Saudi Arabia Sugar Distribution: A Transportation Model Project

This project addresses a sugar distribution problem for a company in Saudi Arabia with four factories and ten distribution centers. The goal is to minimize transportation costs while meeting the demand in each city. The problem is solved using the north-west corner rule method, resulting in a total transportation cost of 5,350 SAR. The solution is identified as degenerate, indicating the existence of other feasible solutions. The report details the problem formulation, the application of the north-west corner rule, and the cost model used to calculate the total transportation cost, providing a comprehensive analysis of the sugar distribution network.

Sugar Distribution Problem 1
SUGAR DISTRIBUTION PROBLEM
Name
Course
Professor
University
City/state
Date

Sugar Distribution Problem 2
Table of Contents
Abstract......................................................................................................................................................3
Summary of Results..................................................................................................................................3
The sugar distribution problem................................................................................................................3
North West corner rule method...............................................................................................................4
The Cost Model..........................................................................................................................................7
References..................................................................................................................................................8

Sugar Distribution Problem 3
Abstract
This report presents a project of formulating a transportation problem involving ten cities in
Saudi Arabia. A sugar manufacturing company with a total production of 1,200 bags of sugar per
day from its four factories wanted to set up ten distribution centres in ten different cities in the
country. The company wanted a solution that would help it deliver all the bags to meet the
demand in each city at the minimum transportation cost. The problem has been solved using
north-west corner rule method. The total transportation cost for the optimal solution developed is
5,350 SAR.
Summary of Results
The total daily demand (1,200 bags of sugar) for the ten cities equals the total daily
supply from the four factories owned by the company. The total cost for the transportation of the
1,200 bags of sugar from the four factories to the ten distribution centres located in ten different
cities across Saudi Arabia is 5,350 SAR. The optimal solution is degenerate because the total
number of cells with allocations is less than m + n – 1 (where m = number of rows and n =
number of columns). This means that there are other feasible solutions.
The sugar distribution problem
A sugar company in Saudi Arabia has four factories with production capacity of 200,
450, 400 and 150 bags every day. The company wants to set up 10 distribution centres (D1, D2,
d3, D4, D5, D6, D7, D8, D9 and D10) located in ten different cities across Saudi Arabia (where
D1 – Dumat Al-Jandal, D2 – Ar’ar, D3 – Sakakah, D4 – Rafha, D5 – Tabuk, D6 – Duba, D7 –
Tayma, D8 – Jubbah, D9 – Ha’il, and D10 – Al’Ula). Every day, the company is required to stock
its ten distribution centres with at least 50, 150, 200, 120, 160, 80, 100, 135, 90 and 115

Sugar Distribution Problem 4
respectively. The cost of transporting one bag of sugar in SAR (Saudi Riyal) from each factory
to every distribution centre is provided in Table 1 below
Table 1: Transportation costs
Factory Distribution Centre Supply
D1 D2 D3 D4 D5 D6 D7 D8 D9 D10
F1 2 5 4 8 6 4 2 3 6 4 200
F2 3 4 3 4 5 8 2 7 3 4 450
F3 4 3 2 3 3 6 4 6 5 3 400
F4 2 3 6 3 2 5 5 4 4 5 150
Demand 50 150 200 120 160 80 100 135 90 115
The company is looking for a transportation model that will enable it distribute the
required amount of sugar bags to each distribution centre every day at the minimum
transportation cost.
North West corner rule method
There are different methods that can be used to solve this problem. The chosen method
for this problem is north-west corner rule method. This method is used for determining the basic
possible solution for different problems specially transportation problems. The method starts by
selecting basic variables from the top left corner hence its name North-West corner (Universal
Teacher Publications, (n.d.)).
Using north-west corner rule, the first step is to allocate 50 bags (maximum feasible
amount) to cell F1D1 thus satisfying the total demand for column one. The next step is to move to
cell F1D2 and allocate 150 bags thus satisfying the total demand for the second column and also
the supply for the first row is fully satisfied.

