Comparison of three flow line layouts with unreliable machines and profit maximization

Abstract

Manufacturing system design is a complex challenge when a new factory is being built. Although some factories produce the same product, the layouts of the factories may be different. Manufacturing systems for automotive engines can be modelled with several types of queueing networks with finite buffers and unreliable machines. In this paper, three types of layout structures which are commonly used in automotive engine shops are compared with respect to maximizing profit that is determined by throughput and the investment cost of buffers. We assume that the service times are constant but inhomogeneous, and the time to failure and the time to repair are exponentially distributed. To solve this problem we used approximation methods which are based on aggregation and overlapping decomposition for computing performance measures, and a gradient search method for finding an optimal buffer allocation.

Appendix

The main idea of overlapping decomposition method is to decompose the complex manufacturing system into a set of serial production lines, with the first or last machines in one serial line overlapped with the first or last machines in another serial line, and to modify the overlapping machines appropriately to accommodate the effects of machines and buffers in other lines. The system performance is obtained when the procedure converges (see Li 2005).

As show in Fig. 11, type C can be decomposed into three overlapped serial lines, where M2 is the overlapping machine. Let’s modify M2 as ${\text{M}}^{\left\langle 1 \right\rangle } 2$ considering the effects when M2 is blocked by B2 or B4, then the first overlapped serial line (Line 1) consists of M1, B1 and ${\text{M}}^{\left\langle 1 \right\rangle } 2$. Then the starving probability of M2 being caused by B1 can be calculated using a two-machine throughput analysis formula (Li and Meerkov 2009). Using this probability, consider machine M2 with capacity allocated to buffer B2 and M3 only, modify M2 to include this starvation and capacity allocation, we obtain ${\text{M}}^{\left\langle 2 \right\rangle } 2$ and the second overlapped serial line, referred to as Line 2 (${\text{M}}^{\left\langle 2 \right\rangle } 2$, B2, M3, B3 and M4). Again, the blocking probability of M2 caused by B2 can be calculated. Analogously, considering the starving probability of M2 caused by B1 and the capacity allocated only to B4, M5, B5 and M6, we modify M2 to ${\text{M}}^{\left\langle 3 \right\rangle } 2$ and obtain Line 3 (${\text{M}}^{\left\langle 3 \right\rangle } 2$, B4, M5, B5 and M6). The blocking probability of M2 due to B4 can be computed. Next, using these blocking probabilities, we conduct the analysis for Line 1 again, and the procedure is repeated until convergence is achieved. Then, the production rates of Lines 1–3 are obtained. In this model, M2 sends parts to buffer B2 and B4 with same probability. If M2 is blocked by one of B2 and B4, the part is sent to the other buffer. If M2 is blocked by both of two buffers, M2 stops working, and waits for available space in buffer B2 or B4.

Fig. 11

Overlapping decomposition of type C

The recursive procedure of type C is shown as follows. The capacity, failure rate, repair rate and the efficiency of machine i are denoted as c _i , p _i , r _i and e _i , respectively. $c_{2}^{\left\langle j \right\rangle } \left( s \right)$ means the capacity of $M^{\left\langle j \right\rangle } 2$ at iteration s. TP(·)is the function for calculating the throughput of the line using the parameters in parentheses. $\widehat{TP}_{i} (s)$ is the approximated throughput of line i at iteration s, and N _i is the buffer size of buffer i. $\widehat{X}_{1,0} (s)$ denotes the estimate of the probability that B1 is empty at iteration s. $\widehat{X}_{{2,N_{2} }} (s)$ and $\widehat{X}_{{4,N_{4} }} (s)$ indicate the estimates of the probabilities that B2 and B4 are full at iteration s.

1.1 Recursive procedure

For iteration s (s = 0, 1, 2,…), we can calculate as follows:

• Line 1

  $$c_{2}^{\left\langle 1 \right\rangle } \left( {s + 1} \right) = c_{2} (1 - \widehat{X}_{{2,N_{2} }} (s)\widehat{X}_{{4,N_{4} }} (s))$$

  (3)
  $$\widehat{TP}_{1} \left( {s + 1} \right) = TP(c_{1} ,p_{1} ,r_{1} ,c_{2}^{\left\langle 1 \right\rangle } \left( {s + 1} \right),p_{2} ,r_{2} ,N_{1} )$$

  (4)
  $$\widehat{X}_{1,0} (s + 1) = 1 - \frac{{\widehat{TP}_{1} \left( {s + 1} \right)}}{{c_{2}^{\left\langle 1 \right\rangle } \left( {s + 1} \right)e_{2} }}$$

  (5)

• Line 2

  $$c_{2}^{\left\langle 2 \right\rangle } \left( {s + 1} \right) = 0.5c_{2} (1 + \widehat{X}_{{4,N_{4} }} (s))(1 - \widehat{X}_{1,0} (s + 1))$$

