Application of Node Based Coincidence Algorithm for Solving Order Acceptance with Multi-process Capacity Balancing Problems

Over the past decade the strategic importance of order acceptance has been widely recognized in practice. This paper presents the application of node based coincidence algorithm to solve the order acceptance problem with multi-process capacity. The results show that Node Based Coincidence Algorithm (NB-COIN) is a potential algorithm which can maximize both profit and can maximize the capacity used at the same time. KEYWORD: Order Acceptance; limited capacity; node based coincidence algorithm; genetic algorithm f!"#$% Time unit that workstation n utilize CTP! for product k in order i by job j at period t g! Cost of unassigned capacity of workstation n CTP! per time unit α! Cost rate of leftover capacity at workstation n d!" Amount of leftover capacity at workstation n Order Constraint p!" Profit of order i q!"# Demand quantity of product k in order i due at period t Decision Constraint R!" = 1, if the order i for product k is accepted = 0, otherwise F!"#$% = 1, if the order i for product k is produced at workstation n by job j at period t = 0, otherwise Model Objective Maximize Z = p!"q!"#R!" ! ! ! − α! ! d!"g! ! (1) Subject to Workstation-level activities Constraint e!"#q!"#× ! ! ! R!"#$ ≤ CTP!" ! ∀n (2) d!" = RT! − CTP!" − e!"#q!"#f!"#$ ! ! ∀n (3) Order-level activities Constraint f!"#q!"# ≥ F!"#$ ∀i, j, k,n, t (4) f!"#q!"# ≤ e!"#q!"#F!"#$ ∀i, j, k,n, t (5) tF! ! !"# ! ≤ D!R!" ∀i, k, t (6) f!" !!! q!"# ́ ! ! + f!"#q!"# ! ≥ e!"(!!!)q!"# ! F!"#$ ∀i, j/ 1 , k, t (7) Binary and non-negativity Constraint R!" = 0or1 ∀i, k (8) F!"#$% = 0or1 ∀i, j, k,n, t (9) f!"#$% ≥ 0 ∀i, j, k,n, t (10) This problem is considered to be a two objectives optimization problem. However, the two objectives are bind into one single objective. The objective function consists of two parts (i) to maximize the total profit and (ii) to minimize the leftover capacity. Generally speaking, the objective is to choose the set and sequence of the profitable orders using as much working capacity as possible. The leftover capacity is considered to have some certain penalty cost. The first set of constraints is established to ensure that the whole capacity of production plant is not disrupted. Constraint (2) was set to calculate the penalty of under capacity utilization. Constraints (3) and (4) sets the Fijkrt decision variables to either 1 or 0. The Fijkrt is the indicator variable; it becomes 1 when fijkrt > 0, indicating that job j of item i is being processed on resource k in period t, otherwise it becomes 0. The Fijkrt variable is used to ensure the precedence relationship. The constraint set (5) ensures that when an order for an item is accepted, the completion time of the final job of that order does not exceed the order due date. The constraint set (6) imposes precedence restrictions to ensure that job j of item i can be processed in period t only after completing job j-1. 2.2 Solution Procedures This work compares the result of Node Based Coincidence Algorithm (NB-COIN) (Waiyapara et al. 2013) with Genetic Algorithm (GA) (Syswerda 1991). The algorithms are modified such that they would consider only the accepted sets of orders. 2.2.1 Node Based Coincidence Algorithm NB-COIN is a permutation based Estimation of Distribution Algorithm (EDA). It generates solution strings in sequences, ensuring that only valid permutations are sampled. NB-COIN is a variation of Coincidence Algorithm (COIN) proposed by Wattanapornprom and others (2013). It uses a data structure called coincidence matrix H to model substructures from absolute positions. The matrix Hxy represents the probability of y found in the absolute position x. The update equation of NB-COIN is H!" t + 1 = H!" t + k n r!" t + 1 − p!" t + 1 + ! ! ! p!" t + 1 !!! − r!" t + 1 !!! (11) where k denotes the learning step, n is the problem size, rxy is the number of xy found in the bettergroup, and pxy is the number of xy found in the worse-group. The incremental and detrimental step is ! !!! , and the term ! !!! ! p!" t + 1 !!! − r!" t + 1 !!! represents the adjustment of all other Hxj, where j ≠ x and j ≠ y. After each population was evaluated and ranked, two groups of candidates are selected according to their fitness values: better-group and worse-group. The better-group is selected from the top c% of the rank and is used as a reward, and Hxy is increased for every pair of xy found in this group. The punishment is a decrease in Hxy for every pair of xy found in the worse group of the bottom c% of the population rank. The pseudo code of NB-COIN is simplified as follows: Step 1 Initialize the model Step 2 Sample the population Step 3 Evaluate the population Step 4 Select candidates Step 5 Update the model Step 6 Repeat steps 2 to 5 until terminated. 2.2.2 Genetic Algorithm The GA used in this research is the permutation based GA with Position-based crossover (PBX) (Syswerda 1991). PBX preserves not only absolute order substructures but also relative order substructures from two parents. Figure 1 illustrates the steps and the example of PBX. The proto offspring 1 mimics the absolute order substructures from the parent 1 and then imitates the relative sequence order of the remaining substructures from the parent 2 and vice versa. For this problem, the chromosomes are sequenced subsets of jobs. The diversity is maintained by ancestor replacement. If a new candidate is better than its ancestors it is used to replace one of its own parents. In this study, the local search is also applied to the new candidates with improvement. The swapping and insertion operations are randomly applied to the candidates until the candidates are no longer improved. The pseudo code of GA is as follows: Step 1 Randomly generate the population. Step 2 Evaluate the population. Step 3 Perform crossover and mutation. If the newly generated candidate is better than its ancestors, then perform the local search until the candidate is no longer improved. Step 4 Repeat Step 3 until the maximum number of generation is reached. Although the encoded solution of GA is a full set of the jobs in the pool, the evaluation process considers only the accepted orders. The evaluation process not only evaluates the orders sequence, but also re-sorts the orders sequences to separate the accepted and rejected orders as illustrated in the Figure 2. The sequence of the accepted orders is kept in the accepted pool while the remaining orders are kept in the rejected pool. The candidate solution is re-sorted by concatenating the accepted pool with the rejected pool. Figure 1. Position-based crossover (PBX). Figure 2. Evaluation with cutting off. Even though, GA and NB-COIN are in the same group of evolutionary algorithms, however, the evaluation process and the updating process of NBCOIN for the order acceptance are slightly different. GA needs to maintain the genetic materials, therefore the whole set of orders need to be maintained. However, NB-COIN can reproduce the missing sequences by itself. In addition, the sequences of the rejected pool are considered to be the useless information, therefore, NB-COIN only updates the models from the accepted sequences of orders. Consequently the evaluation process does not need to concatenate the rejected pool with the accepted pool. The evaluation processes in the figure 2 simply use the accepted pool as the candidate for the NB-COIN. 2 4 1 3 7 6 9 8 5