MODIFIED VOGEL APPROXIMATION METHOD FOR BALANCED TRANSPORTATION MODELS TOWARDS OPTIMAL OPTION SETTINGS

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1 International Journal of Civil Engineering and Technology (IJCIET) Volume 9, Issue 12, December 2018, pp , Article ID: IJCIET_09_12_039 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed MODIFIED VOGEL APPROXIMATION METHOD FOR BALANCED TRANSPORTATION MODELS TOWARDS OPTIMAL OPTION SETTINGS I. D. Ezekiel Department of Mathematics & Statistics, Federal Polytechnic, Ilaro, Nigeria Department of Mathematics, Covenant University, Ota, Nigeria S. O. Edeki Department of Mathematics, Covenant University, Ota, Nigeria Corresponding Author s soedeki@yahoo.com ABSTRACT This paper is built on a study in relation to transportation problem as it affects most organisational decision in a decomposed setting. The case study used in this work is Dangote cement factory (in Ibese, Nigeria) with three sources and four destinations centres. The factory is supported by increasing number of cement delivery trucks. Some models for solving balanced transportation problems (TPs) are considered in order to determine the optimal and initial basic feasible solutions (IBFS). From the analysis, it is observed that Modified Vogel Approximation Method (MVAM) is a better method. This is partly because MVAM considers each unit cost in its solution algorithm and minimises total cost comparatively with Vogel Approximation Method (VAM). The results are further justified and validated using windows version 2.00 Tora package. Key words: Balanced transportation problem, Option pricing, Modified Vogel approximation method, Vogel approximation method, Decomposition method Cite this Article: I. D. Ezekiel and S. O. Edeki, Modified Vogel Approximation Method for Balanced Transportation Models towards Optimal Option Settings, International Journal of Civil Engineering and Technology (IJCIET) 9(12), 2018, pp INTRODUCTION This study contains the data from Dangote cement factory located in Ibese, Nigeria. Dangote Cement operates three manufacturing plants in Obajana, Ibese and Gboko, with longer-term plans to expand its capacity across two sites at Itori and Okpella. However, the Ibese plant is considered herein. The factory supplies cement product to South-South states, Lagos state, editor@iaeme.com

2 I. D. Ezekiel and S. O. Edeki Ogun state and other South west states in Nigerian markets with production capacity of 6.0 MT cement produced from lines 1, 2 and 3 respectively. The factory is supported by increasing number of cement delivery trucks. Hence, the need for enhanced means to resolve transportation problem (TP). This data article therefore aims at making informed decision on effective techniques for finding initial basic feasible solution and applying Modified Vogel Approximation Method (MVAM) to determine the optimal solution to the corresponding balanced TP. Table 1 contains the sources alongside the destinations of the trunk lines, Table 2 contains the General Transportation Problem Tableau (GTPT), while the analysed data are presented in Tables 3-5 and Figures 1-4. Related work on Transportation, economic, and financial problems of relevance to this study can be found in [1-15]. The simplest transportation problem was first presented in 1941 and developed in 1949 and Since then several extensions of transportation models and methods have emerged. Among early pioneers of transportation problems (TPs) are Hitchcock [16] through his paper the Distribution of a Product from Several sources to numerous Localities. Closely related to Hitchcock is Koopman [17] who with his paper Optimum Utilization of Transportation System simplified better understanding of transportation methods involving a number of shipping sources and destinations. Gass [5] in his own contribution explains practical issues for solving transportation problems and offered comments on various aspects of transportation problem (TP) methodologies along with discussions on the computational results by various researchers. Tzeng, Teodorovic, and Hwang, [18] applied time TP to formulate a LP that minimizes cost of transportation to distribute and transport imported Coal to power plants by determining the quantity and quality required amounts under stable supply with least delay. Their model yield optimum results and serves as decision support system to manage coal allocation, voyage scheduling and dynamic fleet assignment problems. Since then, a number of researchers such as Sonia [13] contributed to Transportation Problem in area of decision support. However, their work was restricted on time transportation problem. A number of researchers including Wahead and Lee [19], and Joshi [20], in Optimization Techniques for Transportation Problems of Three Variables proposed for the procedure for full allocation of available total quantity required in each cell. The existing techniques or methods are the minimum cost rule method, Northwest corner rule method, Vogel approximation method, Russell approximation method and Roland Larson s method among others. Resolved TPs in differential model forms can be handled with approximate and numerical methods [21-25]. This paper therefore aims at making comparison of existing techniques for finding initial basic feasible solution and apply modified Vogel approximation method in determining initial basic feasible solutions so as to determine the optimal solution to any given balanced TP. 2. MATERIALS AND METHODS For the purpose of this paper, the table below gives simplified data structure obtained from the cement factory with three sources and four destination centres without service level requirements. The following are considered. There are three sources (lines 1, 2 and 3) and four destinations centres (South-South, Lagos, Ogun, other South west states) represented by A, B, C and D respectively. All the mentioned routes in this work are always accessible. Movement of cement products are always from the sources to the specified destinations. The same means of transportation of the products and constant speed are allowed editor@iaeme.com

