A new novel local search integer-programming-based heuristic for PCB assembly on collect-and-place machines

Abstract

This paper presents the development of a novel vehicle-routing-based algorithm for optimizing component pick-up and placement on a collect-and-place type machine in printed circuit board manufacturing. We present a two-phase heuristic that produces solutions of remarkable quality with respect to other known approaches in a reasonable amount of computational time. In the first phase, a construction procedure is used combining greedy aspects and solutions to subproblems modeled as a generalized traveling salesman problem and quadratic assignment problem. In the second phase, this initial solution is refined through an iterative framework requiring an integer programming step. A detailed description of the heuristic is provided and extensive computational results are presented.

Appendices

Appendix 1: Detailed integer linear program (ILP) model formulation for improvement phase

We formulate herein the integer linear program allowing feeder magazine motions and spindle jumping by using a bi-complete and total asynchronous extraction strategy.

1.1 Parameters

${\mathcal {F}}^B$ :

$=$ set of extracted board nodes

${\mathcal {F}}^F$ :

$=$ set of extracted feeder nodes

${\mathcal {S}}^B$ :

$=$ sequence pool for board node sequences (potentially all board node sequences)

${\mathcal {S}}^F$ :

$=$ sequence pool for feeder node sequences (potentially all feeder node sequences)

${\mathcal {I}}^B$ :

$=$ set of candidate insertion points for board node sequences

${\mathcal {I}}^F$ :

$=$ set of candidate insertion points for feeder node sequences

${\mathcal {S}}_P$ :

$=$ set of spindles

C :

$=$ capacity of the head (vehicle) $=$ $|{\mathcal {S}}_P|$

r :

$=$ route

$r^F$ :

$=$ sequence of nodes corresponding to part of route r covering the feeder nodes

$r^B$ :

$=$ sequence of nodes corresponding to part of route r covering the board nodes

${\mathcal {R}}$ :

$=$ ordered list of all routes

${\mathcal {I}}(r)$ :

$=$ set of insertion points (i.e. arcs) associated with route r

$\tilde{q}(s)$ :

$=$ contribution to capacity usage of sequence s, i.e. number of nodes in node sequence s

$\tilde{c}(s)$ :

$=$ cost of sequence s

$\gamma _{fi}$ :

$=$ additional cost due to insertion of sequence $f \in {\mathcal {S}}^F$ at insertion point i

$\gamma _{bi}$ :

$=$ additional cost due to insertion of sequence $b \in {\mathcal {S}}^B$ at insertion point i

$l_2(r)$ :

$=$ insertion point corresponding to the $G^B-G^F$ arc in the route r

$l_1(r)$ :

$=$ insertion point corresponding to the $G^F-G^B$ arc in the route r

${\mathcal {B}}(s)$ :

$=$ beginning node of sequence s

${\mathcal {L}}(s)$ :

$=$ last node of sequence s

${\mathcal {T}}(s,s_p)$ :

$=$ ${\left\{ \begin{array}{ll} 1 \text { if spindle } s_p \text { is used in sequence } s\\ 0 \text { otherwise} \end{array}\right. }$

$\tilde{t}(s,s_p)$ :

$=$ component type number carried by spindle $s_p$ in the sequence s, returns 0 if none

$F_p(n)$ :

$=$ feeder position at node $n \in G^F$

$\gamma _{fi}$ :

$=$ insertion cost of reinserting a sequence of feeder nodes $f \in {\mathcal {S}}^F$ into a $G^F-G^F$ type insertion point $i \in {\mathcal {I}}^F$

$\gamma ^{L^F_1}_{fl_1(r)}$ :

$=$ insertion cost only on the feeder side of reinserting a sequence $f \in {\mathcal {S}}^F$ of $G^F$ nodes into a $G^F-G^B$ type insertion point $\in {\mathcal {I}}(r)$

$\gamma ^{L^F_2}_{fl_2(r-1)}$ :

$=$ insertion cost only on the feeder side of reinserting a sequence $f \in {\mathcal {S}}^F$ of $G^F$ nodes into a $G^B-G^F$ type insertion point $\in {\mathcal {I}}(r-1)$; note that the cost of the neighboring route of the route containing i is influenced

