Two-Stage Dynamic Programming in the Routing Problem with Decomposition

Abstract

This paper considers an optimal movement routing problem with constraints. One such constraint is due to decomposing the original problem into a preliminary subproblem and a final subproblem; the tasks related to the preliminary problem must be executed before the tasks of the final subproblem begin. In particular, this condition may arise in the tool control problem for thermal cutting machines with computer numerical control (CNC): if there are long parts among workpieces, the cutting process near a narrow material boundary should start with these workpieces since such parts are subject to thermal deformations, which may potentially cause rejects. The problem statement under consideration involves two zones for part processing. The aggregate routing process in the original problem includes a starting point, a route (a permutation of indices), and a particular track consistent with the route and the starting point. Each of the subproblems has specific precedence conditions, and the travel cost functions forming the additive criterion may depend on the list of pending tasks. A special two-stage procedure is introduced to apply dynamic programming as a solution method. The structure of the optimal solution is established and an algorithm based on this structure is developed. The algorithm is implemented on a personal computer and a computational experiment is carried out.

APPENDIX

Proof of Proposition 2. In view of (4.2),

$${{({{\omega }_{t}})}_{t\in \overline{0,\mathbf{n}}}}\,\,\overset{\Delta }{\mathop{=}}\,\,\,{{({{y}_{t}})}_{t\in \overline{0,N}}}\,\square \,{{({{\hat{y}}_{t}})}_{t\in \overline{0,\mathbf{n}-N}}}\in {{\mathcal{Z}}_{\xi \lozenge \eta }}[{{x}^{0}}].$$

Since $\xi \,\lozenge\,\eta$ ∈ P, we obtain (4.5), i.e.,

$$(\xi \,\lozenge\,\eta ,\,\,{{({{\omega }_{t}})}_{t\in \overline{0,\mathbf{n}}}})\in \mathbf{\tilde{D}}[{{x}^{0}}].$$

(A.1)

According to (3.26) and (3.21), $\mathfrak{C}_{\xi }^{\natural }[{{({{y}_{t}})}_{{t \in \overline {0,N} }}}]$ = ${{V}^{\natural }}$[x⁰] = ${{\mathbb{V}}^{\natural }}$, and furthermore, pr₂(y_N) ∈ X⁰⁰ due to (3.19); $\tilde {V}$*[pr₂(y_N)] ∈ ${{\mathbb{R}}_{ + }}$. In addition,

$$\mathfrak{C}_{\eta }^{*}[{{({{\hat {y}}_{i}})}_{{i \in \overline {0,{\mathbf{n}} - N} }}}] = \tilde {V}{\text{*}}[{\text{p}}{{{\text{r}}}_{2}}({{y}_{N}})].$$

(A.2)

Note that by (3.9)–(3.11), however, it follows that

$$\begin{gathered} \mathfrak{C}_{\eta }^{*}[{{({{{\hat {y}}}_{i}})}_{{i \in \overline {0,{\mathbf{n}} - N} }}}] = \sum\limits_{t = 1}^{{\mathbf{n}} - N} {\left[ {{\mathbf{c}}({\text{p}}{{{\text{r}}}_{2}}({{{\hat {y}}}_{{t - 1}}}),\,\,{\text{p}}{{{\text{r}}}_{1}}({{{\hat {y}}}_{t}}),\,\,{{\eta }^{1}}(\overline {t,\,\,{\mathbf{n}} - N} ) \oplus N)} \right.} \\ \left. {\, + {{c}_{{N + \eta (t)}}}({{{\hat {y}}}_{t}},\,\,{{\eta }^{1}}(\overline {t,\,\,{\mathbf{n}} - N} ) \oplus N)} \right] + f({\text{p}}{{{\text{r}}}_{2}}({{{\hat {y}}}_{{{\mathbf{n}} - N}}})). \\ \end{gathered} $$

(A.3)

