Operations Research System Tora Software Free Download
Consuelo Dular
The passion to carry out this research work was ignited by the alarming rate of unemployment in the Nigerian society which mostly result from quest for white-collar jobs among youths and failure of existing industries. The objective of this study is to encourage youths to go into entrepreneurship by helping existing entrepreneurs to maximize their profit and/or minimize their cost irrespective of the constraints militating against them. Both primary and secondary data were collected from the DUNA HOLDINGS PLC ABA. Observation and interview methods were used to source primary data. TORA Optimization system, window version 2.00 was used in maximization and minimization analyses. Maximization analysis suggested that a maximum profit of N3083.7 will be made per day if zero unit of DUNA soft, 2392 unit of DUNA pack and 5211 units of DUNA strip are produced per day. The result of minimization analysis shows that a minimum cost of N66661.14 will be incurred per day if zero unit of DUNA pack, 4784 unit of DUNA soft and 38368 units of DUNA strip are produced per day. It was concluded that the company is neither maximizing profit nor minimizingcost and as such may liquidate in the long run. Recommendations on production combination that will maximize profit and those that will minimize cost were given to the company. It was also recommended that government should organize programs that will educate entrepreneurs on applications of operations research techniques in their production to prevent failure of industries and attract more youths into entrepreneurship so that unemployment will be eradicated.
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 [1] ]. It is therefore an important part of production process. Operations managers are constantly searching for ways of getting their goods/services to their customers that will be cost-effective, meet international best practice and at the same time increase the profit margin of their organization. More so due to demand variability and market uncertainty, achieving this transportation goal requires a lot of flexibility, short response time and development of new innovative solutions for reaching the customers [Noham and Tzur [2] ]. This transportation need becomes more complicated when challenges arise from globalization, increase market competition, and accelerated technology development is considered.
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 [3] ].
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 [4] ]. And the constraints are that the resources to be optimally allocated usually involve a given capacity of goods at each source and a given requirement for the goods at each destination. For the transportation problem, the source/supply points can only send out goods but cannot receive any while the sink or the demand point can only accept goods and not give out. With the diversification commodity type, size, distance to sinks etc., this transportation model becomes limited at some point. To overcome this limitation, a variant of the transportation problem with an intermediate point was introduced. This is known as the Transshipment problem.
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, [5] ]. It can also be seen as transportation of goods/containers to an intermediate destination, before it is shipped to another destination. There may be several reasons for such a change. One of which is to trans-load i.e. changing the means of transport during the journey (either from a land to an air, etc.). Other reasons could be to combine small shipments into a large shipment (consolidation), dividing the large shipment at the other end (deconsolidation). Whatever the reason, this model takes into consideration a multi-phase transport system where the flow of material-raw and/or finished goods and services are taken through the intermediate point which is between the origin and the destination. The whole stock is expected to pass through these points of reloading before the goods are finally sent to their destination [Briš [1] ].
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 [6] ].
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, [1] ]. It further considers that within a given time period each shipping source has a certain capacity and each destination has certain requirements with a given cost of shipping from source to destination. The Objective Function is to minimize total transportation costs and satisfy destination requirements within source requirements [Gupta and Mohan [10] ].
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.
The transshipment problem characterized by the uncertainty relative to customer demands and transfer lead time was studied by Hmiden et al. [13] . For them identifying a transshipment policy that incorporates the fuzziness of customer demands and transfer lead times and determining the approximate replenishment quantities which minimize the total inventory cost was the main focus while using a distribution network of one supplier with several locations selling the product. Their method was centered on the use of expert judgments to evaluate customer demands and the transfer lead time which they represented by fuzzy sets. Among other factors studied on the expert judgments, they used the behavior types of the decision maker (pessimistic and optimistic) to determine the precise transshipment decision moment and the transshipment quantity.
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 [14] ]. Hoppe and Tardos gave the first polynomial-time algorithm for the quickest transshipment problem that provided an integral optimum flow.
Tadei et al. [15] pointed out in their research that the transportation process takes place in two stages for any transshipment problem. These were: 1) from origins to transshipment facilities, 2) from transshipments facilities to final destinations. By economic implication, the process could incure three kinds of cost: 1) the fixed cost of locating a transshipment facility, 2) the transportation cost from an origin to a destination through a transshipment facility, 3) the throughput operation cost at each transshipment facility. Their study sought to reveal the overall cost implication involved in the entire transshipment process that may not be incorporated in the transshipment LPP.
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