$\gamma _{bi}$ :

$=$ insertion cost of reinserting a sequence $b \in {\mathcal {S}}^B$ of $G^F$ nodes into a $G^B-G^B$ type insertion point $i \in {\mathcal {I}}^B$

$\gamma _{bl_2(r)}$ :

$=$ insertion cost of reinserting a sequence $b \in {\mathcal {S}}^B$ of $G^B$ nodes into a $G^B-G^F$ type insertion point $i \in {\mathcal {I}}(r)$

$\gamma ^{L^B_2}_{fl_2(r)}$ :

$=$ insertion cost only on the board side of reinserting a sequence $f \in {\mathcal {S}}^F$ of $G^F$ nodes into a $G^B-G^F$ type insertion point $i \in {\mathcal {I}}(r)$

$\gamma _{bl_1(r)}$ :

$=$ insertion cost of reinserting a sequence $b \in {\mathcal {S}}^B$ of $G^B$ nodes into a $G^F-G^B$ type insertion point $i \in {\mathcal {I}}(r)$

$\gamma ^{L^B_1}_{fl_1(r)}$ :

$=$ insertion cost only on the board side of reinserting a sequence $f \in {\mathcal {S}}^F$ of $G^F$ nodes into a $G^F-G^B$ type insertion point $i \in {\mathcal {I}}(r)$

1.2 Decision variables

$$\begin{aligned} y_{bi}&= {\left\{ \begin{array}{ll} 1 &{} \quad \text {if board node sequence } b \text { is allocated to the insertion point } i \in {\mathcal {I}}^B,\\ 0 &{} \quad \text {otherwise.} \end{array}\right. }\\ x_{fi}&= {\left\{ \begin{array}{ll} 1 &{} \quad \text {if feeder node sequence } f \text { is allocated to the insertion point } i \in {\mathcal {I}}^F,\\ 0 &{} \quad \text {otherwise.} \end{array}\right. }\\ t_r&= \text {time for route } r\\ F^M_r&= \text {amount the feeder can move when the head is executing route } r \end{aligned}$$

1.3 Objective function and constraints

The objective function reads

$$\begin{aligned} \min \sum _{r=1}^{|{\mathcal {R}}|} t_r. \end{aligned}$$

The constraints are as follows.

$$\begin{aligned}&\tilde{c}\left( r^F\right) + \sum _{f \in {\mathcal {S}}^F} \sum _{i \in {\mathcal {I}}^F \cap {\mathcal {I}}(r)} \gamma _{fi} \cdot x_{fi} \nonumber \\&\quad + \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^F_1}_{fl_1(r)} \cdot x_{fl_1(r)} + \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^F_2}_{fl_2(r-1)} \cdot x_{fl_2(r-1)} \nonumber \\&\quad + \max \left\{ \tilde{c}\left( r^B \right) + \sum _{b \in {\mathcal {S}}^B} \sum _{i \in {\mathcal {I}}^B \cap {\mathcal {I}}(r)} \gamma _{bi} \cdot y_{bi} \right. \nonumber \\&\quad + \sum _{b \in {\mathcal {S}}^B} \gamma _{bl_2(r)} \cdot y_{bl_2(r)} + \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^B_2}_{fl_2(r)} \cdot x_{fl_2(r)} \nonumber \\&\quad \left. + \sum _{b \in {\mathcal {S}}^B} \gamma _{bl_1(r)} \cdot y_{bl_1(r)} + \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^B_1}_{fl_1(r)} \cdot x_{fl_1(r)} , ~F^M_r/V_f \right\} = t_r\quad ~r ~\in ~{\mathcal {R}} \nonumber \\&\quad | \sum _{f \in {\mathcal {F}}^F} x_{fl_2(r)} \cdot F_p({\mathcal {B}}(f)) \nonumber \\&\quad + \left( 1 - \sum _{f \in {\mathcal {F}}^F} x_{fl_2(r)}) \cdot F_p({\mathcal {B}}((r+1)^F)\right) \nonumber \\&\quad - \sum _{f \in {\mathcal {F}}^F} x_{fl_1(r)} \cdot F_p({\mathcal {L}}(f)) \end{aligned}$$

