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One dimensional cutting stock problem  1-d-csp_ with second order sustainable
 

One dimensional cutting stock problem 1-d-csp_ with second order sustainable

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    One dimensional cutting stock problem  1-d-csp_ with second order sustainable One dimensional cutting stock problem 1-d-csp_ with second order sustainable Presentation Transcript

    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 136 ONE-DIMENSIONAL CUTTING STOCK PROBLEM (1D-CSP) WITH SECOND ORDER SUSTAINABLE TRIM: A COMPARATIVE STUDY WITH FIRST ORDER SUSTAINABLE TRIM P. L. Powar1 , Vinit Jain2 , Manish Saraf3 , Ravi Vishwakarma4 1 Dept. of Math. & Comp. Sc., R. D. University, Jabalpur 482001, India 2 KEC Int. Company, Panagar, Jabalpur, 482001, India 3 HCET, Dumna Airport Road, Jabalpur, 482001, India 4 Dept. of Math. & Comp. Sc., R. D. University, Jabalpur 482001, India ABSTRACT A method for solving one-dimensional cutting stock problem (1D-CSP) with first order sustainable trim has been studied extensively by many researchers of Economics, Computer Science and Mathematics. The authors have already defined the first order sustainable trim and in this paper by using the second order weighted means of order lengths and demand, a second order sustainable trim has been defined. The cutting plan consists of cutting of at most two order lengths at a time out of the required set of n order lengths ݈ଵ , ݈ଶ, … , ݈௡ from a given set of m stock lengths ܷଵ, ܷଶ, … , ܷ௠ which resolves the problem of space constraint as well as minimization of men power significantly. The main objective of this paper is to study the impact of two different definitions of first and second order sustainable trims on total trim loss for the cutting of same set of data with respect to same pattern of cutting. Keywords: First order sustainable trim, Second order sustainable trim, 1D-CSP, Non-negative integral valued (NIV) linear combination. AMS (2000) subject classification: 90C90; 90C27; 90C10. 1. INTRODUCTION The One-Dimensional stock materials input is a very important criterion in industrial cutting operations. Several cutting plans (cf. [1], [2]) have been designed to obtain required set of pieces from the available stock lengths. The fundamental aim is to minimize the quantity of used stock material or to minimize the wastage. The combination of assortment problem and the trim loss problem is known as the cutting stock problem (CSP). INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & TECHNOLOGY (IJCET) ISSN 0976 – 6367(Print) ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), pp. 136-149 © IAEME: www.iaeme.com/ijcet.asp Journal Impact Factor (2013): 6.1302 (Calculated by GISI) www.jifactor.com IJCET © I A E M E
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 137 The analytic method of optimization proposed by Gilmore and Gomory in 1960’s has turned non-practicable due to sufficiently large number of possible arrangement that render the solution impossible and of no use because of its non-integral solutions (cf. [3]-[6]). Thus, instead of using the analytic methods to obtain the ideal solution, heuristic approaches with acceptable approximation have gained popularity (see [7], [8], [9], [10]). By using the principles introduced by Dikili [11], Dikili et al [11] developed a method to solve 1D-CSP which completely removes the complexity of Gilmore and Gomory method. Using genetic approaches with and without contiguity, Hinterding and khan [12] have studied 1D-CSP. Wagner [13] have studied 1D bundled CSP with contiguity in the lumber industry. In the classical CSP, one wants to minimize the number of stock items used while satisfying the demand of smaller sized items. However, the number of patterns/set ups to be performed on the cutting machine is ignored. In cutting stock problem, with setup cost (CSP-S), considering cost factors for the material and the number of set ups, the total production cost has been minimized in [14]. The main objective of present paper is to minimize the production cost by reducing the area of working and men power. The cutting plan considered in this paper is already proposed in [8] which consist of cutting of at most two order lengths at a time out of the required set of n order lengths ݈ଵ, ݈ଶ, . . . , ݈௡ from a given set of m stock lengths ܷଵ, ܷଶ, … , ܷ௠. This plan resolves the problem of sorting sufficiently large number of order lengths (approximately more than one thousand) after each stage of cutting and keeping them in the form