Metadata

Kegyes, Tamás; Süle, Zoltán; Abonyi, János Disassembly line optimization with reinforcement learning. CEJOR, Cent. Eur. J. Oper. Res. 32, No. 4, 1115-1142 (2024).

Links: doi zbMATH

Authors:

• corresponding: János Abonyi janos@abonyilab.com
• research group
• lead researcher named lab: private or public?

Summary

• Incomplete model

Questions

• Code for real MILP formulation
• Data for test cases in machine readable form

Notes

• data: “precedence graph” is used in transitive reduced form
  • would be nice, if it would be a normal digraph
  • but here it has “AND” and “OR” edge labels
    • I can think of a few ways to reduce this graph, but am not even sure if a transitive reduced form exists, neither if it is uniqe
• Table 2 mixes all kinds of quantities (inputs, decision variables) up
  • $N_a^{ws}$ is never used
• Section 2.3:
  • objective part $F_2$
    • $h$ takes values in Yes/No (see tables 3 and 5)
    • but is described as boolean in table 2 … ok, still compatible
    • but taken as binary value in $F_2$?
      • so the mapping Yes -> 1, No -> 0?
  • 6 kinds of decision variables named $x$. Yeah for readability!
  • variable $x^{Type2}$ is never used
    • is this $C$? … no because of (2)
  • variable $x^{Type5}$ is never used
    • if it equals $T^{ws}$, then a definition of $x^{Type4}k \sum{i} x_{i,k}^{Type3} T_i^{rm}$ seems to be more correct? Difficult to say
  • domains are not given with decision variables, but implicitly given in (13)
  • not defined
    • $N^{ws}$ is the number of workstations
    • $C$ is never defined – is this $x_6$?
    • $T^{ws}$ … is this $x^{Type5}$?
  • index error in (2): $k$ on rhs should be $j$?
  • so in short: problem is not fully defined
• Section 4:
  • 2 cases with data
  • differences between tables 6 and 7 (optimal solutions) not clear
    • are there multiple MIP solutions? Probably …
    • should be one table with differences highlighted
    • text: concordance is 293/300
  • Table 8:
    • benchmarking only on single runs :-( for MILP
      • Gurobi under Matlab
    • but 100 simulations for RL

Models

In ASCI-printable Symplex notation.

Model as in paper

´´´ input N_c input N_ws

input T_rm[i in {1, …, N_c}] input T_c

input P_AND[i in {1, …, N_c}] subseteq {1, …, N_c} input P_OR[i in {1, …, N_c}] subseteq {1, …, N_c}

input h[i in {1, …, N_c}] in {T, F} input d[i in {1, …, N_c}] in [0, infty[

input w_1 in [0, infty[ input w_2 in [0, infty[ input w_3 in [0, infty[

decision x_1[i in {1, …, N_c}][j in {1, …, N_c}] in {0, 1} decision x_2[j in {1, …, N_c}] in [0, T_c] decision x_3[i in {1, …, N_c}][k in {1, …, N_ws}] in {0, 1} decision x_4[k in {1, …, N_ws}] in {0, 1} decision x_5[k in {1, …, N_ws}] in [0, T_c] decision x_6[j in {1, …, N_c}] in {0, …, N_ws}

decision C[k in {1, …, N_c}] in {1, …, N_c} decision N_u_ws

constraint EQ_2[j in {1, …, N_c}] is x_2[j] = T_rm[C[k]]

constraint EQ_6[k in {1, …, N_ws}] is x_5[k] = T_c * x_4[k] * sum[i in {1, …, N_c}] x_3[i][k] * C[i]

constraint EQ_7[i in {1, …, N_c}] is sum[j in {1, …, N_ws}]

constraint EQ_8[k in {1, …, N_ws}] is sum[i in {1, …, N_ws}] T_rm[i] * x_3[i][k] <= T_c

constraint EQ_9[i in {1, …, N_c}][l in P_AND[i]] is sum[j in {1, …, N_c}] j * x_1[l][j] <= sum[j in {1, …, N_c}] j * x_1[i][j]

constraint EQ_10[i in {1, …, N_c}][j in {1, …, N_c}] is x_1[i][j] <= sum[h in {1, …, j}] sum[l in P_OR[i]] j * x_1[l][h]

constraint EQ_11[in in {1, …, N_c}] is sum[k in {1, …, N_ws}] x_3[i][k] = 1

constraint EQ_12_a[j in {2, …, N_c}] is 0 <= x_6[j] - x_6[j - 1]

constraint EQ_12_b[j in {2, …, N_c}] is x_6[j] - x_6[j - 1] <= 1

minimize w_1 * sum[j in {1, …, N_u_ws}] (T_c - T_ws[j]) ^ 2

• w_2 * sum[k in {1, …, N_c}] k * h[C[k]]
• w_3 * sum[k in {1, …, N_c}] k * d[C[k]]

´´´

Model rewritten