Soft Computing Based Procurement Planning of Time-variable Demand in Manufacturing Systems

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1 International Journal of Automation and Computing 04(1), January 2007, DOI: /s x Soft Computing Based Procurement Planning of Time-variable Demand in Manufacturing Systems Kai Leung Yung 1 Wai Hung Ip 1 Ding-Wei Wang 2 1 Department of Industrial and Systems Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, PRC 2 School of Information Science and Engineering, Northeastern University, Shenyang , PRC Abstract: Procurement planning with discrete time varying demand is an important problem in Enterprise Resource Planning (ERP). It can be described using the non-analytic mathematical programming model proposed in this paper. To solve the model we propose to use a fuzzy decision embedded genetic algorithm. The algorithm adopts an order strategy selection to simplify the original real optimization problem into binary ones. Then, a fuzzy decision quantification method is used to quantify experience from planning experts. Thus, decision rules can easily be embedded in the computation of genetic operations. This approach is applied to purchase planning problem in a practical machine tool works, where satisfactory results have been achieved. Keywords: control. Purchase planning, Enterprise Resource Planning (ERP), soft computing, genetic algorithm, fuzzy decision, inventory 1 Introduction Procurement planning is a key functional module in Enterprise Resource Planning (ERP) and Supply Chain Management (SCM) systems [1,2]. The optimization of material purchase planning can lead to savings in operating costs, guarantee material supply to production, and bring greater profits in manufacturing systems [3]. This is most important in regions of low labor cost, where material cost dominates the product cost structure. The procurement of purchase planning is to determine the optimal material stock replenishment strategy to minimize total cost (which consists of purchasing, ordering, and holding costs, and penalties for shortages). Traditional inventory control strategies are mostly based on the famous EOQ model and its various extensions [4]. The basic assumption of EOQ is that demand is independent and described by a mean value with unprotected white noise. In fact, it is assumed that demand is a constant equal to its mean. The issue of inventory control methods for time-varying demand has recently attracted a great deal of research interest. In the studies that have been conducted thus far, the demand rate is usually considered in terms of a linear, exponential, logconcave, or some other time-continuous or stock-dependent function [5 8]. Such studies have mainly focused on the theory of optimal order quantity or order strategy [9]. As a key functional module of ERP, procurement planning faces the situation of the discrete time problem. Its demand is no longer independent. Rather, it strongly depends on production planning or on material requirement planning (MRP). Procurement planning usually cannot be described as a simple analysis function of time. Via MRP, we can obtain the exact requirement of all materials for each day in a planning horizon. It is very popular for ma- Manuscript received November 18, 2005; revised February 25, This work was supported by Hong Kong Polytechnic University (No. G T363), and the National Natural Science Foundation of PRC (No , ). *Corresponding author. address: dwwang@mail.neu.edu.cn terial suppliers to give a big discount on purchase of large amounts of goods. But a larger purchase may lead to higher inventory and holding costs. Therefore, our problem is to study how to carry out optimal procurement planning to meet the material requirements of production and to minimize total cost including purchasing, ordering, and holding costs, and penalties for shortages with discrete time-varying demand and different discount prices. The above problem can be described as a mathematical programming model similar to a production planning model with variable demand [10]. However, owing to the existence of non-analysis factors such as discount prices, ordering costs, and penalties for shortages, the model cannot be solved using common mathematical programming methods. Various problems in procurement planning optimization have been discussed in the literature. The optimal procurement planning of equipment for semiconductor manufacturing and software components for enterprise information systems were introduced in [11,12]. Similar problems dealing with