of heaps till the entire process of cutting is over. Our study is based mostly on the problems of transmission tower manufacturing industry. To control the scrap or the trim loss is one of the basic factor for the sustainability of any industry dealing with cutting of smaller lengths from the given large stock lengths. Powar et al [8] have resolved this problem upto some extent by designing the cutting plan which works under the pre-defined sustainable trim of order one. The mathematical model introduced in [8] involved the classification of data and some recurrence relations. It is quite clear that the computation of total trim loss is data dependent and the sustainable trim of order one defined in [8] is also data dependent which works nicely for some specific set of data. In the present paper, we have defined a sustainable trim of order two by considering the second order weighted means of order length and demand. The impact of these two definitions viz. sustainable trim of order one and two has been explored widely on certain sets of data. It has been noticed that the second order definition of sustainable trim is more effective in some cases to minimize the total trim. Observations and conclusion cover the most important part of this work from practical point of view. 2. NOTATIONS AND PRE-REQUISITES All stock lengths and order lengths, we consider as integers throughout our analysis. According to the requirement, the lengths can be converted into integers by multiplying them by 10௡ ሺ݊ ൒ 1, integerሻ. We use the following notations: ‫ܤ‬ሺ݊ሻ െ Block of integers 0,1, … , ݊ (index set), ݆ ‫א‬ ‫ܤ‬ሺ݊ሻ means ݆ can be any number from the set ሼ0, 1, 2, … , ݊ሽ. ݈௜ െ Order lengths ݅ ൌ 0, 1, 2, … , ݊ arranged in ascending order with respect to length and ݈଴ ൌ 0 by convention. ݀௜ െ Required number of pieces of order length ݈௜, ݀଴ ൌ 0. ܷ௝ െ Stock length ሺ݆ ൌ 1, 2, … , ݉ሻ arranged in ascending order with respect to length. It has been noticed that in particular, in the transmission tower designing industry that most of the required number of order lengths i.e. ݀௜′s are integral multiple of each other. In view of this observation, we classify the order lengths in the following two categories in accordance with their required number of pieces:
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 138 Category I: (C-I) We collect all those order lengths whose required number of pieces are integral multiple of each others. Category II: (C-II) It is the collection of all those order lengths whose required number of pieces are prime numbers (their common multiple is 1). 3. SUSTAINABLE TRIM (ORDER ONE AND TWO) In this section, we give the definition of sustainable trim of order one described in [8] and propose a new definition of sustainable trim of order two. 3.1 Sustainable trim of order one (࢚࢙ ૚ ) In order to cut the linear combination ܽ௜௝ (say) of the two order lengths ݈௜ and ݈௝ from the given stock lengths ܷଵ, ܷଶ, … , ܷ௠, we have to decide upto what extent, we allow the raw material to convert into the scrape. Throughout our cutting process (excluding the last step where it is possible only that few piece of some order length are left to cut), we follow the restriction that 0 ൑ ܷ௞ െ ܽ௜௝ ൑ ‫ݐ‬௦ ଵ , ݇ ൌ 1,2, … , ݉ and ‫ݐ‬௦ ଵ is the sustainable trim of order one and defined as follows: ‫ܮ‬ ൌ ݈଴݀଴ ൅ ݈ଵ݀ଵ ൅ ‫ڮ‬ ൅ ݈௡݀௡ ݀଴ ൅ ݀ଵ ൅ ‫ڮ‬ ൅ ݀௡ ൌ ∑ ݈௜݀௜ ௡ ௜ୀ଴ ∑ ݀௝ ௡ ௝ୀ଴ We next define ‫ܮ‬௞ ൌ |ܷ௞ െ ݅‫|ܮ‬ (݇ ൌ 1, 2, … , ݉ and ݅ is an appropriate positive integer ൒ 1, for which ‫ܮ‬௞ is minimum) where ܷଵ, ܷଶ, … , ܷ௠ are the stock lengths. We finally define ‫ݐ‬௦ ଵ ൌ ∑ ௅ೖ ೘ ೖసభ ௠ (3.1) which is the desired sustainable trim of order one. 3.1 Remark Analytically, it has been noticed that the average value covers the acceptable over all original values. Hence, we have taken the weighted mean of total required lengths. 3.2 Sustainable trim of order two (࢚࢙ ૛ ) Following the same restriction as for ‫ݐ‬௦ ଵ and using the notations from section 2, we define ܵ௡ ൌ l଴d଴ ൅ ‫ڮ‬ ൅ l୬d୬ d଴ ൅ dଵ ൅ ‫ڮ‬ ൅ d୬ ൌ ∑ ݈௜݀௜ ௡ ௜ୀ଴ ∑ ݀௝ ௡ ௝ୀ଴ By convention, ܵ଴ ൌ 0, ܵଵ ൌ ݈ଵ, ܵଶ ൌ ௟బௗబା௟భௗభା௟మௗమ ௗబାௗభାௗమ , … , ܵ௡ ൌ ∑ ௟೔ௗ೔ ೙ ೔సబ ∑ ௗೕ ೙ ೕసబ We next define the second order weighted means ܵ௡ ଵ ൌ ܵ଴ ൅ ܵଵ ൅ ‫ڮ‬ ൅ ܵ௡ ݊ ൅ 1 . Consider ‫ܮ‬௞ ൌ |ܷ௞ െ ݅ ܵ௡ ଵ|, ݇ ൌ 1,2, … , ݉. ݅ is an appropriate positive integers ൒ 1, for which ‫ܮ‬௞ is minimum. ܷଵ, ܷଶ , … , ܷ௠ are stock lengths. 3.2 Remark ܵ௡ ଵ is the average order length which is assumed to be cut from the given stock length ܷ௝ሺ݆ ൌ 1, … , ݉ሻ. The integer ݅ denotes the number of pieces of average stock length to be cut from the stock length ܷ௞.
