Approximating Throughput of Small Production Lines Using Genetic Programming

Abstract

Genetic Programming (GP) has been used in a variety of fields to solve complicated problems. This paper shows that GP can be applied in the domain of serial production systems for acquiring useful measurements and line characteristics such as throughput. Extensive experimentation has been performed in order to set up the genetic programming implementation and to deal with problems like code bloat or over fitting. We improve previous work on estimation of throughput for three stages and present a formula for the estimation of throughput of production lines with four stations. Further work is needed, but so far, results are encouraging.

Appendices

Appendix 1: Exact Formula for Three-Station Line Without Buffers

A K-station serial procuction line is presented in Fig. 7. A system with three stages (see Fig. 8) under the assumptions in Hunt (1956) has eight states in the corresponding state transition rate diagram (see Fig. 9). A state is represented by a node which contains a vector of three numbers. This vector represents the state of the entire line, i.e, first number represents the state of the first station, second number represents the state of the second station and so on. Every number in the vector is interpreted as 0 for idle station, 1 for busy station and 2 for blocked station. The interconnections are represented by the mean service rate μ _i of each station (Muth 1984). According to this analysis the numbers of the states of a K-station line n _K are shown in Table 2 (Muth 1984).

Fig. 7

A K-station production line with K−1 intermediate buffers

Fig. 8

A three station production line with no intermediate buffers

Fig. 9

State transition rate diagram for three station line with no intermediate buffers

Table 2 Number of states for a K-station line without buffers

For the case of K = 3 the throughput is expressed in Eq. (7), (Hunt 1956). In Hunt’s work the term maximum possible utilization ρ_max was used.

$$ {\rho}_{\max }=\frac{N}{D} $$

(7)

Where

$$ \begin{array}{l}N={\mu}_2{\mu}_3\left({\mu}_2+{\mu}_3\right)\Big({\mu_1}^4+2{\mu_1}^3{\mu}_2+3{\mu_1}^3{\mu}_3+{\mu_1}^2{\mu_2}^2\\ {}\kern1.32em +4{\mu_1}^2{\mu}_2{\mu}_3+3{\mu_1}^2{\mu_3}^2+{\mu}_1{\mu_2}^2{\mu}_3+4{\mu}_1{\mu}_2{\mu_3}^2\\ {}\kern1.32em +{\mu}_1{\mu_3}^3+{\mu_2}^2{\mu_3}^2+{\mu_3}^3{\mu}_2\Big)\end{array} $$

$$ \begin{array}{l}D={\mu_1}^5\left({\mu_2}^2+{\mu}_2{\mu}_3+{\mu_3}^2\right)\\ {}\kern1.08em +{\mu_1}^4\left(2{\mu_2}^3+5{\mu_2}^2{\mu}_3+5{\mu}_2{\mu_3}^2+3{\mu_3}^3\right)\\ {}\kern1.08em +{\mu_1}^3\left({\mu_3}^4+5{\mu_2}^3{\mu}_3+8{\mu_2}^2{\mu_3}^2+7{\mu}_2{\mu_3}^3+3{\mu_3}^4\right)\\ {}\kern1.08em +{\mu_1}^2\left({\mu_2}^4{\mu}_3+5{\mu_2}^3{\mu_3}^2+8{\mu_2}^2{\mu_3}^3+5{\mu}_2{\mu_3}^4+{\mu_3}^5\right)\\ {}\kern1.08em +{\mu}_1\left({\mu_2}^4{\mu_3}^2+5{\mu_2}^3{\mu_3}^3+5{\mu_2}^2{\mu_3}^4+{\mu}_2{\mu_3}^5\right)\\ {}\kern1.08em +\left({\mu_2}^4{\mu_3}^3+2{\mu_2}^3{\mu_3}^4+{\mu_2}^2{\mu_3}^5\right)\end{array} $$

Appendix 2: The DECO-2 Algorithm

The DECO-2 algorithm (Diamantidis et al. 2007) is capable of handling saturated lines (with over 1000 stations in series) with exponential service times, parallel identical machines at each station and finite intermediate buffers using a decomposition methodology and estimates the throughput of the specified production line. Figure 10 shows the decomposition scheme for a K-stations line. Details of the algorithm are presented in Papadopoulos et al. (2009). The steps of the algorithm are presented as follow:

Fig. 10

Flow line with K parallel-machine work-stations, K−1 intermediate buffers (Line L) and decomposition scheme (Lines L₁,…,L _K−1) (Papadopoulos et al. 2009). In the present work S = 1