material procurement planning were solved using geometric programming and branch and bound approaches in [13,14] separately. Since Zadeh first introduced the concept of soft computing [15], it has caught the interest of many researchers [16,17]. The basic idea of soft computing is to use hybrid intelligent methods to quickly achieve an inexact solution rather than use an exact optimal solution via a long search [18]. Since GAs are good for adaptive studies and fuzzy logic can be used to solve complex problems using linguistic rule-based techniques [19 22], the combination of GA and fuzzy logic is one of the most promising available hybrid intelligent methods with which to solve complicated optimization problems [23,24]. Wang et al developed a fuzzy decision embedded genetic algorithm for the fuzzy due date bargaining problem and generalized the idea to the quantification of fuzzy rules [24,25]. This method has been successfully applied to some practical problems such as partner

2 K. L. Yung et al./soft Computing Based Procurement Planning of Time-variable Demand in Manufacturing Systems 81 selection, and production planning and scheduling [26,27]. To solve the procurement planning problem, we apply fuzzy decision embedded genetic algorithm again. By summing the experiences of planning makers, we propose new decision rules for the problem. To avoid complex real optimization, we conclude four order strategies for different situations of demand and inventory. These methods are all integrated into a procurement planning module. The results of the computation show us that the module has good potential to be embedded in ERP systems or be directly applied to purchase sections of practical enterprises. The remainder of this paper is organized as follows: A formal description of the problem and model are presented in Section 2. Possible order strategies and fuzzy decision rules are then discussed in Section 3. Section 4 introduces a decision embedded genetic algorithm. Computation examples are presented in Section 5. Finally, concluding remarks are provided in Section 6. 2 Problem description and model Assume the planning horizon is T. A manufacturer usually purchases n kinds of materials from a partner supplier. Via an MRP module in ERP, we know that the requirements for material i at time t are d i(t), i = 1, 2,, n, t = 1, 2,, T. The supplier will charge the manufacturer a constant ordering cost of A, if he/she orders any materials from the supplier. The holding cost for a unit of material i for a given time period is h i, i = 1, 2,, n. The initial inventory level of material i is I i(0), i = 1, 2,, n. Once a shortage of the material occurs, there is a shortage penalty cost of material i for a unit shortage and a unit time s i, i = 1, 2,, n. The purchasing cost for material i is discounted depending on the quantity of the order and is described by a segment function as follows: 8 >< c 1 i, Q 1 i q i Q 2 i c i(q i) = c 2 i, Q 2 i q i Q 3 i, i = 1, 2,, n (1) >: c 3 i, Q 3 i q i where c 1 i c 2 i c 3 i are the discounted costs for different order quantities of Q 1 i, Q 2 i and Q 3 i of material i, i = 1, 2,, n. The warehouse capacity of the manufacturer is limited to the volume V. The space volume of material i for one unit is v i, i = 1, 2,, n. Due to financial control by the manufacturer, there is a limitation of cash flow for purchase each day, of W. The objective of the manufacturer is to achieve optimal procurement planning (q i(t), i = 1, 2,, n, t = 1, 2,, T ) so as to minimize total cost including ordering, holding, and purchasing costs, and a possible shortage penalty. The problem can be described in terms of the following formula: min q Z(q) = T n n A sgn[ q i(t)] + {[c i(q i)q i(t)]+ t=1 h i i=1 i=1 t [q i(τ) d i(τ) + I i(0)] + + τ=1 s.t. n i=1 v i s i t [d i(τ) q i(τ) I i(0)] + } (2) τ=1 t [q i(τ) d i(τ) + I i(0)] + V τ=1 t = 1, 2,, T (3) n c i(q i)q i(t) W, t = 1, 2,, T (4) i=1 q i(t) 0, i, t (5) where [x] + means max{0, x}, and sgn{x} is the signal function, i.e. 8 >< 1, x > 0 sgn{x} = 0, x = 0. (6) >: 1, x < 0 We see that the first term in (2) the is the ordering cost. It applies when there are any order quantities greater than zero. The second is the purchasing cost, and the third and fourth are the holding cost and shortage penalty, respectively. Because the model includes some non-analytic terms such as [x] +, sgn{x}, and the segment function c i(q i), it cannot be solved using common mathematical programming. Therefore, an intelligent algorithm or soft computing method will be the only choice in order to obtain a solution. 