A Heuristic Approach to the Design of Kanban Systems

Transcription

1 A Heuristic Approach to the Design of Kanban Systems Chuda Basnet Department of Management Systems University of Waikato, Hamilton Abstract Kanbans are often used to communicate replenishment requirements in the Just-in-Time pull system. A design consideration is the determination of the container sizes and the number of kanbans to authorise. This problem is intractable on account of the complex flows of production and information. This paper presents a heuristic approach to this problem using decomposition and multiclass queuing approximations. The approach is tested in a simulated environment. 1. Introduction A primary tenet of Just-in-Time (JIT) production is the pull system. In this control scheme, customer demand is directly communicated to the final assembly stages. As the final assembly stations use up the components needed for completing the customer orders, these components are replenished by the workstations responsible for their production. This chain of replenishment extends up to the suppliers supplying the raw materials. This pull system may be contrasted with the push system, where a forecast of future customer demand is communicated to the suppliers and earliest stages of production using back scheduling: the later stages of production are then driven by the earlier stages. Kanbans (literally, cards) are often used to communicate replenishment requirements in a pull system. A kanban authorises production of a specified number of parts. Containers are designated to hold the number of parts authorised by a kanban. On ' receiving a kanban, the producing workstation makes the specified number of parts and places them in the container. The container and the corresponding kanban are then moved to the consuming workstation. When a container load of parts are consumed, the kanban and the container are sent back to the producing workstation. Work-inprocess cannot exceed the authorised number of kanbans (times the container sizes) for each part. See Schonberger [8] or Monden [6] for a description of kanban systems. A decision arising in the implementation of kanban systems is the determination of the container sizes (number of parts for each kanban) and the number of kanbans. The 202

2 container size is the size of a batch of production. Thus it has implications in terms of number of setups and the workload on the system. The number of kanbans reflects the work-in-process. So it affects the ability to satisfy customer demand and the inventory costs. In this paper, we use queuing approximations to develop a heuristic approach that addresses these issues. 2. Literature Review The earliest publications on kanban systems presented a simple heuristic to design a kanban system. This heuristic, as given by Monden [6], is shown below. Number of kanbans = demand * leadtime «(1 + safety coefficient)...(1) container capacity This equation gives the number of kanbans to use, but does not provide us with any guidance on the container size, which has an impact on the leadtime. Bitran and Chang [3] have presented a mathematical programming approach to the design of kanban systems. They consider a deterministic model aimed at optimizing production batch sizes in a multiechelon multiproduct system, incorporating the kanban discipline. In this scenario the future demand is already known in the form of a master production schedule. Thus this formulation models a push system rather than a pull system. Further, the model assumes that the production of upstream workstation is available to the next workstation only at the end of a planning period. Bard and Golany [1] present a similar deterministic model to find the optimum number of kanbans with the objective o f minimizing setup, shortage, and holding costs. Karmarkar and Kekre [5] analyze single-card and dual-card kanban systems and demonstrate that there is a convex relationship between costs of the system and the container size. To use the models, one needs data on back order costs and inventory holding costs, which are not easy to estimate. The models calculate the total number of kanbans at each workstation. It is not clear how this number is to be distributed between the different products associated with the workstation. Wang and Wang [12] present a Markovian model of the interaction between stages of a kanban system. The states correspond to the number of empty and full containers. The transition probabilities are given by the usage rate at the consuming stage and the production rate of the producing stage. This model ignores the issue of container size. Like Karmarkar and Kekre [5], this model needs data on shortage and holding cost. Another problem with this approach is the rise in state space when serial and tree like structures are to be considered. Philipoom et al. [7] describe an analytical and simulation-based investigation of the factors influencing the number of kanbans in a kanban system. They suggest a simulation analysis o f individual workstations to determine the 95th percentile of the 203

3 leadtime at that workstation. This value is then used in Equation (1), above, to calculate the number of kanbans needed. Again, there is no consideration of the container size. Readers are referred to Berkley [2] and Uzsoy and Martin-Vega [11] for a more comprehensive review of the literature pertaining to the design of kanban systems. But it should be clear from the above that there is a considerable need for more research in the area of design of kanban systems Queuing approximations provide an alternative methodology for the design of kanban systems. Even though these models are not very accurate, they can be executed very rapidly on a desktop computer to narrow down the system alternatives [9]. A short list of design alternatives may be created, and then evaluated using the more expensive discrete event simulation [10]. This paper follows this approach. We use simple queuing approximations to estimate the container sizes and the number of kanbans. These parameters may then be fine tuned on the shop floor. In the next section we develop the heuristic approach. Then we present an application o f the approach in a simulated environment. 3. A Heuristic Approach The kanban system is a highly complex discrete event system consisting of a production network with material flows and information (kanban) flows. It is not the intent here to capture all this complexity. On the contrary, we suggest decomposing the system into independent workstations using simple multiproduct queuing network approximations. We assume that : 1. Raw material is always available at every workstation. This decomposes the network into individual nodes. 2. There is no blocking of the workstations. Essentially, the limit on the number of kanbans is ignored. 3. Conveyance times are ignored. 4. For each item, one container load is processed at a time. The demand for each item is derived from the demand for end-products. These assumptions permit modelling the workstations as independent servers with multiple classes of customers. Using approximations, the multiple classes (of products) are merged, and the container sizes and leadtimes are calculated. These are then used in Equation (1), above, to determine the number of kanbans to be used. Each individual workstation may be analyzed in the following manner. For each item i, whose kanban arrives at a workstation, the rate of arrival of kanbans, is Dt / Q;, where D* is the rate of demand of the item, and Qj is the container size. Service time for a kanban, is given by Qj / Pj + Sit where Pj is the rate of production for the item and Sj is the setup time. The arrival and service of kanbans may be merged approximately [4]. The total rate of arrival of kanbans, A., at 204

