The Study of Using LP to Solve Flower Transport Problem

Transcription

1 The Study of Using LP to Solve Flower Transport Problem 1 Chung-Hsin Liu, 2 Le Le Trung 1 Department of Computer Science & Information Engineering, Chinese Culture University, Taiwan, R.O.C. liu3.gold@msa.hinet.net 2 Department of Digital Mechatronic Technology, Chinese Culture University, Taiwan, R.O.C. leletrung88@gmail.com Abstract The main purpose of this paper is to solve Flower Transport Problem (FTP) by using Linear Programming (LP). There are five s can be used to solve Flower Transport Problem by Linear Programming is (1) Northwest Corner (2) Minimum cost (3) Vogel s approximation (4) Row Minimum Method (5) Column Minimum Method. After reviewing the main literature in this area, Mathematical Model of the FTP, this paper presents some examples are solved by five s. This paper introduces some s to improve the results from the s that gave suboptimal results. Those s are (1) Stepping Stone Method (2) Modified Distribution Method. Finally, the paper compare different situation of the examples and propose a comparison table to know which in different situation is the best. In five s, the table shows that most results of Vogel s approximation give an initial solution is optimality. At this time, it is accepted to be the best ; Minimum cost, Row Minima and Column Minima are second s and finally Northwest corner gives an initial solution very far from the optimal solutions. Keywords: Transport Problem, Linear Programming, Northwest Corner Method, Minimum Cost Method, Vogel S Approximation Method, Row Minimum Method, Column Minimum Method 1. Introduction Flower Transport Problem (FTP) is that truckloads of Flower are to be shipped from plants to customers during a particular period of time. Both the available supply at each plant and the required demand by each customer (measured in terms of truckloads) are known. The cost associated with moving one truck load from a plant to a customer is also provided. The objective is to make a least-cost plan for moving the flower such that the demand is met and shipments do not exceed the available supply from each depot [15]. Linear Programming (LP) is a mathematical for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. LP is a specific case of mathematical programming (mathematical optimization) [14]. The data for the problem can be arranged as a matrix of costs of transferring a unit quantity from source i to destination j; a vector which lists the quantities available at each source; and a vector of the demands at each destination. This paper writes about using LP to solve FTP. The remainder of this article is structured as follows. Next sections provide a general overview of FTP formulation and a few words on previous work. An introduction to the FTP Mathematical Model and five s to solve FTP are presented in sections 3 and 4 respectively. Section 5 is devoted to Steps solving FTP. Section 6 presents some examples is solved by five s and AIMMS software. Section 7 is optimal test of the results and introduces some s to improve the suboptimal results. Next, computational results are presented and discussed. Finally, some conclusions, remarks and future research topics are outlined in the last section. Journal of Convergence Information Technology(JCIT) Volume8, Number8, April 2013 doi: /jcit.vol8.issue

2 2. Previous work on FTP Mathematical technique used in computer modeling (simulation) to find the best possible solution in allocating limited resources (energy, machines, materials, money, personnel, space, time, etc.) to achieve maximum profit or minimum cost. However, it is applicable only where all relationships are linear (see linear relationship), and can accommodate only a limited class of cost functions. For problems involving more complex cost functions, another technique called 'mixed integer modeling' is employed [14]. This paper introduces 6 steps to solve FTP is (1) Check for balance of supply and demand (2) Decide the nature of problem Minimization of Transportation-cost and Make an initial basic feasible solution (3) Count the number of filled (or allocated) routes (4) Check the optimality of the initial solution and revise solution if it s not an optimal solution [5]. There are 5 s to make an initial basic feasible solution is Northwest Corner Method, Minimum Cost Method, Vogel s approximation Method, Row Minima Method, Column Minima Method. They are written in section 5 [12] [7] [6]. Checking optimality of the initial solution and revise solution, this paper using two s is (1) Stepping Stone Method (2) Modified Distribution Method (MODI) or (u-v) Method. They are written in section 7 [6]. 3. Mathematical Model of the FTP Transportation model is a special type of networks problems that for shipping a commodity from sources (e.g., factories) to destinations (e.g., warehouse). We can define the Transportation Model as below: Given m sources and n destinations, the supply at i th source is a i and the demand at j th destination is b j. The cost of shipping one unit of goods from i th source to j th destination is c ij. The goal is to minimize the total transportation cost while satisfying all the supply and demand restrictions. Using linear programming to solve transportation problem, we determine the value of objective function which minimize the cost for transporting and also determine the number of unit can be transported from i th source to j th destination. The equivalent linear programming model will be: The objective function Minimize Z= Subject to n m i 1 j 1 cx ij ij (Where x ij is number of units shipped from source i to destination j) n xij ai for i=1, 2 n j 1 m xij bj for j=1, 2 m i 1 xij 0 for all i to j A transportation problem is said to be balanced if the total supply from all sources equals the total demand in all destinations. Otherwise it is called unbalanced. 1117