3 Purchase strategies and fuzzy rules 3.1 Binary encoding scheme The above problem is a nonlinear and non-analysis constrained real optimization problem. The number of real variables q i(t), i = 1, 2,, n, t = 1, 2,, T is equal to n T. If we take these real variables as a gene representation, an optimum search would be very inefficient. Fortunately, planners can sum a limited number of different order strategies drawing on their experiences in a case where the next order point is determined. Therefore, we can encode the chromosomes of a genetic algorithm using a binary string. Let ( 1, period t is an order point x t = (7) 0, period t is not an order point where x t is the t-th gene of chromosome x. x t = 1 means that at least q i(t) > 0, i.e. the enterprise at least orders a kind of material in period t. Therefore, x = [x 1, x 2,, x T ] is the gene representation of the genetic algorithm. example with T = 7, x = [ ] For means there is an order in periods 1, 4, and 6. The binary encoding scheme is much simpler than the original real number encoding scheme. The gene number is just T rather than n T. The possible bit values are only 0 and 1 rather than the whole space of a positive real

3 82 International Journal of Automation and Computing 04(1), January 2007 number. The search process should be much easier than before. However, the problem remains of how to determine optimal order quantities for all materials in the case where the next point is fixed. 3.2 Order strategies in fixed order points By making reference to the experience of procurement planning makers, there are only three choices of possible order strategies for a material: no order, order just to meet demand, or order just discount points. Let t 0 and t 1 be the current and next order points, and D i be the requirement for material i from t 0 to (t 1 1), i.e. D i = k=t 0 d i(k). (8) L = t 1 t 0 is the duration length of this order cycle. I i(t 0 1) is the initial inventory of material i, i = 1, 2,, n. The possible order strategies for a material and its operating cost can be described as follows: 1) No order For a no order strategy, the order quantity is Its operating cost O 1 i is q i(t 0) = 0. (9) Oi 1 = {h i[i i(t 0 1) d i(τ)] + + k=t 0 τ=t 0 s i[ τ=t 0 d i(τ) I i(t 0 1)] }. (10) 2) Order just enough for the period For this strategy, the order quantity is Its operating cost is q i(t 0) = D i I i(t 0 1) (11) Oi 2 = D ic i(d i)+ h i[i i(t 0 1)+D i d i(τ)] +. (12) k=t 0 τ=t 0 We note that there is no shortage penalty. The order quantity is just the MRP order strategy to order the net requirement [1]. 3) Order in discount point Q 2 i For this strategy, no mater how much D i is, the order quantity is q i(t 0) = Q 2 i. (13) Its operating cost is Oi 3 =Q 2 i c 2 i + {h i[i i(t 0 1) + Q 2 i d i(τ)] + + k=t 0 τ=t 0 s i[ τ=t 0 d i(τ) Q 2 i I i(t 0 1)] + }. (14) 4) Order in discount point Q 3 i For this strategy, no mater how much D i is, the order quantity is q i(t 0) = Q 3 i. (15) Its operating cost is Oi 4 =Q 3 i c 3 i + {h i[i i(t 0 1) + Q 3 i d i(τ)] + + k=t 0 τ=t 0 s i[ τ=t 0 d i(τ) Q 3 i I i(t 0 1)] + }. (16) Once the operating cost for each possible order strategy is known, we can determine the best strategy by comparing their operating costs, i.e. if k i = arg min k {O k i, k = 1, 2, 3, 4} (17) the order strategy k i is taken for material i, i = 1, 2,, n. It is evident that the four possible order strategies cannot guarantee a global optimum, because the strategy for selection does not consider the effect of neighboring order cycles. 3.3 Fuzzy decision rules Although a genetic algorithm without a fuzzy rule can be used to solve our problem, its performance of computation is not very good. Therefore, we consider embedding fuzzy rules from people s previous experience so as to improve the performance of the GA. The basic idea of a fuzzy decision embedded genetic algorithm is to quantify fuzzy decision rules from people s experience [24,26,27]. The procedure of the quantification of fuzzy rules is to determine all factors necessary to effect decisions and to describe the rules of decisions through the fuzzy operation of these factors [24,26,27]. For our problem, factors required to effect decisions are listed as follows: Factor 1. A higher ordering cost for a given chromosome. Factor 2. A higher holding cost for a given chromosome. Factor 3. The existence of a penalty if daily purchase exceeds the amount of money allotted to the purchase. Factor 4. The existence of a penalty if total inventory exceeds