Heuristic Approaches for Solving an Integrated One-dimensional Cutting Stock-transportation Problem

Large-scale problems, where the number of variables is large, typically cannot be solved because of time and memory storage limitations. The integration of the one-dimensional cutting stock problem with multiple cutting facilities and the transportation problem was the focus for applications with large-scale problems. In this study, Benders partitioning approach and the column-generation technique with the direct method and the proposed heuristic method for solving IP were developed into three approaches to solve those problems in the reasonable computation time. The computation time and the relative-different percentage are criterions. The results indicated that the approach based on the column-generation technique with the proposed heuristic method is the most efficient method for solving large-scale problems. Hence, this approach could be used in industries; to manage both production and transportation plans simultaneously.


Introduction
In order to solve large-scale problems, the big issue concerned in these problems is how long of the computational time. Absolutely, all industries need to know how to plan the process to reach out the minimal cost and meet all demands. However the computation to find out the solution should be also in reasonable time. More than four decades ago, mathematicians studied intensively involved with large-scale problems. Dantzig and Wolfe (1961) initiated the extensive work on large-scale mathematical programming. The most efficient technique when applied to linear programs is the column generation technique. It is applied for solving the problems with fewer rows and many more columns. There are wide variety of situations could be solved by this approach. Holthaus (2002) proposed the decomposition approaches based on the columngeneration technique for solving the integer one-dimensional cutting stock problem with different types of standard lengths. Benders (1962) proposed Benders decomposition to divide the original problem into subproblems with smaller size. This approach is applied in a variety of applications such as in cellular manufacturing system design (Heragu and Chen, 1998). Benders cuts are effective for fixed-charge capacitated network design problem under heavy traffic load (Sridhar and Park, 2000). Benders decomposition approach is a quick approach and an effective approach for solving problems by resource-directive decomposition (Wu, Hartman, and Wilson, 2003).
The application for large-scale problem is as to industries that have many plants and each plant has to ship the products to many customers both production plans and transportation plans must be produced. In the other words, the one-dimensional cutting stock problem (1D-CSP) with multiple cutting facilities and the transportation problem (TP) are integrated. Simply optimizing the cutting operation may not be the best approach as it may result in excessive transportation costs. Therefore, this study considered the 1D-CSP with multiple cutting facilities and the TP, which are intractable because of the large number of variables involved. Both problems are combinatorial problems, which are solvable in non-polynomial time. They are also known as NP-hard problems.
The 1D-CSP can be addressed by finding the best process with the minimum cost to cut stock material (stock) to meet the demand for small pieces (retail size) of prescribed dimensions. The columngeneration technique is successful for solving the large number of variables in the linear programming formulation of the problem (Gilmore and Gomory, 1961;1963). The 1D-CSP is broadly applied in industries such as in the aluminum industry (Stadtler, 1990) and the clothing industry (Gradišar, Jesenko, Resinovič, 1997). Typical cutting stock problems implicitly assume that the stocks are cut at a single facility. Holthaus (2002) raised Decomposition approaches based on the column-generation technique for solving the 1D-CSP with different types of standard lengths or multiple cutting facilities.
The TP is to find the minimum transportation cost for shipping more units from suppliers to customers so that the demands of all the customers are met. Adlakha and Kowaski (1998) demonstrated the practicality of identifying cases where the paradoxical situation exists in the transportation problems. Sharma and Sharma (2000) proposed a heuristic that runs in polynomial time for enhancing the dual solution of the TP.
In this study, situations where there are multiple cutting facilities that cut stock of various dimension into small retail size pieces are considered. These pieces are shipped to various customers at different locations to satisfy a known demand. In general, it is not an optimum approach to decompose this problem by solving the cutting stock problem and then solve the resulting transportation problem for each retail size. Instead, an efficient procedure is required to make joint cutting and distribution decisions be developed to minimize the total cutting and shipping cost.
The global objectives of this study are to develop three heuristic approaches and to raise the most appropriate approach for solving integrated one-dimensional cutting stock problem-transportation problem, (1D-CSP&TP). The specific objective of this study is to generate an integrated mathematical model for 1D-CSP&TP.