• {Step 1: Initialization}

• for $i=1\kern0.48em \mathbf{t}\mathbf{o}\kern0.48em K-1$ do

• $$ \begin{array}{l}{\mu}_i^u={\mu}_i\\ {}{\mu}_i^d={\mu}_{i+1}\end{array} $$

• ε = small positive number for terminating condition

• end for

• $$ \begin{array}{l}\left\{\mathbf{Step}\;\mathbf{2}:\ \mathbf{Calculate}\;{\mu}_i^u\;\mathbf{and}\;{\mu}_j^d\right\}\\ {}\mathbf{f}\mathbf{o}\mathbf{r}\;i=2\;\mathbf{t}\mathbf{o}\;K-1\;\mathbf{do}\\ {}\mathrm{Calculate}\;{\mu}_i^u\;\mathrm{u}\mathrm{s}\mathrm{in}\mathrm{g}\;\mathrm{the}\;\mathrm{f}\mathrm{o}\mathrm{llowing}\;\mathrm{equation}\\ {}{\mu}_i^u=\frac{1}{\frac{1}{\mu_i}+\frac{s_i}{X_{i-1}}-\frac{1}{\mu_{i-1}^d}},\;i=2,\dots, K-1\\ {}\mathrm{Evaluate}\;\mathrm{the}\;\mathrm{two}\hbox{-} \mathrm{work}\hbox{-} \mathrm{station},\;\mathrm{o}\mathrm{ne}\;\mathrm{b}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{er}\;\mathrm{s}\mathrm{u}\mathrm{b}\hbox{-} \mathrm{line}\;\\ {}{L}_{i-1},\kern0.24em \mathrm{u}\mathrm{s}\mathrm{in}\mathrm{g}\;\mathrm{the}\;\mathrm{most}\ \mathrm{r}\mathrm{ecent}\;\mathrm{values}\;\mathrm{o}\mathrm{f}\;{\mu}_{i-1}^u\;\mathrm{and}\;{\mu}_{i-1}^d\;\mathrm{in}\\ {}\;\mathrm{the}\;\mathrm{algorithm}\;\mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{g}\mathrm{enerating}\ \mathrm{the}\ \mathrm{transition}\ \mathrm{matrix}\\ {}\mathbf{end}\;\mathbf{f}\mathbf{o}\mathbf{r}\end{array} $$
• $$ \begin{array}{l}\mathbf{f}\mathbf{o}\mathbf{r}\;i=2\;\mathbf{t}\mathbf{o}\;\mathbf{K}-\mathbf{1}\;\mathbf{do}\\ {}j=K-i\\ {}\mathrm{Calculate}\;{\mu}_j^d\;\mathrm{u}\mathrm{s}\mathrm{in}\mathrm{g}\;\mathrm{the}\;\mathrm{f}\mathrm{o}\mathrm{llowing}\;\mathrm{equation}\\ {}{\mu}_i^d=\frac{1}{\frac{1}{\mu_{i+1}}+\frac{s_{i+1}}{X_{i+1}}-\frac{1}{\mu_{i+1}^u}},\kern0.24em i=K-2,\dots, 1\\ {}\mathrm{Evaluate}\;\mathrm{the}\;\mathrm{two}\hbox{-} \mathrm{work}\hbox{-} \mathrm{station},\;\mathrm{o}\mathrm{ne}\;\mathrm{b}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{er}\;\mathrm{s}\mathrm{u}\mathrm{b}\hbox{-} \mathrm{line}\;\\ {}{L}_{i+1},\kern0.24em \mathrm{u}\mathrm{s}\mathrm{in}\mathrm{g}\;\mathrm{the}\;\mathrm{most}\ \mathrm{r}\mathrm{ecent}\;\mathrm{values}\;\mathrm{o}\mathrm{f}\;{\mu}_{i+1}^u\;\mathrm{and}\;{\mu}_{i+1}^d\;\mathrm{in}\;\\ {}\mathrm{the}\;\mathrm{algorithm}\;\mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{g}\mathrm{enerating}\ \mathrm{the}\ \mathrm{transition}\ \mathrm{matrix}\\ {}\mathbf{end}\;\mathbf{f}\mathbf{o}\mathbf{r}\end{array} $$
• $$ \begin{array}{l}\left\{\mathbf{Step}\mathbf{3}:\kern0.24em \mathbf{Terminating}\ \mathbf{Conditions}\right\}\\ {}\mathbf{if}\kern0.24em \left|{X}_i^L-{X}_1^L\right|<\varepsilon, \kern0.24em i=2,\dots, K-1\kern0.24em \mathbf{then}\\ {}\kern0.96em \mathbf{GOTO}\kern0.24em Step\kern0.24em 4\\ {}\mathbf{else}\\ {}\kern0.96em \mathbf{GOTO}\kern0.24em Step\kern0.24em 2\\ {}\mathbf{end}\;\mathbf{if}\end{array} $$
• $$ \begin{array}{lll} {\left\{ {{\bf{Step4}}:\:{\bf{Output}}\:{\bf{Results}}} \right\}}\\ {X = X_i^L,i = 1, \ldots ,K - 1} \end{array} $$

The algorithm generates the transition probabilities in three stages (Diamantidis et al. 2007)

