INSERTION SORT APPROACH FOR THE CAR SEQUENCING PROBLEM

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1 195 The 5 th International Conference on Information & Communication Technology and Systems INSERTION SORT APPROACH FOR THE CAR SEQUENCING PROBLEM Bilqis Amaliah 1, Chastine Fatichah 2 1,2 Informatics Department, Faculty of Technology Information Institut Teknologi Sepuluh Nopember (ITS), Surabaya, Indonesia {bilqis 1, chastine 2 }@cs.its.ac ABSTRACT In this paper we present one major approach to solve the car sequencing problem, in which the goal is to find an optimal arrangement of commissioned vehicles along a production line. The method that is used in this paper is insertion sort method. We tasted the method by using example that we have been made. Results highlight the significant impact, we can sort the the car, present cost at change of colour and machine, finally we will know the optimal car sequence. Keyword: Car sequence problem, Insertion sort 1 INTRODUCTION In automobile industry a cost-effective arrangement of commissioned cars along the production line is desired. Although the individual cars are similar, each automobile requires particular components to be installed by different working bays along the assembly line. In addition to the different configurations of cars to be arranged along the production line, each vehicle has to be painted with exactly one color. The arising problem in which the goal is to minimize the number of color changes while considering the constraints defined by various working bays is called car sequencing problem (CarSP). A feasible solution to CarSP is a permutation of all cars to be produced at a certain day taking all constraints into consideration. The production line itself consists of three stages: the body shop, the paint shop, and the assembly shop. In the body shop the chassis of the cars are manufactured, the paint shop workers paint the cars, and in the assembly shop different options like air condition, sun roofs, or sound systems get installed. The constraints defined by the body shop and the assembly shop are similar to each other, whereas the paint shop constraints differ significantly. For the former, we consider restrictions which can be expressed as No more than lc cars are allowed to require component c in any sequence of mc consecutive cars. For the latter, we consider constraints of the form: At most s cars with the same color are allowed to be arranged consecutively. Changing the color after at most s cars is motivated by a more psychological reason: In automobile industry, the paint of a car is applied using an injector. During this spraying the paint slowly agglutinates. To obtain good results the injector has to be cleaned in regular intervals. If the same color would be applied after cleaning the injector again, the staff concerned with the cleaning process would get imprecise, because It s the same color again. This leads to improper painting results, which consequently evoke reclamations. In Section 2 we first give a formal definition of CarSP, and in Section 3 we present a brief literature review. Section 4 proposes shares tell calculation insertion sort having the solution with a purpose to public resolving of car sequence to yield optimal solution evidence. Conclusions complete this article. 2 FORMAL DEFINITION In this section, we present a formal definition of CarSP. The notation introduced here will be used throughout the whole document. Given are a set of possible components C, including a set of colors F C, and a set K of requested configurations K = { k : k C, k F = 1} i.e. each configuration k is a subset of components to be installed, and exactly one color is selected in each configuration. If configuration k contains component c, a corresponding 0 1 constant ack is set to 1, otherwise to 0. Component vector a k = (aok,..,a C k) denotes the incidence vector for configuration k. T T Let function H(a i, a j ) be the Hamming distance of two configurations i and j. All configurations requested in an instance of CarSP T T are pairwise different, i.e. H(a i, a j ) 1, for T

2 196 The 5 th International Conference on Information & Communication Technology and Systems all pairs (i,j) K2, i j, and for each k K there is further given an integer demand δ k 1 indicating how many vehicles of configuration k have to be produced. A solution to CarSP is a mapping X = (x1,..,xn) : {1,..,n} K specifying a sequence of length n= k K δ k, which assigns exactly one configuration k K to each position i=1,..,n. Since only commissioned cars shall be produced, the restriction {xi : xi = k} = δ k, for all k K, has to be fulfilled. Furthermore, the number of consecutive cars painted with the same color f F has to be less than or equal to the maximum color block size s. For each component c C we are given a sliding window length mc Ν and a quota lc Ν. Only lc cars are allowed to require component c in any subsequence of mc consecutive vehicles. Due to the constraints defined by the paint shop lf is equal to s and mf is equal to s + 1 for all colors f F. We denote the total demand of any component c C by dc = k K ack. δ k. The production of the last day needs to be considered. For this purpose additional constants eci {0,1}, c C, I = 1,.,mc 1, are used: eci is set to 1 if I the ith last car of the previous day required component c. Often CarSP is formulated as an optimization problem in which the total costs of color changes and weighted assembly shop constraint violations have to be minimized. For this purpose, we associate constant costs γ f > 0, f F, with a change to color f and costs γ c > 0, c C \ F, with a violation of an assembly shop constraint. The goal is to find a sequence X of commissioned cars minimizing the objective function with the costs for each position i being under the remaining hard constraints ensuring that no more than s cars of the same color are scheduled in a row. Expression change(i) represents the costs for a potential color change at position i of sequence X and viol(i,c); cþ denotes the costs for possible assembly shop constraint violations at position i with respect to component c. The latter are computed as the sum of cars requiring component c within the last mc cars (including the car at position i) minus the quota lc; if this difference is less than 0, the number of violations is set to 0. Note that for computing the costs of color changes considering not only the new color to but also the color of the previous car would be more precise. Since typically color changes are penalized using a constant factor independently of the involved colors, we use this simplified problem description. 3 PREVIOUS WORK Gent showed that the decision problem associated with the car sequencing problem whether there exists an optimal solution without any violations of the constraints defined by the assembly line is NP-hard. Kis proved that this decision problem is NP-hard in a strong sense, and Hu stated that the optimization problem including the color constraints as defined in Section 2 is NPhard, too. Several different approaches have been made to solve CarSP or variants of it. The methods used vary from greedy heuristics to meta-heuristics like ant colony optimization, whereas only a few exact algorithms have been described. a. Exact Methods Gravel et al. proposed an ILP approach for a variant of CarSP without the constraints specifically related to the paint shop. It is able to solve commonly used benchmark instances with about 200 cars and 5 components to proven optimality within practically reasonable time. The main idea applied in this formulation is to group cars with the same configuration into classes to avoid symmetries.

3 B29 Insertion Sort Approach for the Car Sequence Problem Bilqis Amaliah 197 Hu describes another ILP approach that also takes constraints defined by the paint shop into account. Unfortunately, the size of practically solvable instances is limited to about 30 cars with 8 components. Another exact ILP approach is presented in which, in contrast to the approach by Hu and analogously to the formulation of Gravel et al., classifies cars with the same configuration into groups. Thereby, the size of practically solvable instances is enlarged to at most 300 cars with about 8 components. b. Greedy Heuristics Gottlieb et al. proposed greedy heuristics using different evaluation strategies. They construct sequences of cars by always adding the next best car in respect to some evaluation function to a current partially filled sequence. Once a car is placed at a position, it is never reconsidered again. Some of the proposed evaluation functions take the currently available cars and the already existing partial sequence into account, whereas others only compute a global value indicating whether a car is hard to arrange without constraint violations or not. Many other approaches like the following local search based techniques utilize similar greedy heuristics for computing initial solutions. 4 INSERTION SORT FORMULATION This section introduces a new ILP formulation for CarSP. We will later see that for many practical instances it can be significantly faster solved to optimality than previous formulations. One of the algorithm sorting the simplestness is insertion sort. Idea of this algorithm of analogy can by be like sorting the card. Explanation is following explain how do algorithm insertion sort work in card sequence. Let's say you wish to sort a set of card of smallest valuable card [so/till] biggest which. All card put down at desk, call this desk as first desk, compiled from left to right and to the downwards. Then we have the other desk, second desk, where sorted card will be put down. Take the first card laying in the left corner of first desk and be put down at second desk. Take the second card of first desk, compare with the card residing in at second desk, then put down at appropriate sequence after comparison. The process will take place so all card at first desk have been put down successive at second desk. Algorithm insertion sort basically divide the data to be sorted become two shares, what not yet been sorted ( first desk) and which have been sorted ( desk both). First element taken away from by the array shares not yet been sorted and is then put down according to his position at shares of differ from array which have been sorted. This step is done recuring till ly nothing like the element remaining at the array shares not yet been sorted. So that can be told by the algorithm insertion sort take element in left shares which is then compared to his value with previous values. so that his value have to be more a little a few previous values. And if his value is slimmer hence element 4.1 Algorithm of Insertion Sort Insertion-Sort (A) for j 2 to length[a] do key A[ j ] Insert A[ j ] into the sorted sequence A[1.. j 1] i j 1 while i > 0 and A[i ] > key do A[i + 1] A[i ] i i 1 A[i + 1] key example of insertion sort Data to the that is number of 12,9,4,99,120,1,3,10 the is is then sorted become 1,3,4,9,10,12,99,120. Implementation from

4 198 The 5 th International Conference on Information & Communication Technology and Systems insertion sort in this case that is sorting the car, following the example in the the next data. Can seen at picture as follows. then sorted according to his colour and his priority become 4.2 Examination and result Have ever been provable that by using method insertion sort hence car sequence problem can be finished better, as existence of program able to sort the the car. Even the result of from the program can calculate cost from change of colour with wanted priority and cost from change of component or machine. Way usage of application CarSP which have been made is : At picture is following is appearance early application CarSP which have been made, where there are some data inputan to be filled in as according to data of amount of productions to in production. Stages admission filling of application carsp this can seen for following : Figure 2: Design Input Car 2. Both, at menu Priority car colour this function to determine the car colour priority to be produced beforehand. Input as according to required condition or can in random his colour priority by choosing button Random and select Ok. Figure 3: Design Priority Car Colour Figure 1: Design Face 1. First step to be done is filling in the data of amount of car productions, component, colour block, cost change of colour and change of machine as according to the data required by productions amount to be [counted/calculated] or searched for cost biggest to be compared to to choose the amount produce the car is beneficial very. Then click button Next to fill 3. At menu car colour production, process input will walk the phase for the shake of phase. Process input early pursuant to amount produce the car and the components amount filled in input menu car. Fill in the amount produce the car to every n component up to last components amount click button next shall seen button ok.

5 B29 Insertion Sort Approach for the Car Sequence Problem Bilqis Amaliah CONCLUSIONS At solution of this car sequence problem we have shown a formulation by using insertion sort which applicable to sort the car to in paint matching with desire of factory or producer. Besides amount cost that happened at every commutation of colour is and or commutation of countable machine also with also pay attention the colour priority to be used, so that we will be more know how is car sequence is best of time will be painted and later of production cost minimization This formulation significant will be more is easy to comprehended because only using the method insertion sort compared to literature using the integer linier programming (ILP) and hybrid variable neighborhood (VNS) what is felt enough difficult to be comprehended and implementation. Therefore under consideration this method insertion sort can be told enough succeed, because have can finish the sequence problem with good enough. Figure 4: Design Car Colour Production n = amount of components 4. After all admission fillings of data is completed done, hence the next process is by depressing button process shall button the process cannot in click again. Hence will seen the phases process and result of from calculation of amount cost change of colour and cost change of machine. REFERENCES [1] Cormen, T. H., Leiserson, C. E., Rivest, R. L., Stein,C.,2002. Introduction to Algorithms Second Edition [2] I.P. Gent, T. Walsh, CSPLIB: A benchmark library for constraints, Technical Report, APES , Department of Computer Science, University of Strathclyde, UK, [3] I.P. Gent, Two results on car-sequencing problems, Technical Report, APES , Department of Computer Science, University of Strathclyde, UK, April [4] Prandtstetter, M., Raidl,G.R.,2007. An integer linear programming approach and a hybrid variable neighborhood search for the car sequencing problem. European Journal of Operational Research 191, p [5] Prandtstetter, M., Raidl,G.R.,2005. An Integer Linear Programming Approach and a Hybrid Variable Neighborhood Search for the Car Sequencing Problem. p 1-28 [6] Raidl,G.R., Prandtstetter, M.,2005. Exact and heuristic methods for solving the Car Sequencing Problem Figure 5: Cost Hasil Perhitungan

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