3 4. Methods solving FTP Most of these s are only to solve for balanced transportation problem. So if the problem is unbalanced then we can balance a transportation problem by adding a dummy supply center (row) or a dummy demand center (column) as the need arises [5]. The solutions obtained by any of the following s are called an initial basic feasible solution (BFS). In other words, these solutions can not be an optimal solution. The North West Corner Rule is a for computing a basic feasible of transportation where the basic variables are selected from the North-West corner (i.e., top left corner). The minimum-cost finds a better starting solution by concentrating on the cheapest routes. The starts by assigning as much as possible to the cell with the smallest unit cost. The Vogel approximation is an iterative procedure for computing a basic feasible solution of a transportation problem. This is preferred over the two s discussed in the previous sections, because the initial basic feasible solution obtained by this is either optimal or very close to the optimal solution. Row minimum start with first row and choose the lowest cost cell of first row so that either the capacity of the first supply is exhausted or the demand at jth distribution center is satisfied or both. Column minimum starts with first column and chooses the lowest cost cell of first column so that either the demand of the first distribution center is satisfied or the capacity of the ith supply is exhausted or both.three cases arise: 5. Steps solve transportation problem You can read on the paper [5], [12] to get the steps solving transportation problem. Step 1: Check for balance of supply and demand If Supply = Demand. This means that a feasible solution always exists. Go to step If Supply > Demand then add one or some dummy demand centers (column) with zero transportation cost that satisfies Supply = Demand. And then go to step If Supply < Demand then add one or some dummies supply centers (row) with zero transportation cost that satisfies Supply = Demand. And then go to step 2. Step 2: 2.1. Decide the nature of problem: Minimization of transportation-cost Make an initial basic feasible solution (BFS). BFS may be done by using any of the following s: (1) Northwest Corner, (2) Minimum cost, (3) Vogel s approximation, (4) Row minima, (5) Column minima. Step 3: Count the number of filled (or allocated) routes If filled route = (m + n -1) then go for optimality check (step 5) If filled route < (m+n-1) then the solution is degenerate. Hence remove degeneracy and go to step 4. Where m = the number of supplies, including dummy row; n = the number of demands, including dummy column. Step 4: In case of degeneracy, allocate a very-very small quality, (which is zero for all calculation purposes), in the least cost of unfilled cells. Step 5: Check the optimality of the initial solution and revise solution if it s not an optimal solution. We will use one of two s in section 6 in this paper to do that 6. Illustrative example 6.1. Problem description Truckloads of Flower are to be shipped from plants to customers during a particular period of time. Both the available supply at each plant and the required demand by each customer (measured in terms of truckloads) are known. The cost associated with moving one truck load from a plant to a customer is 1118

4 also provided. The objective is to make a least-cost plan for moving flowers such that the demand is met and shipments do not exceed the available supply from each depot. The following table (Table 1) is some examples for the data of the problem described in the previous paragraph. Figure 1. Taiwan Map Example 1... Example 30 Table 1(Summary). Input data of FTP Customers Unit Transport Cost Plants Hsinchu Chunan Changhua Chiai Tai-tung Supply Taipei Tainan Demand Taipei Tainan Demand Review results The example 1: After using one of above s, we easy to see minimum cost is (Vogel s approximation ) and then and final (Figure 2). Results of using the s NorthwestCorner MinimumCost Vogel'sapproximation RowMinima ColumnMinima Figure 2. The results of the example 1 Is an optimal solution? To answer that question, we will go to section 7. The example 2: 1119

