**APPLICATION OF TRANSPORTATION LINEAR PROGRAMMING FOR RICE COMPANY**

Wan Nurshazelin Wan Shahidan<sup>1</sup>, Haydar Hadzori<sup>2</sup>

<sup>1,2</sup>Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA (UiTM) Perlis Branch, 02600 Arau, Perlis, Malaysia

Corresponding author: <shazelin804@uitm.edu.my>

Received date:

Accepted date:

**ABSTRACT**

*Optimization is using available resources in the best possible way. Whether the objective is to maximize profit or minimize cost, optimization of a problem can lead to better planning for an organization. Mathematical tools such as Linear Programming allow such optimization problem to be modelled mathematically to solve it. In this paper, linear programming was used to solve the optimization for transportation problem that involves transporting rice from Padiberas Nasional Berhad (PNB) rice mills to wholesalers and customers in Malaysia. PNB currently deliver rice from five rice mill to 22 different locations in the eight state of Malaysia. Since not all destinations involved in the transportation process, the right amount of rice that should be transport to each destination is important for the company to utilize available resource in the best way. This paper aimed to minimize total transportation cost that was incurred by the company. It also discussed the sensitivity analysis impact of changes in cost incurred at each rice mill to its destination toward the total transportation cost. POM-QM for Windows was used to solve the model for minimization of total transportation cost. The result show that total transportation cost was able to be minimized from RM 3,010,895 to RM 2,187,310.30.*

***Keywords:** linear programming, transportation, minimization, optimization*

**INTRODUCTION**

In today’s highly-competitive marketplace, companies must attempt to increase the efficiency of all their activities and utilize all possible opportunities in order to be successful in their businesses. Cost reduction in logistic is one of many opportunities that a company can implement to increase their profitability (Khan, 2014).

Jonsson (2008) has classified logistic cost into two types, namely direct and indirect costs. Direct cost is defined as any cost that involved physical handling, transportation, and storage of goods in the flow of materials with the administration costs. Meanwhile, indirect costs is said to be the cost for capacity and shortage of the goods. Jonsson also claimed that direct logistic costs differ approximately from 10% to 30% in turnover depending on the type of industry.

Generally in this situation, a reduction in direct logistic cost will also reduce the transportation cost. Ghazali, Abd Majid, and Shazwani (2012) claimed that a five percent drop in transportation cost will impact the sales positively by 30%. Essentially, the view of the aforesaid authors is such that the implementation of optimization to the transportation of goods in order to schedule when and how much to send from each origin to its respective destination over a certain time period may help the company to obtain their objective in profitability. The aim is to transport the goods in the right quantity to the right destination, at the right time, while simultaneously contributing to most of the company's objectives.

The optimization in the transportation of goods is known as transportation problem and it has been broadly studied in real life. In this area of study, quantitative method was used to obtain the optimal solution for the problem. According to Philips, Ravindran and Solberg (1987) quantitative method has been successfully applied in many transportation problems. Through the study, mathematical model for the optimization was developed and it was used to solve the transportation problem.

Transportation problem is a classic problem in operation research where the objective is to transport goods with the lowest possible cost. Mandel (2004) stated that transporting goods at lowest possible cost presents huge potential for cost saving while ideally maximizing the company’s profit. Hence, solving transportation problem is beneficial to a company that seeks to utilize available resources in an optimal way in order to increase their profitability.

Against this background, the objective of this study is to minimize total transportation cost of transporting rice from Padiberas Nasional Berhad rice mills to its customer and wholesaler in the state of Malaysia. In the same time, this study also discussed the sensitivity analysis for impact of changes in transportation cost incurred from each rice mill to its destinations toward total transportation cost. Linear Programming technique is used to model the problem and then it was solve using POM-QM for Windows.

Generally, transportation problem focused on the task of transporting goods from any supply point to any demand destination with the lowest total transportation cost possible ( Ghazali et al, 2012). The problem is to identify the amount of goods to be transported from a supply point to each demanded destination in a way that minimized the total transportation cost. Still, the transportation cost incurred from each supply point to demand destination is varied and ought to change according to situation. So, the change in cost incurred should be analysed to know how much it will impact total transportation cost. Currently, Padiberas Nasional Berhad delivers rice from five rice mills to 22 different locations in the eight state of Malaysia. The issue is not all transportation was made to the destination from the rice mills. So, the company needs to know the optimal quantities of rice that should be allocated to each destination while minimizing the total transportation cost.

