Engineering Applications of Artificial Intelligence
Genetic algorithm for supply planning in two-level assembly systems with random lead times
Introduction
Component delivery planning in supply chains is a crucial issue for companies. By optimizing component supplies, enterprises can save money and increase customer satisfaction. In the literature of production planning and inventory control, most papers examine inventory systems where lead times are supposed to be equal to zero or constant. In reality, lead times are rarely constant; unpredictable events can cause random delays. Most often, lead time fluctuations strongly degrade system performance.
For different reasons (machine breakdowns, transport delays, or quality problems, etc.), the component lead times, i.e. time of component delivery from an external supplier or processing time for the semi-finished product at the previous level, are often uncertain. To minimize the influence of these stochastic factors, firms implement safety stock (or safety lead time), but stock is expensive. In contrast, if there is not enough stock, stockout occurs with corresponding tardiness (backlogging) cost. So, the problem is to minimize the total cost composed of holding and backlogging costs.
In Yano (1987) a problem of planned lead time calculation in a material requirements planning (MRP) environment for multi-level serial systems under lead time uncertainties is investigated. However, Candace Yano considered only the case of two and three stage (level) serial systems. The optimal planned lead times, for a one-level assembly system, are derived by Chu et al. (1993). Both Yano (1987) and Chu et al. (1993) proposed continuous inventory control models.
The objective of this paper is to extend these approaches for two-level assembly (several components are assembled at each level) systems. Taking into account the fact that MRP approach uses planning buckets (discrete time), a discrete inventory control model where decision variables are integer is developed.
In literature, few works model lead times as discrete random variables. In Dolgui and Louly (2002), a one-level inventory control problem with random lead times and fixed demand is considered, for a dynamic multi-period case. The authors provide a Markov model for this problem. In Ould Louly and Dolgui (2002), under the additional restrictive assumptions that the lead times of the different types of components follow the same probability distribution, and the unit holding costs per period are the same for all types of components, the optimal solution is obtained as a generalized Newsboy model. In Ould Louly et al. (2008), a Branch & Bound (B&B) procedure is proposed to calculate planned lead times for the same type of one-level assembly system, but where diverse components can have different probability distributions and the holding costs can also differ.
In Dolgui et al., 1995, Dolgui et al., 1996 and Dolgui (2001), the authors consider a similar one-level multi-period problem but with several types of finished products. No restrictive hypothesis is made on lead time random variables; the only assumption is that the distribution probabilities are known. An approach is suggested, based on the coupling of a linear programming model with a simulation using some heuristics. The same problem is also studied by Proth et al. (1997). However, they do not seek an explicit mathematical solution and so use simulations in order to estimate the costs and to find the parameters of an algorithm for optimization. The products to be assembled are selected on the basis of heuristics, generating a set of priorities for the finished products. The quantities of components to be ordered at the beginning of each period are also computed.
In this paper, a more complex two-level assembly system with discrete random lead times at each level is considered. The same assumptions as in Yano (1987) and Chu et al. (1993) are used: one-period model, one type of finished product, infinite capacity, fixed demand (due date is known, i.e., the end of the period), component lead times (delivery time for next level) are independent random variables (with known probability distributions). The problem is then to find optimal release dates for items at level 2 in order to satisfy the finished product demand for a given due date. In an MRP environment this corresponds to planned lead time calculation where planned lead time=due date—release date. If the release date is too early, the safety lead time is increased uselessly, and consequently surplus stocks are obtained and therefore additional holding costs. In contrast, with late release dates there is tardiness (backlogging), and so the corresponding costs are incurred for the finished product. Hence, the objective is to minimize the total cost which is composed of holding and backlogging costs.
The difficulties to solve this problem exactly reside in the following principal factors: (i) the dependence among the stocks of components at each level, (ii) the dependence among the levels, i.e. the propagation of delays and advances along the supply chain.
In Hnaien et al. (2008), lower and upper bounds on the objective function are suggested. By applying a B&B approach based on these lower and upper bounds, it is possible to solve exactly this problem. Nevertheless, the results of tests for the developed B&B have shown that this approach is limited to small size problems. In order to solve large scale problems, a genetic algorithm (GA) is proposed and tested here. Its performances among a wide range of instances are reported.
Section snippets
Problem description
The supply planning for two-level assembly systems is considered: the finished product is produced from components themselves obtained from other components (see Fig. 1).
This study is motivated by the problem of MRP parameterization under lead time uncertainties. More precisely, our endeavour is focused on the following: for each product, to calculate the planned lead times for its components, where the actual component lead times are uncertain.
The finished product demand D for a given due date
Mathematical model
Taking into account the fact that the different components on the same level do not arrive at the same time, there are stocks at level 1 and 2. If the finished product is assembled after the due date there is backlog and therefore a backlogging cost (see Fig. 2).
Then, the objective is to find, for each ci,2, the release dates −Xi,2, i=1,…, N2, in order to minimize the total of the holding costs for the components and backlogging cost for the finished product. Proposition 1 An explicit form for the total cost
Genetic algorithm
The problem considered in this paper has a non linear objective function with integer variables (see Eq. (10)). Due to the combinatorial explosion, the explicit enumeration of the entire search space becomes impossible when the size of the problem is significant.
The function EC(X) is polynomially computable in O(N2*U) time. It does not imply that the problem is polynomially solvable. The question whether problem is polynomially solvable or NP-hard is open. Nevertheless, the B&B based on our
Computational experiments
The GA described in Section 4 has been implemented in C++. The experiments were executed on a SUN UltraSPARC IIIi with 1593 MHz CPU and 16 GB of memory.
The experiment consisted in running the algorithm for 120 randomly generated instances. For theses instances, the number of components at level 2 varies within the range [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120]. For each number of components, 10 different instances were generated. The input data for each instance were: the distribution
Conclusion and perspectives
A problem, dealing with MRP parameterization for supply planning for two-level assembly systems under random actual lead times was studied. A new model was proposed with as criterion the sum of backlogging and inventory costs. In contrast with the known approaches which are usually based on continuous time models, the suggested method uses discrete optimization techniques with integer decision variables. This is more appropriate for MRP environment where the planning horizon is divided into
Acknowledgement
This text has been checked by native English speaker Chris Yukna.
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