THE APPLICATIONS OF TIME INTERVAL ALIGNMENT POLICIES IN SUPPLY CHAIN SYSTEM

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1 THE APPLICATIONS OF TIME INTERVAL ALIGNMENT POLICIES IN SUPPLY CHAIN SYSTEM Meimei Wang Decision Support, Beth Israel Deaconess Medical Center, 1135 Tremont Street, Boston, MA 02120, Abstract- This paper discusses the applications of interval alignment (IA) scheduling policies in supply chain management. IA policies effectively smoothen part flows, improve performance and decrease aver- age Work-in-Process (WIP) and lead times by adding intermediate delays to the system. We also discussed the applications of IA policy as kanban control in supply chain systems which result in the achievement of Just-in-Time (JIT). The results of the this paper may be used to facilitate factory layout and shop-floor design, using the non-traditional idea that adding machines (test, rework, etc.) in appropriate places will reduce end-to-end lead times. Keywords- Scheduling policies, supply chain management, interval alignment policies, integer linear programming, Optimization and lean system. 1 Introduction Any manufacturing system can be thought of as supply chain system. Stadtler [1] defines supply chain management as, the task of integrating organizational units along a supply chain and coordinating materials, information, and finance flows, in order to fulfill (ultimate) customer demands with the aim of improving competitiveness of the supply chain as a whole. As the result of globalization, companies face everincreasing competition and, if there are stagnant, a slowing rate of growth in profits. Thus, over the past decade, there has been an enhanced focus on improving the entire supply chain. A supply chain system with excessive inventory is ineffective. Excessive inventory typically results from, and masks, poor planning and inefficient operation. Such inefficiencies result in unacceptable lead times and erode the customer base, opening the door for competitors. Thus, there is continued interest, both by researchers and practitioners, in discovering new methodologies and mechanisms to improve every aspect of supply chains. In order to maintain market competitiveness, companies continually 1

2 must increase productivity, improve quality of service, lower costs, and add value to their supply chains. Typically, control policies for manufacturing and supply chain systems are classified either as push or pull implementations. According to Venkatesh et al. [2], in a push system, a machine produces parts without waiting for a request from any of the downstream machines; however, in a pull system, the up- stream machine commences activity only after receiving a request from the downstream machine. Thus, in a pull system the flow of control information is in the opposite direction to the material flow, whereas in a push system the control information flow is in the same direction as the material flow. The most common example of a push system is the use of material requirements planning (MRP), under which production is driven by forecasted demand. The most common pull system is the use of a kanban control, in order to implement a Just-in-Time (JIT) philosophy. However, as is discussed in Bonney et al. [3], most manufacturing production systems contain both push and pull components. In fact, it is not difficult to demonstrate that, for the vast majority of real-world systems, a combination push-and-pull system will outperform either a push system or a pull system. This is because a push system, which is driven by forecasted demands, often builds up excessive inventory; while a pull system, often driven by actual customer demands, is susceptible to machine failures, bursts of demand, and other variability. As Hodgson, King, O Grady and Savva discussed in [4], which investigated both the advantage and disadvantage of kanbanbased pull system and MRP principled push system and proposed a hybrid push/pull approach, their hybrid push/pull approach achieved both a lower inventory level and less fluctuation in the total inventory compared with the pure pull or push approach. With the ever-increasing competitiveness of a global marketplace, a lean supply chain has become the goal of many manufacturers. The overall objectives are to have high levels of output, low levels of inventory, and responsive customer service. As discussed in Womack, Jones, and Roos [5], a lean manufacturing system strives to achieve these objectives, by removing waste from every aspect of production and the supply chain, including product design, factory management, customer relations, and supplier net- works. Thus, lean manufacturing is a systematic approach to identifying 2