**METHODOLOGY**

All the information for this transportation problem is illustrated in Table 1.Therefore in order to determine the optimal quantity of rice that should be transported to each destination while minimizing total transportation cost, the following information was collected from the Operation Department of Padiberas Nasional Berhad company:

1.  Cost incurred for transport of rice from each rice mill

2.  Amount of supply of rice available at rice mill

3.  Total demand from destination.

4.  
Table 1: Shipping cost and actual total cost from rice mill to destination

<table>
<thead>
<tr class="header">
<th>Rice mill</th>
<th>Destination</th>
<th>Transportation cost from PNB rice mill (TC), (RM)</th>
<th><p>Actual total transportation cost incurred</p>
<p>(RM)</p></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>KBB Simpang 4 (A)</td>
<td>Johor</td>
<td>90</td>
<td>142740.00</td>
</tr>
<tr class="even">
<td></td>
<td>Kedah</td>
<td>15</td>
<td>55800.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Kedah</td>
<td>20</td>
<td>68640.00</td>
</tr>
<tr class="even">
<td></td>
<td>Selangor</td>
<td>55</td>
<td>238700.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Perak</td>
<td>40</td>
<td>15920.00</td>
</tr>
<tr class="even">
<td>KBB Rompin (B)</td>
<td>Selangor</td>
<td>80</td>
<td>284880.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Pahang</td>
<td>25</td>
<td>17250.00</td>
</tr>
<tr class="even">
<td></td>
<td>Johor</td>
<td>50</td>
<td>430800.00</td>
</tr>
<tr class="odd">
<td>KBB Paya Keladi (C)</td>
<td>Wilayah Persekutuan (WP)</td>
<td>60</td>
<td>47940.00</td>
</tr>
<tr class="even">
<td></td>
<td>Johor</td>
<td>70</td>
<td>81410.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Kedah</td>
<td>22</td>
<td>6160.00</td>
</tr>
<tr class="even">
<td></td>
<td>Selangor</td>
<td>60</td>
<td>69720.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Perak</td>
<td>30</td>
<td>50730.00</td>
</tr>
<tr class="even">
<td></td>
<td>Penang</td>
<td>15</td>
<td>22185.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Johor</td>
<td>65</td>
<td>346840.00</td>
</tr>
<tr class="even">
<td>KBB STJ (D)</td>
<td>Wilayah Persekutuan (WP)</td>
<td>20</td>
<td>11920.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Selangor (S)</td>
<td>20</td>
<td>119400.00</td>
</tr>
<tr class="even">
<td></td>
<td>Melaka (M)</td>
<td>30</td>
<td>4050.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Johor (J)</td>
<td>50</td>
<td>245000.00</td>
</tr>
<tr class="even">
<td>KBB Kuala Perlis (E)</td>
<td>Johor (J)</td>
<td>90</td>
<td>451735.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Kedah(K)</td>
<td>15</td>
<td>22215.00</td>
</tr>
<tr class="even">
<td></td>
<td>Selangor (S)</td>
<td>55</td>
<td>276860.00</td>
</tr>
<tr class="odd">
<td>Total transportation cost (RM)</td>
<td>3010895.00</td>
<td></td>
<td></td>
</tr>
</tbody>
</table>

Refer to Table 1, the shipping cost incurred is based on the schedule of transport of rice from PNB rice mills. Since the transports of rice from the mills are scheduled to the specific destinations only, the shipping costs incurred are only available for certain destination only. The actual total transportation cost originally obtained from data which is RM 3,010,895.00 since the amount of transported rice is not yet optimized.

***Model formulation***

From the information given, mathematical model was formulated using Linear Programming technique. The model consists of the following component:

*Objective function:* The aimed solution that wants to be obtained from the model which is to minimize total transportation cost.

*Decision variable:* The variable that represent the unknown quantities of rice that should be transport in the model for it to obtain minimal cost.