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 139 We finally define ‫ݐ‬௦ ଶ ൌ ∑ ௅ೖ ೙ ೖసభ ௠ (3.2) which is the sustainable trim of order two viz. ‫ݐ‬௦ ଶ . 4. MATHEMATICAL FORMULATION OF THE PROBLEM We first consider C-I and define the following ratios: ௗ೔ ௗೕ ൌ ఈ೔ൈ௥೔ೕ ఉೕൈ௥೔ೕ (4.1) (where ‫ݎ‬௜௝ is a positive integer ݅, ݆ ‫א‬ ‫ܤ‬ሺ݊ሻ, ݅ ൐ ݆ ) Note: It is not necessary to consider always the largest common factor between ݀௜ and ݀௝. Any other factor ‫ݎ‬௜ (if exists) may be selected according to the length of stock to minimize the trim. In view of (4.1), define the following set: ‫ܣ‬ ൌ ሼܽ௜௝ ൌ ߙ௜݈௜ ൅ ߚ௝݈௝ ‫׷‬ ܽ௜௝ ൑ ܷ௠, ߙ௜, ߚ௝ ൒ 0, integer ൫݅ ൏ ݆, ݅, ݆ ‫א‬ ‫ܤ‬ሺ݊ሻ൯ሽ (4.2) We are now in a position to define the sets ‫ܣ‬௞ ‫ك‬ ‫ܣ‬ ሺ݇ ൌ 1, 2, … , ݉ሻ as follows: ‫ܣ‬௞ ൌ ሼܽ௜௝: 0 ൑ ܷ௞ െ ܽ௜௝ ൑ ‫ݐ‬௦ ఒ , ݇ ‫א‬ ‫ܤ‬ሺ݉ሻ, ݅, ݆ ‫א‬ ‫ܤ‬ሺ݊ሻ, ܽ௜௝ ‫א‬ ‫,ܣ‬ ߣ ൌ 1,2ሽ (4.3) where ‫ݐ‬௦ ఒ is defined by (3.1) and (3.2) respectively for ߣ ൌ 1 and λ ൌ 2. At this stage, we may come across with the following situations: • ‫ܣ‬௞ ൌ ߶, ‫׊‬ ݇ ‫א‬ ‫ܤ‬ሺ݉ሻ, in this case, all the order lengths have to shift in C-II. • In view of the definition of ‫ݐ‬௦ ఒ , the sets ‫ܣ‬௞ ሺ݇ ‫א‬ ‫ܤ‬ሺ݉ሻሻ may or may not cover all order lengths belonging to Category-I. In view of above observations and the definition of the sets ‫ܣ‬௞, we redefine our categories I and II as follows: Category-I (C-I) Let ݈ఈଵ , ݈ఈమ , … , ݈ఈ௣ order lengths have been covered by the sets ‫ܣ‬௞ ሺ݇ ൌ 1, 2, . . , ݉ሻ. For convenience, we denote these order lengths by ݈ଵ, ݈ଶ, … , ݈௣ arranged in ascending order with respect to the length. 4.1 Remark There may exist some order lengths ݈௜ and ݈௝ (say) such that ݀௜ and ݀௝ ofcourse are multiple of each others but the length of combination ܽ௜௝ exceeds the largest stock length ܷ௠ or ሺܷ௝ െ ܽ௜௝ሻ exceeds the sustainable trim loss. We shift all such order lengths to Category-II and finally, we assume that the order lengths ݈ଵ, ݈ଶ, … , ݈௣ have been covered by Category-I. Category-II (C-II) The remaining all order lengths ܰ ൌ ݊ െ ‫݌‬ denoted by ݈ଵ, ݈ଶ, … , ݈ே arranged in ascending order with respect to the length. 4.2 Remark (i) The real numbers ‫ݐ‬௦ ఒ ሺߣ ൌ 1, 2ሻ defined by ሺ3.1ሻ and ሺ3.2ሻ play a crucial role in the computation of total trim loss. It is natural to expect that the trim loss can be minimized by considering the minimum value lying between 0 and ‫ݐ‬௦ ఒ , but it has been experienced practically in the industries that by increasing the value of ‫ݐ‬௦ ఒ , the impact on the total trim loss results in a significantly acceptable range in some particular cases. But we are strict to ‫ݐ‬௦ ఒ only.
