One-Dimensional Cutting Stock Problem with Cartesian Coordinate Points

: The cutting stock problem is used in many industrial processes and recently has been considered as one of the most important research topics. It is basically describes in two ways, One –dimensional and Two-dimensional Cutting Stock Problems (CSP). An optimum cutting stock problem can be defined as cutting a main sheet into smaller pieces while minimizing the total wastage of the raw material or maximizing overall profit obtained by cutting smaller pieces from the main sheet. In this study, One-dimensional CSP is discussed. Modified Brach and Bound algorithm for One-dimensional cutting stock problem is coded and programmed in the Matlab programming environment to generate feasible cutting patterns. At the same time, Cartesian coordinate points are derived from the developed algorithm. In our approach, the case study is pertained to the real data used at L.H. Chandrasekara & Brothers (Pvt Ltd) in Sri Lanka for its production


Introduction
Efficiency of the production process has become a key factor of the success in today's competitive manufacturing environment in industries. Productivity can be improved by minimizing wastage, lead time and hence reducing the cost of production. Throughout the process of production, wastage can occur in many ways at any stage of the production process. Cutting stock problem can be defined as optimizing the raw material wastage. Therefore, techniques in Operations Research play a major role in minimizing wastage of raw material or maximizing usage of the raw material. In early days, in order to achieve the goal of (1D-CSP), researchers used different methodologies. The linear programming based procedure was the first approach which may be traced back to the column generation procedure proposed by Gomary (1961, 1963). Moreover, Sequence Heuristic Procedure (SHP) described by Haesseler (1971) and then Hybrid Solution Procedure which is a combination of the LP based procedure and SHP or some other procedures (Haesseler et al., 1991). Basically CSP is describes in two ways, One-dimensional and Twodimensional CSPs and it can be defined as cutting a raw material which is available in stock into smaller required items, which are known in size while minimizing the total wastage of the raw material or maximizing the overall profit obtained by cutting smaller items from the main sheet. Many researchers have worked on the CSP and developed different algorithms to solve the problem. Among them, the very first formulation of the CSP was formulated by Kantorovinch in 1939, although it was published 1960 (Kantorovinch, 1960). Gilmore et al [1] has proposed the linear programming approach, which solves the one-dimensional cutting stock problem. Unlimited number of raw materials with different lengths are available in stock is a one of assumptions of this study. A mathematical model has been developed to minimize the total cost of the raw material. Column Generation Algorithm has been developed to generate feasible cutting patterns. As a result of increasing the number of cutting items, the number of cutting patterns also increases. Linear programming problems which arise from cutting stock problems become intractable because of the large number of variables are involved (i.e. Linear programming technique is not applicable to solve mathematical model with too many variables). Gilmore [2] has made an approach for one-dimensional cutting stock problem as an extended of early study and cutting stock problem has been described as a NP-hard problem 1 . Gilmore [2] has described a new and faster Knapsack method, and mathematical formulation changes. The following formulation changes have been discussed; limitation on the number of cutting knives available, balancing of multiple machine usage when orders are being filled from more than one machine, and introduce a tolerance in the customers' orders. Gilmore has explained the procedure of Knapsack Method using a test problem 2 .
Addition to that, Morbito [3] has reported the problem of cutting rectangular plates into smaller ones in Brazilian hardboard industry. The problem was determine the best patterns to be cut by an automated machine composed of a set of circular saws, device to move and hold the plates and loading and unloading stations. A particular two-phase column generation procedure was described for the cutting stock formulation of hardboard industry. Each phase of the procedure was modeled as an integer program and solved by two alternative methods. The first was based on dynamic programming and the second was a simple extension of the algorithm presented in Gilmore [2]. Also, Saad M. A. Suliman [4] has developed a simple pattern generating procedure for solving the auxiliary problem which solves the one-dimensional CSP to minimize the trim loss or minimize the total amount of stock material. Branch and Bound Algorithm has been modified to find feasible cutting patterns for one-dimensional CSP and a search tree is used to develop the pattern generation method. Modified Branch and Bound Algorithm is illustrated using a case study, where standard steel coil which is 130 cm in length is cut into four different lengths of coils with similar width of main coil 4 . The onedimensional Cutting Stock Problem becomes more difficult when there are a large number of objects of different sizes available. A similar problem is addressed by Belov [5], and a method based on combination of some enumerative methods is proposed. In this work the more general case was studied where different stock types with supply bounds are available. The total material cost minimization has been modeled as an integer linear optimization problem 5 . Puchinger [6] has described a combined genetic algorithm/branch & bound approach for solving a real world glass-cutting problem. The Genetic Algorithm uses an order-based representation, which is decoded using a greedy heuristic. The Branch & Bound algorithm was applied with a certain probability enhancing the decoding phase by generating locally optimal subpatterns. Reported results indicate that the approach of occasionally solving sub patterns to optimality may increase the overall solution quality 6 . Fernandez [7] has presented a method to find Integer Solutions to One-dimensional CSP to minimize the trim loss or to minimize the number of master rolls needed to satisfy the orders. Two Integer linear programming models for the One-dimension CSP with different objective cost functions have been considered. Fernandez has studied an approach based on the classical column-generation procedure by Gilmore and Gomary for solving the linear programming (LP) relaxations. Also to obtain an integer solution, a final integer CSP after using an extra column-generation procedure has been solved 7 .
In addition to above studies, Rodrigo [8] has presented modified Branch and Bound Algorithm and a computer program using Matlab software package to generate feasible cutting patterns. As a case study, 1200 cm long steel bars are considered to be cut into eight types (different lengths) of pieces according to the customer requirements. Optimum cutting schedule has been presented by solving the integer linear programming problem which is given in the paper 8 . Rodrigo [9] has proposed a method to discover the feasible cutting patterns for One-Dimensional Cutting Stock Problem with varying dimensions. Pattern Generation Technique has been used to nest the pieces of cutting items within the main sheet by minimizing the raw material wastage. Computer program using Matlab software package has presented to generate feasible patterns. In the case study, three different paper rolls of widths 1000mm, 500mm and 800mm are used as raw materials to cut different sizes of paper items according to the given specifications 9 . In addition to that Rodigo [10] has developed an algorithm based on Modified Branch And Bound to generate the cutting patterns for twodimensional triangular shaped cutting items. As a case study, four different sizes of triangular shaped items with their demands are selected to cut from a raw material with known dimensions 10 .

