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This research work seeks to model the distribution of 50 cl Pepsi soft drink as a transhipment problem. The transshipment problem is an extension of the traditional transportation problem which takes into account a multi-phase transport system in which the flows of goods and services are taken through an intermediate point (transhipment points) between the origin and the destination with varying objective functions. The main focus in this research was to obtain the minimum cost of transporting 10,000 crates of the product from the Benin plant (source) through deports (transshipment points) to the Sapele-Warri region (sinks) where the product is demanded. Data collected were analyzed using TORA Windows Version 2.00 software. The analysis shows that the minimum cost of transporting the product can be achieved if the product is shipped directly from the source to the sink. This forms that basis for the conclusions and recommendations of the research.

Transportation is a very important subsystem of logistics in terms of value and its cost takes a large portion of costs in such logistics system [Briš [

With the introduction of Operations Research and in particular linear programming and networking to subject areas like statistics and management, operations managers can now deal with this challenging need. Among the many linear programming problems introduced by Operations Research is the Transportation Problem. The transportation problem introduced as far back as the 1940s has received wide acceptance over the years with many researchers making several improvements to suit their peculiar/present needs.

The typical transportation problem deals with the distribution of goods from several points of supply to a number of points of demand. This problem usually arises when a cost-effective pattern is needed to ship/transport items from origins that have limited supply to destinations that have demand for the goods. It also refers to a class of linear programming problems that involve selection of most economical shipping/transportation routes for transfer of a uniform commodity from a number of sources to a number of destinations [Khurana [

Like all linear programming problems (LPP), the transportation problem has its objective function and constraints. The most common objective function is to schedule shipments from sources to destinations so that total production and transportation costs are minimized [Slide Share [

A transshipment is defined as the transfer of stock between two locations at the same level of the inventory/distribution system. The problem is to determine replenishment quantities and how much to transship each period so as to satisfy deterministic dynamic demand at each location at minimal cost. The planning horizon is finite and no back orders are allowed [Herer and Tzur, [

Though the transshipment problem is an extension or improvement to the transportation problem its optimum solution is found by easily converting the transshipment problem into an equivalent transportation problem and solving using the usual transportation techniques. The availability of such a conversion procedure significantly broadens the applicability of the algorithm for solving transportation problems. The conventional Transportation Problem can be represented as a mathematical structure which comprises an Objective Function subject to certain Constraints. In classical approach, transporting costs from M sources or wholesalers to N destinations or consumers are to be minimized. It is an Optimization Problem which has been applied to solve various NP-Hard Problems [Chaudhuri and De [

To obtain the optimum solution for the converted transshipment problem the LPP will be solved in 2 two stages. In first stage involves obtaining the Initial Basic Feasible Solution (IBFS) using available methods such as North West Corner, Least Cost Method, and Vogel’s Approximation Method (VAM). The second stage employs the use of Modified Distribution Method (MODI) and the Stepping Stone Method from the IBFS to finally obtain the optimal solution [Patel and Bhathawala [

As previously stated, the foremost concern of every manufacturer is the need to get his products from the plants/warehouse to the consumer/final destination where it is most demanded. For the manufacturer therefore transporting and/or transshipping is the key to achieving this. The question therefore is the best way to go about this considering the costs involved and also to make profits. The aim of the study is to model the transportation/distribution of the product as a transshipment problem in other to obtain the minimum cost.

The transshipment problem is dated back to the medieval times when trading started becoming a mass phenomenon. The concept takes into account a transportation model in which any of the origin and destination can serve as an intermediate point through which goods can be temporarily received and then transshipped to other points or to the ﬁnal destination [Gass, 1969: cited by Briš, [

The major focus in the transshipment problem was to initially obtain the minimum-cost and/or shortest transportational route, however due to technological development the minimum-durational transportation problems are now being studied. In recent times other researchers have extensively studies this Linear Programming Problems and developed different variants based on the needed objective function.

Among recent literatures, researchers like Khurana et al., [

The transshipment problem characterized by the uncertainty relative to customer demands and transfer lead time was studied by Hmiden et al. [

Transshipment problem is characterized by a dynamic network with several sources and sinks and that there were no polynomial-time algorithms known for most of transshipment problems [Hoppe and Tardos [

Tadei et al. [

Still dealing with the overall cost implication, Kestin and Uster [

The multi-objective transshipment problem (MOTP) is another model that was studied by Das et al. [

Apart from the different objective functions describe above, other researchers had focused on the transshipment center problem instead of the transshipment itself. Previous studies have shown that the transshipment center problem is important in dealing with transshipment in a supply chain management. It was noted that most of the relevant existing research focused on networking and do not consider the dynamics of the configurations in transshipment center units [Perng and Ho, [

Ameln and Fuglum [

Agarwal and Ergun [

Whatever the reason for the study of the transshipment problem, the basic mathematical formulation is still the same though with slight variations. The typical transshipment model is given below.