*Constraint:* The limitation and requirement of the problem; expressed in equalities for decision variable.

Hence the model is formulated as shown:

> (1)
> 
> (2)

(3)

(4)

The decision variable for the above linear programming model are defined as shown in Table 2 until Table 6.

**Table.2: Decision variables for KBB Simpang 4**

| Destination | Decision variable |
| ----------- | ----------------- |
| Johor       | *X<sub>AJ</sub>*  |
| Kedah       | *X<sub>AK1</sub>* |
| Kedah       | *X<sub>AK2</sub>* |
| Selangor    | *X<sub>AS</sub>*  |
| Perak       | *X<sub>APk</sub>* |

**Table 3: Decision variable for KBB Rompin**

| Destination | Decision variable |
| ----------- | ----------------- |
| Selangor    | *X*<sub>BS</sub>  |
| Pahang      | *X*<sub>BPh</sub> |
| Johor       | *X*<sub>BJ</sub>  |

**Table 4: Decision variable for KBB Paya Keladi**

| Destination              | Decision variable |
| ------------------------ | ----------------- |
| Wilayah Persekutuan (WP) | *X*<sub>CW</sub>  |
| Johor                    | *X*<sub>CJ1</sub> |
| Kedah                    | *X*<sub>CK</sub>  |
| Selangor                 | *X*<sub>CS</sub>  |
| Perak                    | *X*<sub>CPk</sub> |
| Penang                   | *X*<sub>CP</sub>  |
| Johor                    | *X*<sub>CJ2</sub> |

**Table 5: Decision variable for KBB STJ**

| Destination              | Decision variable |
| ------------------------ | ----------------- |
| Wilayah Persekutuan (WP) | *X*<sub>DW</sub>  |
| Selangor                 | *X*<sub>DS</sub>  |
| Melaka                   | *X*<sub>DM</sub>  |
| Johor                    | *X*<sub>DJ</sub>  |

**Table 6: Decision variable for KBB Kuala Perlis**

| Destination | Decision variable |
| ----------- | ----------------- |
| Johor       | X<sub>EJ</sub>    |
| Kedah       | X<sub>EK</sub>    |
| Selangor    | X<sub>ES</sub>    |

Since this study is aimed to minimize total transportation cost from the rice mills to demand destinations. POM- QM for Window version 4 was used to analyze the linear model formulated to obtain the optimal solution. For the sensitivity analysis of varied changes in transportation cost incurred per unit to total transportation cost, non-probabilistic method adapted from (Ghazali et. al 2012) was used. The varied of changes are set over the range of 20% and 50%.

**FINDINGS AND DISCUSSION**

The optimal solutions for the model are obtained using QM for Window. The summarizations of the result obtained are shown in Table 7. According to the solution in QM for Windows, from KBB Simpang 4, the amount of rice should be transported is only to three destinations which is to Kedah (K1), Selangor (S) and Perak (Pk). The amount should be transport to Kedah (K1) is 8,913 tons while 310.69 tons to Selangor (S) and 936.09 tons to Perak (Pk). To Johor (J) and Kedah (K2) there are no amount of rice that should be transported as it will not give optimal solution for the model. Results in the table also shows that for KBB Rompin, the solution suggested that 690 tons should be sent to Pahang (Ph) while 12,035.78 tons of rice should be satisfied at Johor (J). To Selangor (S) the model showed zero value, indicated that no transport of rice should be scheduled.

From the table also shows that three out of seven destinations was selected by the model for the transport of rice from KBB Paya Keladi (C). The amount is 1,152.91 tons to Perak (Pk), 1,479 tons to Penang (P) and 8,530.70 tons to Johor (J2). Other four destinations excluded from the transportation process are Wilayah Persekutuan (W), Johor (J1), Kedah (K) and Selangor (S). From KBB STJ, the amount of rice should be transported is 1,395 tons toWilayah Persekutuan (W), 9,993.13 tons to Selangor (S), 135 to Melaka (M) while to Johor (J) there are no amount should be transport to the location. At KBB Kuala Perlis (E), the model concluded that only Selangor (S) should receive the amount of rice from the rice mill. The amount is 9,763 tons.