One-Dimensional (1D) Cutting Stock Problem
In most industries, the cost of the materials consists of a high percent of total price (More than 80%). The CSP is one of the most famous operation research problems which are defined to improve the quality of using materials. This CSP can be described as follow; there is a collection of rectangular shaped stock materials with different widths (but with the same length L) that we are going to divide them into smaller pieces of desired widths (with the same length L). First, the desired widths has been arranged in descending order ( , , … ..) and then divide the original sheets into small pieces having the width-as much as possible. After then if there is a loss raw material, we go to the next required width-and divide that loss into small pieces having the width-. By continuing the same procedure up to the smallest required width we can generate all the cutting patterns and finally calculate the minimum trim loss.

Mathematical Model (Gilmore and Gomary, 1961)
Any firm's main objective is to maximize the annual contribution margin accruing from its production and sales. By reducing wastages and maximizing sales, productivity can be improved. Wastage can occur in many ways and at any step of production line cutting stock problem can be described under the raw material wastage. According to the selection, a mathematical model to minimize the raw material wastage is formulated as follows:
Step 4:Set 1 1 Where W k is the length of the k th main sheet. Here, ijk a is the number of pieces of the th i item in the th j pattern along the width of the k th main sheet and [ ] y     is the greatest integer less than or equal to y.
Step 5: find the cut loss resulting from cut pattern 1.

Case Study
Modified branch and bound algorithm is tested to determine feasible and optimal cutting patterns.
A paper company (Chandrasekara Group Of Companies in Sri Lanka) uses Bank papers of widths 80cm and 60cm with same length (90cm) as raw materials to cut different sizes paper items according to the given specifications. The company has received an order for papers with the following specifications: Bank Papers are produced by the company in two standard widths; 60cm and 80cm that are slit to the sizes specified by the orders.

Results and Discussion
Modified Branch and Bound Algorithm is used to the above example to generate feasible cutting patterns as given below; Step 1: For = 1, 2 widths of rawmaterials = 80cm and 60 cm.
Step 4: Allocating cutting items for cutting pattern 1, Item 1 is selected and nested along the width of the raw material.
Following equation states the maximum number of pieces that can be allocated for Item 1.
Residual width of the raw material calculated by the formula bellow: Residual width = ( ) where 2 w is the width of the Item 2. Then Item 2is nested within the residual width of the raw material.
Nesting of Item 2 is as follows: This process continues till the last cutting item (Item 5) while calculating residual width of the raw material.

Nested cutting items in
Step 2 is defined as first cutting pattern Set pattern1 Step 5: Total cut loss Cutloss is calculated as follows: Step 6: At the end of each pattern, Seti= n-1 (> 0). Set i = 4.
Step 7: Value of a ikj is checked. If a ikj = 0, setting i = i-1 and again value of a ikj is checked. When a ikj > 0, set j = j+1 and determine the number of pieces of each cutting item along the length of the raw material. Nested cutting items of pattern 2 as follows: Step 5: Total cut loss = 6 − 7 5

= 0 9
Repeat the procedure until i = 5 (n). This process ends up with 44 different cutting patterns to allocate 5 items which is exhibited in the Table 2.  There are 44 feasible cutting patterns available to cut raw materials with the dimensions 80cm and 60cm into required items. By considering the standard lengths of the main sheets (length of the main sheets are same (90cm)) and the other items; (No. of items* -number of items that can be cut from the relevant strip after cutting the main sheet width-wise). Then the total number of cutting patterns can be listed as follows:

Conclusion
In this study, a cutting stock problem is formulated as a mathematical model based on the concept of cutting patterns. As shown in the Table 2, forty four feasible cutting patterns were generated and after solving the model, only four patterns were selected as given in Table 5 to cut the main sheets according to the requirements. In this case study, the plant assumes that all extra pieces from each item were stocked for the next production. Since the company cuts only one item from one main sheet, they need minimum 500 main sheets from 80 × 90cm 2 and 1642 main sheets from 80 × 90cm 2 to satisfy the given demand. According to our process we need only 250 main sheets from 80 × 90cm 2 and 1917 main sheets from 80 × 90cm 2 to satisfy the demand. The total cutloss of the company= 2525200cm 2 Rate of utilized area (of the company)= 79.77% Total cutloss (obtained from our model) = 2208484.6cm 2 Rate of utilized area=81.83% (obtained from our model) Clearly we can see that the optimum solution we obtained through our process is more profitable than the existing one.