3 Modified Vogel Approximation Method for Balanced Transportation Models towards Optimal Option Settings Table1 specifies sources ( ), destination centres ( ), where ; and the unit transportation cost from to. Table 1 Case study of Dangote Cement (Ibese-Station) 3. GENERAL TRANSPORTATION PROBLEM AND APPLICATIONS Mathematically, we state TP compactly as given below: subject to: S/D Destination Source A B C D , and where is the capacity of supply at source, is capacity of demand at destination, is the amount shipped from source to destination, and is the total shipping cost and it is non-negative. is the condition for existence of feasible solution for standard TP. In other words, total units demanded must match total units supplied. A balanced transportation problem having sources and destinations is usually represented in tabular form as shown below: Table 2 General Transportation Problem Tableau S/d Destination Source Here, the methods applied include: Northwest-Corner Rule (NCR), Least-Cost Method (LCM), Vogel Approximation Method (VAM), and Modified Vogel Approximation Method (MVAM) editor@iaeme.com

4 Sources and supply-stages I. D. Ezekiel and S. O. Edeki Table 3 Northwest Corner Rule S/D Destination Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Source A B C D Supply Stage Stage Stage Stage Stage Stage Using Northwest-corner rule, we have the following initial basic feasible solution for as:,,,,,. The optimal minimization cost is given by ( ) ( ) ( ) ( ) ( ) ( ) Supply:1 Supply:2 Supply:3 dj Stage:1 Stage: 2 Stage: 3 Stage: 4 Stage: 5 Stage: 6 0 A B C D SS S1 S2 S3 S4 S5 S6 Destinations and stages Figure 1 Northwest Corner Rule Table 4 Least Cost Rule S/D Destination Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Origin A B C D Supply Stage Stage Stage Stage Stage Stage editor@iaeme.com

5 Sources and supply-stages Modified Vogel Approximation Method for Balanced Transportation Models towards Optimal Option Settings For, we have:,,,,, The optimal minimization cost is given by ( ) ( ) ( ) ( ) ( ) ( ) A B C D SS S1 S2 S3 S4 S5 S6 Destinations and stages Supply: 1 Supply: 2 Supply: 3 dj Stage: 1 Stage: 2 Stage: 3 Stage: 4 Stage: 5 Stage: 6 Figure 2 Least Cost Rule Table 5 Vogel Approximation Method S/D Destination Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Source A B C D Supply Stage Stage Stage Stage Stage Stage Using Vogel approximation rule, where is the row difference at stage and is the column difference at stage. Thus the following initial basic feasible solution for becomes:,,,,, The optimal minimization cost is given by ( ) ( ) ( ) ( ) ( ) ( ) editor@iaeme.com

