Pergamon

Computers in& Engng VoL 31, No. 1/2, pp. 193 - 196,1996 Copyright O 1996 El~vier Science Lid

Printed in C-~a Brita~ All fish~ tmerved S0360-835~(96) 00109-X o36o- s352~6 Sl5.oo + o.oo

GLOBAL MANUFACTURING SYSTEMS: A MODEL SUPPORTED BY GENETIC ALGORITHMS TO OPTIMIZE PRODUCTION PLANNING

A. Claudio GARAVELLI ~, O. Geoffrey OKOGBAA 2, Nicola VIOLANTE I

1University of Basilicata, Via della Teenica 3, Potenza 85100, Italy 2University of South Florida, 4202 E. Fowler Avenue, ENB 118, Tampa 33620-5350, Florida, USA

Abstract Global competition emphasizes the benefits of a more coordinated production planning among the subsidiaries of a multinational company (MNC). In this paper, a model for the production assignment of the global demand to the MNC plants is proposed. The solution search, based on total cost optimization, is pursued by genetic algorithms. A simple example is provided to show the model implementation.

Keywords: global manufacturing planning, assignment problem, genetic algorithms, optimization

1. In troduct ion World-wide competition drives manufacturing companies to be global in both production and marketing activities. From the manufacturing pomt of view, the complexity of logistics and planning activities in a global environment, due to the management of remote production plants, is extremely important and frequently plays a strategic role for the company efficiency and effectiveness [ 1, 2]. Supply and product flows among plants and markets, often located in different countries, need then to be optimized in order not to waste the company's efforts to gain competitive advantages.

Despite in many cases the benefits of a network of highly connected and coordinated subsidiaries are emphasized for a multinational company [2, 3] and the general problem of activity coordination is widely discussed, literature does not provide a unified framework for this subject [4]. In a global manufacturing system, different product mix and quantities have to be assigned to plants in different countries, according to customer orders or demand forecasts. Besides the accomplishments of scheduling and technical requirements, the production assignments are based on economic considerations that cannot be easily handled. In fact, different costs (and currencies) varying dependmg on plant locations, product mix and quantities and specific contingencies have to be considered [5].

In this paper, a production planning model is proposed concerning the assignment of the global demand of a product to a MNC characterized by subsidiaries located in different countries. It provides a first step in the development of a more complex global manufacturing system planning (GMSP). In particular, the model is concerned with the optimization of the total cost associated with the product quantities to be manufactured in the different plants. In this model, that addresses an integer optimization problem, a genetic algorithm (GA) is used to find good solutions, since GAs have been emerging as a powerful tool to handle complex optimization problems [6]. An example of application is also provided to show the model implementation and test the solution search accuracy of GA in the particular problem.

2. G l o b a l m a n u f a c t u r i n g sys t em planning The configuration of a world-wide distributed network of plants allows a MNC to exploit the opportunities offered by both local and global markets. For instance, lower labor cost, better customer service and local technological resources [7] are some of the exploitable local factors, while the international gathering of small national market segments and the settlement of focused plants characterized by high product volumes (economy of scale), the transfer of organizational practices and know-how among the subsidiaries, the possibility of production shifting towards the most favorable locations [8] are some of the profitable global factors.

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Market trends continuously change global economic scenarios. For instance, many factors stress the importance for an MNC to effectively connect and coordinate its network of subsidiaries geographically dispersed, thus increasing the relevance of the "transnational" companies among MNCs [9]. An important role in this trend is played by communication technology innovations, which allow a stronger interrelation among remote plants [2].

A more coordinated and centralized production planning can then be achieved in order to increase MNC efficiency. In particular, Flaherty [10] points out how a centralized production planning allows to manage the local demand peaks that would saturate the capacity of plants serving usually that market. To a broader extent, this coordination is intended to support a set of decisions related to the operative activities of a MNC, such as production allocation, scheduling, sourcing, delivering. In this paper, however, it is considered mainly addressed to provide more dynamic and economically efficient production assignments to the different plants of a MNC, in terms of short-range planning [11]. The production planning is then aimed to pursue a better awareness of the economic opportunities offered by a global manufacturing system. Providing the supervision of real time plant costs and capacities, it is addressed to find the best set of feasible product and supply flows among plants and markets.

Among the advantages of the global production planning process, it can be stressed how supplier offers in different markets can be shared by subsidiaries, so that a centralized supply management is allowed with considerable economies of scale, and the demand from the different markets can be handled as a whole and re, allocated to production plants according to appropriate optimization criteria. Moreover, this process can allow MNC management to better evaluate the economic impact of shifting production among subsidiaries depending on specific contingencies, which are particularly frequent in a multinational environment.

3. A model for GMSP In this section, a model to support the production planning of a global manufacturing system is described In particular, the model concerns the problem of product demand assignment to the different plants of a MNC. Despite the importance that production and transport le~l times assume in the global manufacturing context, the model does not consider time as a decision variable, which will be investigated in further steps of the study. The model is also focused on the production assignment of only one type of product.

In the model, the relationship between cost and product quantity in the manufacturing system is represented by discrete functions. Among the main considerations that have lead to this choice, it can be observed that, as far as the production costs are concerned, the approximation of the variable production cost to a linear function of the product quantity, even if widely applied in both practice and literature, is often not accurate, while its representation by other continuous functions (such as quadratic ones) is often complex to be defined. Besides, as far as transportation costs are concerned, different combinations of transport means and product quantities to be delivered can be requested by the production plan (because of economic and/or schedule requirements, transport means availability, weather conditions etc.), so that complex computations can be required to define the total transport cost as a continuous function of the product quantity. Finally, also other costs, such as taxes and custom fees, increase the number of combinations of different costs to be added to obtain the final required cost [5]. Every discrete product quantity that can be assigned to the plants is then characterized by a specific cost value which can be up- to-dated by the subsidiaries when required by the plan setting, thus allowing to easily take into account specific contingencies (such as those related to transport availability, currency rate exchange fluctuation, etc.).