Based on the choice of ξ and ${{({{y}_{t}})}_{{t \in \overline {0,N} }}}$, we have

$$\begin{gathered} \mathfrak{C}_{\xi }^{\natural }[{{({{y}_{i}})}_{{i \in \overline {0,N} }}}] = \sum\limits_{t = 1}^N {\left[ {{\mathbf{c}}({\text{p}}{{{\text{r}}}_{2}}({{y}_{{t - 1}}}),\,\,{\text{p}}{{{\text{r}}}_{1}}({{y}_{t}}),\,\,{{\xi }^{1}}(\overline {t,N} ) \oplus \overline {N + 1,\,\,{\mathbf{n}}} )} \right.} \\ \left. { + {{c}_{{\xi (t)}}}({{y}_{t}},\,\,{{\xi }^{1}}(\overline {t,\,\,N} ) \cup \overline {N + 1,{\mathbf{n}}} )} \right] + \tilde {V}{\text{*}}[{\text{p}}{{{\text{r}}}_{2}}({{y}_{N}})] \\ \end{gathered} $$

(A.4)

(see (3.21) and (3.22)). Finally, considering (2.16) and (A.1) leads to the equality

$$\begin{matrix} {{\mathfrak{C}}_{\xi \,\lozenge\,\eta }}[{{({{\omega }_{t}})}_{t\in \overline{0,\mathbf{n}}}}]=\sum\limits_{t=1}^{N}{\left[ \mathbf{c}(\text{p}{{\text{r}}_{2}}({{\omega }_{t-1}}),\,\,\text{p}{{\text{r}}_{1}}({{\omega }_{t}}),\,\,{{(\xi \,\lozenge\,\eta )}^{1}}(\overline{t,\,\,\mathbf{n}}))+{{c}_{(\xi \,\lozenge\,\eta )(t)}}({{\omega }_{t}},\,\,{{(\xi \,\lozenge\,\eta )}^{1}}(\overline{t,\,\,\mathbf{n}})) \right]} \\ +\sum\limits_{t=N+1}^{n}{\left[ \mathbf{c}(\text{p}{{\text{r}}_{2}}({{\omega }_{t-1}}),\,\,\text{p}{{\text{r}}_{1}}({{\omega }_{t}}),{{(\xi \,\lozenge\,\eta )}^{1}}(\overline{t,\,\,\mathbf{n}}))+{{c}_{(\xi \,\lozenge\,\eta )(t)}}({{\omega }_{t}},\,\,{{(\xi \,\lozenge\,\eta )}^{1}}(\overline{t,\,\,\mathbf{n}})) \right]}+f(\text{p}{{\text{r}}_{2}}({{\omega }_{\mathbf{n}}})). \\ \end{matrix}$$

(A.5)

Due to (4.1), (A.5), and the definition of (ω_t${{)}_{{t \in \overline {0,{\mathbf{n}}} }}}$, we obtain

$$\begin{matrix} {{\mathfrak{C}}_{\xi \,\lozenge\,\eta }}[{{({{\omega }_{t}})}_{t\in \overline{0,\mathbf{n}}}}]=\sum\limits_{t=1}^{N}{\left[ \mathbf{c}(\text{p}{{\text{r}}_{2}}({{y}_{t-1}}),\,\,\text{p}{{\text{r}}_{1}}({{y}_{t}}),\,\,{{\xi }^{1}}(\overline{t,N})\cup \overline{N+1,\,\,\mathbf{n}})+{{c}_{\xi (t)}}({{y}_{t}},\,\,{{\xi }^{1}}(\overline{t,\,\,N})\cup \overline{N+1,\,\,\mathbf{n}}) \right]} \\ +\sum\limits_{t=N+1}^{N}{\left[ \mathbf{c}(\text{p}{{\text{r}}_{2}}({{{\hat{y}}}_{t-N-1}}),\,\,\text{p}{{\text{r}}_{1}}({{{\hat{y}}}_{t-N}}),\,\,{{\eta }^{1}}(\overline{t-N,\,\,\mathbf{n}-N})\oplus N) \right.} \\ \left. +\,{{c}_{\eta (t-N)+N}}({{{\hat{y}}}_{t-N}},\,\,{{\eta }^{1}}(\overline{t-N,\,\,\mathbf{n}-N})\oplus N) \right]+f(\text{p}{{\text{r}}_{2}}({{{\hat{y}}}_{\mathbf{n}-N}})). \\ \end{matrix}$$

