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Represent a schedule by a permutation $v_1,\dots ,v_{40}$, with $v_0=v_{41}=0$. Choose $1\le p<q\le 40$ and reverse the block $v_p,\dots ,v_q$. This is a 2-opt move on the dummy tour. Its exact setup-cost change is the asymmetric 2-opt reversal costThe internal sum matters because the specified changeovers need not be symmetric. When the block begins at the first job, the boundary terms correctly update the initial setup; the dummy return costs are zero. Processing times cancel.

At temperature $T>0$, simulated annealing accepts a move with probabilityWith uniformly sampled reversal pairs the proposal is symmetric, so this is the appropriate Metropolis–Hastings acceptance probability. Improvements and ties are always accepted; occasional uphill moves escape local minima. Repeatedly sample moves, gradually lower $T$, and retain the best schedule seen even if the current schedule worsens. Reversing a block of length two swaps adjacent jobs, so these moves connect the whole permutation space. A cooling schedule, iteration budget and restarts control practical performance. This is a heuristic at a finite budget; it does not certify global optimality unless paired with an exact bound or an appropriate limiting convergence argument.

The total processing time ${\sum }_{i=1}^{40}{t}_i$ is constant for a single machine, so only the initial setup and changeover sum depends on the order. Use the dummy-job reduction for sequence-dependent setup times. Add vertex $0$, and setLet the binary variable $x_{ij}$ indicate that $j$ follows $i$ in the augmented tour. A Hamiltonian cycle through $0$ specifies exactly one schedule, read starting after $0$. An exact integer programming formulation isThe degree equations alone describe a cycle cover of a directed graph, possibly several separate graph cycles. The subtour elimination constraints exclude every proper directed graph cycle; the cover is therefore a single tour through all jobs. Conversely every schedule supplies a feasible tour with exactly its completion time as objective. Exponentially many written inequalities are permitted in this exact formulation; an implementation may add them only when violated.

For three parallel machines, the relevant objective is the parallel-machine makespan with sequence-dependent setups, not simply total setup cost. Partition jobs into three ordered lists ${\pi }_r$. For a nonempty list definean empty machine has $C_r=0$. Minimize $C_{\max }$ subject to $C_r\le C_{\max }$ for each machine and each job assigned once. This can be written as a mixed-integer extension with assignment variables, one dummy tour per machine and conditional subtour cuts.

A practical strategy starts with a balanced assignment, sequences each machine using the single-machine methods, and improves by relocating or swapping jobs between machines together with within-machine 2-opt moves. Score each move by its change in ${\max }_r{C}_r$. Branch and bound can combine machine-assignment decisions with within-machine assignment bounds. For nonnegative times, the total processing time divided by three and the largest individual processing time are valid lower bounds, supplemented by setup-dependent route bounds. Balancing processing and changeovers jointly is essential; independently minimizing the three tour costs does not by itself minimize completion time.