4 a workstation is X X,. The average service time, x, may be approximated by (E ^tj) / (X X,). The utilization p of the workstation is p = Xz. Using exponential distribution assumptions, the leadtime for this queue (with a single server) is: L»... (2) 1 - p Using Equation (1), the number of kanbans for item i is given by:. Di La - f.) Ni (J) 1 Q, ' where f* is the safety coefficient. The maximum inventory (when all the containers are full) for all the items, I, is: 1 5>iQi E d + W L For the purpose of this heuristic, we set the batch size Qj proportional to the setup time Sj, i.e., Qj = x where x is the factor of proportionality. After some algebra, the above expression may be written as: I = K, where y a + q d, d, _ K = = ! a = and b = J'D. D. ^ P, L " 1 A little algebra shows that d2i/dx2 > 0 as long as p < 1. function of x, and to minimise I, set dl/dx = 0. This gives Hence I is a convex a(l - a)x2-2abx - b2 = 0 Solving this quadratic equation, x is obtained. From this value of x, the container sizes Qj may be worked out. Obviously, rounding will be required. The leadtime L may then be calculated from Equation (2). These may be substituted in Equation (3) to calculate the number of kanbans. A similar logic may be followed when there are multiple machines in a single department: the multiserver (M/M/c) formula should be used. When there is only one item visiting a workstation, Sj = 0, and there is no need to use the merging approximations. To minimise I, Qs = 1, and the formulas for M/M/1 or M/M/c queues may be used to calculate the leadtime, L, and the number of kanbans Ns. 205

5 4. Simulation Results The above approach was used on a simulated hypothetical production network. The bill of materials is shown in Figure 1 and the product flows are depicted in Figure 2. Figure 1. Bill of Materials Figure 2. The Production Network Two demand patterns were simulated. The first scenario had customer demands for A of 0.1 units per hour and for L of 0.2 units per hour. This resulted in utilizations (p) in the range of to The second scenario had half of this demand rate, with utilizations in the range of to A safety coefficient of 1.0 yielded a customer service level of 99.99% in each case. 206

6 5. Conclusion A heuristic approach to the determination of container sizes and number of kanbans in the design of a kanban system was presented. Queuing approximations permitted rapid estimations of these parameters. These estimates, however, need to be refined, on the shop floor, or using a simulation model of the production system. There is much scope for further research in this area, for example: refinements for limited number of kanbans, considerations of different distributions of customer demand and service time, the choice o f the safety coefficient, and the effect of limited transport resources. References [1] J.F. Bard and B. Golany, Determining the Number of Kanbans in a Multiproduct, Multistage Production System, Int. J. o f Production Res., 29 (1991), pp [2] B.J. Berkley, A Review of the Kanban Production Control Research Literature, Production and Operations Management, 1 (1992), pp [3] G.R. Bitran and L. Chang, A Mathematical Programming Approach to a Deterministic Kanban System, Management Science, 33 (1987), pp [4] G.R. Bitran and D. Tirupati, Multiproduct Queuing Networks with Deterministic Routing: Decomposition Approach and the Notion of Interference, Management Science, 34 (1988), pp [5] U.S. Karmarkar and S. Kekre, Batching Policy in Kanban Systems, Journal of Manufacturing Systems, 8 (1989), pp [6] Y. Monden, Toyota Production Systems, Industrial Engineering and Management Press, Atlanta, U.S.A. (1983). [7] P.R. Philipoom, L.P. Rees, B.W. Taylor, and P.Y. Huang, An Investigation of the Factors Influencing the Number of Kanbans Required in the Implementation of the JIT Technique with Kanbans, Int. J. Of Production Res., 25 (1987), pp [8] R.J. Schonberger, Applications of Single Card and Dual-Card Kanban, Interfaces, 13 (1983), pp [9] R. Suri, Lead Time Reduction through Rapid Modelling, Manufacturing Systems, July 1989, pp [10] R. Suri and G.W. Diehl, Rough-Cut Modelling: an Alternative to Simulation, CIM Review, 3 (1987) pp [11] R. Uzsoy and L.A. Martin-Vega, Modelling Kanban-based Demand-pull Systems: a Survey and Critique, Manufacturing Review, 3 (1990), pp [12] H. Wang and H.-P. Wang, Determining the Number of Kanbans: a Step Toward Non-stock-production, Int. J. o f Production Research, 28 (1990), pp