(A.6)

This expression involves the chain of equalities pr₂(ω_N) = pr₂(y_N) = pr₂(${{\hat {y}}_{0}}$) (see (3.4), (3.5), and (3.14)). From (A.3), (A.4), and (A.6) it follows that

$$\begin{matrix} {{\mathfrak{C}}_{\xi \,\lozenge\,\eta }}[{{({{\omega }_{t}})}_{t\in \overline{0,\mathbf{n}}}}]=\mathfrak{C}_{\xi }^{\natural }[{{({{y}_{t}})}_{t\in \overline{0,N}}}]-\tilde{V}\text{*}[\text{p}{{\text{r}}_{2}}({{y}_{N}})] \\ +\,\sum\limits_{\tau =1}^{\mathbf{n}-N}{\left[ \mathbf{c}(\text{p}{{\text{r}}_{2}}({{{\hat{y}}}_{\tau -1}}),\,\,\text{p}{{\text{r}}_{1}}({{{\hat{y}}}_{\tau }}),\,\,{{\eta }^{1}}(\overline{\tau ,\,\,\mathbf{n}-N})\oplus N)+{{c}_{\eta (t)+N}}({{{\hat{y}}}_{\tau }},\,\,{{\eta }^{1}}(\overline{\tau ,\,\,\mathbf{n}-N})\oplus N) \right]} \\ +\,f(\text{p}{{\text{r}}_{2}}({{{\hat{y}}}_{\mathbf{n}-N}}))=\mathfrak{C}_{\xi }^{\natural }[{{({{y}_{t}})}_{t\in \overline{0,N}}}]-\tilde{V}\text{*}[\text{p}{{\text{r}}_{2}}({{y}_{N}})]+\mathfrak{C}_{\eta }^{\natural }[{{({{{\hat{y}}}_{i}})}_{t\in \overline{0,\mathbf{n}-N}}}]. \\ \end{matrix}$$

Hence, considering (A.2), we can write

$${{\mathfrak{C}}_{\xi \,\lozenge\,\eta }}[{{({{\omega }_{t}})}_{t\in \overline{0,\mathbf{n}}}}]=\mathfrak{C}_{\xi }^{\natural }[{{({{y}_{t}})}_{t\in \overline{0,N}}}],$$

(A.7)

where $\mathfrak{C}_{\xi }^{\natural }[{{({{y}_{t}})}_{{t \in \overline {0,N} }}}]$ = ${{V}^{\natural }}$[x⁰] = ${{\mathbb{V}}^{\natural }}$ by the choice of (ξ, (y_t${{)}_{{t \in \overline {0,N} }}}$) and (3.26). Using (4.4) and (A.7), we derive the chain of equalities

$${{\mathfrak{C}}_{\xi \,\lozenge\,\eta }}[{{({{\omega }_{t}})}_{t\in \overline{0,\mathbf{n}}}}]={{\mathbb{V}}^{\natural }}=\mathbb{V};$$

according to the definition of (ω_t${{)}_{{t \in \overline {0,{\mathbf{n}}} }}}$, the property (4.6) is the case, where (4.5) holds due to (A.1).

Proof of Proposition 4. Let ${{x}_{*}}$ ∈ $X_{{{\text{opt}}}}^{\natural }$, i.e., ${{x}_{*}}$ ∈ X⁰ and ${{V}^{\natural }}$[${{x}_{*}}$] = $\mathbb{V}$ (see (4.4)). Using (3.27), we choose

$$(\xi ,{{({{y}_{i}})}_{{i \in \overline {0,N} }}}) \in {{({\text{sol}})}^{\natural }}[{{x}_{*}}],$$