(3)

$$\begin{aligned}&\displaystyle \left( 1 - \sum _{f \in {\mathcal {F}}^F} x_{fl_1(r)}\right) \cdot F_p({\mathcal {L}}(r^F)) | = F^M_r/W_s\quad r ~\in ~{\mathcal {R}} \end{aligned}$$

(4)

$$\begin{aligned}&\displaystyle \sum _{f \in {\mathcal {S}}^F} x_{fi} \le 1 \qquad i ~\in ~{\mathcal {I}}^F \end{aligned}$$

(5)

$$\begin{aligned}&\displaystyle \sum _{b: B(v) \in b} \left( ~ \sum _{r \in {\mathcal {R}}} (y_{bl_2(r)} + y_{bl_1(r)}) + \sum _{i \in {\mathcal {I}}^B} y_{bi} \right) = 1 \qquad v ~\in ~{\mathcal {F}}^B \end{aligned}$$

(6)

$$\begin{aligned}&\displaystyle \sum _{b \in {\mathcal {S}}^B} y_{bi} \le 1 \qquad i ~\in ~{\mathcal {I}}^B \end{aligned}$$

(7)

$$\begin{aligned}&\sum _{f \in {\mathcal {S}}^F} x_{fl_1(r)} + \sum _{b \in {\mathcal {S}}^B} y_{bl_1(r)} \le 1 \qquad r ~\in ~{\mathcal {R}} \end{aligned}$$

(8)

$$\begin{aligned}&\sum _{f \in {\mathcal {S}}^F} x_{fl_2(r)} + \sum _{b \in {\mathcal {S}}^B} y_{bl_2(r)} \le 1 \qquad r ~\in ~{\mathcal {R}} \end{aligned}$$

(9)

$$\begin{aligned}&\tilde{t}\left( r^F, ~s_p\right) - \tilde{t}\left( r^B, ~s_p\right) + \sum _{f \in {\mathcal {S}}^F} \tilde{t}(f, s_p) \cdot \left( \sum _{i \in {\mathcal {I}}(r) \cap {\mathcal {I}}^F} x_{fi} + x_{fl_1(r)} + x_{fl_2(r-1)} \right) \nonumber \\&\quad - \sum _{b \in {\mathcal {S}}^B} \tilde{t}(b, s_p) \cdot \left( \sum _{i \in {\mathcal {I}}(r) \cap {\mathcal {I}}^B} y_{bi} + y_{bl_1(r)} + y_{bl_2(r)} \right) = 0 \qquad r ~\in ~{\mathcal {R}},\quad s_p ~\in ~{\mathcal {S}}_P \nonumber \\\end{aligned}$$

(10)

$$\begin{aligned}&{\mathcal {T}}(r^F, s_p) + \sum _{f \in {\mathcal {S}}^F} {\mathcal {T}}(f, ~s_p) \cdot \left( \sum _{i \in {\mathcal {I}}(r) \cap {\mathcal {I}}^F} x_{fi} + x_{fl_1(r)} + x_{fl_2(r-1)} \right) \le 1 \nonumber \\&\quad r ~\in ~{\mathcal {R}},\quad s_p ~\in ~{\mathcal {S}}_P \end{aligned}$$

(11)

$$\begin{aligned}&{\mathcal {T}}(r^B, s_p) +\sum _{i \in {\mathcal {I}}(r) \cap {\mathcal {I}}^B} \sum _{b \in {\mathcal {S}}^B} {\mathcal {T}}(b, s_p) \cdot \left( \sum _{i \in {\mathcal {I}}(r) \cap {\mathcal {I}}^B} y_{bi} + y_{bl_1(r)} + y_{bl_2(r)} \right) \le 1 \nonumber \\&\quad r ~\in ~{\mathcal {R}},\quad s_p ~\in ~{\mathcal {S}}_P \end{aligned}$$