Problem formulation
The 1D-CSP&TP can be stated as a distribution system in which there are I suppliers and J customers. Each supplier has known supply of stocks of length L 1 , L 2 , …, L K available. Each customer has a demand for small size items of length l 1 , l 2 , …, l M . The joint cutting stock-transportation problem is to decide the cutting patterns and the associated frequency to be used at each supplier and the number of units of each retail size items to ship from each supplier to each customer to minimize the total cost. Here the total cost includes the cost of stocks used as well as the transportation cost. To develop a mathematical model for this problem, some notations used throughout this paper are defined as follows: i = index for suppliers (i = 1, 2, …, I), j = index for customers (j = 1, 2, …, J), k = index for stock lengths (k = 1, …, K), m = index for retail items (m = 1, …, M), L k = the length of stock k; k = 1, …, K, l m = the length of item m; m = 1, …, M, , 0

Column-Generation Technique
Consider the procedure to apply the column-generation technique to the [1D-CSP&TP] so that at least i kp x pieces of standard length L k are furnished. First, the problem is relaxed in order to provide the simplex multipliers. After that the problem is decomposed to I subprblems, called [NPP i ]: a new-pattern generated subproblem at supplier i. These subproblems applied the column-generation technique to search the essential cutting patterns for the problem.
The model (1)-(6) is rewritten in the relaxation form (see also (7)-(12)) as PRIMAL model. Following the rules to convert from the primal problem to the dual problem, the DUAL model is written in (13)-(18).
To develop the DUAL model, the simplex multipliers used throughout this paper are defined as follows: e ik = the simplex multipliers of supplier i and used stock k from constraints (8), r im = the simplex multipliers of supplier i and item m from constraints (9), and b m = the simplex multipliers of the retail item m from constraints (10). , Subject to , and q k = the used stock length for stock k; q k ∈{0,1}, if q k = 1 then the stock length k is used. Otherwise, the stock length k is not used. Every supplier is solved to obtain the effective patterns. If V ik* = max {V ik | k=1,..., K }> 0 then the new column ., I, the current column is proved to be globally optimal. For a profound description of the column-generation procedure for cutting stock problems, see Lasdon (1970, section 4.1).

Benders Decomposition
Consider the procedure to apply the Benders decomposition to the [1D-CSP&TP] so that a largescale problem would be separated to multi subproblems. The relaxed [1D-CSP&TP] (see also (7) -(12)) is decomposed into i subproblems [SP i ] and a master problem [MP]. The [SP i ] is corresponded to the 1D-CSP, whereas [MP] is corresponded to the TP. The models of them are shown as follows: [SP i ]: Constraints (25) correspond to the limitation of capacities of each standard lengths at supplier i. Constraints (26) correspond to the difference between the amount of the products and the amount of shipped products should greater than or equal to zero. In the other word, the quantity of the products should greater than the quantity of the shipped products from the supplier i to the customer j for all required lengths.
The simplex multipliers of constraints (25) ∑ ∑ ∑ ∑∑ (31) The [SP i ], as known in 1D-CSP with different types of standard lengths and the limitation of capacities, is solved by column-generation technique for exactly solving the continuous relaxation of the problem. The simplex multipliers of Constraints (25) and (26)  To work with hybrid Benders decomposition and column-generation techniques, it is involved with doing loop between the master problem and subproblems. Obviously, two loops; column-generation loop and Benders loop, are overlapped. If any loop gets struck such as; sluggish convergence or occurred degeneracy situation, the algorithm will take too long computation time. Hence, some rules are stipulated; 1) if the different of UB and LB value between adjacent iterations has less than and equal to 0.5 occurred more than 10 times, the program will stop. 2) the number of new generated patterns by the column-generation technique is limited at 6 times the number of required lengths (M).
The first rule is raised for protection the sluggish convergence because the [MP] of Benders decomposition may add the slowly convergent Benders cuts in (30). This rule helps to save the lose computation time; however, it makes the objective value be greater than the optimal solution. The second one is raised for prevent the degeneracy event. If this sub procedure gets struck, Benders approach will also get stuck. This stopping rule is created under the idea that for every iterations of Benders approach, the right hand side values are modified. So some efficient patterns, in which the column-generation technique provided, may not be the efficient patterns in the next iteration. To reduce the computation time, this rule does work and not much effect occurs in the objective value in case using 6*m limitation.
Furthermore, the first rule as mentioned above is one of the stopping rules too. There are two stopping rules in this algorithm; 1) UB=LB, and 2) The number of iterations that UB-LB ≤ 0.5 occurred more than 10 times.