Every market considered in the model is characterized by a specific product demand for the company, in terms of forecasts or customers' orders. These demands are given inputs for the production planning process. Every plant has no constraints in satisfying market demands, except its production capacity and costs, the latter being subject to the other subsidiaries' competition. As a consequence, a flow of products between each couple of final plant and market is allowed. The product quantities which characterize each of these flows represent the unknown variables of the problem.

According to the assumptions made, a binary integer optimization problem can be addressed, where the objective function to be minimized is the total cost CT, sum of production and transportation costs. In particular, the binary unknown variables X,jk can be associated to the product quantities qijk delivered from a plant i of capacity Ci to a market j of demand Dj at the cost Cijk. The X~jk values (0, 1) stand for either non-activated or activated flow, respectively, with the indices i, j and k representing the plant (i = 1, ...,

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Np), the market ( /= 1, ..., Nm) and the product quantity level (k = 1 ..... Nq), respectively. No horizontal product flow between plants or markets is allowed. As an example, in Figure 1 a model of two final plants (i = 1, 2), two markets (j = 1, 2) and two quantity levels (k = 1, 2) is reported. Inputs and outputs of the production planning model of the global manufacturing system are reported in Table 1.

Plant 1 " ~ xm ~ ( " Market 1 1

X122 ~ X212

Figure 1. A model scheme with 2 markets and 2 plants.

Table 1. Inputs and outputs of the optimization model.

Inputs characterizing the manufacturing system

Number of markets Number of plants

Inputs characterizing the specific planning

Markets' demands Plants' capacities Number and values of discrete quantity levels Costs associated with each product quantity

Outputs

Product quantities to be assigned to each plant

Total cost performed by the manufacturing system

In order to search good solutions for the production planning problem, a genetic algorithm is considered. This choice is due to the usual size of the optimization problem in the global manufacturing systems environment, that often requires a rapid solution as close as possible to the optimal one, as well as to the high potentialities shown by GA in this field, as reported by the literature [6]. In particular, the main aspects which make GA particularly interesting in combinatorial integer programming are both the capacity (basically due to the mutation process) of searching the optimal solution in a wide range of solutions, not allowing a premature convergence towards a local solution, and, at the same time, the possibility of searching new solutions without losing the best solution features already found, due to a peculiar mechanism of solution generation (the crossover process), allowing thus to increase the search speed [12]. For the aims of this paper, the basic aspects of GA are not discussed, since a quite wide literature is available for references (for instance, [13, 14]). However, the sequential steps of the solution search with some of the main characteristics of the GA model are here shown for the specific application.

The main steps of the optimal solution search by the GA model can be synthesized as follows. After the definition of the main model parameters (population size S, i.e. number of solutions (chromosomes) concurrently handled, crossover probability Pc and mutation probability Pro), the first population of chromosomes is randomly generated. A random selection of the population chromosomes then takes place in order to form couples of chromosomes (parents) for the crossover operation. By the application (with probability Pc) of crossover to the couples of parents, a new population is chosen among all the parents and sons, according to the lowest total cost CT associated to the chromosomes. This phase defines a form of "population elitism" aimed not to destroy the best solutions of both the old and the new generations. After crossover, a fraction Pm of the binary variables (genes) of each chromosome is varied (mutation phase). A new population of chromosomes is then selected choosing the best solutions among the "ante- mutation" and the "post-mutation" populations. This selection, guided by the same criterion of the post- crossover phase, represents another population elitism aimed to set a high mutation rate Pm which, despite causing the destruction of good and feasible solutions, also provides a broad new generation of different solutions. A new cycle is then run, starting with the last obtained population, until G cycles are run (i.e., G populations are generated).

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4. A simple example of application In order to test the GA effectiveness in the specific problem, a GA model has been built to be applied to a scenario characterized by 2 markets' demands of a product and a global manufacturing company characterized by 3 plants. The example has been modeled with 5 levels of product quantities between each plant and market. The main problem data and model parameters are reported in Table 2. In particular, 5 quantity levels qi of 10 units each have been considered. For the plant P2, having capacity C=40, a high cost Cijs = 1000 has been set in correspondence of the level q5=50, in order to inhibit the selection of that level during the simulation. The simulation was run 20 times, each for G=500 generations. The model found the best solution, that is CT=142, 18 times over 20, with a mean time of G=80. In the other two cases, at the end of the simulation the solution found by the model was Cr=143.

Table 2. Main model parameters and problem data.

Ciik S = 60 Pc = 1.0 P~ (50) Pm = 0.1 P2 (40) G = 500 P3 (50)

Mt (70) I Ms (60) ql q2 q3 q4 q5 I q~ q2 q3 q4 q5 1 0 2 1 32 40105300 1 2 2 5 37 49 61 11 22 35 46 10 21 33 41 1000 12 26 39 48 56 13 23 35 48 62

5. Conclusions A GA model has been proposed for the production assignment of the global demand of a product to the subsidiaries of a MNC. A simple example of application has shown the effectiveness of GA in providing good solutions for the problem. Due to the simple assumptions made and to the complexity of the global environment in which MNCs usually operate, the model can be considered a first step towards the definition of a broader manufacturing system planning. Further developments are addressed to the production assignment modeling of more complex environments, characterized by a larger number of products, markets, subsidiaries and suppliers, to the test of GA (and/or other techniques) in those contexts, and to lead times consideration in the model as a fundamental element of the planning process.

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