5 After using one of above s, we easy to see minimum cost is 2560 (Minimum Cost ) and then 2600 and final 3290 (Figure 3). Results of the s Series Northwest Corner MinimumC ost Vogel'sapp roximation RowMinim a ColumnMi nima The s Figure 3.The results of the example 2 Is 2560 an optimal solution? To answer that question, we will go to section 7. The example 30: After using one of above s, we easy to see minimum cost is 2848 (Vogel s approximation) and then 2790 (Figure 4). Results of the s Series Northwe stcorner Minimu mcost Vogel'sa pproxim ation RowMini ma Column Minima The s Figure 4. The results of the example 30 Is 2487 an optimal solution? To answer that question, we will go to section Optimality test and optimize solution After using any of the above s to get an initial basic feasible solution, we must now proceed to determine whether the solution obtained is optimal or not. If the solution is optimal, ok. Otherwise we will optimize the solution until it is optimal solution. So this section introduces two s to be used for testing and optimize solution. They are: 1. Stepping Stone Method 2. Modified Distribution Method (MODI) 7.1. Stepping stone Step 1: Make sure that the number of occupied cell is exactly equal to m+n-1, where m is the number of rows and n is the number of columns. 1120

6 Step 2: Select an unoccupied cell. Beginning at this cell, trace a closed path, starting from the selected unoccupied until finally returning to that same unoccupied cell. The cells the turning points are called Stepping Stones on the path. Step 3: Assign plus (+) and minus (-) alternatively on each corner cell of the closed path just traced, beginning with the plus sign at unoccupied cell to be evaluated. Step 4: Add the unit transportation costs associated with each of the cell traced in the closed path. This will give net change in terms of cost. Step 5: Repeat step 3 to 5 until all unoccupied cells are evaluated. Step 6: Check the sign of each of the net change in the unit transportation costs If all the net changes computed are greater than or equal to zero, an optimal solution has been reached If not, it is possible to improve the current solution and decrease the total transportation cost, so move to step 7. Step 7: 7.1. Select the unoccupied cell having the most negative net cost change and determine the maximum number of units that can be assigned to this cell The smallest value with a negative position on the closed path indicates the number of units that can be shipped to the entering cell Add this number to the unoccupied cell and to all other cells on the path marked with a plus sign Subtract this number from cells on the closed path marked with a minus sign Modified Distribution Method (MODI) or (u-v) The modified distribution, also known as MODI or (u-v) provides a minimum cost solution to the transportation problem. In the stepping stone, we have to draw as many closed paths as equal to the unoccupied cells for the evaluation. To the contrary, in MODI, only closed path for the unoccupied cell with highest opportunity cost is drawn. MODI is an improvement over stepping stone : Step 1: Determine the values of dual variables, ui and vj, using ui + vj = cij Step 2: Compute the opportunity cost using cij (ui + vj). Step 3: Check the sign of each opportunity cost If the opportunity costs of all the unoccupied cells are either positive or zero, the given solution is the optimal solution If one or more unoccupied cell has negative opportunity cost, the given solution is not an optimal solution and further savings in transportation cost are possible. Step 4: Select the unoccupied cell with the smallest negative opportunity cost as the cell to be included in the next solution. Step 5: Draw a closed path or loop for the unoccupied cell selected in the previous step. Please note that the right angle turn in this path is permitted only at occupied cells and at the original unoccupied cell. Step 6: Assign alternate plus and minus signs at the unoccupied cells on the corner points of the closed path with a plus sign at the cell being evaluated. Step 7: 7.1. Determine the maximum number of units that should be shipped to this unoccupied cell. The smallest value with a negative position on the closed path indicates the number of units that can be shipped to the entering cell Now, add this quantity to all the cells on the corner points of the closed path marked with plus signs, and subtract it from those cells marked with minus signs. In this way, an unoccupied cell becomes an occupied cell. Step 8: Repeat the whole procedure until an optimal solution is obtained Optimality test of Flower Transportation problem We will use one of above the s to test if the above solutions are optimal or not. 1121

