reserved for these
constraints.
Cell I7= H7-G7 (Copy to I8:I13)
Right-Hand Sides (RHS) – The RHS
of the constraints represent the
supply at each node. Cells K7:K13
are reserved for these values. (Note
the negative supply at the three
demand nodes)
Excel Solution of Transshipment Problem
Select Solver from the Analysis Group in
the Data Ribbon. When the Solver
Parameters dialog box appears, enter the
proper values for the constraints and the
objective function, select Simplex LP,
and click the checkbox for Make
Unconstrained Variables Nonnegative. Then click Solve.
Excel Solution of Transshipment Problem
The excel solution displays that
the optimal number of units to
ship over each arc are shown in
cells C5:C14. Also, the total
minimum cost is 1150.
Assignment:
1. Solve the given transportation problem using NWC, LCM and MODI
(show the complete solution and label the final answer).
2. Also, use the excel solver to find the solution.
Store 1
Store 2
Store 3
Plant 1
Plant
Capacity
8 Php/unit 5 Php/unit 4 Php/unit 250
Plant 2
9 Php/unit
6 Php/unit
3 Php/unit
Store
Demand
80
140
200
Plant
200
450
420
3. Answer #11(a, b and c) on page 298 in your prescribed textbook.
Note: For 11c, solve it using the excel solver (computer solution).
References
Anderson, D. R., Sweeney, DJ., Williams, T.A., Camm, J.D., Cochran, J.J.,
& Ohlmann, J.W. (2019). An Introduction to Management Science:
Quantitative Approaches to Decision Making. Singapore: Cengage
Learning Asia Pte Ltd.
Cengage
Learning
Asia
Philippines
(2016).
Chapter
Distribution and Network Models [PowerPoint slides].
Uy, C., et al, (2013). Quantitative Techniques in Business. S.M.A.R.T.
Innovations.
6A: