International Journal of Mathematical and Computational Methods W. Wongsinlatam et al. http://www.iaras.org/iaras/journals/ijmcm

A Mixed Integer Linear Programming for Maximizing Effectiveness of Case Assignment in Court of Justice using Optimization

W. WONGSINLATAM1, K. PIMPILA2, A. WONGPHAT3, AND S. BUCHITCHON4*

1 Faculty of Applied Science and Engineering, Khon Kaen University, Nong Khai Campus, Nong Khai, 43000, THAILAND E-mail address : [email protected]

2 Faculty of Education, Rajabhat Phetchabun University, Phetchabun, 67000, THAILAND E-mail address : [email protected]

3 Faculty of Science and Technology, Rajabhat Mahasarakham University, Mahasarakham, 44000, THAILAND E-mail address : [email protected]

4 Faculty of Integrated Social Sciences, Khon Kaen University, Nong Khai Campus, Nong Khai, 43000, THAILAND *Corresponding author E-mail address : [email protected]

Abstract: - It is known that litigation is time consuming. However, justice delayed is justice denied. The effectiveness of judicial system depends on the efficient and timely manner of court case operation. Court case assignment in Thailand is currently operated in a random assignment norm. However, the increasing number of cases in the court of justice contributes as a building block to the reputation of time consuming operating system. Selecting the right cases to be assigned to the available efficient judge in the area is a vital need for judicial operation. Therefore, major challenges with respect to court case assignment are to determine the right case to be assigned to the right efficient judge in the field and to allocate appropriate time for delivering the court decision in order to maximize the effectiveness of the judicial system. In this article, the selected math model is a mixed integer linear programming that is developed to analyze and solve the problem. A metaheuristic with polynomially bounded computational complexity is proposed to address the issue. Furthermore, results of extensive computational experiments to empirically evaluate its effectiveness to find an optimal solution are reported.

Key-Words: - Justice administration, Case assignment, Mixed , , Metaheuristic method, NP-hard problem

1 Introduction However, the increasing number of cases in the Justice administration and case assignment court of justice contributes as a building block to the are routine works in judicial system. However, the reputation of time consuming operating system. In effectiveness of such work represents as a goal of the last decade, caseload in the court of Thailand has judicial system which is to ensure the rule of law. dramatically increased. This is due to the change in The ‘rule of law’ means that the administration of constitutional law which requires a full quorum of justice and other exercise of public authority must judges – generally a team of two - to be seated at a be predictable and consistent [1]. hearing. It is known that litigation is time Several proposals have been made to consuming. The effectiveness of judicial system increase system efficiency including case depends on the efficient and timely manner of court management, court-annexed mediation, and digital case operation. Court case assignment in Thailand is audio testimony recording system [2]. Case currently operated in a random assignment norm. assignment is one of the most significant tools in

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case management to improve efficiency of judicial high quality calcium carbonate slurry, supplied to system. It is proposed in this paper that selecting the European paper manufacturers. Christodoulos A. right cases to be assigned to the available efficient Floudas [6] applied the scheduling of chemical judge in the area is a vital need for judicial processing systems based on MILP approaches operation. Therefore, major challenges with respect which improved the computational efficiency in the to court case assignment are to determine the right solution of MILP problems. However, the use of case to be assigned to the right efficient judge in the MILP model in juridical problem is still limited. field and to allocate appropriate time for delivering Metaheuristic optimizations purpose is to the court decision in order to maximize the find optimal solution for NP-hard problem which effectiveness of the judicial system. solves the local minima problem to find the global In mathematics, mathematical modelling minima that solve several practical optimization transforms the real problem into the objective problems. The algorithm of metaheuristic methods with constrained or unconstrained is made of a random number that moves in a wide optimization problems to find an exact solution or feasible solution to solve several optimization optimal solution. The mathematical modelling is problem. The metaheuristic method reduces the time being used in several fields of science and in finding the exact solution or the good solution in technology. The objective function is generally the problem. Popular methods for assignment being formed in convex programming, nonlinear problem includes; Ant Colony Optimization (ACO) programming, and . The [7]; (TS) [8]; Particle Swarm new trend of objective function is interested in Optimization (PSO) [9]; and (FA) forming the mixed integer linear programming. The [10]. mixed integer linear programming (MILP) is In this paper, the practical problem in popular in the problem of assignment because the juridical area is being used as assignment problem. objective function of minimize or maximize Selecting the right cases to be assigned to the optimization in MILP is more effective than the available efficient judge in the area is a vital need linear or . MILP is a mixed for judicial operation. Therefore, major challenges of the several forms of programming which with respect to court case assignment are to combined the binary integer programming with determine the right case to be assigned to the right linear programming. efficient judge in the field and to allocate This research uses the metaheuristic method appropriate time for delivering the court decision in for assignment problem of the court case and order to maximize the effectiveness of the judicial coordination of justice team. The objective of system. The objective function is used the MILP assignment problem includes N cases that must be optimization model of case analysis and assigned to N teams where each team has the examination. The goal of this proposed model is to competence to do all cases. However, due to maximize the overall effectiveness in identifying the personal ability, case specification or other reasons, perpetrator of the crime. each team may spend different time for minimize or The rest of the paper proceeds as follows: maximize with the objective of assignment problem. Section 2 is an introduction to assignment problem In planning the working system in an of the case assignment in court of justice. The organization, the coordination of workers and tasks objective function uses the MILP optimization by is a major consideration that reflects the efficiency using two metaheuristic optimizations. The of the company. The Mixed integer programming computational results and discussion about the (MIP) was used in the examination proctor effectiveness of the proposed metaheuristic assignment problem by Takeshi Koide where staffs in finding an optimal are provided in are assigned for examinations as proctors in the Section 3. Finally, section 4 summarizes our regular examination period [3]. Julia Rieck et al. [4] conclusions. applied the project scheduling problems subject to general temporal constraints using MILP model and domain-reducing processing techniques. The models 2 The Case Assignment in the Court used CPLEX 12.1 software to solve lower and upper bounds for resource requirements at particular of Justice points in time. David Bredström et al. [5] used MILP model for barge transport planning on the 2.1 The definition of problem river Rhine. The planning solved the supply chain management of Omya’s production of Norwegian

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Nomenclature 2.2 The formulation of problem C index set for cases assignment. {1,2,...,n } The MILP model formulation for the above P index set for justice panel. {1,2,..., m} problem is as follows: nm c each case assignment. cC∈ . maximize ZT=∑∑µi ⋅ ij (1) effective rate for case assignment. ij=11 = µc L lower bounds on justice time. Constraints: c n upper bounds on justice time. Uc ∑Tij ≤ MT; ∀ jm≤ (2) MT maximum time for justice deadline. i=1 T− XU ⋅≤0; ∀ i≤≤ nj, m (3) T assignment of case c to the right justice. ij ij i cj T− XL ⋅≥0; ∀ i≤≤ nj, m (4) X an binary decision variable. ij ij i ij m ∑ X ij ≤1; ∀ in≤ (5) To define the problem, consider the j=1 following scenario: A set Cn= {}1,2,.., of n cases 1, if case i is selected to justice j; X ij =  has been filed to the court of justice where a set 0, otherwise; Pm= {}1,2,.., of m are available justice panels ∀≤ i nj, ≤ m (6) forming justice teams. If case cC∈ is selected for We assume that the effectiveness rate of a justice team, at least Lt time units must be spent µµ µ to deliver the court decision. Also, the maximum cases administration are 12, ,.., n , respectively. amount of time to be spent on each case cC∈ must The objective function according equation (1) for not exceed U be time units since spending more the above problem is defined to the overall t effectiveness of the judicial procedure. The than this time would distort justice. in equation (2) is the total time spent on a To simplify the presentation of our approach sequential of case does not exceed the maximum to develop and solve a for this allowable time MT for each justice team. The problem, the mathematical model that we have constraint of upper bound is the time spent on adopted assumes that: Ui . There are fixed numbers of justice panels in each selected case i is no more than upper bound in

one court and one judge can only be in one equation (3). The constraint of lower bound Li is panel. the time spent on each selected case i is no less . All of the justice panels have the same than lower bound in equation (4) and define X as a expertise. ij . One case is being assigned to only one binary decision variable which let a value of 1 if panel. case i is selected and assigned to justice team j . No interdependencies exist between each and let a value of 0 if otherwise in constraint case. according equation (5) and (6). The total number of . Each case required difference time limit binary variables is nm and the total number of depending on the nature of the case. constrains is 2nm++ n m . . The minimum effective time for delivering a decision for one case is one day. Whereas 2.3 Metaheuristic method for problem the maximum effective time for delivering a The optimization method of maximizing decision is 90 days which represents the effectiveness of case assignment in court of justice longest time period where a person can be is used two technique metaheuristic method i.e. held in prison on remand. Firefly algorithm (FA) and Ant Colony Optimization (ACO). The FA technique of In this paper, the problem is one of finding metaheuristic algorithms was firstly proposed by the assignment of case c to the justice team j . The Yang in 2008 [11,12]. The FA algorithm is based on n ≤ the flashing characteristics and behavior of fireflies. amount of time ∑ = Tcj MT and the total c 1 The FA is especially used as a metaheuristic effectiveness of all cases is maximized and the technique for solving continuous optimization in bound of time, LTUc≤≤ cj c .

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NP-hard problems. Algorithm 1 illustrates the FA In applying the method, the cases are pseudo code. assigned to justice panels considering the maximum time available. The cases are sorted into a list with Algorithm 1. Pseudo code of the FA the amount of time allocated for each case. Then the 1: Define the objective function fx() ; first case in the list is assigned to the justice panel. If the minimum time required for this case is greater 2: Initialize the firefly population x= xx12, ,... xn ; than the total time available of the justice team, this 3: Define the light absorption coefficient γ ; case is skipped to the next case in the list. This 4: for each firefly x in the population do process is carried out until the total time available i for all justice panels is utilized or all cases have 5: Initialize light intensity Ii ; been assigned to justices. 6: end 7: repeat 2.3.1 The metaheuristic optimization to solve problem 8: for each firefly xi in the swarm do The proposed metaheuristic optimization to 9: for each other firefly in the swarm do x j solve the problem is described as follows:

10: if IIij> then Input: Index set of C and P . For each case cC∈

,effectiveness rate is µc and maximum time for 11: Move firefly xi toward x j ; 12: end justice team deadline is MT . Lower and upper 13: Attractiveness varies bounds on decision times are Lc and Uc , 14: with distance via exp(−γ r ); respectively. Assumption, effectiveness rates for

15: Evaluate new solutions light and each cases are in a decreasing order ( µµ12, ,..., µn ). 16: update intensity; Step 1: Set c =1, and set ZT**= = 0, MT= MT, 17: end c ci i 18: end for each cC∈ and iP∈ . 19: Rank the fireflies and find the current best; Step 2: Find justice iP∈ , and MTi if Uci≤ MT 20: until termination criterion reached; then TU* = and update MT= MT − T * . 21: Rank the fireflies and return the best one; ci c i i ci Next, go to step 4 and go to step 3 where The second metaheuristic method that is being otherwise. * used in this paper is ACO. Different from the FA, Step 3: If Lci≤ MT then Tci= MT i and the ACO principles are based on the behaviour of MT= MT − T * and go to step 4. ants [13,14]. Despite the fact that finding the i i ci shortest path is not the goal of ants, at the end their Step 4: If MTz = 0 for all zM∈ or cn= then go achievement is creating the shortest path between to step 5. Otherwise, let cc= +1 and loop their nest and food source. During the commuting = c and return go to step 2. between their nest and food source, ants deposit a n Step 5: Calculate ZT**= µ and stop. chemical substance called pheromone on the path. ∑ c=1 c ci The algorithm of ACO is presented in Algorithm 2. Output: The best solution of total effectiveness is Z * and T * is the global maximize point of Algorithm 2. Pseudo code of ACO ci 1: Initialize pheromone trials; problem used by FA and ACO. 2: while termination criterion 3: for each ant do 3 Results and Discussion 4: Construct a solution based on ant The computational and effectiveness results 5: behavior and pheromone trial; are generated using FA and ACO. For empirically 6: end evaluating the effectiveness and the efficiency, the 7: Update pheromone trials; workstation used for carrying out the result is 8: Apply evaporation strategy; equipped with 2.40 GHz processor, 4.00 GB of 9: Apply reinforcement strategy; RAM, and the system used Microsoft 7. Matlab 10: end verison R2013b software was used to carry out 11: Return best found solution; simulation of the proposed algorithm. The number of cases for experimental parameters was 20, 30 and 50 and the number of justice team for experimental

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parameters was 3, 4 and 5. The total time available In the results, it can be seen that when the for justice decision is MT = 90 . Therefore, we number of cases was 20 and the number of justice team was 3 for experimental parameters, P1(20,3), generated parameters µii,,U and Li by following the average of percent deviation of FA is 0.01 and two problem with a uniform distribution e.t. in from total available time of FA is 103.81 which is better P1: µii=()()()1, 90 ,LU = 1, 30 , i = 1, 90 and than the fast ACO. Furthermore, in the P2(50,5) the P2:µ =()()()1, 90 ,LU = 1, 90 , = 1, 90 . average of percent deviation of FA is 0.02 and total ii i available time of FA is 976.25 which is also better Problem hardness is likely to depend on the than fast ACO for the measures of performance. values and ranges of the parameters of the problem. Table 1 shows the total available time Table 2 Parameter settings for the metaheuristic (second) in finding an optimal solution and 20 optimization problem instances were generated, which is Methods Parameter name Parameter value sufficient to produce steady state empirical result for FA Maximum number of 1000 the measures of performance. Percent deviation is iterations 100*[Optimal solution - FA or ACO Number of Fireflies 20 solution]/[Optimal solution] and average of percent Light absorption 1 20 deviation of 20 instances is [percent ∑ i=1 coefficient deviation]/20. Attraction coefficient 2 base value Table 1 Comparing the results of FA and ACO for Mutation coefficient 0.2 the problem Mutation coefficient 0.98 P n m FA ACO damping ratio Effectiveness Effectiveness ACO Maximum number of 1000 iterations Avg Time Avg Time Number of ants 50 P1: 20 3 0.01 103.81 0.61 258.47 Initial pheromone 10 4 0.27 222.88 0.97 331.12 Pheromone 0.3 5 0.13 372.01 0.58 472.56 exponential weight 30 3 0.52 436.05 0.78 548.01 Evaporation rate 0.1 4 0.54 554.50 0.82 642.70 5 0.19 640.51 0.54 714.40 In this research, in order to maximize the 50 3 0.08 733.30 0.81 812.31 overall effectiveness, a MILP model is used to 4 0.04 862.47 0.30 928.27 analyse the problem of allocating optimal time for 5 0.02 918.17 0.35 1005.35 selected cases and the team of justice panels. The P2: 20 3 0.05 115.97 0.63 218.56 research shows that even the idealized simple 4 0.02 275.46 0.34 379.35 version of the sequential problem in administrative 5 0.37 393.27 0.58 495.32 law with multiple justice teams is NP-hard. The 30 3 0.29 458.45 0.52 558.54 results of the empirical experiments strongly 4 0.74 516.37 0.82 696.18 validate that FA is quite effective in finding an 5 0.19 647.64 0.24 724.64 approximately optimal solution to the general 50 3 0.13 752.46 0.48 858.51 problem. 4 0.14 853.54 0.62 952.67 5 0.02 976.25 0.30 1065.64 4 Conclusion In this paper, selecting the right cases to be Two techniques of metaheuristic method are assigned to the available efficient judge in the area the valid and efficient methods in numeric is a vital need for judicial operation. The selected programming and have been employed due to their math model is a mixed integer linear programming strong convergence properties. Specific parameter that is developed to analyze and solve the problem. settings for the algorithms are described in Table 2. A metaheuristic algorithm with polynomially Experimental results show in Table 1 that bounded computational complexity is proposed to time of optimal solutions and average of percent address the issue. The results, simulation deviation for maximize effectiveness of FA are experiments are implemented on MILP assignment better than those of fast ACO. problem. Experimental results show that time of

