A Cut Approach for Solving the Single-Row Linear Layout Problem

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A Cut Approach for Solving the Single- Row Linear Layout Problem

SIAM 2014: Session CP-30 July 07~11, 2014

Chicago, IL, USA

Shine-Der Lee

Department of Industrial & Information Management

National Cheng Kung University

1 University Road, Tainan, 70101, Taiwan

sdlee@mail.ncku.edu.tw

1. Problem Description

 Scenario:

 A family of parts is processed in the system

 The routing sequence of parts are known and fixed

 Input/output stations are located at two ends of the single- row linear layout

 Forward flow: part moves from left (i.e., Input station is at the left side) to right (Output station is at the right side)

 Backward flow: part moves from right (Output station) to left (Input station), also called backtracking

 Per unit backtracking movement cost can be different from that of forward flow

 Decision model:

 Find the layout sequence or design (a permutation of

workstation sequence) so that the total material flow cost for a given routing requirement is minimized.

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1. Problem Description (continued)

WS n

Linear flow path

WS 1 WS i

Input station 0

Output station n+1 WS j

S0

( ) j 1 α = + β

 Material flow cost is the sum of part flow multiplied by the distance between any two workstations

 Equal-distance between any two adjacent locations

 Bi-directional material handling system, e.g., AGV, conveyor, guided rail, etc.

 n workstations are to be assigned

backtracking

2. Some Established Results

 Classical layout models: Quadratic Assignment Problem (QAP)

 Koopmans and Beckman (1957): QAP

 Lawler (1963): Integer programming model

 Sahni and Gonzalez (1976): NP-complete

 Optimal Linear Ordering problem (OLO)

 Adolphson and Hu (1973): Network flow property

 Garey and Johnson (1976): NP-complete

 Single-row linear layout

 Kouvelis et al. (1996): Linear integer programming model, minimize the times of backtracking, branch & bound

 Palubeckis (2012): Improved lower bound, optimization approach with Tabu search

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2. Some Established Results

 Single-row linear layout (Alternative objective functions)

 Picard and Queyranne (1981): Dynamic programming

 Houshyar and McGinnis (1990): Minimizing WIP, cut tree based heuristic

 Sarker et al. (1995): Multi-Pass Heuristic and Depth-first Insertion Heuristic

 Samarghandi et al. (2010): Continuous version of the linear layout, Particle optimization heuristic

 Open issues:

 Minimizing the total material flow cost

 Exact procedures with improved computational efficiency

 Solving realistic-size problems, e.g., over 100 work stations for assembling iphones, and nearly 300 work stations for ipads.

3. The network flow Model

 Objective function:

 (1)

 : The set of workstations that are located before the (q+1)

sequence in the linear layout

 : The set of workstations that are located after the q

sequence in the linear layout

 : The cut value (total material flow cost) for the corresponding q-locale network, e.g., see Picard

& Ratliff (1978), it can be expressed as:

(2)

0 , 1

1 1 1 1

( ) ( ) ( ) ( 1) 1 ( ) ( )

2

n n n n

i i n ij

i i i j

F α t α i t ₊ α i n w α i α j

= = = =

∑ ∑ ∑ ∑

= + − + + −

{

}

Sq = j α j ≤ q

{

}

Sq = j α j > q ( , )

q q q

C S S

( ^, )

q q q q

q q q oi i n ij

i S i S

i S j S

C S S t t

w

∈ ∈

∈∑ ∑ ∑ ∑∈

= + +

3.1 The q-locale network

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An example: The equivalent transformation for q = 1 locale network

 q-locale network: The sub-network where the minimum cut is used to determine the q^th assignment (location vector) of the workstation

Output station (n+1) Input

Station(0)

1

2

n i

, 1

tn n₊

t01

t02

t0,i

t0, n

1,n 1

t ₊

2,n 1

t ₊

, 1

ti n₊

w12

w2, n

,

wi n

w1, n

w1,i

w2, n





( )

1 1 1 0 1, 1 1

2 2

, = ⁿ _{i n} ⁿ _i

i i

C S S t t ₊ w

= =

∑ + + ∑

3.2 Equivalence properties

 Proposition 1: Let be one of the minimal cut in the q-locale network and be the corresponding

location vector, there exists a minimum cut and layout sequence such that:

for any workstation j, 1 ≤ j ≤ n, and

 Proof: By the definition of cut value on the q-locale network and using term-by-term comparison.

(S S_q′ ′, _q)

(S S_q, _q) α^′

α *

* ( ) ( ) 1, ,

* ( ) ( ) 1, ,

* ( ) ( ),

q q

q q

j j if j S S

j j if j S S

j j otherwise

α α

α α

α α

′ ′

= − ∈ ∩

′ ′

= + ∈ ∩

= ′

(

) ( )

(

,

)

(

,

)

F α − F α ^′ ≤ C S ^{′ ′} S − C S S

3.2 Equivalence properties

 Proposition 2: There exists a set of q-locale networks

with the minimum cut , and , for q = 1,. . . , n-1, where α is the

optimum layout sequence with the minimum total flow cost, and vice versa.