  (6)
  $$\widehat{TP}_{2} \left( {s + 1} \right) = TP(c_{2}^{\left\langle 2 \right\rangle } \left( {s + 1} \right),p_{2} ,r_{2} ,c_{3} ,p_{3} ,r_{3} ,c_{4} ,p_{4} ,r_{4} ,N_{2} ,N_{3} )$$

  (7)
  $$\widehat{X}_{{2,N_{2} }} (s + 1) = 1 - \frac{{\widehat{TP}_{2} \left( {s + 1} \right)}}{{c_{2}^{\left\langle 2 \right\rangle } \left( {s + 1} \right)e_{2} }}$$

  (8)

• Line 3

  $$c_{2}^{\left\langle 3 \right\rangle } \left( {s + 1} \right) = 0.5c_{2} (1 + \widehat{X}_{{2,N_{2} }} (s + 1))(1 - \widehat{X}_{1,0} (s + 1))$$

  (9)
  $$\widehat{TP}_{3} \left( {s + 1} \right) = TP(c_{2}^{\left\langle 3 \right\rangle } \left( {s + 1} \right),p_{2} ,r_{2} ,c_{5} ,p_{5} ,r_{5} ,c_{6} ,p_{6} ,r_{6} ,N_{4} ,N_{5} )$$

  (10)
  $$\widehat{X}_{{4,N_{4} }} (s + 1) = 1 - \frac{{\widehat{TP}_{3} \left( {s + 1} \right)}}{{c_{2}^{\left\langle 3 \right\rangle } \left( {s + 1} \right)e_{2} }}$$

  (11)

• Initial Conditions

  $$\widehat{\text{X}}_{{ 2 , N_{ 2} }} ( 0 ) { = }\widehat{\text{X}}_{{ 4 , N_{ 4} }} ( 0 ) { = 0}$$

  (12)

• Convergent Conditions

  $$\mathop {\lim }\limits_{s \to \infty } \widehat{TP}_{i} \left( s \right) = \widehat{TP}_{i}, i = 1,2,3$$

  (13)

Serial line evaluation method with exponential parameters is described as follows. bl _i (s) is the blocking probability of machine i due to the downstream buffer at iteration s, and st _i (s) is the starving probability of machine i caused by the upstream buffer at iteration s. K denotes the number of machines.

1.2 Aggregation procedure

For iteration s (s = 0,1,2,…), we can calculate equations as follows:

$$bl_{i} (s + 1) = \frac{{e_{i} c_{i}^{f} \left( s \right) - TP(c_{i}^{f} \left( s \right),p_{i} ,r_{i} ,c_{i + 1}^{b} \left( s \right),p_{i + 1} ,r_{i + 1} ,N_{i} )}}{{e_{i} c_{i}^{f} \left( s \right)}},\quad i = 1, \ldots ,{\text{K}} - 1$$

(14)

$$c_{i}^{b} (s + 1) = c_{i} [1 - bl_{i} (s + 1)],\quad i = 1, \ldots ,{\text{K}} - 1$$

(15)

$$st_{i} (s + 1) = \frac{{e_{i} c_{i}^{b} \left( s \right) - TP(c_{i - 1}^{f} \left( {s + 1} \right),p_{i - 1} ,r_{i - 1} ,c_{i}^{b} \left( {s + 1} \right),p_{i} ,r_{i} ,N_{i - 1} )}}{{e_{i} c_{i}^{f} \left( {s + 1} \right)}}, \quad i = {\text{K}}, \ldots ,2$$

(16)

$$c_{i}^{f} \left( {s + 1} \right) = c_{i} \left[ {1 - st_{i} \left( {s + 1} \right)} \right],\quad i = 2, \ldots ,{\text{K}}$$

(17)

• Initial Conditions

  $$c_{i}^{f} \left( 0 \right) = c_{i} , \quad i = 2, \ldots ,{\text{K}} - 1$$

  (18)

• Boundary Conditions

  $$c_{1}^{f} \left( s \right) = c_{1} , \;c_{K}^{b} \left( s \right) = c_{K}$$

  (19)

• Convergent Conditions

  $$\mathop {\lim }\limits_{s \to \infty } bl_{i} \left( s \right) = bl_{i} \quad i = 1, \ldots ,{\text{K}} - 1$$

  (20)
  $$\mathop {\lim }\limits_{s \to \infty } st_{i} \left( s \right) = st_{i} \quad i = 2, \ldots ,{\text{K}}$$

  (21)
  $$\mathop {\lim }\limits_{s \to \infty } c_{i}^{b} \left( s \right) = c_{i}^{b} \quad i = 1, \ldots ,{\text{K}}$$

  (22)
  $$\mathop {\lim }\limits_{s \to \infty } c_{i}^{f} \left( s \right) = c_{i}^{f} \quad i = 1, \ldots ,{\text{K}}$$

  (23)
  $$c_{K}^{f} e_{K} = c_{1}^{b} e_{1} = TP\left( {c_{i}^{f} ,p_{1} ,r_{1} ,c_{i + 1}^{b} ,p_{i + 1} ,r_{i + 1} ,N_{i} } \right) ,\quad i = 1, \ldots ,{\text{K}} - 1$$

  (24)