Enabling flexibility on a dual head placement machine by optimizing platform-tray-feeder picking operations

  • Wilbert E. Wilhelm
  • Xiaoyan Zhu


The typical circuit card assembly line includes one or more dual head placement machines (DHPMs), which are capable of highly accurate placement and promise the flexibility to assemble a broad variety of circuit card types, each in minimal time. Each DHPM stages components for picking from feeder racks and platform tray feeders (PTFs), which are well suited for components that are large or odd-shaped, contributing to DHPM flexibility. The purpose of this paper is to present a model that can optimize operations that pick from a PTF and/or a feeder rack with the goal of enabling the flexibility needed to assemble a variety of circuit card types efficiently. The primary objective of this paper is a model to optimize operations that pick from a PTF and/or a feeder rack; a secondary objective is computational testing to evaluate the solvability of the model within reasonable run times.


Flexible assembly Circuit card assembly Dual head placement machines Platform tray feeders Picking operations Column generation 

List of symbols


G = (N, A)

Graph on which sub-problem SPhmr(o) solves a CSPP to generate a column

GE = (NE, AE)

Expanded graph that is constructed by the solution method



Index of source node in graph G(N, A) associated with sub-problem, SPhmr(o)


Arc, a = (i, j)




CT picked if arc a = (i, j) is traversed, c j  = CT(μj,d j )


Platform associated with the PTF, d ϵ {1,…,  D}


Feeder slot in which CT c j is registered in the feeder rack


Head, hϵ{1,2}


Node in graph G(N, A)


Node in graph G(N, A)


Nozzle type


Level of nodes in graph G(N, A), l = 0,1,2,3,4,5




Index of sink node in graph G(N, A) associated with sub-problem SPhmr(o)


Sub-problem, o = 1,…,24




Rack, rϵ{1,2}


Spindle, sϵ{1,2,3,4}

s1, s2, s3 and s4

spindles 1, 2, 3, and 4, respectively


MTF staging location on a platform, μϵ{1,2}


(MTF staging location, platform) in the PTF


CT picked if arc a = (i, j) is traversed


\( \bar{C}_{c} \)

Number of components of type c required to populate the circuit card


Number of guides provided by the PTF, typically, D = 29


Number of components of CT c picked in CTPC p, e cpϵ{0,1,2,3,4}

\( {\underline {e}}_{p} \)

Vector of e cp values for cϵC


Number of levels of nodes in graph G(N, A) that represent CT and location, L = 4


Total number of columns of nodes graph G(N, A), K = K 1 + K 2 + 1


Number of columns of nodes in the PTF subgraph of G(N, A), K 1 = 2D


Number of columns of nodes in the feeder rack sub-graph of G(N, A)


Reduced cost associated with arc a (updated each column generation iteration)

\( \bar{r}_{a} \)

Amount of composite resource required to traverse arc a


Total amount of composite resource available


Minimum composite resource requirement on all paths from node 0 to node n


Maximum composite resource requirement on all paths from node 0 to node n


Time associated with traversing arc a

\( \hat{T}_{p}^{{}} \)

Time to pick all components in CTPC p, travel to the camera, and display each for viewing

\( \tilde{T} \)

Upper bound on the total time \( \hat{T}_{p}^{{}} \) that a picking step can take


Coordinate of an individual component on the circuit card

\( \bar{x}_{p}^{{}} \)

Upper bound for decision variable x p


Coordinate of an individual component on the circuit card

\( Z_{SP}^{*} \)

Optimal objective function value of sub-problem (4)–(8), SPhmr(o)

(α, β, γ)

Tuple that identifies a test instance (i.e., a randomly generated graph), 0 < α, β, γ ≤ 1


Probability arc a = (i, j) appears in generated graph if i and j are in the PTF subgraph


Probability arc a = (i, j) appears in generated graph if i and j are in the feeder rack subgraph


Probability arc a = (i, j) appears in generated graph if i is in the PTF sub-graph and j is in the feeder rack sub-graph


a PTF picking step must pick from platforms within a neighborhood of Δ adjacent platforms


Number of nozzles of type k provided to the head by the process planner


1 if arc a uses spindle s, 0 otherwise,


1 if arc a employs nozzle k, 0 otherwise


Specifies the tightness of the resource limitation, determining R for constraint (9)



Arcs in graph G(N, A) associated with sub-problem SPhmr(o), aϵA


CTs, cϵC


Nodes in graph G(N, A) associated with sub-problem SPhmr(o), N = {0,1,…, n}


CTPCs, pϵP


Nozzles provided to the head by the process planner


Arcs on an optimal path through graph G(N, A)

Decision variables: master problem


Number of times CTPC p is used to pick all components that populate the circuit card


Value of the dual variable associated with row cϵC in constraint (2) of the master problem

Decision variables: sub-problem SPhmr(o)


If a unit of flow is prescribed on arc aϵA, equivalently,if an arc is on the constrained shortest path (0 otherwise).



This material is based in part upon work supported by the Texas Advanced Technology Program on Grant Numbers ATP-036327-140 and 000512-0248-2001 and by the National Science Foundation on Grant DMI-9500211. The authors gratefully acknowledge industrial collaborators Luis Giraldo, Gordon O’Hara, Richard Evans and Shelli Farr, who provided information that allowed us to model the system realistically and to devise test instances of realistic size and scope.


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringTexas A&M UniversityCollege StationUSA
  2. 2.Department of Industrial and Information EngineeringUniversity of TennesseeKnoxvilleUSA

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