obtaining (ξ, ${{({{y}_{i}})}_{{i \in \overline {0,N} }}}$) ∈ ${{{\mathbf{D}}}^{\natural }}$[${{x}_{*}}$] with the property $\mathfrak{C}_{\xi }^{\natural }[{{({{y}_{i}})}_{{t \in \overline {0,N} }}}]$ = ${{V}^{\natural }}$[${{x}_{*}}$] (the optimal FS in the (${{\mathcal{M}}_{1}}$, ${{x}_{*}}$)-problem). Then, see (4.4), $\mathfrak{C}_{\xi }^{\natural }[{{({{y}_{i}})}_{{t \in \overline {0,N} }}}]$ = $\mathbb{V}$. In view of (3.15), we choose

$$(\eta ,\,\,{{({{\hat {y}}_{i}})}_{{i \in \overline {0,{\mathbf{n}} - N} }}}) \in ({\text{sol}}){\text{*}}[{\text{p}}{{{\text{r}}}_{2}}({{y}_{N}})],$$

obtaining (η, ${{({{\hat {y}}_{i}})}_{{i \in \overline {0,{\mathbf{n}} - N} }}}$) ∈ D*[pr₂(y_N)] with the property

$$\mathfrak{C}_{\eta }^{*}[{{({{\hat {y}}_{i}})}_{{i \in \overline {0,{\mathbf{n}} - N} }}}] = \tilde {V}{\text{*}}[{\text{p}}{{{\text{r}}}_{2}}({{y}_{N}})].$$

By Proposition 2, ($\xi \,\lozenge\,\eta$, ${{({{y}_{i}})}_{{i \in \overline {0,N} }}}\,\square \,{{({{\hat {y}}_{i}})}_{{i \in \overline {0,{\mathbf{n}} - N} }}}$) ∈ ${\mathbf{\tilde {D}}}$[${{x}_{*}}$] is such that (4.6) holds. Considering (2.18), we have the inequality

$$\tilde{V}[{{x}_{*}}]\le {{\mathfrak{C}}_{\xi \,\lozenge\,\eta }}[{{({{y}_{i}})}_{t\in \overline{0,N}}}\,\square \,{{({{\hat{y}}_{i}})}_{t\in \overline{0,\mathbf{n}-N}}}]=\mathbb{V},$$

where $\mathbb{V}$ ≤ $\tilde {V}$[${{x}_{*}}$] due to (2.21). As a result, $\tilde {V}$[${{x}_{*}}$]= $\mathbb{V}$ and consequently, ${{x}_{*}}$ ∈ $X_{{{\text{opt}}}}^{0}$ (see (2.25)). Thus,

$$X_{{{\text{opt}}}}^{\natural } \subset X_{{{\text{opt}}}}^{0}.$$

(A.8)

Let x* ∈ $X_{{{\text{opt}}}}^{0}$, i.e., x* ∈ X⁰ and $\tilde {V}$[x*] = $\mathbb{V}$. In view of (2.19), we choose the optimal FS

$$(\alpha ,{{({{z}_{i}})}_{{i \in \overline {0,{\mathbf{n}}} }}}) \in ({\text{sol}})[x{\text{*}}];$$

then ${{\mathfrak{C}}_{\alpha }}[{{({{z}_{i}})}_{{i \in \overline {0,{\mathbf{n}}} }}}]$ = $\tilde {V}$[x*] = $\mathbb{V}$. In addition, α ∈ P, which implies α = α₁ α₂, where α₁ ∈ ${{\mathcal{A}}_{1}}$ and α₂ ∈ ${{\mathcal{A}}_{2}}$. Therefore, see (2.11), ${{({{z}_{i}})}_{{i \in \overline {0,{\mathbf{n}}} }}}$ ∈ $\mathcal{Z}_{{{\alpha }_{1}}{{\alpha }_{2}}}^{{}}$[x*]. Then ${{({{z}_{i}})}_{{i \in \overline {0,N} }}}$ ∈ $\mathcal{Z}_{{{{\alpha }_{1}}}}^{\natural }$[x*] and consequently,

$$({{\alpha }_{1}},{{({{z}_{i}})}_{{i \in \overline {0,N} }}}) \in {{{\mathbf{D}}}^{\natural }}[x{\text{*}}]$$

