Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 74 Volume 3 Issue 1 February (2023) DOI: 10.47540/ijias.v3i1.729 Page: 74 – 85 The Multi-Objective Transportation Problem Solve with Geometric Mean and Penalty Methods K.P.O.Niluminda1, E.M.U.S.B.Ekanayake1 1Department of Physical Sciences, Rajarata University of Sri Lanka, Sri Lanka Corresponding Author: K.P.O. Niluminda; Email: kponiluminda@gmail.com A R T I C L E I N F O A B S T R A C T Keywords: Best Solution, Geometric Mean, Linear Programming, Multi- Objective Transportation Problem, Penalty Method. Received : 27 November 2022 Revised : 20 February 2023 Accepted : 25 February 2023 The traditional (classical) Transportation Problem (TP) can be viewed as a specific case of the Linear Programming (LP) problem, as well as its models are used to find the best solution for the problem of predetermined how many units of a good or service need to be shipped from one source to multiple locations, with the goals being to reduce time or expense. Classical TP has one objective but when there are two or more objectives to be optimized for in a TP, the strategies used to optimize a single objective are inapplicable. The term “Multi-Objective Transportation Problem (MOTP)” refers to situations in which there are two or more objectives in a TP. The specific extension of the transportation problem is the MOTP. This work provided a novel alternative algorithm that uses geometric means along with the penalty technique to address MOTP. Specifically, analyzed data by comparing our method with numerical examples and presenting the results in a line graph. Our analysis shows that our approach yields better solutions than existing methods, demonstrating the novelty and effectiveness of our approach. The comparison with numerical examples provides a clear and intuitive way of presenting the superiority of our method, making it accessible to practitioners and researchers in the field. These findings have important implications for improving the accuracy and reliability of solutions to the problem at hand. Overall, our study contributes to the advancement of the field by providing a novel and effective method for solving the problem. INTRODUCTION Transportation of products and services from several supply locations to several demand centers is an important field in which linear programming is used. The simplex approach may also be used to resolve a TP that is formulated in terms of an LP model. It requires a lot of time to solve a TP using simplex methods, even though it has many variables and constraints. There are several shipping routes from various supply locations to various demand locations that make up the structure of the TP (Kankanam Pathiranage Oshan Niluminda, 2022; Niluminda & Ekanayake, 2022). The goal is to establish shipping routes between supply and demand hubs to fulfill the demand for a certain amount of products or services at each destination location with the supply of those same goods or services at each supply location at the lowest possible transportation expense. There are several types of TP with different cases. One of the special types of TP is called MOTP. It is referred to as a multi-objective transportation issue when it incorporates numerous objective functions (Ekanayake et al., 2022). When dealing with real-world issues, every business aims to convey commodities while also achieving multiple objectives such as minimizing cost, time, distance, risk, etc. The first TP model was created by Hitchcock (Hitchcock, 1941) in 1941. In real- world scenarios, classical TP can be reformulated as MOTP models due to the complexity of the social and economic context, which necessitates explicit consideration of factors other than expense. Various methods for resolving management-level issues with many competing objectives were initially INDONESIAN JOURNAL OF INNOVATION AND APPLIED SCIENCES (IJIAS) Journal Homepage: https://ojs.literacyinstitute.org/index.php/ijias ISSN: 2775-4162 (Online) Research Article mailto:kponiluminda@gmail.com https://ojs.literacyinstitute.org/index.php/ijias http://issn.pdii.lipi.go.id/issn.cgi?daftar&1587190067&1&&2020 Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 75 described by Charnes and Cooper (Tjalling C. Koopmans, 1941) in 1961. M. Zangiabadi (Zangiabadi & Maleki, 2007) use fuzzy goal programming to address MOTP in 2007. Lushu Li (Lohgaonkar & Bajaj, 2009) proposed a fuzzy compromise programming technique for MOTP. For the linear MOTP, Lakhveer Kaur (Ahmed et al., 2016) suggests a straightforward method for obtaining the optimal compromise solution. Osuji George (George A., 2014) proposed a method using a fuzzy programming algorithm. The three objective linear TPs were solved by Doke D.M. (Doke, 2015) using the arithmetic mean of the global assessment. Evolutionary algorithms were used for the MOTP by K. Bharathi (Bharathi & Vijayalakshmi, 2016) in 2016. Khilendra Singh (Singh & Rajan, 2020) used geometric means to address MOTP under a fuzzy environment. Using an S-type membership function, M.A.M. Khan (Khan & Kabeer, 2015) examines the multi-objective transportation issue. Kavita (Goel, 2021) suggested the novel row maxima method to MOTP utilizing c-program and fuzzy methodology. To solve a MOTP using Pareto Optimality Criteria, Khilendra Singh (Singh & Rajan, 2019) created a novel technique called the Matrix Maxima Method. The geometric mean method and ant colony optimization algorithm were proposed by E.M.U.S.B.Ekanayake (Ekanayake, 2022) in 2022 to address MOTP in fuzzy environments. Moreover, many researchers proposed several approaches to solve MOTP. M. Afwat A.E. (M. Afwat et al., 2018), T. Karthy (Karthy & Ganesan, 2018), Ekanayake E.M.U.S.B. (E. M. U. S. B. et al., 2021; Ekanayake, 2022; Ekanayake et al., 2020, 2021, 2022) Rakesh Verma (Zangiabadi & Maleki, 2013), Abouzar Sheikhi (Pandian & Anuradha, 2011), Sanjay R. Ahir (College, 2021), Kirti Kumar Jain (Jain et al., 2019), and M.A.Nomani (Nomani et al., 2017) were proposed a different type of algorithms to resolve MOTP. This work focus on building a new alternative algorithm to solve multi-objective transportation problem using geometric mean combined with the penalty method. In the end, the proposed method compares with existing different methods using illustrative examples of MOTP with several objectives. METHODS Geometric mean By calculating the multiplication of the values of a collection of numbers, the Geometric Mean (GM) is the average value or mean that denotes the central tendency of the data. The nth root of the multiplication of n numbers is another way to determine the geometric mean. GM average formula can be shown below in equation (1): √ ∏ ⁄ (1) Mathematical Formulation of MOTP Amount in the MOTP is to be carried from sources to destinations at an expense . Here i = 1, 2, „ m and j = 1, 2, „, n and also might represent shipping costs/times/ distances, transport risk, or power consumption, among other things. If the problem has a “t” number of objectives to minimize transport expenses, then it can be written as . The mathematical formulation of MOTP with constraints is given in below: Objective functions: bj no 1= 1 x = ∑ ∑ cij 1 nj=1 m i=1 xij 2 bj no 2= 2 x = ∑ ∑ cij 2 nj=1 m i=1 xij 3 ⋮ ⋮ ⋮ bj no t= t x = ∑ ∑ cij t nj=1 m i=1 xij 4 Constraints: Supply constraint: ∑ xij= ai , i=1, 2, „, m n j=1 5 Demand constraint: ∑ xij= bj , j=1, 2, „, n 6 m i=1 xij i=1, 2, „, m j=1, 2, „, n Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 76 Table 1. General Tabular representation of the multi-objective transportation problem with notations Destination → D1 D2 „ Dn Supply (ai) Source ↓ S1 C11 1, C11 2, „ C11 t C12 1, C12 2, „ C12 t „ C1n 1, C1n 2, „ C1n t a1 S2 C21 1, C21 2, „ C21 t C22 1, C22 2, „ C22 t „ C2n 1, C2n 2, „ C2n t a2 ⋮ ⋮ ⋮ „ ⋮ ⋮ Sm Cm1 1, Cm1 2, „ Cm1 t Cm2 1, Cm2 2, „ Cm2 t „ Cmn 1, Cmn 2, „ Cmn t am Demand (bj) b1 b2 „ bn Here; CijMAX = Each column's or row's highest Cij value CijMIN2 = Each column's or row's second-minimum Cij value The Proposed Novel Algorithm This section presents the proposed novel algorithm with steps. This method can apply both balanced and unbalanced multi-objective transportation problems and the best solution or near-best solution can be obtained. The steps of the proposed method can illustrate as follows: Step 