Sensitivity Analysis of Road Freight Transportation of a Mega Non-Alcoholic Beverage Industry

Re-optimization can be very costly for gathering and obtaining more data for a particular problem, to curb this very expensive investment. Sensitivity analysis has been used in this work to determine the behaviour of input parameters of the formulated problem. The main goal of the study is to respectively provide, derive, observe, compare and discuss the sensitivity analysis of data that has been optimized using different methods of the optimal solution. The best method, saving the highest percentage of transportation cost, for the formulated problem is determined to be the North-West Corner method. This was carried out by arbitrarily assigning values to the available warehouses to determine the best possible demand and supply cases rather than the initial cases. Thus, more cases are advised to be supplied to FID from the Asejire plant for the optimum reduced value of transportation cost. DOI: https://dx.doi.org/10.4314/jasem.v24i3.8 Copyright: Copyright © 2020 Latunde et al. This is an open access article distributed under the Creative Commons Attribution License (CCL), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Dates: Received: 16 November 2019; Revised: 11 January 2020; Accepted: 22 February 2020

The best application of real-life problem that has been successful over the century as numerous studies suggested in the optimization field is the transportation problem, Shraddha (2017) solved the transportation problem of the Millennium Herbal Company using different methods and comparing the results yielded. Shraddha gave several methods of solving the transportation problem and obtain the objective of evaluating the similarities and differences of these methods. Dantzig (1963) used the simplex method to determine the solution of transportation as the primal simplex transportation. He proposed that initial basic feasible solution for the transportation problem can be determined through the Column minima method, Row minima, Matrix minima, Vogel's approximation method and the North-West Corner Rule. For the optimal solution, he used the Modified Distribution (MODI). Rao (2009) explained the concept of linear programming, the revised simplex method, duality in Linear Programming under which he worked on duality theorems and dual simplex method. The most interesting of part of this fourth chapter is the Decomposition Principle, Sensitivity of Postoptimality Analysis especially the Transportation Problem. Rao, however, concluded that there are two major methods of obtaining solutions for practical aspects of optimisation, firstly is the sensitivity equation using Kuhn-Tucker conditions and secondly sensitivity equation using the Concept of Feasible Direction. Pursulai and Niittymaki (2001) worked on the functionality of the simulation program Simu+++ and Dispo+++, developed at the Institute of Transport, Railway Construction and Operation in the University of Hanover, Germany. In their book Mathematical Methods on Optimization in Transportation Systems shared the book to two distinct parts, (i) Public Transport Models and (ii) General Transport Models. In the first part they carefully dealt with the prevention of delay in railway traffic by optimization and simulation, then the heuristic for scheduling buses and drivers for an ex-urban public transport computing with bus-driver dependencies. In the research work Optimization Techniques for Transportation Problem of Three Variables, Joshi (2013) used four methods, namely; the North-West Corner Method, the Least Cost Method, the Vogel and the MODI method, in the process of considering the optimization techniques of transportation for three variables, she vividly explain the steps to each method and the steps to determine the optimal solution and comparison between the MODI method and every other method. The book aims at getting the shortest, best and cheapest route to satisfying the demand from any destination. Latunde et al. (2016), Latunde and Bamigbola (2018), Latunde et al. (2019) and Latunde et al. (2020) also analysed the model design by sensitizing some model parameters in the approach of optimal control models to asset management and transportation problems. Therefore, the objective of this paper is to present the sensitivity analysis of road freight transportation of a mega non-alcoholic beverage industry in Nigeria.