(12)

$$\begin{aligned}&\tilde{q}\left( r^F\right) + \sum _{f \in {\mathcal {S}}^F} \sum _{i \in {\mathcal {I}}^F \cap {\mathcal {I}}(r)} \tilde{q}(f) \cdot \left( \sum _{i \in {\mathcal {I}}(r) \cap {\mathcal {I}}^F} x_{fi} + x_{fl_1(r)} + x_{fl_2(r-1)} \right) \le C \nonumber \\&\quad r ~\in ~{\mathcal {R}} \end{aligned}$$

(13)

$$\begin{aligned}&\tilde{q}\left( r^B\right) + \sum _{b \in {\mathcal {S}}^B} \sum _{i \in {\mathcal {I}}^B \cap {\mathcal {I}}(r)} \tilde{q}(b) \cdot \left( \sum _{i \in {\mathcal {I}}(r) \cap {\mathcal {I}}^B} y_{bi} + y_{bl_1(r)} + y_{bl_2(r)} \right) \le C \nonumber \\&\quad r ~\in ~{\mathcal {R}} \end{aligned}$$

(14)

$$\begin{aligned}&\displaystyle x_{fi} ~ \qquad \text {binary} \qquad f ~\in ~{\mathcal {S}}_f,~i ~\in ~{\mathcal {I}}^F \end{aligned}$$

(15)

$$\begin{aligned}&\displaystyle y_{bi} ~ \qquad \text {binary} \qquad b ~\in ~{\mathcal {S}}_b,~i ~\in ~{\mathcal {I}}^B \end{aligned}$$

(16)

$$\begin{aligned}&\displaystyle t_r \ge 0 \qquad r ~\in ~{\mathcal {R}}\end{aligned}$$

(17)

$$\begin{aligned}&\displaystyle F^M_r \ge 0 \qquad r ~\in ~{\mathcal {R}} \end{aligned}$$

(18)

Constraint (3) models $t_r$, the time to complete a route. The first term represents the cost on the feeder side of the short-cutted solution. The cost of inserting a $G^F$ node sequence into a $G^F-G^F$ type insertion point is given by the second term, into a $G^F-G^B$ type insertion point by the third term, and into a $G^B-G^F$ type insertion point by the fourth term. The maximum function captures the waiting cost due to the moving feeder and the two terms in it represent the cost of moving the feeder and the cost of the vehicle’s trip to the customers before returning to the feeder. The cost incurred by the vehicle’s trip is computed as the sum of the cost of the short-cutted solution, the cost of inserting a $G^B$ node sequence into a $G^B-G^B$ type insertion point, and the cost of inserting $G^F$ and $G^B$ node sequences into $G^F-G^B$ and $G^B-G^F$ type insertion points. Constraint (4) models the cost of the feeder movement. The terms capture the two possibilities each for determining the last feeder node to be visited in a given route and the first feeder node to be visited in its subsequent route. Constraints (5) and (7) specify that no more than one sequence can be inserted at an insertion point. Constraint (6) states that every board node can belong to exactly one re-inserted string. Constraints (8) and (9) state that only one of a feeder or board node can be reinserted at route junction points in order to reconstruct a solution. Constraint (10) balances the component types between the board and feeder route segments, whereas constraints (11) and (12) ensure no spindle reusage within the board and feeder route segments. Constraints (13) and (14) enforce capacity limitations on the head on the feeder and board segments, respectively.

1.4 Linearization of constraints

We note that (1) and (2) are non linear constraints. Rewriting the constraints in a linearized form and using,

$$\begin{aligned} k_{1fr}&= F_p({\mathcal {L}}(r^F)) - F_p({\mathcal {L}}(f))\\ k_{2fr}&= F_p({\mathcal {B}}(f)) - F_p({\mathcal {B}}((r+1)^F))\\ k_{3r}&= F_p({\mathcal {B}}((r+1)^F)) - F_p({\mathcal {L}}(r^F)) \end{aligned}$$