1: Verify whether your MOTP is balanced. If the MOTP table is unbalanced, add a dummy row or column to make it balanced. Step 2: Calculate the Geometric Mean (GM) of every cell using the following equation (7) and create a new table using GM values. ∏ ⁄ √ Here; is the unit cost of each objective in each cell Step 3: Use the formula below (8) to determine the penalty value for each row and column: (8) Step 4: Assign the relevant min (Supply, Demand) to the minimum Cij value cell, selecting the maximum penalty value for each row and column. Step 5: If there is a tie in the maximum penalty value for a column or row, choose the penalty value that corresponds to the minimum Cij value of that rows or columns. Step 6: If the allocation in the previous row meets the supply at the origin, cross out the corresponding row. If it fulfills the requirement there, cross out the corresponding column. Step 7: The procedure should be terminated if demand is satisfied at each destination and supply is enough at each origin. If not, repeat the preceding steps. Step 8: Utilizing a MOTP allocation table, determine the relevant effective cost values for each objective. RESULTS AND DISCUSSION This section will investigate both balanced and unbalanced MOTP and compare the newly recommended method to an optimal solution. Example 1 (Ekanayake et al., 2022). This example represents a multi-objective transportation problem with three objectives such as cost, time, and distance. Table 2. Step 1 (Initial multi-objective transportation table with cost, time, and distance) (Cost, Time, Distance) D1 D2 D3 D4 Supply S1 21 1 11 16 2 13 15 1 17 13 4 14 11 S2 17 3 16 18 3 18 24 2 14 23 1 10 13 S3 32 4 21 27 2 24 18 5 13 41 9 10 19 Demand 6 10 12 15 Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 77 Table 3. Steps 2 – 3 (Geometric mean of objectives (Cost, Time, Distance)) Geometric Mean D1 D2 D3 D4 Supply S1 6.13 7.46 6.34 8.99 11 S2 9.34 9.90 8.75 6.13 13 S3 13.90 10.90 10.53 15.45 19 Demand 6 10 12 15 Table 4. Steps 4 – 7 (Penalty method for example 1) D1 D2 D3 D4 Supply P1 P2 P3 P4 P5 P6 S1 6.13 (6) 7.46 6.34 (3) 8.9 (2) 11 2.65 2.65 1.53 1.12 - - S2 9.34 9.90 8.75 6.13 (13) 13 1.15 - - - - - S3 13.90 10.90 (10) 10.53 (9) 15.45 19 4.55 4.55 4.55 0.37 0.37 10.53 Demand 6 10 12 15 P1 4.56 1.00 1.78 6.46 P2 7.77 3.44 4.19 6.46 P3 - 3.44 4.19 6.46 P4 - 3.44 4.19 - P5 - 10.90 10.53 - P6 - - 10.53 - Table 5. Step 8 (Final allocation table of example 1) (Cost, Time, Distance) D1 D2 D3 D4 Supply S1 (21, 1, 11) [6] (16, 2, 13) (15, 1, 17) [3] (13, 4, 14) [2] 11 S2 (17, 3, 16) (18, 3, 18) (24, 2, 14) (23, 1, 10) [13] 13 S3 (32, 4, 21) (27, 2, 24) [10] (18, 5, 13) [9] (41, 9, 10) 19 Demand 6 10 12 15 Minimum Cost = (6 × 21) + (3 × 15) + (2 × 13) + (13 × 23) + (10 × 27) + (9 × 12) = 874 Minimum Time = (6 × 1) + (3 × 1) + (2 × 4) + (13 × 1) + (10 × 2) + (9 × 5) = 95 Minimum Distance = (6 × 11) + (3 × 17) + (2 × 14) + (13 × 10) + (10 × 24) + (9 × 13) = 632 Table 6. Comparison analysis of example 1 with different methods Comparison Analysis Minimum Cost Minimum Time Minimum Distance New Row Maxima Method (Goel, 2021) 938 117 457 Product Approach (M. A. E. Afwat et al., 2018) 938 132 552 Geometric Mean Method (Singh & Rajan, 2020) 904 107 587 Ekanayake’s Method (Ekanayake, 2022b) 904 107 587 Proposed Method 874 95 632 LINGO 796 89 527 Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 78 Figure 1. Comparison Analysis of example 1 with different methods Example 1 shows the MOTP, which has three objectives. The objective of that problem is to minimize the cost, time, and distance. Table 6 represents a comparison analysis of the proposed method with the New Row Maxima Method (Goel, 2021), Product Approach (M. A. E. Afwat et al., 2018), Geometric Mean Method (Singh & Rajan, 2020), Ekanayake’s Method (Ekanayake, 2022b), and Optimum solution obtained by LINGO. Figure 2 shows the line graph representation of that comparison. The proposed method gives a better solution in cost, time, and distance objectives by comparison to the other existing methods. Example 2 (Singh & Rajan, 2019). This example represents a multi-objective transportation problem with two