stock capacity. Factor 5. A higher shortage cost for a given chromosome. Factor 6. A higher purchasing cost for a given chromosome. Factor 7. A lower total cost for a given chromosome. Within these factors, Factors 3 and 4 are crisp. Once the penalties for exceeding a purchase quota or warehouse capacity exist, the membership functions will be 1, otherwise 0. As we know, ordering cost and holding cost are contradictory; a higher ordering cost means more order times and a lower holding cost. Let Fmin 1 and Fmax 2 be the ordering cost and holding cost in a feasible solution with the fewest order times, respectively. Let Fmax 1 and Fmin 2 be the ordering cost and holding cost of the feasible solution with maximum allowed order times. Under these conditions, a pair of fuzzy factors can be described by the following membership functions: 8 1, z Fmax >< 1 (z F µ F1 (z) = min) 1 (Fmax >: 1 Fmin 1 ), F min 1 z < Fmax 1 (18) 0, z < Fmin 1

4 K. L. Yung et al./soft Computing Based Procurement Planning of Time-variable Demand in Manufacturing Systems 83 and 8 1, z Fmax >< 2 (z F µ F2 (z) = min) 2 (Fmax >: 2 Fmin 2 ), F min 2 z < Fmax 2 0, z < Fmin 2 (19) where F 1 and F 2 stand for fuzzy factors 1 and 2. The curves of the two membership functions are shown in Fig. 1. so as to reach feasibility. Rule 2 means that if there is a higher shortage or lower ordering cost and higher holding cost, then we should take Decision 2, i.e. add one more order. Rule 3 means that if there is a higher purchasing cost (no discount) or lower holding cost and higher ordering cost, then we should take Decision 3, i.e. delete one time of order. Rule 4 is simple; it states that if the total operating cost is lower, the chromosome is a better chromosome and should remain unchanged. These decision rules can be described in terms of the following formulas: µ D1 = max{µ F3, µ F4 } (20) µ D2 = max{µ F5, µ F2 (1 µ F1 )} (21) µ D3 = max{µ F6, µ F1 (1 µ F2 )} (22) µ D4 = µ F7. (23) Therefore, according to the decision selection method of fuzzy decision quantification, if k = arg max{µ D1, µ D2, µ D3, µ D4 } (24) (a) The membership function of Factor 1 to improve the given chromo- then we select Decision k some. 4 Fuzzy decision embedded genetic algorithm (b) The membership function of Factor 2 Fig. 1 The membership functions of Factors 1 and 2 Similarly, we can determine membership functions for the other fuzzy factors 5, 6, and 7. By making reference to the experience of experts, four possible decisions for improving purchase planning were acquired as follows: Decision 1. Repair the feasibility of a chromosome. Decision 2. Add one time of order to a chromosome. Decision 3. Delete one time of order from a chromosome. Decision 4. Keep a chromosome unchanged. These decision rules can be described using IF..., THEN.... statements as follows: Rule 1. IF (F 3 F 4), THEN D 1. Rule 2. IF ( F 5 F C 1 F 2), THEN D 2. Rule 3. IF ( F 6 F C 2 F 1), THEN D 3. Rule 4. IF ( F 7), THEN D 4. where F C is the complementary factor of Factor F, is the operation to select the maximum value, and is the multiplication operation of the factors. Rule 1 means that penalties for exceeding money quota or warehouse capacity exist. Therefore, we need to take Decision 1 to improve the purchase planning or chromosome, Fig. 2 A flowchart of GA/ FD The basic procedure for the recommended fuzzy decision embedded genetic algorithm is to select the best order point

5 84 International Journal of Automation and Computing 04(1), January 2007 (a) Material 1 (b) Material 2 (c) Material 3 Fig. 3 The demand, purchases and inventories of three materials

6 K. L. Yung et al./soft Computing Based Procurement Planning of Time-variable Demand in Manufacturing Systems 85 combination by the genetic process. For each chromosome with fixed order points, we determine the best order strategy with a lower operating cost for each order cycle. After the order strategies for all order cycles of the chromosome are fixed, we calculate its ordering, holding, and purchasing costs, and shortage penalty, and check its feasibility. Then, we call the fuzzy decision subroutine to improve the chromosome. Therefore, the computation performance of the genetic algorithm can be greatly advanced. For the genetic algorithm, we adopt two cutting crossover and an altering mutation as genetic operators, proportional selection as a selection strategy, and give a maximum generation number as a stopping criterion [20,22]. The linear scaled objective value of (2) is taken as a fitness function [20,22]. The step-by-step