we obtain

$$\begin{aligned}&\tilde{c}\left( r^F\right) {+} \sum _{f \in {\mathcal {S}}^F} \sum _{i \in {\mathcal {I}}^F \cap {\mathcal {I}}(r)} \gamma _{fi} \cdot x_{fi} {+} \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^F_1}_{fl_1(r)} \cdot x_{fl_1(r)} {+} \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^F_2}_{fl_2(r-1)} \cdot x_{fl_2(r-1)} \nonumber \\&\quad +\, \tilde{c}\left( r^B\right) + \sum _{b \in {\mathcal {S}}^B} \sum _{i \in {\mathcal {I}}^B \cap {\mathcal {I}}(r)} \gamma _{bi} \cdot y_{bi} + \sum _{b \in {\mathcal {S}}^B} \gamma _{bl_2(r)} \cdot y_{bl_2(r)} + \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^B_2}_{fl_2(r)} \cdot x_{fl_2(r)} \nonumber \\&\quad + \sum _{b \in {\mathcal {S}}^B} \gamma _{bl_1(r)} \cdot y_{bl_1(r)} + \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^B_1}_{fl_1(r)} \cdot x_{fl_1(r)} \le t_r \qquad r ~\in ~{\mathcal {R}} \end{aligned}$$

(19)

$$\begin{aligned}&- V_f \cdot \left( t_r - \tilde{c}(r^F) - \sum _{f \in {\mathcal {S}}^F} \sum _{i \in {\mathcal {I}}^F \cap {\mathcal {I}}(r)} \gamma _{fi} \cdot x_{fi} \right. \nonumber \\&\quad \left. - \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^F_1}_{fl_1(r)} \cdot x_{fl_1(r)} - \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^F_2}_{fl_2(r-1)} \cdot x_{fl_2(r-1)} \right) \nonumber \\&\qquad \le \sum _{f \in {\mathcal {F}}^F} x_{fl_1(r)} \cdot k_{1fr} + \sum _{f \in {\mathcal {F}}^F} x_{fl_2(r)} \cdot k_{2fr} + k_{3r} \qquad r ~\in ~{\mathcal {R}} \end{aligned}$$

(20)

$$\begin{aligned}&\sum _{f \in {\mathcal {F}}^F} x_{fl_1(r1)} \cdot k_{1fr} + \sum _{f \in {\mathcal {F}}^F} x_{fl_2(r)} \cdot k_{2fr} + k_{3r} \nonumber \\&\quad \le V_f \cdot \left( t_r - \tilde{c}(r^F) - \sum _{f \in {\mathcal {S}}^F} \sum _{i \in {\mathcal {I}}^F \cap {\mathcal {I}}(r)} \gamma _{fi} \cdot x_{fi} \right. \qquad \nonumber \\&\qquad \left. - \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^F_1}_{fl_1(r)} \cdot x_{fl_1(r)} - \sum _{f \in {\mathcal {S}}^F} \gamma ^{L^F_2}_{fl_2(r-1)} \cdot x_{fl_2(r-1)} \right) \qquad r ~\in ~{\mathcal {R}} . \end{aligned}$$

(21)

We split constraints (3) and (4) into their linearized forms, thereby getting rid of the troublesome maximum and absolute value functions.