objectives such as cost and time. Table 7. Step 1 (Initial multi-objective transportation table with cost and time) (Time, Cost) D1 D2 D3 Supply S1 (13, 14) (15, 15) (16, 10) 17 S2 (7, 21) (11, 13) (2, 19) 12 S3 (19, 17) (20, 26) (9, 9) 16 Demand 14 8 23 Table 8. Step 2 – 3 (Geometric mean of objectives (Cost, Time)) Geometric Mean D1 D2 D3 Supply S1 13.49 15.00 12.65 17 S2 12.12 11.96 6.16 12 S3 17.97 22.80 9.00 16 Demand 14 8 23 938 938 904 904 874 796 117 132 107 107 95 89 457 552 587 587 632 527 0 200 400 600 800 1000 1200 1400 1600 1800 New Row Maxima Method Product Approach Geometric Mean Method Ekanayake’s Method Proposed Method LINGO O b je ct iv e F u n ct io n V a lu e Name of the Method Comparitive Analysis of Example 1 Minimum Cost Minimum Time Minimum Distance Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 79 Table 9. Steps 4 – 7 (Penalty method for example 2) D1 D2 D3 Supply P1 P2 P3 P4 P5 S1 13.49 (14) 15.00 12.65 (3) 17 1.51 0.84 0.84 0.84 12.65 S2 12.12 11.96 (8) 6.16 (4) 12 0.16 5.96 5.96 - - S3 17.97 22.80 9.00 (16) 16 4.83 8.97 - - - Demand 14 8 3 P1 4.48 7.80 3.65 P2 4.48 - 3.65 P3 1.37 - 6.49 P4 13.49 - 12.65 P5 - - 12.65 Table 10. Step 8 (Final allocation table of example 2) (Time, Cost) D1 D2 D3 Supply S1 (13, 14) [14] (15, 15) (16, 10) [3] 17 S2 (7, 21) (11, 13) [8] (2, 19) [4] 12 S3 (19, 17) (20, 26) (9, 9) [16] 16 Demand 14 8 23 Minimum Time = (14 × 13) + (3 × 16) + (8 × 11) + (4 × 2) + (16 × 9) = 470 Minimum Cost = (14 × 14) + (3 × 10) + (8 × 13) + (4 × 19) + (16 × 9) = 550 Table 11. Comparison analysis of example 2 with different methods Comparison Analysis Minimum Time Minimum Cost New Row Maxima Method (Goel, 2021) 656 652 Product Approach (M. A. E. Afwat et al., 2018) 440 583 Matrix Maxima (Singh & Rajan, 2019) 470 550 Proposed Method 470 550 Example 2 TP has two objectives (Time, Cost). The goal is to reduce the overall time and expense. Table 11 shows the comparative analysis of example 2 and figure 2 represents its graphical representation. By comparing the New Row Maxima Method (Goel, 2021), Product Approach (M. A. E. Afwat et al., 2018), and Matrix Maxima Method (Singh & Rajan, 2019) our proposed method gives a better solution. Both Matrix Maxima and the proposed method give the same results. Figure 2. Comparison Analysis of example 2 with different methods 652 583 550 550 656 440 470 470 0 500 1000 1500 1 2 3 4 O b je ct iv e F u n ct io n V a lu e Name of the Method Comparitive Analysis of Example 2 Minimum Cost Minimum Time Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 80 Example 3 (Singh & Rajan, 2019). This example represents a multi-objective transportation problem with two objectives such as cost and time. Table 12. Initial multi-objective transportation table with cost and time (Time, Cost) D1 D2 D3 D4 Supply S1 (6, 1) (4, 2) (1, 3) (5, 4) 14 S2 (8, 4) (9, 3) (2,2) (7, 0) 16 S3 (4, 0) (3, 2) (6, 2) (2, 1) 5 Demand 6 10 15 4 Table 13. Comparison analysis of example 3 with different methods Comparison Analysis Minimum Time Minimum Cost New Row Maxima Method (Goel, 2021) 162 83 Product Approach (M. A. E. Afwat et al., 2018) 114 62 Matrix Maxima (Singh & Rajan, 2019) 115 57 Proposed Method 121 54 The MOTP with two objectives such as time and cost shown in example 3. The objective of that example is to minimize the total time and cost. This example compares with New Row Maxima Method (Goel, 2021), the Product Approach (M. A. E. Afwat et al., 2018), Matrix Maxima Method (Singh & Rajan, 2019), and the proposed algorithm. The proposed method gives the most minimum cost value compared to the other methods in this example. Table 13 shows the comparative results of that example and figure 3 represents the line graph analysis with the existing three methods. Figure 3. Comparison Analysis of example 3 with different methods Example 4 (Ahmed et al., 2016). This example represents a multi-objective transportation problem with two objectives. Table 14. Initial multi-objective transportation table with objectives Z1 and Z2 (Z1 , Z2) D1 D2 D3 Supply S1 (3, 5) (4, 2) (5, 1) 8 S2 (4, 3) (5, 4) (2, 3) 5 S3 (5, 2) (1, 3) (2, 1) 2 Demand 7 4 4 162 114 115 121 83 62 57 54 