MATERIALS AND METHODS
The methods used in the model formulation of the road freight transportation of a mega non-alcoholic beverage industry in Nigeria is Linear programming (LP). LP is applied to the problems related to the study of efficient transportation routes, that is, how efficiently the product from different sources of production is transported to the different destinations such that the total transportation cost is minimised.
Here, two major demand warehouses are considered: Asejire and Ikeja; and 11 major supply destinations are considered: FID, Akin, Oniyele, MGR, Adhex, FDR, Vero, BnB, Mimx, Nuhi and Ile-Iwe. The available data are utilised in the formulated model and solved using four different methods of solving transportation problems: North-West corner method (NWC), Least Cost method (LCM), Vogels Approximation method (VAM) and directly through a computer software called Maple. The model design is therefore analysed using Sensitivity analysis as a post optimality tool where the model designed are analysed to determine the behaviour of these different methods of solving the identified problem, thus recommend a possible improvement on different cases of demand and supply.
Model Formulation: Let's consider a condition in which nine distinct source i.e , , , , , , , , and have to meet the request of also nine destinations, say , , , , , , , , . The gods available at each source is specified , , , , , , , , and the goods requested at each destinations , , , , , , , , . The cost of moving goods from the source to destination can be represented on tables of say where the subscripts (1-9) indicate the cell , given the cost of moving from the source (origin) i to destination j, therefore is the cost of moving goods from source to destination .
The Linear Programming method is a useful tool for dealing with such a problem as a transportation problem. Each source can supply a fixed number of units of products, usually called the capacity or availability, and each destination has a fixed demand, usually known as a requirement. The nature of the and its application in solving problems involving several products from sources to several destinations, this type of problem is frequently generally called " The Transportation Problem ".
Suppose a company has x warehouse and the number of retailers to be y, we can only ship one product from x to y. We can build a mathematical model for the following transportation problem. For example: Consider the table 1 Where implies the supply to the warehouses and is the demand by the retailer outlets.
Let companies producing goods at different places (factories) say "m" factories, from i=1,2,3,...,m. Also, it supplies to different distributors or warehouses, we have this to be , i=1,2,3,...,n. The demand from the factory reaches all requested places (say, wholesalers). The demand from the last wholesaler is the jth place, we call this . The problem of the company is to get goods from factory i and supply to the wholesaler j, the cost is and this transportation cost is linear. By formulation, if we transport numbers of goods from factory i to wholesaler j, then the cost is .
The problem is to find the minimum cost of transporting those goods. The condition that must be satisfied here is that we must meet the demand at each of the wholesalers' request and supply cannot exceed. Therefore, linearly, the cost of this program is The number of goods transported from the factory i is ∑ .
(2) Meanwhile recall, is the good transported i to j. From the factory, you can transport any goods to any of the wholesalers j=1,2,...,n. Here (2) above is the addition of all goods supplied by the factory i from the first wholesaler to the last. The goods cannot be more than the request to be supplied to the wholesaler, we, therefore, have it that ∑ ≤ ∀ = 1,2, . . . , .
(3) Similarly, the constraints to make sure demand is met at all wholesalers point is.
There would be excess demand if the sum of all supply is not more than the demand as such, the request from the wholesalers will be much after supplies have been made to avoid this, therefore, have that ∑ ≤ ∑ , (5) if this is not holding, then the demand cannot be met. Hence, there must be enough possibly excess supply to be sure that demand is met. It is also fair to assume that the quantities demand is exactly equal to the quantities supplied.
When this happens it means the plan for transportation cost is perfect and the supply meets the wholesalers' need at every point and disposed of all goods that left the factory. Therefore, at the cost , m supplies for i=1,2,3,...,m, and n demands for j=1,2,3,...,n. The major work is finding a transportation schedule denoted by to get a solution to min ∑ ∑ (7) subject to ∑ = ∀ = 1,2, . . . , .
Optimal Solutions to the Problem: This subsection shows the data gathered, the result of the optimal feasible solution obtained and the sensitivity analysis of the parameters from two major plants, namely: The Asejire and the Ikeja plant.  The study of the graph and the chart above in Figure 1 shows that the North-West Corner Method produces the optimum transportation cost which is 517,040. For the analysis here, we increase the number of cases in demanded by each warehouse from both plants (Asejire and Ikeja), we do this by adding 50 cases each. We run every addition by the Maple software to determine the outcome and the optimized cost that will be generated, it was studied that some warehouses had more cost value than the others. In Table 3 above is the amount of cases demanded by FID from the Asejire plant, is the amount of cases demanded by Akin also from the Asejire plant, it continues as represented in Table 2, Table 3 and Table 4 and continues on the row until which is the number of cases demanded by Nuhi and lastly which is Ile-Iwe.
is the amount of cases demanded by FID from the Ikeja plant, is the amount of cases demanded by Akin also from the Ikeja plant, it continues as represented in Table 2, Table 3 and Table 4 and continues on the row until which is the number of cases demanded from Ikeja by Nuhi and lastly which is Ile-Iwe.

RESULTS AND DISCUSSION
The problem was solved with three distinct methods namely; the North-West Corner Method, The Least Cost Method and the Vogel's Approximation and then compared with the result computed by the linear programming module called MAPLE. The study of the graph and the chart above in Figure 1 shows that the North-West Corner Method produces the optimum transportation cost which is 517,040. It is noted that Optimized Result is the cost computed by MAPLE and RCV is the Reduced Cost Value in percentage. The result of the Sensitivity Analysis studied shows that more cases of drinks can be should be supplied to which is the FID warehouse from Asejire as it has the largest optimum reduced cost value almost twice of the others. The implication of this is that by priority of supply, will get more cases and will still minimize cost.