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 140 (ii) In order to implement the algorithm smoothly, the data of more than one tower (preferably of same pattern) may be clubbed. Now consider Category-II and order lengths ݈ଵ, ݈ଶ, … , ݈ே with the required number of pieces ݀ଵ, ݀ଶ, … , ݀ே respectively. For ݅ ് ݆, ݅, ݆ ‫א‬ ‫ܤ‬ሺܰሻ, define: ݀௜ ൌ ݊ଵߙ௜ଵ ൅ ݀௜ଵ (4.4) ݀௝ ൌ ݊ଵߚ௝ଵ ൅ ݀௝ଵ (4.5) 0 ൑ ܷ௞ െ ൫݈௜ߙ௜ଵ ൅ ݈௝ߚ௝ଵ൯ ൌ ሺ‫ݓ‬௞ sayሻ ൑ ‫ݐ‬௦ ఒ ሺߣ ൌ 1, 2ሻ for at least one value of k (k=1,2,…,m). The number ݊ଵ has been chosen in such a way that ‫ݓ‬௞ attains a minimum value lying between 0 and ‫ݐ‬௦ ఒ . Similarly, choose a number ݊ଶ satisfying the following condition: ݀௜ଵ ൌ ݊ଶߙ௜ଶ ൅ ݀௜ଶ (4.6) ݀௝ଵ ൌ ݊ଶߚ௝ଶ ൅ ݀௝ଶ (4.7) Proceeding this way, we finally define ݀௜,௦ିଵ ൌ ݊௦ߙ௜௦ ൅ ݀௜௦ (4.8) ݀௝,௦ିଵ ൌ ݊௦ߚ௝௦ ൅ ݀௝௦ (4.9) The process would be continued till either ݀௜௦ ൌ 0 or ݀௝௦ ൌ 0 and in view of (4.4) - (4.9), we have ݀௜ ൌ ∑ ݊௞ߙ௜௞ ൅ ݀௜௦ ௦ ௞ୀଵ ߙ௜௞ ൐ ߙ௜,௞ାଵ (4.10) ݀௝ ൌ ∑ ݊௞ߚ௝௞ ൅ ݀௝௦ ௦ ௞ୀଵ ߚ௝௞ ൐ ߚ௜,௞ାଵ (4.11) ݀௨௦ ൌ ݊௦ାଵߜ௨,௦ାଵ ൅ ݀௨,௦ାଵ ሺ݀௨,௦ାଵ ൏ ߜ௨,௦ାଵሻ (4.12) where ‫ݑ‬ ൌ ݅ or ݆, ߜ ൌ ߙ or ߚ for ݅ or ݆ respectively. Also ݊௞, ߙ௜௞, ߚ௝௞ are positive integers, may be selected according to the length of stock in order to minimize the trim. Referring relation (4.10)-(4.12), we now define the set ‫ܤ‬ ൌ ሼܾ௜௝ ௥ , ܾ௨ ௦ାଵ , ܾ௨ ௦ାଶ : ܾ௜௝ ௥ ൌ ߙ௜௥݈௜ ൅ ߚ௝௥݈௝, ܾ௨ ௦ାଵ ൌ ߜ௨,௦ାଵ݈௨, ܾ௨ ௦ାଶ ൌ ݀௨,௦ାଵ݈௨ where ܾ௜௝ ௥ , ܾ௨ ௦ାଵ , ܾ௨ ௦ାଶ ൑ ܷ௠, ‫ݑ‬ ൌ ݅ ‫ݎ݋‬ ݆, ߜ ൌ ߙ ‫ݎ݋‬ ߚ according to ‫ݑ‬ ൌ ݅ ‫ݎ݋‬ ݆ repectively, ‫ݎ‬ ൌ 1, 2, … , ‫.ݏ‬ ݅, ݆ ൌ 0, 1, … , ܰሽ (4.13) Define |c୧| ൌ max c୧୨ , where c ൌ a or b for fixed ݅ and arbitrary ݆ ܿ௜௝ ൑ ܷ௠ (4.14) In view of relation (4.13), we now define ‫ܤ‬௞ ௥ ൌ ሼܾ௜௝ ௥ , ܾ௨ ௦ାଵ , ܾ௨ ௦ାଶ : 0 ൏ หܷ௞ െ ൫ܾ௜௝ ௥ |ܾ௨ ௦ାଵ|ܾ௨ ௦ାଶ ൯ห ൏ ‫ݐ‬௦ ఒ ሺߣ ൌ 1, 2ሻሽ (4.15) ‫ݎ‬ ൌ 1, 2, … , ‫.ݏ‬ 5. CUTTING PLAN It has been noticed practically that with the preference of starting from the largest order lengths to the smaller ones, the cutting process has been executed in general as the smaller order lengths left behind can be adjusted easily amongst them and results in less trim loss (see Figure 7.1).