(see (3.18)). We introduce a tuple ${{({{\tilde {z}}_{i}})}_{{t \in \overline {0,{\mathbf{n}} - N} }}}$ in X × X by the rule

$$({{\tilde {z}}_{0}}\,\,\mathop = \limits^\Delta \,\,({\text{p}}{{{\text{r}}}_{2}}({{z}_{N}}),{\text{p}}{{{\text{r}}}_{2}}({{z}_{N}})))\& ({{\tilde {z}}_{t}}\,\,\mathop = \limits^\Delta \,\,{{z}_{{N + t}}}\,\,\,\forall t \in \overline {1,{\mathbf{n}} - N} ).$$

Obviously ${{({{\tilde {z}}_{t}})}_{{t \in \overline {0,{\mathbf{n}} - N} }}}$ ∈ $\mathcal{Z}_{{{{\alpha }_{2}}}}^{*}$[pr₂(z_N)] (see (4.3)). Therefore, see (3.5), we have

$$({{\alpha }_{2}},{{({{\tilde {z}}_{t}})}_{{t \in \overline {0,{\mathbf{n}} - N} }}}) \in {\mathbf{D}}{\text{*}}[{\text{p}}{{{\text{r}}}_{2}}({{z}_{N}})].$$

In addition, (z_t${{)}_{{t \in \overline {0,{\mathbf{n}}} }}}$ = ${{({{z}_{t}})}_{{t \in \overline {0,N} }}}\,\square \,{{({{\tilde {z}}_{t}})}_{{t \in \overline {0,{\mathbf{n}} - N} }}}$. Hence, according to Proposition 3 and (4.4),

$${{\mathbb{V}}^{\natural }} = {{\mathfrak{C}}_{\alpha }}[{{({{z}_{t}})}_{{t \in \overline {0,{\mathbf{n}}} }}}] = \mathfrak{C}_{{{{\alpha }_{1}}}}^{\natural }[{{({{z}_{t}})}_{{t \in \overline {0,N} }}}] - \tilde {V}{\text{*}}[{\text{p}}{{{\text{r}}}_{2}}({{z}_{N}})] + \mathfrak{C}_{{{{\alpha }_{2}}}}^{*}[{{({{\tilde {z}}_{t}})}_{{t \in \overline {0,{\mathbf{n}} - N} }}}],$$

where $\tilde {V}{\text{*}}$[pr₂(z_N)} ≤ $\mathfrak{C}_{{{{\alpha }_{2}}}}^{*}[{{({{\tilde {z}}_{t}})}_{{t \in \overline {0,{\mathbf{n}} - N} }}}]$ (see (3.14)). Now, we obtain

$$\mathfrak{C}_{{{{\alpha }_{1}}}}^{\natural }[{{({{z}_{t}})}_{{t \in \overline {0,N} }}}] = {{\mathbb{V}}^{\natural }} - \mathfrak{C}_{{{{\alpha }_{2}}}}^{*}[{{({{\tilde {z}}_{t}})}_{{t \in \overline {0,{\mathbf{n}} - N} }}}] + \tilde {V}{\text{*}}[{\text{p}}{{{\text{r}}}_{2}}({{z}_{N}})] \leqslant {{\mathbb{V}}^{\natural }}.$$

Then ${{\mathbb{V}}^{\natural }}$ ≤ $\mathfrak{C}_{{{{\alpha }_{1}}}}^{\natural }[{{({{z}_{t}})}_{{t \in \overline {0,N} }}}]$ ≤ ${{\mathbb{V}}^{\natural }}$. As a result, $\mathfrak{C}_{{{{\alpha }_{1}}}}^{\natural }[{{({{z}_{t}})}_{{t \in \overline {0,N} }}}]$ = ${{\mathbb{V}}^{\natural }}$[x*] = ${{\mathbb{V}}^{\natural }}$ and consequently, see (3.29), x* ∈ $X_{{{\text{opt}}}}^{\natural }$. This finally verifies the property $X_{{{\text{opt}}}}^{0}$ ⊂ $X_{{{\text{opt}}}}^{\natural }$ and, see (A.8), the equality $X_{{{\text{opt}}}}^{0}$ = $X_{{{\text{opt}}}}^{\natural }$ as well.