# An Adaptive Kanban and Production Capacity Control Mechanism

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4 An Adaptive Kanban and Production Capacity Control Mechanism 455 servers. Then, adapting flexible servers only after adapting additional kanbans is considered for suppressing the cost. This system is equivalent to the AKS if M=0, and to the traditional kanban system (TKS) if M=0 and E=0. Let i be the amount of inventories minus backorders, s the number of active flexible servers and x the number of additional kanbans released at the time. The proposed control rule for adjusting the number of kanbans and flexible servers is shown as follows. Control rule for additional kanbans: If R s and when a demand arrives; release a kanban x=x+1 If C s and 0 when a demand arrives; capture a kanban x=x 1 Control rule for flexible servers: If i R s and x =E and s <M when a demand arrives; switch on a server s=s+1 If i > C s and x =0 and s > 0 when a part exits MP; switch off a server s=s 1 If i =K+x s and s > 0 when a part exits MP; switch off a server s=s 1 Fig. 2. Markov Chain for E=2,M=2, C>R

5 456 L.L.P. Marand et al. Under the assumption of exponential processing times and Poisson demand arrivals, we can model this system as a Markov Chain in terms of states (i, x, s). An example of the Markov Chain of the proposed AKCS is shown in Fig. 2. If the maximum throughput of MP is greater than the average demand rate then the steady state probability P(i, x, s) of each state exists and can be calculated. Summing the balanced equations of all states (i, E, M) where i < R E M, we can prove Eq. (1).,,,,/ 1,, 1 Therefore the balanced equations of all states (i, x, s) where K s+x < i < R s x form a finite linear system of N independent equations with N unknown variables P(i, x, s). This system is easy to solve and once the steady probability of each state (i, x, s) has been calculated, and we can estimate the following total cost Z of the system. 2 Because finished parts and WIP are attached to a kanban, the inventory holding cost I is equal to the average number of cards in the ordering system multiplied by the unit cost h set to 1.,, The backorders cost B is equal to the average amount of backorders multiplied by the unit cost b.,, The cost F is equal to the average number of times production capacity is adjusted, multiplied by the unit cost f. The first part of the formula is the number of times a server is switched on. The second and last parts are the number of times a server is switched off when i >C s and x=0 or when i=k+x s. 3 4,,1, 0,,0,,,,, 5 The cost O is equal to the average number of active servers, multiplied by the unit cost d.,, 6

6 An Adaptive Kanban and Production Capacity Control Mechanism Performance Study 3.1 Expected Performance under Stable Demand We consider the single stage, single product manufacturing process described in section 3, with μ P = 0.15, λ D = 0.2 and the following cost parameters: b=1000, f=1000, d=1. Table 1 presents the best results obtained with the AKCS for S+M varying from 4 to 6 compared to the best result obtained using the TKS and AKS. Each time, parameters K, E, R, and C were chosen for fixed values of S and M after conducting an exhaustive search over the space of possible values. Here, the optimum number of servers with the AKS and TKS is 4. The AKCS yields better performances only if more than 4 servers are available and the gains are very small. However, cost parameters have a strong impact on performance and this is only a result under the condition of d=1 and f=1000. Cost parameters could be changed to advantage the AKCS (smaller f, bigger d). Also Table 1 shows results under stable environment while the AKCS performs much better in unstable environments where there is a great need for adjusting both WIP and capacity (see Section 3.2). Table 1. Performance under stable demand; d=1, f=1000, b=1000 K,E,R,C S M S+M I+B F O Total Gain TKS 7,0,0, AKS 6,1,5, ,0,0, AKCS 7,1,0, % 7,1,0, % For a fixed total number of servers, using more flexible servers leads to bigger inventories, WIP and backorder costs but smaller operating costs. Using Markov analysis we study the impact of the cost ratios f and d on the performance of the AKCS. Demand is considered stable with λ D =0.2. Fig. 3 presents the space (f, d) where the local optimum configuration of the AKCS uses M flexible servers, when b=1000 and S+M=5. As d increases Op increases and using more flexible servers becomes more interesting. As f increases the cost for adjusting capacity increases and performance is reduced. Therefore, if the penalty for switching servers is not prohibitive and d is expensive using more flexible servers may reduce the total cost and improve performance, but if f is too high and d too small, then increasing M reduces performance. cost ratio d M = 3 M = 2 M = 1 M = Switching cost ratio f Fig. 3. Impact of the cost ratios f and d on the choice of M, for S+M=5 and b=1000

7 458 L.L.P. Marand et al. Fig. 4 illustrates the impact of parameter M on the total cost, for b=1000 and various values of f and d, when the total number of servers is constant. We observe that the expected gain from the AKCS compared to the AKS is very dependent on the cost ratios. For f=900 and d=0.5, using more flexible servers is counterproductive as the gains from the reduced O are too small to balance F and the increased I and B. As d increases so does O but the expected gain from using more flexible servers becomes significant and the effectiveness of increasing M increases. When d=3.5, if f varies from 900 to 300 then F decreases and the effectiveness of using more flexible servers increases. We observe that the AKCS outperforms the AKS only if the reduced O outweighs the increase in I and B and if F is not too high, but if these conditions are met, it is able to reduce the total cost. Total Cost Z Op F B 0 M= f = 900, d = 0.5 M= f = 900, d = 3.5 M= f = 300, d = 3.5 IW Fig. 4. Impact of M on the total cost for S+M=5, b=1000 and various f and d. 3.2 Expected Performance under Unstable Demand The objective of this paper is to propose a Production Planning and Control system more resistant to changes in the demand than existing mechanisms. To assess its performance under unstable demand we conducted a discrete event simulation of a random walk through the Markov Chain. We considered the same manufacturing process and cost parameters as in the preceding study but unstable demand was introduced as follows: the mean of the demand inter-arrival times is set to 3 during the first 25 time units, 5 for the next 25 time units and 7 for the next 25 time units. This pattern repeats itself for the duration of the simulation. We considered a simulation length of Table 2. Simulation under unstable demand mean; b=1000, f=1000, d=1 TKS AKS AKCS K,E,R,C S M S+M I+B F O Total Gain 7,0,0, ,0,0, ,1,5, ,1,6, ,0,0, ,1,0, ,1,0, ,0,6, % 9,0,4, %

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