procedure for GA/FD can be described as follows: Step 1. Specify the parameters: population size NP, the maximum number of generations NG, crossover probability P c, and mutation probability P m. Step 2. Generate an initial population x(j), j = 1, 2,, NP, randomly. Set the iteration index K = 0. Step 3. Based on the initial population and on the experience of decision-makers, determine parameters to describe the fuzzy factors: Fmin, k Fmax, k k = 1, 2, 5, 6, 7. Step 4. For chromosome x(j), j = 1, 2,, NP, determine the best order strategy for each order cycle (by comparing their costs calculated using formulas (10), (12), (14), and (16)). Calculate the ordering, holding, purchasing, and shortage costs, and total costs Z(j). Step 5. K K +1, if K > NG, go to Step 10; otherwise, perform the following Steps 6 9. Step 6. Call the fuzzy decision subroutine to calculate the membership values of all factors and select the best decision to improve the chromosome. Calculate the total cost of the revised chromosome z (j). Step 7. If z (j) < z(j), replace the original chromosome by the revised one; otherwise, maintain the original unchanged. Step 8. According to the probabilities P c and P m, select chromosomes to perform crossover and mutation. Step 9. Use proportional selection to update the population. Update the best achieved solution x and best value z. Go to Step 4. Step 10. Output the best achieved solution x and best value z, then stop. A flowchart of the fuzzy decision embedded genetic algorithm is shown in Fig. 2. Note that Steps 1 3 initialize the algorithm and generate the initial population of the GA. Step 4 is the stopping criterion. Steps 5 9 are the backbone of the GA/FD. Among them: Step 5 determines the order strategies for all order cycles; Step 6 uses a fuzzy decision to improve the chromosome; Step 7 checks if the improvement is better than the original; Steps 8 and 9 are the common steps of a genetic algorithm for genetic operations and updating the population, while the final Step 10 outputs the achieved result. 5 Numerical analysis The algorithm was coded via FORTRAN 90 and executed on a Pentium IV/2.60G computer. Satisfactory results were achieved. A practical example from a machine tool works was conducted as follows. An enterprise was contracted to a material supplier for three kinds of materials. The planning horizon was 30 days. The fixed ordering cost for each order was 50 (the unit of currency was the RMB of China). The money quota for daily purchase was 500. The warehouse capacity was 480. The discount levels and prices for all materials are shown in Table 1. Table 1 Discount levels and prices of purchasing materials Material i Level Q 1 i Price c 1 i Level Q 2 i Price c 2 i Level Q 3 i Price c 3 i The holding costs, shortage penalties, volumes of unit materials and initial inventory levels are shown in Table 2. Table 2 Data for holding cost, shortage penalty, volume, and initial inventory Material i Holding Cost h i Shortage Penalty s i Unit Volume v i Initial Inventory I i(0) From the MRP module, we obtained the daily demands for each material as shown in Table 3. Table 3 The daily demands for all materials from ERP Day t d 1(t) d 2(t) d 3(t)

7 86 International Journal of Automation and Computing 04(1), January 2007 The parameters of the genetic algorithm were: NP = 100, NG = 800, P c = 0.80, and P m = The best solution achieved is shown in Table 4. The daily purchase money requirement and daily total volume of the inventory are shown in Table 5. The best value was Table 4 Best achieved purchase planning Order Point t q 1(t) q 2(t) q 3(t) From Table 5, it can be seen that the daily money requirement and total volume are both not over quota or warehouse capacity. The demands, purchases, and inventory levels of the three kinds of materials are shown in Fig. 3. From the figure, it can be seen that there are eight orders in the planning horizon. Each purchase makes an order period. For each order period, the final inventory is always zero. Due to a higher shortage penalty, a shortage never occurs in this kind of planning. When comparing the three kinds of materials, the purchase amount of material 3 is not usually large. This is because its holding cost is higher than those of the others. Since a heuristic order strategy and fuzzy decision rules are used in this recommended approach, the achieved result is not guaranteed to be optimal. But the recommended approach can quickly and efficiently reach the best solution in a short time. This is simply a characteristic of soft computing. Table 