The amount of rice that should be transported to each destination was optimized and the total transportation cost obtained from the result is RM 2,187,310.30 which lower that actual transportation total cost (RM 3,101,895) obtained by the company with decrement of RM 823,584.74 in the total transportation cost. Thus, the model in this study have achieved the study objective because the total transportation cost was able to be minimized using this linear model.

**Table 7: Summarization of result from POM-QM for Windows**

<table>
<thead>
<tr class="header">
<th>Rice mill</th>
<th>Destination</th>
<th>Transportation cost from PNB rice mill (TC), (RM)</th>
<th>QM solution for transport amount (TA), (Tons)</th>
<th><p>Total transportation cost incurred</p>
<p>(TCTA),</p>
<p>(RM)</p></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>KBB Simpang 4 (A)</td>
<td>Johor (J)</td>
<td>90</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="even">
<td></td>
<td>Kedah (K1)</td>
<td>15</td>
<td>8913.00</td>
<td>133695.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Kedah (K2)</td>
<td>20</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="even">
<td></td>
<td>Selangor (S)</td>
<td>55</td>
<td>310.69</td>
<td>17087.95</td>
</tr>
<tr class="odd">
<td></td>
<td>Perak (Pk)</td>
<td>40</td>
<td>936.09</td>
<td>37443.60</td>
</tr>
<tr class="even">
<td><p>KBB Rompin</p>
<p>(B)</p></td>
<td>Selangor (S)</td>
<td>80</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="odd">
<td></td>
<td>Pahang (Ph)</td>
<td>25</td>
<td>690.000</td>
<td>17250.00</td>
</tr>
<tr class="even">
<td></td>
<td>Johor (J)</td>
<td>50</td>
<td>12035.78</td>
<td>601789.00</td>
</tr>
<tr class="odd">
<td>KBB Paya Keladi (C)</td>
<td>Wilayah Persekutuan (W)</td>
<td>60</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="even">
<td></td>
<td>Johor (J1)</td>
<td>70</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="odd">
<td></td>
<td>Kedah (K)</td>
<td>22</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="even">
<td></td>
<td>Selangor (S)</td>
<td>60</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="odd">
<td></td>
<td>Perak (Pk)</td>
<td>30</td>
<td>1152.91</td>
<td>34587.30</td>
</tr>
<tr class="even">
<td></td>
<td>Penang (P)</td>
<td>15</td>
<td>1479.00</td>
<td>22185.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Johor (J2)</td>
<td>65</td>
<td>8530.70</td>
<td>554495.5</td>
</tr>
<tr class="even">
<td><p>KBB STJ</p>
<p>(D)</p></td>
<td>Wilayah Persekutuan (WP)</td>
<td>20</td>
<td>1395.00</td>
<td>27900.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Selangor (S)</td>
<td>20</td>
<td>9993.13</td>
<td>199862.60</td>
</tr>
<tr class="even">
<td></td>
<td>Melaka (M)</td>
<td>30</td>
<td>135.00</td>
<td>4050.00</td>
</tr>
<tr class="odd">
<td></td>
<td>Johor (J)</td>
<td>50</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="even">
<td>KBB Kuala Perlis (E)</td>
<td>Johor (J)</td>
<td>90</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="odd">
<td></td>
<td>Kedah(K)</td>
<td>15</td>
<td>0</td>
<td>0</td>
</tr>
<tr class="even">
<td></td>
<td>Selangor (S)</td>
<td>55</td>
<td>9763</td>
<td>536965.00</td>
</tr>
<tr class="odd">
<td>Total transportation cost (RM), ∑(TCTA)</td>
<td><strong>2187310.30</strong></td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>

The sensitivity of total transportation cost is analyzed by varying the cost incurred for one rice mill at a time while others remain the same. The sensitivity analyses are shown in Table 8.