3 and eliminating waste through continuous improvement. Over the past decade, many control strategies for supply chains have been proposed which attempt to produce a lean manufacturing system. Both in the literature and in practice, kanban controls have become the prevalent method, due as much to their simplicity as to their effectiveness. Kanban is a Japanese word meaning ticket or signal. In a manufacturing system, a kanban is a replenishment signal used to transmit information which is related to the movement or processing of material. Figure 1: Snapshot of a Kanban System from Martinich [6]. The following example, from Martinich [6], provides a straightforward illustration of the kanban con- trol. A snapshot of the two-stage system is shown in Figure 1. Suppose a company decided to produce products in lot sizes of 100 units. Whenever 100 units of a product are sold, Machine 3 receives a signal to produce another 100 units to replace those sold. The signal is issued by sending a kanban to Machine 3. Machine 3 receives its material from Machine 2, its upstream machine. Whenever Machine 3 uses all of the material required to make 100 units, it signals Machine 2 to produce another batch of material to re- place the consumed units. This process continues upstream to Machine 1, and, in many implementations, up the supply chain system to the company vendors as well. There is a large, and growing, research community searching for efficient detailed scheduling policies in order to improve supply chain performance. The objective of many scheduling policies that combine part release with detailed scheduling is bottleneck avoidance. Instead of focusing on the bottleneck ma- chines in the system, Lu, Ramaswamy, and Kumar [7] proposed a scheduling policy that attempts to smooth 3

4 fluctuations in material flow. As illustrated by the examples in Wang and Perkins [8],[9] and [10] for the applications of IA policies in the work cell design in the supply chain systems, the TIA policies can have an immediate impact in this area. In this paper, we introduce a new class of scheduling policy, called the time-based interval alignment (IA) policy, which can effectively reduce system delay. Moreover, queueing network models typically are Markovian and use either FCFS or bufferlevel-based policies. But, many real-world systems do not have the Markovian structure. For example, in the supply chain, processing-time distributions often are bounded within specification levels, such as, uniform or triangular distributions. The solution to combat this obstacle is to use embedded or aggregated Markov chains. Unfortunately, often the level of aggregation necessary to capture the system dynamics leads to exponentially large system states that are computationally prohibitive to analyse. Thus, this modelling approach is not practical for many real-world systems. The following are several specific scheduling policies, which will be referred to in this paper. First Come, First Serve (FCFS) gives priority to parts in the order which they arrived at the machine; Last Buffer, First Serve (LBFS), at each machine, gives priority to parts in the most downstream buffer; First Buffer, First Serve (FBFS), at each machine, gives priority to parts in the most upstream buffer; Fastest Service Time, First Serve (FSFS), at each machine, the part with the fastest processing time is processed first; Fluctuation Smoothing, Mean Cycle-Time (FSMCT) is a form of Least Slack (LS) policy introduced by Lu, Ramaswamy, and Kumar [7]. In the remainder of this paper, we will discuss IA class of policies and formulate an integer linear programming which yields the optimal transportation delay for 1-IA and 2-IA in Section 2; In Section 3, we will discuss how the class of interval alignment policies can be implemented as kanban controls for supply chain systems and how to determine number of kanbans through IA policies. We discuss the application of IA policies in Uniform distribution system. Finally, in Section 4, we summarize our main work and propose directions for future research. 2 Time Interval Alignment (IA) Policy Consider a single part-type, re-entrant system consisting of m machines, M = {1, 2, 3,..., m}. Parts arrive to the system periodically with constant rate λ. 4

5 Assume that there are L buffers, labeled b1, b2,..., bl. The sequence M (b1), M (b2),..., M (bl ) is then the route followed by parts through the system. Since two buffers may serve the same machine, i.e., M (bi ) = M (bj ) for some i and j with i = j, the system is called a re-entrant line. Parts in buffer bi require a constant, but bufferdependent, processing time pi from machine M (bi ). It is assumed that a machine may work on at most one part at any given time. Clearly, there is no benefit to having a part arrive at a machine before it is able to be processed at that machine. Based on this fact, the underlying idea of the interval alignment (IA) class of scheduling policies is to incorporate structural properties of a system with dynamic timing information, in order to balance the workload and improve system performance by decreasing average WIP and lead times in the supply chain An IA policy that also minimizes the average system lead time will be called a Tight Interval Alignment (TIA) policy. Note that, since there is no non-processing waiting time under a TIA policy, minimizing the sum of the delay parameters also will minimize the average system lead time. 2.1 Deterministic Time 1-IA Policy Consider a single part-type manufacturing system with periodic part arrivals and constant, but buffer- dependent, processing times at the machines. Insert artificial Transportation Delays following each buffer, with Di being the constant delay after parts leave buffer i. The 1-IA policy chooses the delay parameters Di such that no part will have to wait for processing; that is the part flows are synchronized so that there is at most one part at any machine at any given time and parts from each class arrive to the downstream machine periodically. The optimal IA policy, we also call it 1-TIA, imposes the additional restriction that the lead time be minimized, which is equivalent to minimizing the sum of the delay parameters. The 1-IA policy treats every part at a given stage of the system as identical. Thus, for the ideal system, it will induce the same periodic behavior for every part. In order to describe the periodic behavior induced by the 1-IA policy in more detail, consider the two- server, four-buffer system shown n Figure 2. Each part makes two 5