6 A B C D SS S1A S1B S2A S2B S3A S3B S4A S4B S5A S5B S6A S6B Sources and supply-stages I. D. Ezekiel and S. O. Edeki Destinations and stages Supply: 1 Supply: 2 Supply: 3 dj Stage 1 Stage: 1 Stage 2 Stage: 2 Stage 3 Stage: 3 Stage 4 Stage: 4 Stage 5 Stage: 5 Stage 6 Stage: 6 Figure 3 Vogel Approximation Method Table 6 Modified Vogel Approximation Method (MVAM) S/D Destination Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Sour. A B C D Sup Stage Stage Stage Stage Stage Stage Using modified Vogel approximation rule method, where is the row mean deviation at stage and is the column mean deviation at stage. Therefore, the initial basic feasible solution for becomes:,,,,, The optimal minimization cost is given by: ( ) ( ) ( ) ( ) ( ) ( ) editor@iaeme.com

7 A B C D SS S1A S1B S2A S2B S3A S3B S4A S4B S5A S5B S6A S6B Sources and supply-stages Modified Vogel Approximation Method for Balanced Transportation Models towards Optimal Option Settings Destinations and stages Supply: 1 Supply: 2 Supply: 3 dj Stage 1 Stage 1 Stage 2 Stage 2 Stage 3 Stage 3 Stage 4 Stage 4 Stage 5 Stage 5 Stage 6 Stage 6 Figure 4 Modified Vogel Approximation Method (MVAM) Table 7 Comparison of the Methods S/N Methods IBFS 1 Northwest corner rule 2 Least cost rule 3 Vogel Approximation 4 Modified Vogel Approximation 4. DISCUSSION OF RESULTS, CONCLUDING REMARKS, AND RECOMMENDATIONS Table 7 shows the result for selected existing approaches available for solving balanced transportation problem and their order of efficiency. MVAM is applied as well as other existing methods for solving balanced transportation problem and compared the efficiency of the methods at reducing total cost using the same case study. From the analysed data in Table 7, it is obvious that the MVAM gives better improvement than any other selected techniques in determining the initial basic feasible solution (IBFS) in terms of reduction in total cost of production in the cement factory. The results of the solution obtained in Table 7 was further justified and validated using windows version 2.00 Tora package. The use of the modified VAM for solving transportation problem gives a systematic and transparent solution when compared with other existing methods. Thus for more scientific transportation problem, the modified VAM gives a better result. Hence the modified VAM guarantees optimal solution. The researcher, therefore, recommends that the modified VAM should be adopted and encouraged by companies, producers and other agencies involved in transportation business. The researcher also recommends further research in this area ACKNOWLEDGEMENTS The authors are indeed grateful to Covenant University for the provision of resources, and enabling working environment editor@iaeme.com