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 141 • Cutting of the largest order length ࢒࢖ from category-I Referring relation (4.14), we consider |ܽ௣|. In view of ‫ܣ‬௞ [cf. relation (4.10)], there exist sets ‫ܣ‬௤, ‫ܣ‬௥, ‫ܣ‬௦, … ሺ1 ൑ ‫,ݍ‬ ‫,ݎ‬ ‫,ݏ‬ … ൑ ݉ሻ containing |ܽ௣| along with some other ܽ௜௝’s. Corresponding to each set ‫ܣ‬௤, ‫ܣ‬௥, ‫ܣ‬௦, … respective fixed stock lengths ܷ௤, ܷ௥, ܷ௦, … have been assigned. We select the combination ܽ௣௤ corresponding to the smallest stock length ܷ௣ and focus our attention on it for the first step of cutting. Let หܽ௣ห ൌ ܽ௣௤(say) for ‫ݍ‬ ‫א‬ ‫ܤ‬ሺ‫݌‬ሻ~௣ where ܽ௣௤ ൌ ߙ௣݈௣ ൅ ߚ௤݈௤ satisfying the condition: ௗ೛ ௗ೜ ൌ ఈ೛ൈ௥೛೜ ఉ೜ൈ௥೛೜ (5.1) In view of (5.1), it may be noted that by cutting ‫ݎ‬௣௤ bars of stock length ܷ௣, total number of required pieces of order lengths ݈௣ and ݈௤ are cut. Define ‫ݐ‬௣ ଵ ൌ ‫ݎ‬௣௤൫ܷ௣ െ หܽ௣ห൯ ൑ ‫ݎ‬௣௤‫ݐ‬௦ ఒ ሺߣ ൌ 1, 2ሻ (5.2) • Cutting of other stock lengths from the set ࡭࢘ For ݅, ݆ ് ‫,݌‬ ‫,ݍ‬ we next consider the largest order length ݈௨ (say) contained in ‫ܣ‬௥ and consider |ܽ௨| for ܽ௨௩ ‫א‬ ‫ܣ‬௞ corresponding to the stock length ܷ௨ satisfying the condition: ݀௨ ݀௩ ൌ ߙ௨ ൈ ‫ݎ‬௨௩ ߚ௩ ൈ ‫ݎ‬௦௩ for some ‫ݒ‬ ‫א‬ ‫ܤ‬ሺ‫݌‬ሻ~ሼ௣,௤,௨ሽ Referring relation (5.1), it is clear that by cutting ‫ݎ‬௨௩ െbars of the stock length ܷ௨, total number of required pieces of order lengths ݈௨ and ݈௩ have been cut. Define ‫ݐ‬௨ ଵ ൌ ‫ݎ‬௨ ሺܷ௨ െ |ܽ௨|ሻ ൑ ‫ݎ‬௨‫ݐ‬௦ ఒ ሺߣ ൌ 1, 2ሻ (5.3) Proceeding this way, for ݅, ݆ ് ‫,݌‬ ‫,ݍ‬ ‫,ݑ‬ ‫,ݒ‬ we consider the next largest order length out of the remaining once and applying the same technique as before, the trim loss with respect to corresponding stock lengths ܷ௝’s has been computed. The process is continued till all order lengths belonging to category I are totally exhausted. ܶଵ ൌ ∑ ‫ݐ‬௣ ଵ ௣‫א‬஻ሺ௡ሻ (5.4) If this cutting process covers all the order lengths ݈ଵ, ݈ଶ, … , ݈௡, then STOP. • Cutting of the largest order length ࢒ࡺ from Category-II Referring definition of ‫ܤ‬௞ ௥ [cf. relation (4.15)], we first set ‫ݎ‬ ൌ 1 and consider ܾே௝ ଵ for fixed ܰ and arbitrary ݆ and select ܾே௝ ଵ as follows: |ܾே| ൌ max ௝‫א‬஻ሺ௡ሻ~ಿ ܾே௝ ଵ Such that |ܾே| ൑ ܷ௝ for some ݆ ‫א‬ ‫ܤ‬ሺ݉ሻ. Now, corresponding to |ܾே|, there exists sets ‫ܤ‬௤ ଵ , ‫ܤ‬௥ ଵ , ‫ܤ‬௦ ଵ , … associated with the stock lengths ܷ௤, ܷ௥, ܷ௦, … respectively containing |ܾே|. We select the set ‫ܤ‬௤ ଵ corresponding to the smallest stock length ܷ௤. In view of the relation (4.15), we have ܾே௝ ଵ ൌ ߙேଵ݈ே ൅ ߚ௝ଵ݈௝ for ݆ ‫א‬ ‫ܤ‬ሺܰሻ~ே. It is clear from relations (4.10) and (4.11), that by cutting ݊ଵ bars of stock length ܷ௤, we cut ݊ଵ. ߙேଵ pieces of order length ݈ே and ݊ଵ. ߚ௝ଵ pieces of order length ݈௝. Our aim is to finish cutting of only two order lengths first ݈ே and ݈௝ (fixed) at a time. Following cases may arise: Case1. Either ݀ே െ ݊ଵ. ߙேଵ ൏ ߙேଵ or ݀௝ െ ݊ଵߚ௝ଵ ൏ ߚ௝ଵ or both the inequalities hold together. Case 2. Either ݀ே ൌ ݊ଵ. ߙேଵ or ݀௝ ൌ ݊ଵߚ௝ଵ.