6 passes through the system, with the processing time at buffer bi being pi, for i = 1, 2, 3, 4. From Figure 3, it can be seen that, by adding positive delays after buffer b2 and buffer b3 (in this example, the delay after buffer b1 is 0, i.e., D1 = 0), the flow of parts will be synchronized so that every part commences processing at the moment it arrives at a machine. In order to align the arrival times of parts to each buffer, and avoid overlapping between parts processed at the same machine, the completion time of parts in buffer b1 should be less than arrival time of parts to buffer b3, and the completion time of parts to buffer b3 should be less than next arrival time at buffer b1, i.e., parts to buffer b 3 should be less than next arrival time at buffer b 1, i.e., 1 λ + p 1 p 1 + p 2 + D 2 2 λ p 3, where p 1 + p 2 + D 2 is the arrival time of the part to buffer b 3. Figure 3 illustrates the fact that, under the 1-IA policy, the delay following buffer b2 will be D2 = λ +p1 (p1 +p2). Figure 3 also illustrates the periodic behavior induced by the IA policy. It can be seen that every part arrives to machine M1 periodically, i.e., parts arrive to buffer b1 at times 0, λ 1, 2λ 1, 3λ 1,..., and parts arrive to buffer b3 at times λ 1 + p1, 2λ 1 + p1, 3λ 1 + p1,... Clearly, the method discussed above can not guarantee the delay is optimal. The following algorithm will allow us to determine the optimal delays Integer Linear Program Approaches to 1-TIA In order to obtain the optimal value of the delay parameters {Di : i = 1, 2,..., L 1}, under a 1-IA policy, it is possible to construct a mixed integer linear program (MILP). As discussed earlier, for the 1-IA policy, there will be at most one part at each machine at any moment. Thus, the problem can be formulated as the following MILP, called Problem P1: 6

7 (P1) min s.t. P N Dk k=1 λ j 1 j 1 n ij + p i P (p k + D k ) λ i, j M (α) k=i (1) P (pk + D k ) (n ij +1) p j k = i D k 0 k i, j M (α) n ij 0 n ij is integer In the above, Di is the delay after buffer bi ; pi denotes the processing time at buffer bi ; nij is an integer chosen in order to shift the parts being compared into the same interval of length λ 1. Figure 2: Four-Buffer Two-Server System Figure 3: Periodic Property of 1-IA 2.2 Deterministic Time 2-IA Policy Motivated by 1-IA, what would happen if we keep at most two parts at each machine and allow alter- nating jobs to follow different periodic trajectories? However, suppose that there now are two fundamental periods, each of length 2λ 1 i. Specifically, parts arriving to the system at times 0, 2λ 1, 4λ 1,... will be called Class 1 parts, and will endure delays of length {D 1 : i = 0, 1, 2,..., L 1}; similarly, parts arriving at λ 1, 3λ 1, 5λ 1 will be called Class 2 parts. The resulting policy is called the 2-IA policy. 7