**Table 8: Sensitivity analysis for total transportation cost**

| Change in unit of cost incurred (%) | Total transportation cost (RM) |            |            |            |            |
| ----------------------------------- | ------------------------------ | ---------- | ---------- | ---------- | ---------- |
|                                     | Rice Mills                     |            |            |            |            |
|                                     | A                              | B          | C          | D          | E          |
| 50                                  | 2281424.00                     | 2496831.00 | 2485924.00 | 2303217.00 | 2277534.00 |
| 20                                  | 2224956.00                     | 2311119.00 | 2309565.00 | 2233674.00 | 2223400.00 |
| 0                                   | 2187310.30                     | 2187310.30 | 2187310.30 | 2187310.30 | 2187310.30 |
| \-20                                | 2078362.00                     | 2063503.00 | 2065057.00 | 2140949.00 | 2079918.00 |
| \-50                                | 1905577.00                     | 1877792.00 | 1880665.00 | 2066724.00 | 1916488.00 |

Table above shows that the impact of variation of cost incurred at one rice mill to total transportation cost. All the values are obtained from solution in the computer software through with linear model in this research. The zero value indicated that the total transportation cost obtained from the model. From the model B have the highest and the lowest value among all other value in model A, C, D and E. The impact of the changes to the total transportation cost is made clear by drawing the sensitivity graph as shown by Figure 1.

<span class="chart">\[CHART\]</span>

**Figure 1: Sensitivity graph**

From figure above, the relative degree of sensitivity for changes in variation of cost incurred at each rice mill to total transportation cost is indicated by slope of curves. The steeper the slope of the curve, the sensitive the total transportation cost when there is changes in unit of cost incurred. In the graph, B has the greatest slope. Thus, it can be said that variation of cost incurred at KBB Rompin has the greatest impact to total transportation cost.

**CONCLUSION AND RECOMMENDATION**

**Linear Programming technique was used to solve the optimization for transportation problem. It is also used in various types of field such as economics, business, agriculture and health. Often, it is used for optimization of a problem in the real world. For that purpose, linear model is developed to represent the problem in mathematical formulation. Linear model are subjected to limitations from the real world known as the constraint. These constraints had to be satisfied for the model to obtain the optimized solution. By analyzing the data of transportation of rice from Padiberas Nasional Berhad rice mills, transportation problem component are identified for the optimization model. Linear model was developed, and the optimized solutions are obtained by allocating the right amount of rice to be transported from the rice mills to its respective destinations. Though, the availability of computer software such as POM-QM for Windows makes the model easier to be solved. Based on analysis and discussion carried out on the model, the total transportation cost obtained is RM 2,187,310.30. Which mean there are decreased** of RM 823,584.70 in the total transportation cost in the study **if the company transported the rice as suggested by the model. Thus, the study objective is achieved in this study because the linear model was able to minimize the total transportation cost. From the sensitivity analysis it can be seen that variation of change in cost incurred at rice mill KBB Rompin gives the most impact to the total transportation cost.**

For future study, there are some additional variable that could be added to the model. For example, the types of rice demanded at each destination should be considered as another variable in the model. There are probably considerably many scenarios and therefore equations, variable and constraint to go along with these scenarios. The problem also can be solved using another method such as Integer Programming for a better result. From the sensitivity analysis result, the company can take precautions on every possible outcome from the changes that might occur. This study also hopes to encourage continuous research in transportation problem using linear model especially in Malaysia.

**REFERENCES**

Ghazali, Z., Abd Majid , M.A. and Shazwani, M. (2012). Optimal Solution of Transportation Problem Using Linear Programming: A case of a Malaysian Trading Company. *Journal of Applied Science*, 12(23), pp. 2430-2435. <https://scialert.net/abstract/?doi=jas.2012.2430.2435>

Jonsson, P. (2008). *Logistics and supply chain managements*. London: McGraw-Hill.

Khan, Muztoba Ahmad (2014).*Transportation cost Optimization Using Linear Programming*. International Conference on Mechanical, Industrial and Energy Engineering 2014 (25-26 Dec.), Khulna, Bangladesh.

Mandel, J.K., 2004*. *Cutting your distribution cost. Growing Your Business, Price Water House Coopers.

Philip, D. T., Ravindran, A. and Solberg, J.J. (1987). *Operation* *Research: Principles and Practice* (2nd Ed.). New York: John Wiley and Sons. Pages: 637.