5 Daily purchase money and inventory volume of the solution Day t Purchase Inventory Day t Purchase Inventory money volume money volume Conclusions This research in modeling and optimization for purchase planning in ERP has led us to the following conclusions: 1) The procurement planning problem with a discrete time varying demand is a typical peculiarity of ERP. The proposed mathematical model can describe this kind of problem very well. 2) The proposed methods for order strategy selection and fuzzy decision rules can simplify planning optimization procedures. Therefore, a fuzzy decision embedded genetic algorithm can solve problems of a practical size quickly and efficiently. 3) Since heuristics are adopted in the recommended approach, the algorithm cannot guarantee the optimality of its solution. We think that it is a characteristic of soft computing that searches for inexact solutions quickly and efficiently. References [1] S. Avraham. Enterprise Resource Planning (ERP): the Dynamics of Operations Management, Kluwer Academic Publishers, Boston, [2] S. Biswas, Y. Narahari. Object Oriented Modeling and Decision Support for Supply Chains. European Journal of Operational Research, vol. 153, no. 3, pp , [3] W. H. Ip, Y. Li, K. F. Man, K. S. Tang. Multi-product Planning and Scheduling Using Genetic Algorithm Approach. Computers & Industrial Engineering, vol. 38, no. 3, pp , [4] E. A. Silver, R. Peterson. Decision Systems for Inventory Management and Production Planning, John Wiley & Sons, New York, [5] Z. T. Balkhi, L. Benkhrouf. 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8 K. L. Yung et al./soft Computing Based Procurement Planning of Time-variable Demand in Manufacturing Systems 87 [20] M. Gen, R. Cheng. Genetic Algorithms and Engineering Design. Wiley, New York, [21] J. H. Holland. Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, MI, [22] Z. Michalewicz. Genetic Algorithms + Data + Structures = Evolution Programs. 2nd ed., Springer-Verlag, New York, [23] H. J. Zimmermann. Fuzzy Set Theory and Its Applications. Kluwer-Nijhoff, Hinghum, Netherlands, [24] D. Wang, S. C. Fang, H. L. W. Nuttle. Soft Computing for Multi-customer Due-date Bargaining. IEEE Transactions on Systems, Man, and Cybernetics Part C, vol. 29, no. 4, pp , [25] Y. Yi, D. Wang. Soft Computing for Scheduling with Batch Setup Times and Earliness-tardiness Penalties on Parallel Machines. Journal of Intelligent Manufacturing, vol. 14, no. 3-4, pp , [26] D. Wang, K. L. Yung, W. H. Ip. A Heuristic Genetic Algorithm for Subcontractor Selection in a Global Manufacturing Environment. IEEE Transactions on Systems, Man, and Cybernetics Part C, vol. 31, no. 2, pp , [27] D. Wang, K. L.Yung, W. H. Ip. Partner Selection Model and Soft Computing Approach for Dynamic Alliance of Enterprises. Science in China: Series F, vol. 45, no. 1, pp , Kai Leung Yung received his B.Sc. degree from Brighton University, UK, in 1975, the M.Sc degree from Imperial College of Sci. & Tech., University of London, UK, in 1976, and the Ph.D. degree from Plymouth University, UK, in He has worked for BOC Advanced Welding Co. Ltd., British Ever Ready Group, and the Cranfield Unit for Precision Engineering, in UK. He is currently the associate head and professor of the Department of Industrial and Systems Engineering, the Hong Kong Polytechnic University. His research interests include precision motion control, systems aspects of computer integrated manufacturing and management, and logistic planning and optimization. Wai Hung Ip received his M.Sc degree in industrial engineering from Cranfield University, U. K., and the MBA degree from Brunel University, U. K. He was awarded Ph. D. degree in manufacturing engineering from Loughborough University, UK, in He is currently an associate professor at the Department of Industrial and Systems Engineering, the Hong Kong Polytechnic University. His research interests include AI-based optimization and decision making, information systems and decision support systems. Ding-Wei Wang received his B.Sc. degree from Northeastern University, China, in 1982, the M.Sc degree from Huazhong University of Science and Technology, China, in 1984, and the Ph.D. degree from Northeastern University, China, in He had worked as a postdoctor with North Carolina State University, USA. He is currently a professor of the Institute of Systems Engineering, Northeastern University, China. His research interests include ERP/MRP- II/JIT, modeling and optimization, production planning and scheduling, fuzzy optimization, and soft computing.