The formulation of the transshipment problem requires certain initial basic assumptions these are:

• The system consists of m origins and n destinations, where i = 1 , ⋯ , m , j = 1 , ⋯ , n

• Homogenous set of goods are available for shipping.

• The required amount of good at the destinations equals the produced quantity available at the origins.

• Transportation simultaneously starts at the origins and is possible from any node to any other (also to an origin and from a destination).

• Transportation costs are independent of the shipped amount.

Notations

• c_{r,}_{s}: cost of transportation from node r to node s.

• a_{i}: goods available at node i.

• b_{m}_{+}_{j}: demand for the good at node (m + j).

• x_{r,}_{s}: actual amount transported from node r to node s.

The goal is to minimize

∑ i = 1 m ∑ j = 1 n c i x i , j

subject to:

• x r , s ≥ 0 ; ∀ r = 1 , ⋯ , m , s = 1 , ⋯ , n

• ∑ s = 1 m + n x i , s − ∑ r = 1 m + n x r , i = a i ; ∀ i = 1 , ⋯ , m

• ∑ r = 1 m + n x r , m + j − ∑ s = 1 m + n x m + j , s = b m + j ; ∀ j = 1 , ⋯ , n

• ∑ i = 1 m a i = ∑ j = 1 n b m + j

The Seven-Up bottling company Plc is one of the largest independent manufacturer and distributor of the well-known and widely consumed brands of soft drinks in Nigeria. They are the producers of Pepsi, 7 UP, Mirinda, Teem and Mountain Dew brand. The Benin production plant is one among the many plants owned by the company in Nigeria. A simple random sampling of the weekly demand and supply chain for the plastic Pepsi 50 cl soft drink was used for the analysis. This sample is necessary as it will reflect the typical operations of the demand and supply of the company in question.

The data collected from the Benin production plant was properly summarized into tables (

From | To | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Swift | Frendo | Clement Vents | Life Dew | Osoro | Afoke | Oriemu Vents | T & O | Best Beer | Stella Chukwu | |

Benin | N9.60 | N9.90 | N10.20 | N13.10 | N13.70 | N13.88 | N13.92 | N13.20 | N13.83 | N13.65 |

The general linear programming model of a transshipment problem is

Min ∑ a l l a r c s c i j x i j

Subject to

∑ a r c o u t x i j − ∑ a r c i n x i j = s i Original nodes i

∑ a r c o u t x i j − ∑ a r c i n x i j = 0 Transshipment nodes

∑ a r c i n x i j − ∑ a r c o u t x i j = d i Destination nodes j

where

x_{ij} = Amount of unit shipped from node i to node j.

c_{ij} = Cost per unit of shipping from node i to node j.

s_{i} = Supply at origin node i.

d_{j} = Demand at sink node j.

This section is concerned with the presentation of data, its analysis and summary of the findings. For clarity the various key points in the transshipment model are listed out below;

• The Benin plant serves as the source of the 50 cl Pepsi soft drink and its weekly supply is 10,000 crates to Sapele and Warri environs.

• The transshipment points are Swift, Frendo, Clement Vents, Life Dew, Osoro, Afoke, Temi & Oris, Best Beer, Stella Chukwu and Oriemu Vents (These are actual mega distributors within Sapele and Warri LGAs of Delta State).

• While the demand points are in the respective towns/location in Delta State of Nigeria. They include Jesse, Mosogar, Koko, Oghara, Gana, Amukpe, Jakpan, Effurun, Osubi, Igbudu and Adeje towns. These demand points have varying capacities based on their consumption of the crates per week.