• 1. transition probabilities of the lower boundary states

• 2. transition probabilities of the internal states

• 3. transition probability of the upper boundary state

The steps of the algorithm are presented as follow:

• $$ \begin{array}{l}\left\{\mathbf{Lower}\;\mathbf{boundary}\;\mathbf{states}\right\}\\ {}{P}_{0,0}=1-{S}_1{\mu}_1\\ {}{P}_{0,1}={S}_1{\mu}_1\\ {}\mathbf{f}\mathbf{o}\mathbf{r}\;c=2\;\mathbf{t}\mathbf{o}\;C\;\mathbf{do}\\ {}\kern0.24em {P}_{0,c}=0.0\\ {}\mathbf{end}\;\mathbf{f}\mathbf{o}\mathbf{r}\end{array} $$

• {Internal states}

• $$ \begin{array}{l}\mathbf{f}\mathbf{o}\mathbf{r}\;i=1\;\mathbf{t}\mathbf{o}\;C-1\;\mathbf{do}\\ {}\kern0.5em \mathbf{f}\mathbf{o}\mathbf{r}\;j=0\kern0.36em \mathbf{t}\mathbf{o}\;j=C\;\mathbf{do}\end{array} $$
• $$ \begin{array}{l}\mathbf{if}\;i>j\;\mathbf{and}\;i-j=1\;\mathbf{and}\;i<{S}_2\;\mathbf{then}\\ {}{P}_{i,j}=i{\mu}_2\\ {}\mathbf{end}\;\mathbf{if}\\ {}\mathbf{if}\;i>j\;\mathbf{and}\;i-j=1\;\mathbf{and}\;i\ge {S}_2\;\mathbf{then}\\ {}{P}_{i,j}={S}_2{\mu}_2\\ {}\mathbf{end}\;\mathbf{if}\end{array} $$
• $$ \begin{array}{l}\mathbf{if}\;i=j\;\mathbf{and}\;j<{S}_2\;\mathbf{and}\;i<{S}_2+B+1\;\mathbf{then}\\ {}{P}_{i,j}=1-{S}_1{\mu}_1-j{\mu}_2\\ {}\mathbf{end}\;\mathbf{if}\\ {}\mathbf{if}\;i=j\;\mathbf{and}\;j\ge {S}_2\;\mathbf{and}\;i<{S}_2+B+1\;\mathbf{then}\\ {}{P}_{i,j}=1-{S}_1{\mu}_1-{S}_2{\mu}_2\\ {}\mathbf{end}\;\mathbf{if}\end{array} $$
• $$ \begin{array}{l}\mathbf{if}\;i=j\;\mathbf{and}\;j\ge {S}_2\;\mathbf{and}\;i\ge {S}_2+B+1\;\mathbf{then}\;\\ {}K=C-i\\ {}{P}_{i,j}=1-K{\mu}_1-{S}_2{\mu}_2\\ {}\mathbf{end}\;\mathbf{if}\\ {}\mathbf{if}\;j>i\;\mathbf{and}\;j-i=1\;\mathbf{and}\;i<{S}_2+B+1\;\mathbf{then}\\ {}{P}_{i,j}={S}_1{\mu}_1\\ {}\mathbf{end}\mathbf{if}\end{array} $$
• $$ \begin{array}{l}\mathbf{if}\;j>i\;\mathbf{and}\;j-i=1\;\mathbf{and}\;i\ge {S}_2+B+1\;\mathbf{then}\\ {}m=C-i\\ {}{P}_{i,j}=m{\mu}_1\\ {}\mathbf{end}\;\mathbf{if}\end{array} $$
• $$ \begin{array}{l}\mathbf{if}\;i>j\;\mathbf{and}\;i-j>1\;\mathbf{then}\;\\ {}{P}_{i,j}=0.0\\ {}\mathbf{end}\;\mathbf{if}\\ {}\mathbf{if}\;j>i\;\mathbf{and}\;j-i>1\;\mathbf{then}\\ {}{P}_{i,j}=0.0\\ {}\mathbf{end}\;\mathbf{if}\end{array} $$
• $$ \begin{array}{l}\kern0.24em \mathbf{end}\;\mathbf{f}\mathbf{o}\mathbf{r}\\ {}\mathbf{end}\;\mathbf{f}\mathbf{o}\mathbf{r}\end{array} $$
• $$ \begin{array}{l}\left\{\mathbf{Upper}\;\mathbf{boundary}\;\mathbf{states}\right\}\\ {}{P}_{C,C-1}={S}_2{\mu}_2\\ {}{P}_{C,C}=1-{S}_2{\mu}_2\\ {}\mathbf{f}\mathbf{o}\mathbf{r}\;c=0\;\mathbf{t}\mathbf{o}\;C-2\;\mathbf{do}\\ {}{P}_{C,c}=0.0\\ {}\mathbf{end}\;\mathbf{f}\mathbf{o}\mathbf{r}\end{array} $$

Table 3 Glossary for decomposition approach