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 142 5.1 Remark Here two cases will not hold together because in that case ݈ே and ݈௝ will belong to Category-I. We first deal with the case 1. In view of the relation (4.13), we next consider ܾேೕ ଶ ൌ ߙேଶ݈ே ൅ ߚ௝ଶ݈ଶ (݆ fixed as given by (4.13) ) Now, corresponding to ܾேೕ ଶ , there exist sets ‫ܤ‬௤ ଶ , ‫ܤ‬௥ ଶ , ‫ܤ‬௦ ଶ , … containing it. The sets ‫ܤ‬௤ ଶ , ‫ܤ‬௥ ଶ , ‫ܤ‬௦ ଶ , … are associated with the stock length ܷ௤, ܷ௥, ܷ௦, … respectively. We select the set ‫ܤ‬௥ ଶ (say) corresponding to the smallest stock length ܷ௥. It is clear from relations (4.10) and (4.11) that by cutting ݊ଶ bars of stock length ܷ௥, we cut ݊ଶ. ߙேଶ more pieces of order length ݈ே and ݊ଶ. ߚ௝ଶ more pieces of order length ݈௝. We continue the process till either ݀ே ൌ ∑ ݊௞ߙ௜௞ ௦ ௞ୀଵ or ݀௝ ൌ ∑ ݊௞ߚ௝௞ ௦ ௞ୀଵ . Let if possible ݀ே ൌ ∑ ݊௞ߙ௜௞ ௦ ௞ୀଵ holds, then ݀௝ would be of the form ݀௝ ൌ ∑ ݊௞ߚ௝௞ ௦ ௞ୀଵ ൅ ݀௝௦ where we express ݀௝௦ ൌ ݊௦ାଵߚ௝,௦ାଵ ൅ ݀௝,௦ାଵ ሺ݀௝,௦ାଵ ൏ ߚ௝,௦ାଵሻ. Referring the relation (4.13), we now consider ܾ௝ ௦ାଵ ൌ ߚ௝,௦ାଵ݈௝ Now, corresponding to ܾ௝ ௦ାଵ , there exists sets ‫ܤ‬௤ ௦ାଵ , ‫ܤ‬௥ ௦ାଵ , ‫ܤ‬௦ ௦ାଵ , … containing it. The sets ‫ܤ‬௤ ௦ାଵ , ‫ܤ‬௥ ௦ାଵ , ‫ܤ‬௦ ௦ାଵ , … associated with the stock lengths ܷ௤, ܷ௥, ܷ௦, … respectively. We select the set ‫ܤ‬௦ ௦ାଵ corresponding to the smallest stock length ܷ௦(say). It is clear from relations (4.10) and (4.11) that by cutting ݊௦ାଵ. ݀௝,௦ାଵ pieces of order length ݈௝. Now ݀௝,௦ାଵ pieces of order length ݈௝ are left to cut out of ݀௝. We now consider ܷ௞ െ ݀௝,௦ାଵ݈௝ for all ݇ ൌ 1, 2, … , ݉ and select the minimum difference corresponding to the stock length ܷ௧(say) ܷ௧, all pieces of order length ݈௝ have been cut. 5.2 Remark At this last step of cutting ܷ௧ െ ݀௝,௦ାଵ. ݈௝ may exceed the sustainable trim ‫ݐ‬௦. We now compute the trim loss corresponding to the order lengths ݈ே and ݈௝ belonging to the Category-II. ‫ݐ‬ ൌ ቀܷ௤ െ ܾேೕ ଵ ቁ ݊ଵ ൅ ቀܷ௥ െ ܾேೕ ଶ ቁ ݊ଶ ൅ ‫ڮ‬ ൌ ∑ ቀܷ௟ െ ܾேೕ ௥ ቁ݊௥ ൅ ሺܷ௧ െ ݀௝,௦ାଵ݈௝ሻ௦ ௥ୀଵ Order lengths ݈ே and ݈௝ belonging to category-II have been cut completely. Remaining order lengths we again arrange in increasing order ݈ଵ, ݈ଶ, … , ݈ெ (say). We first consider |ܾெ| ൌ max ௝‫א‬஻ሺெሻ~ಾ ܾெೕ ଵ ݆ ‫א‬ ‫ܤ‬ሺ‫ܯ‬ሻ~ெ such that |ܾெ| ൑ ܷ௝ for some ݆ ‫א‬ ‫ܤ‬ሺ݉ሻ. Proceeding in a similar manner, we get ‫ݐ‬௥మ ′ ൌ ∑ ቀܷ௟ െ ܾெೕ ௥ ቁ ݊௥ ௧ ௥ୀଵ ൅ ൫ܷ௧ െ ݀௝,௦ାଵ݈௝൯ ݈ ‫א‬ ‫ܤ‬ሺ݉ሻ We continue the process till all order lengths are exhausted and get ܶଶ ൌ ‫ݐ‬௥భ ′ ൅ ‫ݐ‬௥మ ′ ൅ ‫ڮ‬ Finally, we get total trim ܶ ൌ ܶଵ ൅ ܶଶ. The percentages of trim lose with respect to ‫ݐ‬௦ ଵ and ‫ݐ‬௦ ଶ have been computed in accordance with the total stock length used.