7 After using one of above the s, we will see: The example 1: is optimal solution and other solutions are not optimality. So we will continue to use one of two the s to edit the solutions that are not optimality into optimal solutions. The example 2: 2560 is not optimal solution. So we can continue to use one of two the s to edit the solutions that are not optimality into optimal solutions. And finally, we can get optimal result is The example 30: 2487 is optimal solution. So other solutions are not optimality. Hence we can continue to use one of two the s to edit the solutions that are not optimality into optimal solutions. 8. Solving FTP by using AIMMS AIMMS (Advanced Interactive Multidimensional Modeling System) is a software system designed for modeling and solving large-scale optimization and scheduling-type problems. We use AIMMS to solve FTP and we can get the results as follows: (Table 2) Example 1... Example 30 Table 2 (Summary). Distribution quality of goods of FTP Customers Unit Transport Cost Plants Hsinchu Chunan Changhua Chiai Tai-tung Supply Taipei Tainan Demand Taipei Tainan Demand And final, total transport cost of the example 1 is 27499; total transport cost of the example 2 is 2530 and total transport cost of the example 30 is Results Solution is called to be Near to optimal solution if it is less than optimal solution and greater than all the remaining solutions. A lot of papers have been written that: Northwest corner gives an initial solution very far from optimal solution; Minimum cost gives an initial solution or optimal solution or very near to optimal solution; Vogel s approximation also gives an initial solution or optimal solution or very near to optimal solution and it is preferred over the two previous s [7][13][15]. Look at results of the 30 examples. We can see (Table 10): Northwest corner often gives initial solutions very far from the optimal solutions; Minimum Cost, Row Minima, Column Minima often give initial solutions or optimal solutions (over 20%) or very near to optimal solutions (over 50%); and Vogel s approximation also often gives initial solutions or optimal solutions (90%) or very near to optimal solutions and it is preferred over the four previous s very much. 10. Conclusions and Future Work In five s, we can see that: Most results of Vogel s approximation give an initial solution is optimality. At this time, it is accepted to be the best ; and then Minimum cost, Row Minima, Column Minima and finally Northwest corner gives an initial solution very far from the optimal solutions. 1122

8 Time Optimal Advantage Table 3(Summary). Comparison between the five s Vogel s Northwest corner Minimum cost Row minimum approximation Computations take short time The solution is very far from optimal solution (about 80%) It is only convenient for programming. Not certain, sometimes shortest Give an initial solution or optimal solution (over 20%) or near to optimal solution (over 50%). It can choose the initial solution to save time. Computations take the longest time Give an initial solution or optimal solution (90%) or very near to optimal solution. It can choose the initial solution to save time. Computations take short time Give an initial solution or optimal solution (over 20%) or near to optimal solution (over 50%). Useful in small number of supply and when the cost of transportation on supply. Column minimum Computations take short time Give an initial solution or optimal solution (over 20%) or near to optimal solution (over 50%). Useful in small number of demand and when the cost of transportation on demand. After using any of the above s to get an initial basic feasible solution, we must now proceed to determine whether the solution so obtained is optimal or not. So we will use Stepping Stone Method or Modified Distribution Method (MODI) to test and find optimal solution (MODI is an improvement over stepping stone ). If Models have a lot of variables and constraints then we should use some software to get results faster and more accurate. E.g.: ABQM, QSB, LINGO, EXCEL, and AIMMS. Using AIMMS with optimization technology offers a complete versatile modeling environment, reduces development time, risk and cost, provides flexibility and scalability, offers the highest possible performance, is easy to learn, has proven its benefits in a wide range of applications. Future Work will be The study of Solving Capacity Routing Problem. The Capacity Routing Problem has been divided into subproblems, concerning customers allocation and routing optimization separately. The subproblems are (1) Capacity Problem and (2) Routing Problem. 11. References [1] [2] [3] Thomas S.Ferguson, ebook Linear Programming [4] AIMMS, ebook AIMMS Optimization Modeling [5] Transportation Models, [6] Transportation Problems, [7] Transportation Problem: A special case for Linear Programming Problem, [8] Thomas S. Ferguson, Linear Programming [9] Transportation Problems, [10] The Transportation Problem: LP Formulations, /documents/TP1-Formulation.pdf [11] Transportation Problems, [12] Transportation Problems, [13] How to solve Transportation problem? [14] An Introduction to Linear Programming df [15] The Transportation Problem [16] Shen Lin, Computer Solutions of the Traveling Salesman Problem, Journal of The Bell System Techical, December,

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