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optimal solutions and average of percent deviation Journal of Annals of , Vol. for maximize of FA are better than those of fast 139, 2005, pp. 131–162. ACO. The performance of FA is better than that of [7] Charbel Rizk and Jean-Paul Arnaout, ACO for ACO for maximizing effectiveness of case the Surgical Cases Assignment Problem, assignment in court of justice. In the future work, it Journal of Medical Systems, Vol. 36, 2012 will be interesting and useful to conduct empirical pp.1891–1899. and theoretical studies to develop appropriate [8] Juan A . D ı́ a z and Elena Fernández, A Tabu measures and methodologies to determine the search heuristic for the generalized assignment effective time of various types of cases and problem, European Journal of Operational minimizing the total time of case assignment in Research, Vol. 132, 2001, pp. 22–38. court of justice. Moreover, it will be useful to [9] Hongbo Liu, Ajith Abraham, and Jianying improve metaheuristic optimization for the fast Zhang, A Particle Swarm Approach to computing and defining parameter optimization for Quadratic Assignment Problems, Soft the assignment problem in legal fields. Computing in Industrial Applications, Vol. 39, 2007, pp. 213–222. Acknowledgments: [10] Anil Goyal, Shiv Krishan Joshi , and Surbhi The authors would like to thank Faculty of Gupta, A Heuristic Method of Cell to Switch Applied Science and Engineering and Faculty of Assignment in Mobile Communication Integrated Social Sciences Khon Kaen University, Networks Using Firefly Algorithm, Nong Khai Campus, Faculty of Education, Rajabhat International Journal of Scientific & Phetchabun University, and Faculty of Science and Engineering Research, Vol. 4, No. 7, 2013, pp. Technology, Rajabhat Mahasarakham University, 1971-1977. Thailand for funding this research. [11] X. S. Yang, Nature-Inspired Metaheuristic Algorithms, Luniver Press, 2008. References: [12] X. S. Yang, Firefly algorithm, stochastic test [1] Stephen C. Lubkemann, Deborah H. Isser, and functions and design optimisation, Philip A. Z. Banks III, Unintended International Journal of Bio-Inspired Consequences: Constraint of Customary Justice Computation, Vol. 2, No. 2, 2010, pp. 78-84. in Post-Conflict Liberia, in Deborah H. Isser, [13] M. Dorigo and T. StÄutzle. Ant Colony ed. Customary Justice and The Rule of Law in Optimization. MIT Press, 2004. War-Torn Societies, United States Inst. of [14] Lin-yu Tseng, and Shyi-Ching liang, A Hybrid Peace, 2011. Metaheuristic for the Quadratic Assignment [2] Thammanoon Phitayaporn, Strengthening the Problem, International Journal of Independence and Efficiency of the Judiciary in Computational Optimization and Applications, Thailand, Chulalongkorn University Law Vol. 34, pp. 85–113, 2006. Journal, 2004, pp.45-53. [3] Takeshi Koide, Mixed integer programming approach on examination proctor assignment problem, Procedia Computer Science, Vol. 60, 2015, pp. 818 – 823. [4] Julia Rieck, Jürgen Zimmermann, and Thorsten Gather, Mixed-integer linear programming for resource leveling problems, European Journal of Operational Research, Vol. 221, 2012, pp. 27–37. [5] David Bredström, Kjetil Haugen, Asmund Olstad, and Jan Novotnýc, A mixed integer linear programming model applied in barge planning for Omya, Journal of Operations Research Perspectives, Vol. 2, 2015, pp. 150– 155. [6] Christodoulos A. Floudas, Mixed Integer Linear Programming in Process Scheduling: Modeling, Algorithms, and Applications

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