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 143 6. DESIGN OF ALGORITHMS
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 6367(Print), ISSN 0976 – 6375(Online) Volu International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 144 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- August (2013), © IAEME
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 145 7. NUMERICAL EXAMPLES 7.1 Example Consider the following data for our analysis S.No. Order lengths (in cm.) Required no. of pieces S.No. Order lengths (in cm.) Required no. of pieces 1. 801 03 6. 498 16 2. 748 24 7. 492 39 3. 733 46 8. 471 21 4. 641 23 9. 327 40 5. 548 39 10. 303 32 Table 7.1 Available stock lengths S.No. Stock lengths (in cm.) S.No. Stock lengths (in cm.) ‫܍ܔ܊܉ܖܑ܉ܜܛܝ܁‬ ‫ܕܑܚ܂‬s ࢚࢙ ૚ ൌ ૟૟.૜૞૛૚ ࢉ࢓. ࢚࢙ ૛ ൌ ૟ૡ.૚ૢ૛ૡ ࢉ࢓. 1. 2110 4. 3883 2. 2210 5. 4177 3. 3120 6. 4239 Table 7.2 Cutting Plan by using second order sustainable trim ࢚࢙ ૛ S.No. Order lengths (in cm) Pieces to cut Trim loss (in cm.) Used Stock lengths (in cm.) Category-I 1. 492 548 ቅ 3 3 ቅ 0 ൈ 13 ൌ 0 3120 ൈ 13 ൌ 40560 2. 303 498 ቅ 4 2 ቅ 2 ൈ 8 ൌ 16 2210 ൈ 8 ൌ 17680 3. 327 748 ቅ 5 3 ቅ 4 ൈ 8 ൌ 32 3883 ൈ 8 ൌ 31064 4. 641 733 ቅ 1 2 ቅ 3 ൈ 23 ൌ 69 2110 ൈ 23 ൌ 48530 Category-II 5. 471 9 0 ൈ 2 ൌ 0 4239 ൈ 2 ൌ 8478 6. 471 801 ቅ 3 3 ቅ 67 ൈ 1 ൌ 67 3883 ൈ 1 ൌ 3883 Total 184 150195 Total Trim loss (%) 0.1225% Table 7.3
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 146 Cutting Plan by using first order sustainable trim ࢚࢙ ૚ :::: S.No. Order lengths (in cm) Pieces to cut Trim loss (in cm.) Used Stock lengths (in cm.) Category-I 1. 492 548 ቅ 3 3 ቅ 0 ൈ 13 ൌ 0 3120 ൈ 1 ൌ 40560 2. 303 498 ቅ 4 2 ቅ 2 ൈ 8 ൌ 16 2210 ൈ 8 ൌ 17680 3. 327 748 ቅ 5 3 ቅ 4 ൈ 8 ൌ 32 3883 ൈ 8 ൌ 31064 4. 641 733 ቅ 1 2 ቅ 3 ൈ 23 ൌ 69 2110 ൈ 2 ൌ 48530 Category-II 5. 471 9 0 ൈ 2 ൌ 0 4239 ൈ 2 ൌ 8478 6. 471 801 ቅ 1 2 ቅ 37 ൈ 1 ൌ 37 2110 ൈ 1 ൌ 2110 7. 471 801 ቅ 2 1 ቅ 367 ൈ 1 ൌ 367 2210 ൈ 1 ൌ 2110 Total 521 150532 Total Trim loss (%) 0.3461% Table 7.4 7.2 Figure and Screen shots of programming Figure 7.1
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 147 Figure 7.2 Figure 7.3 7.3 Conclusion Referring tables 7.3 and 7.4, it may be noted that for this particular set of data ‫ݐ‬௦ ଶ is reducing the total trim significantly in comparison with the trim obtained by using ‫ݐ‬௦ ଵ .