Appendix 2: Pricing problem formulation for improvement phase

The dual of the above linear program has the objective

$$\begin{aligned}&\max \left\{ \sum _{r \in {\mathcal {R}}} \left( \tilde{c}\left( r^F\right) + \sum _{r \in {\mathcal {R}}} \tilde{c}\left( r^B\right) \right) \cdot w^A_r + \sum _{r \in {\mathcal {R}}} \left( V_f \tilde{c}\left( r^F\right) - k_{3r}\right) \cdot w^B_r \right. \nonumber \\&\quad + \sum _{r \in {\mathcal {R}}} \left( V_f \tilde{c}\left( r^F\right) + k_{3r}\right) \cdot w^C_r \nonumber \\&\quad + \sum _{i \in {\mathcal {I}}^F} w^D_i + \sum _{v \in {\mathcal {F}}^B} w^E_v + \sum _{i \in {\mathcal {I}}^B} w^F_i + \sum _{r \in {\mathcal {R}}} w^G_r + \sum _{r \in {\mathcal {R}}} w^H_r \nonumber \\&\quad + \sum _{r \in {\mathcal {R}}} \sum _{s \in {\mathcal {S}}_P} \left( \tilde{t}\left( r^F, s\right) - \tilde{t}\left( r^B, s\right) \right) \cdot w^I_{rs} \nonumber \\&\quad + \sum _{r \in {\mathcal {R}}} \sum _{s \in {\mathcal {S}}_P} \left( 1 - {\mathcal {T}}\left( r^F, s\right) \right) \cdot w^J_{rs} + \sum _{r \in {\mathcal {R}}} \sum _{s \in {\mathcal {S}}_P} \left( 1 - {\mathcal {T}}\left( r^B, s\right) \right) \cdot w^K_{rs} \nonumber \\&\quad + \sum _{r \in {\mathcal {R}}} \left( C - \tilde{q}\left( r^F\right) \right) \cdot w^L_r + \sum _{r \in {\mathcal {R}}} \left( C - \tilde{q}\left( r^B\right) \right) \cdot w^M_r \nonumber \\&\quad \left. + \sum _{i \in {\mathcal {I}}^F} w^E_i + \sum _{v \in {\mathcal {F}}^B} w^F_v + \sum _{i \in {\mathcal {I}}^B} w^G_i \right\} \end{aligned}$$

(22)

subject to constraints

$$\begin{aligned}&w^A_r + V_f \cdot w^B_r + V_f \cdot w^C_r \le 1 \qquad r ~\in ~{\mathcal {R}} \end{aligned}$$

(23)

$$\begin{aligned}&\quad \sum _{r \in {\mathcal {R}}} \left\{ - \gamma _{fi} \cdot w^A_r - (V_f \cdot \gamma _{fi} - k_{1fr}) \cdot w^B_r - \sum _{s \in {\mathcal {S}}_P} w^J_{rs} \cdot {\mathcal {T}}(f, s) \right. \nonumber \\&\quad \left. (- V_f \cdot \gamma _{fi} + k_{1fr}) \cdot w^C_r - w^D_i - \sum _{s \in {\mathcal {S}}_P} w^I_{rs} \cdot \tilde{t}(f, s) - w^L_r \cdot \tilde{q}(f) \right\} \le 0 \nonumber \\&\qquad f ~\in ~{\mathcal {S}}^F,\quad i ~\in ~{\mathcal {I}}^F \cap {\mathcal {I}}(r)\end{aligned}$$

(24)

$$\begin{aligned}&\sum _{r \in {\mathcal {R}}} \left\{ - \left( \gamma ^{L_1^F}_{fi} + \gamma ^{L_1^B}_{fi}\right) \cdot w^A_r - V_f \cdot \gamma ^{L_1^F}_{fi} \cdot w^B_r - \sum _{s \in {\mathcal {S}}_P} w^J_{rs} \cdot {\mathcal {T}}(f, s) \right. \nonumber \\&\quad \left. - V_f \cdot \gamma ^{L_1^F}_{fi} \cdot w^C_r - w^G_r - \sum _{s \in {\mathcal {S}}_P} w^I_{rs} \cdot \tilde{t}(f, s) - w^L_r \cdot \tilde{q}(f) \right\} \le 0 \nonumber \\&\qquad f ~\in ~{\mathcal {S}}^F,\quad i=l_1(r)\end{aligned}$$

(25)

$$\begin{aligned}&\sum _{r \in {\mathcal {R}}} \left\{ - (\gamma ^{L_2^F}_{fi} - V_f \cdot \gamma ^{L_2^F}_{fi} \cdot w^B_r - \sum _{s \in {\mathcal {S}}_P} w^J_{rs} \cdot {\mathcal {T}}(f, s) \right. \nonumber \\&\quad \left. - V_f \cdot \gamma ^{L_2^F}_{fi} \cdot w^C_r - \sum _{s \in {\mathcal {S}}_P} w^I_{rs} \cdot \tilde{t}(f, s) - w^L_r \cdot \tilde{q}(f) \right\} \le 0 \nonumber \\&\qquad f ~\in ~{\mathcal {S}}^F,\quad i=l_2(r-1)\end{aligned}$$