Methodology
In order to solve the [1D-CSP&TP], the concept of this study could be divided into two phases. One, the relaxed [1D-CSP&TP] is solved by the column-generation technique or the hybrid Benders decomposition and the column-generation technique. And two, the integer [1D-CSP&TP] is solved by the direct method and the proposed heuristic method.
In the first phase, the idea is to reduce the number of variables by using the two techniques as mentioned above. Instead of solving the model with all possible patterns, the column-generation approach and the hybrid Benders decomposition and the column-generation provide only essential patterns.
In the second phase, the idea is:-if the optimal solution and the computation time cannot go together, we have to choose which one is the most important. This study aims at the computation time, so the heuristic method is proposed to solve IP. By the way, the direct method also uses to solve IP for computational time comparison.
First of all, the necessary feasible initial-patterns are generated for forming the initial IP. Concern with various lengths of stock and various costs of stock of each supplier, the suitable stock length is selected under that it can cut into all retail sizes and that its length has the lowest price for all suppliers. This procedure is called "Initial-pattern Generator" as described in the steps below.
Initial-pattern Generator Step1: Find the maximum length of the retail item.
Step2: Find the stock length, which is longer than the one in Step1. Ensure that its cost is the cheapest one from all suppliers Step3: Find the maximum number of pieces that the selected stock length from Step2 can cut into each retail item. Step4: Generate the initial patterns by using the numbers from Step3. The relaxed IP is formulated by using obtained initial-patterns. All decision variables are nonnegative values. Now three approaches are designed for solving this problem 7.1 A0 Approach The first approach is called "A0". This approach applied the Column-Generation technique (CG) to generate appropriated patterns more (see also in Section 3). These patterns are necessary to reduce the objective value. The i kp x variables are added more and more until the reduced cost couldn't decrease in the relaxed [1D-CSP&TP]. Profoundly understanding, Lasdon (1970, section 4.1) described the columngeneration procedure for cutting stock problems.
After that, all patterns are gathered for solving Integer Problem (IP) by added nonnegative integer variable constraints. This method is called the direct method. In this study, all approaches are is limited the computational time at 6 hours. The steps of A0 algorithm are as follows:
Step2: Solve IP by the direct method.
Step3: Test for the stopping rule. If the computational time > 6 hours, stop. The otherwise the optimal solutions are found.

A1 Approach
The second one is called "A1". This approach applied the Column-Generation technique (CG) to generate appropriated patterns more. These patterns are necessary to reduce the objective value. The i kp x variables are added more and more until the reduced cost couldn't decrease in the relaxed . After that, all patterns are gathered for solving IP by the proposed heuristic method. The steps of A1 algorithm are as follows:
Step1: Apply CG into Step0. Step2: Solve IP by the proposed heuristic method.
Step3: Test for the stopping rule. If the computational time > 6 hours, stop. The otherwise the optimal solutions are found.

A2 Approach
The last one is called "A2". This approach applied the Hybrid Benders decomposition (see also in Section 4) and the Column-Generation technique (HB&CG) to generate patterns and cutting plane simultaneously. Therefore, the new i kp x variables and the new constraints are added more and more until meet the optimal solutions. After that, all patterns are gathered for solving IP by the proposed heuristic method. The steps of A2 algorithm are as follows:
Step2: Solve IP by the proposed heuristic method.
Step3: Test for the stopping rule. If the computational time > 6 hours, stop. The otherwise the optimal solutions are found.
To comparison the efficiency of each approach, the IP computation time and the relative-different percentage are raised for judgment. Due to three-approach observation, the IP computation times are collected. Nevertheless, only two approaches with the proposed heuristic method: A1&A2 are considered to compute the relative-different percentage. Let R1&R2 be the relative-different percentage of the approach A1&A2 respectively. The formulas for computing the relative-different percentage of each approach are as follows: where IP1 represents the objective value, solved by A1. application (see also in Section 9) demonstrates the way to design the production plan and the transportation plan. This application stated at I = 2, J = 9, K = 3, and M = 50. The name of the special application is P [2,9,3,50]. Totally, there are 64+1 = 65 instances in this study. In order to generate all instances, the problem generator is coded with the feasible linear problem guaranty. The total length of all suppliers is greater than 1.25 times of the total need length of all demands. The assumptions of the problem generator are the longer stock, the higher price and adequate quantities of stocks for distributions. The algorithm is shown as follows: Problem Generator Algorithm Step0: Input a seed number and the parameters I, J, K, and M.
Step1: Randomize K standard stock lengths (L k ). Then sort by descending.
Step2: Randomize K costs of the standard stock lengths ( i k g ), then sort by descending, and match them with L k from Step 1.
Step3: If i ≤ I then repeat Step 2.
Step4: Randomize the retail items (l m ) from 1 to the longest L k .
Step5: Randomize D mj , m ij c , and i k R .
Step6: Check the feasibility of the problem.
6.1) Classify l m into K groups where the longest l m in each group is less than or equal to L k . 6.2) If 1.1 times the total needed length of group k is less than the total stock length of group k, the capacity will be added up until the total stock length is greater than or equal to 1.1 times the needed length. 6.3) Consider the overall, if 1.25 times the total needed length then the longest L k of the first supplier will be added up until the total stock length is greater than or equal to 1.25 times the total needed length.