8 A 2-IA policy which minimizes the average system lead time will be called a 2-TIA policy. In order to formulate the MILP for 2-TIA policies, it is necessary to consider the possible scenarios under which two parts simultaneously could be at a given machine. Consider the following example for the system, shown in Figure 5. We will use the notation i(j) : k(l) to compare a part from Class j that is at buffer bi with a part from Class l that is at buffer bk. Note that two parts belonging to the same class will not simultaneously be in the same buffer under a 2-IA policy, since consecutive parts belonging to the same class will travel through the system exactly 2 λ 1 time units apart. There are five possible cases in which two parts can be at machine Mi simultaneously: Case 1: The two parts are both from Class 1, but are in different buffers; Case 2: The two parts are both from Class 2, but are in different buffers; Case 3: The two parts are from different classes, but both are in the same buffer; Case 4: The two parts are from different classes, and are in different buffers, with the Class 1 part upstream from the Class 2 part; Case 5: The two parts are from different classes, and are in different buffers, with the Class 2 part upstream from the Class 1 part. Figure 4 illustrates the periodic part flow behavior induced by the 2-IA policy. There are two different periods, each of which resembles a 1- IA period. Figure 4: Periodic Property of 2-IA Utilizing the 5 cases discussed above, the 2-TIA problem can be formulated as the following MILP, called Problem P2: 8

9 3 Application of IA Policies for Kanban Control A kanban control passes a ticket or signal upstream which provides authorization either to produce additional product, or to transfer material from a supplying location to the consuming location. A kanban control is the primary component of most (pull) JIT systems. A shortcoming of kanban controls is that a mechanism must be established to provide the requisite information flow. It should be noted that IA policies, which can be envisioned as a realization of push JIT control, do not require this additional layer of information flow. However, since kanban control implementations are omnipresent in current factories and supply chains, in this section we will discuss how to devise special kanban control systems that will achieve the perfor- mance of IA policies. 3.1 Kanban Control and the 1-IA Policy Under a 1-IA policy, when a part completes its production at an upstream machine, it undergoes some delay, and then proceeds to the next downstream buffer. As discussed in Section 2, for systems with no variability, by adding delays, the 1-IA class of policies guarantees that there is no non-processing waiting time for any part. As illustrated by the following example, it is possible to construct a special kanban control to mimic the 1-IA policy. Again consider the four-machine, twelve-buffer system shown in Figure 5. In order to apply a kanban control which replicates the IA policy for this system, assume that the delay D i following buffer b i is replaced by a virtual machine with processing time D i. Then, whenever a part in buffer b i completes its processing, a ticket requesting (possibly virtual) production is sent either to the machine serving the upstream buffer b i 1 or to the virtual machine with processing time D i 1, depending on whether D i 1 > 0 or D i 1 = 0, respectively. In addition, a virtual machine, call it M 0, is placed before buffer b 1, with processing time λ 1 p 1. Assume that, under the kanban control, each machine processes the tickets in the order that they were received, i.e., First-Come-First-Serve (FCFS). Also, whenever a part at machine 9

10 M 0 completes its processing, a ticket is sent to the raw material supplier (or system input controller), requesting that a new part enter the system. Figure 6 illustrates the flow of parts (white regions) and tickets (shadow regions) under this kanban control for the system in Figure 5. Note that, under the kanban control, parts follow a synchronized Figure 6: Kanban Control for 1-IA Policy. periodic trajectory through the system. Thus, the performance of the resulting kanban control will be identical to that of the 1-IA policy so that the JIT philosophy can be achieved. In the kanban control process above, it is noticed that kanban works to either authorize production when upstream is a real machine, or delay sending parts to the buffer when upstream is a virtual machine, in which case, kanban works as a transportation kanban. Transportation kanbans also are utilized by Suri [11] and [12], where the POLCA policy, a kanban-like signal, authorizes the movement of parts between work cells. 3.2 Kanban Control and the 2-IA Policy The 2-IA policy can be implemented as a kanban control as well. To do this, consider a kanban control which has a separate ticket for each of the two classes of parts. Thus, the two possible delays following buffer bi are treated as virtual machines, with processing times D 1 for parts of Class 1 and D 2 for parts of Class 2. Each of these virtual machines has an associated kanban ticket. In addition, two virtual with 10