8 REFERENCES I. D. Ezekiel and S. O. Edeki [1] Masoud, E. M., Kozan E., Kent, G., and Liu, S. Q., Experimental dataset for optimising the freight rail operations, Data in Brief, 9 (2016), [2] Akpan, S., Ugbe, T., Usen, J., & Ajah, O. (2015). A Modified Vogel Approximant Method for Solving Balanced Transportation Problems. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS), 14 (3) [3] Denis, H., Leo, G. K., Ramon, M. L., & Vromans, J. C. M. (2002). Operations Research in Passenger Railway Transportation. Econometric Institute Report E [4] Sulymon, N., Ofuyatan, O., Adeoye, O.,, Bamigboye, G., Jolayemi, J. Engineering properties of concrete made from gravels obtained in Southwestern Nigeria, Cogent Engineering, 4(1), (2017), [5] Gass, S. I. (1990). On solving the transportation problem. The journal of the operational research society, 41 (4) doi: / [6] Ofuyatan, O.M., Edeki, S.O. Dataset on predictive compressive strength model for selfcompacting concrete, Data in Brief, 17, (2018), [7] Pawan, T., Bhagivah, P., & Dhami, H. S. (2011). Development of an Algorithm for all Types of Transportation Problems. International Journal of Computer Applications, 30 (6) [8] Omuh, I.O., Mosaku, T.O., Joshua, O.,, Afolabi, A.O., Arowolo, A.O. Data on mixing and curing methods effects on the compressive strength of concrete, 2018 Data in Brief, 18, [9] Seshan, C. R., & Tikekar, V. G. (1980) On the Sharma and Swarup algorithm for time minimizing transportation problem. Proceeding of Indian Academy of Sciences Mathematical Science, [10] Edeki, S.O., Ugbebor, O.O., Owoloko, E.A. He's polynomials for analytical solutions of the Black-Scholes pricing model for stock option valuation, Lecture Notes in Engineering and Computer Science, Volume 2224, 2016, World Congress on Engineering 2016, WCE 2016, Code [11] Sharma, J. K., & Swarup, K. (1977). Time minimizing transportation problem, Proceedings of Indian Academy of Sciences-Mathematical Sciences, [12] Edeki, S.O., Adeosun, M.E., Owoloko, E.A., Akinlabi, G.O., Adinya, I. (2016). Parameter Estimation of Local Volatility in Currency Option Valuation, International Review on Modelling and Simulations 9 (2), [13] Sonia, & Puri, M. C. (2004). Two level hierarchical time minimizing transportation problem. Top, 12 (2) [14] Edeki, S.O., Owoloko, E.A., Ugbebor, O.O. The modified Black-Scholes model via constant elasticity of variance for stock options valuation, AIP Conf. Proc., 1705 (2016), p [15] Surapati, P., & Roy, T. K. (2008). Multi-objective transportation model with fuzzy parameters: Priority based fuzzy goal programming approach. Journal of Transportation Systems Engineering and Information Technology, [16] Hitchcock, F. L. (1941). The distribution of product from several source to numerous localities. Journal of Mathematics and Physics, [17] Koopman, T. C. (1947). Optimum utilization of transportation system. Proc. International Statistics Conference, Washington D. C editor@iaeme.com

9 Modified Vogel Approximation Method for Balanced Transportation Models towards Optimal Option Settings [18] Tzeng, G., Teodorovic, D., & Hwang, M. (1996), Fuzzy bicriteria multi-index transportation problems for coal allocation planning of Taipower. Europe journal of operational research, 95 (95) doi: / (95) [19] Wahead, W. F. & Lee, S. M. (2006). Interactive fuzzy goal programming problem for multi-objective transportation problem, Omega, [20] Joshi R. V. (2013), Optimization Techniques for Transportation Problems of Three Variables, IOSR Journal of Mathematics, 9 (1), [21] Agbolade, O. A. and Anake T. A. (2017), Solutions of First-Order Volterra Type Linear Integrodifferential Equations by Collocation Method, Journal of Applied Mathematics, 2017, Article ID: [22] S. A. Bishop, and T. A. Anake, Extension of Continuous Selection Sets to Non- Lipschitzian Quantum Stochastic Differential Inclusion, Stochastic Analysis and Applications, 31 (6), 2013, [23] Akinlabi, G.O. and Adeniyi R. B. (2018). Sixth- order and fourth- order hybrid boundary value method for systems of boundary value problems, WSEAS Transactions on Mathematics, 17, 2018, [24] Edeki, S.O. Owoloko, E. A., Ugbebor, O. O. (2015). The modified Black-Scholes model via constant elasticity of variance for stock options valuation, AIP Conference Proceedings, Volume 1705, 1 February 2016, Article number , 1st Progress in Applied Mathematics in Science and Engineering, PIAMSE 2015; Bali; Indonesia. [25] Akinlabi, G.O. and Edeki, S. O. (2017). The Solution of Initial-value Wave-like Models via Perturbation Iteration Transform Method, Lecture Notes in Engineering and Computer Science, Volume 2228, 2017, editor@iaeme.com

UNIT 4 TRANSPORTATION PROBLEM

UNIT 4 TRANSPORTATION PROBLEM UNIT 4 TRANSPORTATION PROLEM Structure 4.1 Introduction Objectives 4.2 Mathematical Formulation of the Transportation Problem 4.3 Methods of Finding Initial asic Feasible Solution North-West orner Rule