 Proof:

Hence, the set of minimum cuts of the q-locale network, for q = 1, . . , n-1, n, corresponds to the minimum cost for the optimum layout. Conversely, using Proposition 1, we can find a minimum cut with a reduced cut value, which implies that the minimum

cut is the solution corresponding to the optimal layout. ₉ ( , )

q q q

C S S Sq =

{

} {

}

Sq = j α j > q

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4. An Efficient Solution Procedure

 The cut approach for linear row layout

1. Let S

= {0}, i.e., it includes only the input station.

2. For q = 1, find the minimum cut for the 1-locale network as follows:

 Generate a network (graph) using the input station as source node and the output station as sink node. All n

workstations are placed in between, i.e., every workstation is a candidate to be assigned to the first location. The arc capacity for any two nodes i and j is the part flow in

between.

 Compute n cut values:

for n workstations. Find the minimum cut value.

 Assign workstation i* in this minimum cut to the q^th layout sequence. Tie breaking is arbitrary.

(

)

q q q q

q q q oi i n ij

i S i S

i S j S

C S S t t ₊ w

∈ ∈

∈∑ ∑ ∑ ∑∈

= + +

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4. An Efficient Solution Procedure

 The cut approach for linear row layout (continued) 3. Update . Increase q by 1 and repeat steps

2 and 3 until q = n-1. Go to the next step.

4. The optimum layout sequence has been found;

and the total minimum flow cost is

 Complexity analysis: To find the optimum layout, the total computations for the q-locale networks are: . The complexity of the cut approach is , where n is the problem size.

{ }*

q q

S = S  i

1

0 , 1

1 1 1

( ) ( , )

n n n

i i n q q q

i i q

F

α

= = =

=

∑ ∑

∑

α

( )

1

2 1

n

i

i i

=

⋅ −

∑

1 1

( 1) (2 1) ( 1)

2 6 2

n n

i i

n n n n n

i i

= =

⋅ + ⋅ + ⋅ +

   

= ⋅∑ −∑ =    − 

( 3) Ο n

4.1 An illustration

 Consider a bi-directional single-row linear layout with 6

workstations to be assigned. Flow between workstations is given below, where 0 is the input station and 7 is the output station.

0 1 2 3 4 5 6 7

0 -- 20 5 0 12 0 14 0

1 0 -- 24 12 0 15 0 0

2 0 0 -- 10 0 0 15 14

3 0 19 0 -- 0 12 0 10

4 0 12 10 14 -- 0 0 15

5 0 0 0 0 29 -- 10 12

6 0 0 0 5 10 24 -- 0

7 0 0 0 0 0 0 0 --

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4.1 An illustration

The flow matrices for the q-locale network are as follows:

1. Flow matrix T from/to I/O

2. Flow matrix W

1 2 3 4 5 6 0 20 5 0 12 0 14 7 0 14 10 15 12 0

1 2 3 4 5 6

1 -- 24 31 12 15 0

2 24 -- 10 10 0 15

3 31 10 -- 14 12 5

4 12 10 14 -- 29 10

5 15 0 12 29 -- 34

6 0 15 5 10 34 --

4.1 An illustration

 Computational details of the cut approach for the 1- locale network

0

1

q = S¹={2, 3, 4, 5, 6}

1 {1, 3, 4, 5, 6}

S =

1 {1, 2, 4, 5, 6}

S =

1 {1, 2, 3, 5, 6}

S =

1 {1, 2, 3, 4, 6}

S =

*

1 {1, 2, 3, 4, 5}

S =

0 6 7

4.1 An illustration

 Computational details for q = 4 and 5

15 Workstation 1 is assigned to the 4^th sequence

Workstation 3 is assigned to the 5^th sequence. Thus, the only unassigned workstation 2 is assigned to the last

sequence

4.1 An illustration

 The optimum layout vector is α = (6, 5, 4, 1, 3, 2)

 The minimum flow cost of the layout is the sum of:

 Flow cost from the input station to every workstation, = 20+5+12+14 = 51 (From flow matrix T)

 Flow cost from every workstation to the output station, = 14+10+15+12 = 51 (From flow matrix T), and

 Flow cost from the minimum cuts, = 101 + 135 + 135 + 143 + 101= 615 (Between workstations)

 = 717

.

 Verification of total flow cost ,

= 160 + 154 + (149 + 95 + 70 + 79 + 10) = 717

6 0 1

i i

t

∑=

6 7 1

i i

t

∑

1

( , q)

q q

q

C S S

∑

( )

1 1 1 ,

1

i i i i ij ij

i i i j i

j

F α t d t d w d

= = = ≠

=

∑ ∑ ∑ ∑

= + +

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5. Concluding remarks

 Some findings:

 The single-row linear layout problem can be transformed into the equivalent network optimization problem of

finding minimum cuts for the corresponding q-locale networks.

 The minimum cut in the q-locale network is the optimal assignment of workstation in the q^th location sequence, and vice versa.

 An efficient solution procedure, i.e., a cut approach, of complexity is developed to solve this layout

problem.

 Extension:



The cut approach can be extended to solve the non- equal distance linear layout problem.

( )3

Ο n