Sugar Distribution Problem 5
The next step is to allocate 200 bags to cell F2D3 thus satisfying the total demand for the third
column.
The next step is to allocate 120 bags to cell F2D4 thus satisfying the total demand for the fourth
column.
The next step is to allocate 130 bags to cell F2D5 thus satisfying the supply for the second row
but the demand for the fifth column has a deficit of 30 bags. Next is to allocate 30 bags to cell
F3D5 so as to fully satisfy the demand for the fifth column.
The next step is to allocate 80 bags to cell F3D6 thus satisfying the total demand for the sixth
column.
The next step is to allocate 100 bags to cell F3D7 thus satisfying the total demand for the seventh
column.
The next step is to allocate 135 bags to cell F3D8 thus satisfying the total demand for the eighth
column.
The next step is to allocate 55 bags to cell F3D9 thus satisfying the supply for the third row but
the demand for the ninth column has a deficit of 35 bags. Next is to allocate 35 bags to cell F4D9
so as to fully satisfy the demand for the ninth column.
The last step is to allocate 115 bags to cell F4D10 thus satisfying the total demand for the tenth
column and also fully satisfying the supply for the fourth row.
The final table showing the demand and supply allocation for all the bags produced by the four
factories and supplied to all the ten distribution centres is as follows:
Factory Distribution Centre Supply

Sugar Distribution Problem 6
D1 D2 D3 D4 D5 D6 D7 D8 D9 D10
F1 2
(50)
5
(150)
4 8 6 4 2 3 6 4 200
F2 3 4 3 (200) 4 (120) 5
(130)
8 2 7 3 4 450
F3 4 3 2 3 3 (30) 6
(80)
4
(100)
6 (135) 5 (55) 3 400
F4 2 3 6 3 2 5 5 4 4 (35) 5 (115) 150
Demand 50 150 200 120 160 80 100 135 90 115
Therefore the initial possible solution is as follows:
Factor
y
Distribution Centre Suppl
y
D1 D2 D3 D4 D5 D6 D7 D8 D9 D10
F1 2
(50
)
5
(150
)
4 8 6 4 2 3 6 4 200
F2 3 4 3
(200)
4
(120)
5
(130)
8 2 7 3 4 450
F3 4 3 2 3 3 (30) 6
(80)
4
(100)
6
(135)
5
(55)
3 400
F4 2 3 6 3 2 5 5 4 4
(35)
5
(115)
150
Dema
nd
50 150 200 120 160 80 100 135 90 115
To calculate the minimum total transportation cost, the cost in each cell is multiplied by the
corresponding number of bags allocated in that cell then the values are summed as follows
(Business Jargons, 2018):
Minimum total transportation cost = (2 x 50) + (5 x 150) + (3 x 200) + (4 x 120) + (5 x 130) + (3
x 30) + (6 x 80) + (4 x 100) + (6 x 135) + (5 x 55) + (4 x 35) + (5 x 115)
= 100 + 750 + 600 + 480 + 650 + 90 + 480 + 400 + 810 + 275 + 140 + 575
= 5,350 SAR

Sugar Distribution Problem 7
The no. of cells that have been allocated units of bags = 12
The no. of basic variables is calculated as follows:
No. of factories, rows, m = 4
No. of distribution centres, columns, n = 10
No. of basic variables = m + n – 1 = 4 + 10 – 1 = 13
This means that the no. of cells allocated units of bags and the no. of basic variables is not equal
(12 ≠ 13) meaning that the solution is degenerate, i.e. there are other feasible solutions.
The Cost Model
When using north-west corner rule, the total cost of transportation is obtained using equation 1
below (Klinz & Woeginger, 2011)
∑
1
N
DiCi ……………………………………..…………………… (1)
Where Di = demand of bags to the city, Ci = transportation cost per bag to the city, and N =
number of cities.
The minimum transportation cost is calculated as follows:
The minimum total transportation cost = (2 x 50) + (5 x 150) + (3 x 200) + (4 x 120) + (5 x 130)
+ (3 x 30) + (6 x 80) + (4 x 100) + (6 x 135) + (5 x 55) + (4 x 35) + (5 x 115)
= 100 + 750 + 600 + 480 + 650 + 90 + 480 + 400 + 810 + 275 + 140 + 575
= 5,350 SAR

Sugar Distribution Problem 8
But since the feasible solution obtained is degenerate, there is need to determine other possible
solutions and find whether the total transportation cost is less than the one fund in the first
solution.
References
Business Jargons, 2018. North-West Corner Rule. [Online]
Available at: https://businessjargons.com/north-west-corner-rule.html
[Accessed 24 November 2018].
Klinz, B. & Woeginger, G., 2011. The Northwest corner rule revisited. Discrete Applied Mathematics,
159(12), pp. 1284-1289.
Universal Teacher Publications, (n.d.). North West Corner Rule. [Online]
Available at: http://www.universalteacherpublications.com/univ/ebooks/or/ch5/nw.htm
[Accessed 24 November 2018].