11 processing time λ 1 p1. Each machine processes its accumulated kanban tickets in FCFS order. To illustrate, again consider the system in Figure 5. Whenever a Class k part is completed at a machine, a Class k ticket is sent to the (possibly virtual) 0machine upstream from the buffer serviced to request (possibly virtual) production. In addition, whenever a part at machine M k completes its processing, a Class k ticket is sent to the raw material supplier (or system input controller), requesting that a new Class k part enters the system. Under this kanban control, the tickets at each machine will be processed in the same order as the arrival times of parts under the 2-IA policy. Thus, the performance of the resulting kanban control will be identical to that of the 2-IA policy. Clearly, this approach may be extended to construct a kanban policy that replicates any N -IA policy. 3.3 IA Applications in Closed Systems The primary focus of this paper has been on open systems, i.e., systems in which each part arrives to and departs from the system. However, IA policies can also be utilized on closed systems, the systems with no arrivals and no departures, only a fixed number of parts. In a closed system, the primary concern is the inverse relationship between WIP and throughput rate. Consider the closed network shown in Figure 7. If delays are added to the system shown in Figure 7, the result is a system similar to that considered in Section 2, except it is a closed system. Figure 7: Closed System and With Delays 11

12 3.4 Determination of Number of Kanbans A kanban control system is a pull system. When the demand rate is set, the number of kanbans flowing in the system is a critical control variable to be determined in order to satisfy the customer demand. The IA policies can be used to determine the minimum number of kanbans required by the system through solving the integer linear program, instead of by numerical results like other policies. If the desired throughput, λ, for the system is known, the integer linear program P1 in Section 2 can be utilized to determine the optimal delays. After the delays are obtained from solving the integer linear programming P1, the system lead time, T, is determined by L T = X(pi + Di ). According to Little s Law, the number of parts N p in system will be N p = λt. The above analysis may then be used to decide how many kanbans are required in a i=1 system, in order to achieve the desired throughput λ. Suppose the closed system is converted to an open system by removing the connection between pro- cesses 1 and 4. Then, whenever there is an exit from the system, simultaneously release an arrival to the system. Thus, although the system is an open system, it works as a closed system. Such a policy is called a Constant WIP (CONWIP) policy. See Figure 8. The above analysis, determining the appropriate number of kanbans, clearly will work for both the open system and the closed system. 4 Conclusions In this paper, a class of scheduling policies called Interval Alignment (IA) policies were developed. It was shown that IA policies are efficient and effectively improve system performance by decrease average WIP and lead times. The 12

13 implementations of IA policies as kanban control in the supply chain systems to achieve JIT are discussed and the number of kanbans is determined. Although IA policies are designed to utilize the structural properties of systems with deterministic processing times, it was also for systems with bounded, lowvariability processing times. So the possible directions of future research include the examination of the application of IA policies; improving the performance of modified IA policies for systems with variability; and the application of IA policies to other areas, such as project management and master-level decision-making. References [1] H. Stadtler, Supply chain management and advanced planning - basic, overview and challenges, European Journal of Operational Research, vol. 163, pp , June [2] M. K. K. Venkatesh, M. C. Zhou and R. Caudill, A Petri net approach to investigating push and pull paradigms in flexible factory automated systems, International Journal of Production Research, vol. 34, pp , September- December [3] M. C. Bonney, Z. Zhang, M. A. Head, C. C. Tien, and R. J. Barson, Are push and pull systems really so different, International Journal of Production Economics, vol. 59, pp , January- March [4] T. Hodgson, R. King, P. O Grady, and A. Savva, Integrating kanban type pull systems and MRP type push systems: Insights from a Markovian model, IIE Transactions, vol. 24, pp , March [5] J. P. Womack, D. T. Jones, and D. Roos, The Machine that Changed the World: The Story of Lean Production. Rawson Associates, Harper Perennial, New York, NY, [6] J. Martinich, Production and Operations Management. John Wiley and Sons, NewYork, NY, [7] S. C. H. Lu, D. Ramaswamy, and P. R. Kumar, Efficient scheduling policies to reduce mean and variance of cycle-time in semiconductor manufacturing plants, IEEE Transactions on Semiconduc- tor Manufacturing, vol. 7, pp , August [8] M. Wang and J. Perkins, Using interval alignment policies for efficient production control of supply chain systems, Int. Journal of Industrial and System Engineering, vol. 1, pp , January [9] M. Wang and J. Perkins, The time interval alignment (ia) policies, boundary and applications with multiple stream arrivals, Journal of Systems Science and Systems Engineering, vol. 20, No 4, pp , December [10] M. Wang, The work cell design through ia policy to achieve a jit system, American Journal of Engineering and Technology Research, vol. 11, October

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