Results
Of the three approaches just described, 64 instances have been tested on [1D-CSP&TP]. All instances were put in order of I, J, K and M. The computational performances of all instances were shown in Appendix Table 1.
A0: There were 17 instances, in which took the computation time more than 21,600 seconds (> 6 hours). From now on, only 47 instances were considered. To compare with A0 and A1, we found that there were 34 instances, in which took the computation times less than those of A1 with the maximum gap 10 seconds, whereas the 13 instances left took the computation times greater than those of A1 with the maximum gap 18,789 seconds. To compare with A0 and A2, we found that there were 43 instances, in which took the computation times less than those of A2 with the maximum gap 7,016.76 seconds, whereas the 4 instances left took the computation times greater than those of A2 with the maximum gap 18,689.28 seconds. A1: 64 instances could solve within 219.855 seconds. The computation time of A1 were less than those of A2 for all instances with the maximum gap 55,377.8 seconds.
A2: 63 instances could solve within 10,726.3 seconds, but the left one took more than 21,600 seconds (or > 6 hours). The computation time of this case, P[3-4-3-35], went up to 55,425.2 seconds. The computation time of A2 did not direct-change with the computation time of A0 and A1. For example, P[2-4-3-20]: the computation times of A0, A1, and A2 were 11. 878, 12.849, and 291.18, respectively, whereas P[2-4-3-25]: the computation times of A0, A1, and A2 were 18,802.7, 13.699, and 113.422. Note: All computation times of 64 instances by A1 were less than those of A2 with the maximum gap 55,377.8 seconds. It is obvious that A1 could solve the [1D-CSP&TP] within the reasonable time. This approach was fast enough for the processors who would like to design some proficient production and transportation plans.
From the table in the appendix, the maximum relative-different percentage of R1 was 0.140256%, whereas the maximum relative-different percentage of R2 was 11.41063% (see also Figure 6). There were 37 instances that the percentages of R1 were less than those of R2 with the maximum gap 11.41058%, 18 instances that the percentages of R1&R2 were both equal and 9 instances that the percentages of R1 were greater than those of R2 with the maximum gap 0.05163%. In summary, the approach A1 could solve [1D-CSP&TP] with the relative-different gap 0.140256%, whereas the approach A2 could solve [1D-CSP&TP] with the relative-different gap 11.41063%. For planning the production and transportation strategies, the processors need to know the relative-different percentage as less as any approach could solve. With the reasonable computation time, A1 is an alternative approach to solve [1D-CSP&TP].

Conclusion and Extensions
In this study, three approaches: A0, A1, and A2, are proposed. From the experiments, A1 is the most efficient approach for solving large scale problems. This approach can solve [1D-CSP&TP] in the reasonable computation time; in addition, the range of the relative-different percentage of this approach is quite narrow too. Moreover, the solutions that obtained from A1 are very useful for the processors to design the proficient production and transportation plan simultaneously.
However, three approaches are the alternative approaches to solve [1DCSP&TP]. They have their own advantages and disadvantages. Each approach can be discussed as follows: For A0: This approach could not conduct the optimal IP objective value and frequently could not solve the problem in the reasonable time. As to the column-generation technique, this technique obtained only the optimal LP objective value. It is not guaranteed that the IP objective value will also be optimal. Although the IP objective value of A0 is not optimal, but with the direct method for solving the IP, A0 obtained the best IP objective value. And from the experiments, 34 out of 64 instances took the computation times less than those of A1. Therefore, if the problems are not too tough to solve, A0 is the best alternative approach in which obtained the best IP objective value within the reasonable time.
For A1: Again, this approach could not conduct the optimal IP objective value. As to the columngeneration technique, this technique obtained only the optimal LP objective value, not IP objective value. With the proposed heuristic method, this approach could solve the large-scale problems within a reasonable time. But the IP objective value of A1 is worse than the one of A0. However, the experiments show that all instances could solve within 219.855 seconds (or less than 4 minutes) and the range of the relative-difference percentage of A1 is quit narrow (less than 0.15%).
For A2: This approach could not conduct the optimal LP objective value for all problems. There are one rule for the degeneracy protection in the column-generation technique and two stopping rules in this algorithm. Hence, the obtained LP objective value by A2 will be the best if the operation runs step by step without jumping with the rule for the degeneracy protection and stop with the first stopping rule (UB=LB). Otherwise, the obtained LP objective value by A2 will be not optimal. Furthermore, the column-generation technique was applied in benders iteration. For example, suppose A2 obtained the LP objective value after 10 Benders iterations. It means the column-generation technique was applied 10 times too. Therefore, the computation time of A2 takes more than the computation time of A1. However, the IP objective values of A2 in some instances were less than those of A1. That means this approach is not all worse. Maybe there is another algorithm that could manage this approach better.
For the further study, the algorithm for managing A2 will be developed. Since the [1D-CSP&TP] considered is NP-hard, there is a need to develop heuristic approaches to solve it. Nonato and Scutellà (1998) proposed a linear programming algorithm based on hyper flows for solving the one-dimensional cutting stock problem but not limited the number of stocks. Therefore, their algorithm will be applied to a new alternative approach for solving the [1D-CSP&TP] in the future study.