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 148 7.4 Example Consider the following data for our analysis S.No. Order lengths (in cm.) Required no. of pieces S.No. Order lengths (in cm.) Required no. of pieces 1. 801 03 6. 498 16 2. 748 24 7. 492 39 3. 733 46 8. 471 21 4. 641 23 9. 327 40 5. 548 39 10. 303 32 Table 7.5 Available stock lengths S.No. Stock lengths (in cm.) S.No. Stock lengths (in cm.) ‫܍ܔ܊܉ܖܑ܉ܜܛܝ܁‬ ‫ܕܑܚ܂‬ssss ࢚࢙ ૚ ൌ ૡ૙. ૚૙૜ૡ ࢉ࢓. ࢚࢙ ૛ ൌ ૠૢ. ૜૚૚ૡ ࢉ࢓. 1. 2110 4. 3883 2. 2210 5. 4170 3. 3120 --- --- Table 7.6 7.5 Conclusion It may be verified that in example 7.4, the trim loss obtained by using ‫ݐ‬௦ ଶ is exceeding the value of trim loss obtained by ‫ݐ‬௦ ଵ . REFERENCE 1. Gradisar M., Resinovic G., Kljajic M., Evaluation of algorithms of one-dimensional cutting, Computer & Operations Research 29, 1207-1220, 2002. 2. Powar P.L., Jain Vinit, Saraf Manish, Vishwakarama Ravi, One-Dimensional Cutting Stock Problem (1D-CSP): A New Approach, International Journal of Advanced Computing, 46(2), 1223-1228, 2013. 3. Gilmore P. C., Cutting stock, linear programming, knapsacking, dynamic programming and integer programming; some interconnections, Annals of Discrete Mathematics, 4, 217-235, 1979. 4. Gilmore P., Gomory R., A linear programming approach to the cutting stock problem, Operations Research 9(6), 848-859, 1961. 5. Gilmore P., Gomory R., A linear programming approach to the cutting stock problem, part-II, Operations Research 11, 863-888, 1963. 6. Gilmore P.C., Gomory R.E., Multistage cutting stock problems of two and more dimensions. INFORMS.Operations Research 13(1), 94-120, 1965. 7. Haessler R. W. and Vonderembse M. A., A procedure for solving the master slab cutting stock problem in the steel industry, AIIE Transactions, 11, 160-165, 1979. 8. Powar P.L., Jain Vinit, Saraf Manish, Vishwakarama Ravi, One-Dimensional Cutting Stock Problem (1D-CSP) with First Order Sustainable Trim: A Practical Approach, International Journal of Computer Science Engineering and Information Technology Research (IJCSEITR), 3(3), 227-240, August 2013. 9. Mobasher A., Ekici A., Solution approaches for the cutting stock problem with setup cost, Computers & operations research 40, 225-235, 2013.
    • International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME 149 10. Gradisar M., Erjavec J., Tomat L., One-dimensional cutting stock optimization with usable leftover: Acase of low stock-to-order ratio, International journal of decision support system technology, 3(1), 54-66, 2011. 11. Dikili A.C., A generalized approach to the solution of one-dimensional stock-cutting problem for small shipyard, Journal of marine science and technology, 19(4), 368-376, 2011. 12. Hinterding R., Khan L., Genetic algorithms for cutting stock problems: with and without contiguity, In Yao, X.(Ed), Lecture Note in Artificial Intelligence, Springer, New York, 187- 205, 1997. 13. Wagner B.J., A genetic algorithm solution for one-dimensional bundled stock cutting, European Journal of Operational Research, 117(2), 368-381, 1999 14. Feifei G.E., Lin Liu. Jun Pei, Xiazi Zhang, Study of one-dimensional cutting stock problem with multi-objective optimization, American journal of engineering and technology research, 11(9), 2011. 15. K. V. Sujatha and S. Meenakshi Sundaram, “Regression, Theil’s and Mlp Forecasting Models of Stock Index”, International Journal of Computer Engineering & Technology (IJCET), Volume 1, Issue 1, 2010, pp. 82 - 91, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375. 16. Rajesh V. Argiddi and Sulabha S. Apte, “A Study of Association Rule Mining in Fragmented Item-Sets for Prediction of Transactions Outcome in Stock Trading Systems”, International Journal of Computer Engineering & Technology (IJCET), Volume 3, Issue 2, 2012, pp. 478 - 486, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.