0 50 100 150 200 250 300 1 2 3 4 O b je ct iv e F u n ct io n V a lu e Name of the Method Comparitive Analysis of Example 3 Minimum Time Minimum Cost Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 81 Table 15. Comparison analysis of example 3 with different methods Comparison Analysis Z1 Z2 Kaur’s Method (Ahmed et al., 2016) 55 40 Proposed Method 56 39 Figure 4 Comparison Analysis of example 4 with Kaur’s method The proposed method was compared with Kaur’s Method (Ahmed et al., 2016) in this example. This TP has two objectives (Z1 and Z2). The comparative evaluation of that example is shown in table 15 and the graphical evaluation shows in figure 4. By looking at above table 15 and figure 4, the proposed method gives the most minimum value in the Z1 objective. Example 5 (Ekanayake, 2022). This example represents a multi-objective transportation problem with three objectives such as cost, time, and distance. Table 16. Initial multi-objective transportation table with cost, time, and distance (Cost, Time, Distance) D1 D2 D3 D4 Supply S1 (6, 13, 6) (4, 11, 3) (1, 15, 5) (5, 20, 4) 14 S2 (8, 17, 5) (9, 14, 9) (2, 12, 2) (7, 13, 7) 16 S3 (4, 18, 5) (3, 18, 7) (6, 15, 8) (2, 12, 6) 5 Demand 6 10 15 4 Table 17. Comparison analysis of example 5 with different methods Comparison Analysis Minimum Cost Minimum Time Minimum Distance New Row Maxima Method (Goel, 2021) 112 461 130 Product Approach (M. A. E. Afwat et al., 2018) 114 425 128 Geometric Mean Method (Singh & Rajan, 2020) 114 425 118 Ekanayake’s Method (Ekanayake, 2022b) 114 425 118 Proposed Method 114 425 118 LINGO 114 424 106 55 56 40 39 0 50 100 1 2 O b je ct iv e F u n ct io n V a lu e Name of the Method Comparitive Analysis of Example 4 Z1 Z2 Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 82 Figure 5. Comparison Analysis of example 5 with different methods The TP with three objectives (Cost, Time, Distance) is represented in this example 5. The main objective of this example is to minimize the total cost, time, and distance. The comparison analysis of this example shows in table 17. Compare to the other existing method proposed method gives the optimal solution to the cost objective and a near-optimal solution to the time and distance. Figure 5 presents the comparison result of three objectives using a line graph. Compare to the Geometric Mean Method (Singh* & Rajan, 2020) and Ekanayake’s Method (Ekanayake, 2022b), the proposed method also gives the same outcomes. Example 6 ( Doke et al., 2015). This example represents a multi-objective transportation problem with three objectives. Table 18. Initial multi-objective transportation table with cost, time, and distance (Z1, Z2, Z3) D1 D2 D3 D4 Supply S1 (3, 2, 8) (2, 5, 4) (5, 7, 3) (7, 9, 2) 10 S2 (4, 4, 5) (3, 4, 3) (3, 5, 4) (5, 6, 2) 20 S3 (2, 3, 7) (1, 2, 2) (4, 6, 6) (3, 8, 8) 40 Demand 15 15 20 20 Table 19. Comparison analysis of example 6 with different methods Comparison Analysis Z1 Z2 Z3 Doke’s Method (Doke et al., 2015) 205 335 305 Proposed Method 235 325 265 112 114 114 114 114 114 461 425 425 425 425 424 130 128 118 118 118 106 0 100 200 300 400 500 600 700 800 1 2 3 4 5 6 Comparison Analysis of Example 5 Minimum Cost Minimum Time Minimum Distance Indonesian Journal of Innovation and Applied Sciences (IJIAS), 3 (1), 74-85 83 Figure 6. Comparison Analysis of example 6 with Doke’s method In example 6 TP has three different objectives (Z1, Z2, and Z3). The main objective of this example is to minimize the objective function values Z1, Z2, and Z3. The proposed method results compare with Doke’s Method (A Solution to Three Objective Transportation Problems Using Fuzzy Compromise Programming Approach, n.d.) in this example. The comparative results between the proposed method and Doke’s Method are represented in table 19 and it line graph evaluation shows in figure 6. By looking at the comparison table, both values of objective functions 2 and 3 (Z2 and Z3) are best when using our proposed method. CONCLUSION Multiple supply stations to multiple demand stations are connected by a massive number of transport routes in the structure of the TP. 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