(26)

$$\begin{aligned}&\sum _{r \in {\mathcal {R}}} \left\{ - k_{2fr} \cdot w^B_r - k_{2fr} \cdot w^C_r - \gamma ^{L_2^B}_{fi} \cdot w^A_r - w^H_r \right\} \le 0 \qquad f ~\in ~{\mathcal {S}}^F,\quad i=l_2(r)\nonumber \\ \end{aligned}$$

(27)

$$\begin{aligned}&\sum _{r \in {\mathcal {R}}} \left\{ - \gamma _{bi} \cdot w^A_r - \sum _{s \in {\mathcal {S}}_P} w^K_{rs} \cdot {\mathcal {T}}(b, s) \right. \nonumber \\&\quad \left. - w^F_i - \sum _{s \in {\mathcal {S}}_P} w^I_{rs} \cdot \tilde{t}(b, s) - w^M_r \cdot \tilde{q}(b) \right\} \le 0 \qquad b ~\in ~{\mathcal {S}}^B,\nonumber \\&\qquad i ~\in ~{\mathcal {I}}^B \cap {\mathcal {I}}(r)\end{aligned}$$

(28)

$$\begin{aligned}&\sum _{r \in {\mathcal {R}}} \left\{ - \gamma _{bi} \cdot w^A_r - \sum _{s \in {\mathcal {S}}_P} w^K_{rs} \cdot {\mathcal {T}}(b, s) \right. \nonumber \\&\quad \left. - w^F_i -w^G_r - w^H_r - \sum _{s \in {\mathcal {S}}_P} w^I_{rs} \cdot \tilde{t}(b, s) - w^M_r \cdot \tilde{q}(b) \right\} \le 0 \qquad b ~\in ~{\mathcal {S}}^B,\nonumber \\&\qquad i ~\in ~\{l_1(r), l_2(r)\} \end{aligned}$$

(29)

$$\begin{aligned}&w^A_r, w^B_r, w^C_r, w^G_r, w^H_r, w^L_r, w^M_r \ge 0 \qquad r ~\in ~{\mathcal {R}} \end{aligned}$$

(30)

$$\begin{aligned}&w^E_v \text { unrestricted } \qquad v ~\in ~{\mathcal {F}}^F \cup ~{\mathcal {F}}^B \end{aligned}$$

(31)

$$\begin{aligned}&w^I_{rs} \text { unrestricted } \qquad r ~\in ~{\mathcal {R}},\quad s ~\in ~{\mathcal {S}}_P \end{aligned}$$

(32)

$$\begin{aligned}&w^J_{rs}, w^K_{rs} \ge 0 \qquad r ~\in ~{\mathcal {R}},\quad s ~\in ~{\mathcal {S}}_P \end{aligned}$$

(33)

$$\begin{aligned}&w^D_i, w^F_i \le 0 \qquad i ~\in ~{\mathcal {I}}^F \cup {\mathcal {I}}^B \end{aligned}$$

(34)

$$\begin{aligned}&w^I_r, w^J_r \le 0 \qquad r ~\in ~{\mathcal {R}} \end{aligned}$$

(35)

where $w^A, w^B, \ldots , w^M$ are dual variables corresponding to the 13 constraint sets labelled (19)–(21) and (5)–(14). The column generation subproblem (i.e. finding the column with the minimum reduced cost) reduces to one with the following form for a given insertion point i:

$$\begin{aligned} \max _{r \in R} \left\{ \max _{f \in {\mathcal {S^F}}} \left\{ - \beta _1 \cdot \text {(TSP cost for } f \text { at } i) + \beta _2 \cdot \sum _{v \in f} \alpha _v + \beta _3 \right\} \right\} \end{aligned}$$

for $x_{fi}$ variables, and, similarly, for $y_{bi}$ variables we have

$$\begin{aligned} \max _{r \in R} \left\{ \max _{b \in {\mathcal {S^B}}} \left\{ - \beta _1 \cdot \text {(TSP cost for } b \text { at } i) + \beta _2 \cdot \sum _{v \in b} \alpha _v + \beta _3 \right\} \right\} . \end{aligned}$$

In both cases, we have to solve $|{\mathcal {R}}|$ maximum weight circuit problems.