A source is a point that can only send products to other points but cannot receive any products. Similarly, a sink is defined as a point that can only receive products from other points but cannot send products. Then, a transshipment point is defined as a point that can both receive products and also send products to other points. Therefore, this model consists of a problem with 1 source, 11 sinks, and 10 transshipment points as shown in

The crates of Pepsi 50 cl soft drinks are transported by road. Full trailer load consists of 800 crates of the products this gives above 12 trailer loads weekly. The cost of transportation per crate of the product is presented in

The figures from the above tables can be converted into the transportation tableau with their respective transportation costs as shown in

The mathematical representation of X_{ij} = amount to be transported from i^{th} node

to j^{th} node; i = 1 , 2 , 3 , ⋯ , 11 ; j = 1 , 2 , 3 , ⋯ , 21 . Note: All costs are in Naira (N)

Minimize Z = N 9.11 X 112 + 9.73 X 113 + 9.30 X 114 + 9.11 X 115 + 9.11 X 116 + 9.00 X 117 + 11.01 X 118 + 11.23 X 119 + 12.40 X 120 + 12.98 X 121 + 10.24 X 122 + 9.60 X 123 + 9.90 X 124 + 10.20 X 125 + 13.10 X 126 + 13.70 X 127 + 13.88 X 128 + 13.92 X 129 + 13.20 X 130 + 13.23 X 131 + 13.65 X 132 + ⋯ + 0.05 X 1129 + 1.00 X 1130 + 1.00 X 1131 + 0.00 X 1132

Subject to

X 112 + X 113 + X 114 + X 115 + X 116 + ⋯ + X 132 = 10000 (Supply point constraint)

X 112 + X 212 + X 312 + X 412 + ⋯ + X 1112 = 740 X 113 + X 213 + X 313 + X 413 + ⋯ + X 1113 = 820 X 114 + X 214 + X 314 + X 414 + ⋯ + X 1114 = 920 ⋮ X 122 + X 222 + X 322 + X 422 + ⋯ + X 1122 = 720 } (Demand point constraints)

X 123 − X 212 − X 213 − X 214 − X 116 − ⋯ − X 132 = 0 X 112 − X 113 − X 114 − X 115 − X 116 − ⋯ − X 132 = 0 X 112 − X 113 − X 114 − X 115 − X 116 − ⋯ − X 132 = 0 ⋮ X 112 − X 113 − X 114 − X 115 − X 116 − ⋯ − X 132 = 0 } (Transshipment point constraints)

The data was imputed and analyzed with the TORA Windows Version 2.00 software. The software made 14 iterations to obtain the optimum solution with a total cost of N103,350.00 as the minimum cost that satisfies the objective function. The result is shown in

As can be seen from the analysis, the total cost = N9.11 (720) + 9.73 (820) + 9.30 (320) + 9.11 (730) + 9.11 (730) + 9.00 (1200) + 11.01 (1300) + 11.23 (1200) + 12.40 (820) + 12.40 (820) + 12.98 (820) + 10.24 (720) = N103,350.00.

As can be seen from the analysis, the best route to obtain the minimum cost can be summarized in

This can be achieved if Benin Plant transports the product directly to the various demand points namely Jesse, Koko, Oghara, Mosogar and Gana (in the Sapele region) and Jakpan, Effurun, Osubi, Igbudu and Adeje (in the Warri region). The analysis from the TORA software is shown in Appendix B.

We wish to recommend that the Seven-up Bottling Company adapts this result of

Source | Sink | Quantity | Amount in Naira (N) |
---|---|---|---|

Benin | Jesse | 720 | 9.11 |

Benin | Koko | 820 | 9.73 |

Benin | Oghara | 920 | 9.30 |

Benin | Mosogar | 730 | 9.11 |

Benin | Gana | 730 | 9.11 |

Benin | Amukpe | 1200 | 9.00 |

Benin | Jakpan | 1300 | 11.01 |

Benin | Effurun | 1200 | 11.23 |

Benin | Osubi | 820 | 12.40 |

Benin | Igbudu | 820 | 12.98 |

Benin | Adeje | 720 | 10.24 |

Total Cost | N103,350.00 |

transporting the product obtained by the analysis so that their cost of transporting 50 cl Pepsi soft drinks within the Benin-Sapele/Warri region can be drastically minimized.

This study was conducted using primary data and with the findings derived, it can be a source of more information to other researchers who may wish to improve on the limitations of this study to cover areas like minimum duration problems, transshipment facility layout problem/labour and man-power related issues, the shortest route problem and so on. Therefore, more research and studies can still be carried out to look into these other areas on the transshipment model.

The authors declare no conflicts of interest regarding the publication of this paper.

Agadaga, G.O. and Akpan, N.P. (2019) A Transshipment Model of Seven-Up Bottling Company, Benin Plant, Nigeria. American Journal of Operations Research, 9, 129-145. https://doi.org/10.4236/ajor.2019.93008

Screen shots from the TORA Windows Version 2.00 software.

(a) Initial data before analysis.

(b) Continuation of the Initial data before analysis.

(c) The optimum solution and respective allocations.

(d) Continuation of the optimum solution and respective allocations.