Zhu J., Cook W.D. (Eds.) Modeling Data Irregularities and Structural Complexities in Data Envelopment Analysis

Chapter 11

DEA MODELS FOR SUPPLY CHAIN OR MULTI-

STAGE STRUCTURE

Wade D. Cook

1

, Liang Liang

2

, Feng Yang

2

, and Joe Zhu

3

1

Schulich School of Business, York University, Toronto, Ontario, Canada,

M

3J 1P3,

wcook@shulich.yorku.ca

2

School of Business,University of Science and Technology of China, He Fei, An Hui Province,

P.R. China 230026 lliang@ustc.edu.cn

3

Department of Management, Worcester Polytechnic Institute, Worcester, MA 01609,

j

zhu@wpi.edu

Abstract: Standard data envelopment analysis (DEA) models cannot be used directly to

measure the performance of a supply chain and its members, because of the

existence of the intermediate measures connecting those members. This

observation is true for any situations where DMUs contain multi-stage

processes. This chapter presents several DEA-based approaches in a seller-

buyer supply chain context. Some DEA models are developed under the

assumption that the relationship between the seller and buyer is treated as

leader-follower and cooperative, respectively. In the leader-follower (or non-

cooperative) structure, the leader is first evaluated using the standard DEA

model, and then the follower is evaluated by a new DEA-based model which

incorporates the DEA efficiency information for the leader. In the cooperative

structure, one maximizes the joint efficiency that is modeled as the average of

the seller’s and buyer’s efficiency scores, and both supply chain members are

evaluated simultaneously.

Supply chain; Efficiency; Best practice; Performance; Data envelopment

analysis (DEA); Buyer; Seller.

Key words:

190

1. INTRODUCTION

It has been recognized that lack of appropriate performance measurement

systems has been a major obstacle to effective management of supply chains

(Lee and Billington, 1992). While there are studies on supply chain

performance using the methodology of data envelopment analysis (DEA),

the research has been focused on a single member of the supply chain. For

example, Weber and Desai (1996) employed DEA to construct an index of

relative supplier performance.

Note that an effective management of the supply chain requires knowing

the performance of the overall chain rather than simply the performance of

the individual supply chain members. Each supply chain member has is own

strategy to achieve efficiency, however, what is best for one member may

not work in favor of another member. Sometimes, because of the possible

conflicts between supply chain members, one member’s inefficiency may be

caused by another’s efficient operations. For example, the supplier may

increase its raw material price to enhance its revenue and to achieve an

efficient performance. This increased revenue means increased cost to the

manufacturer. Consequently, the manufacturer may become inefficient

unless it adjusts its current operating policy. Measuring supply chain

performance becomes a difficult and challenging task because of the need to

deal with the multiple performance measures related to the supply chain

members, and to integrate and coordinate the performance of those

members.

As noted in Zhu (2003), although DEA is an effective tool for measuring

efficiencies of peer decision making units (DMUs), it cannot be applied

directly to the problem of evaluating the efficiency of supply chains. This is

because some measures linked to supply chain members cannot be simply

classified as “outputs” or “inputs” of the supply chain. In fact, with respect

to those measures, conflicts between supply chain members are likely

present. For example, the supplier’s revenue is an output for the supplier,

and it is in the supplier’s interest to maximize it; at the same time it is also an

input to the manufacturer who wishes to minimize it. Simply minimizing the

total supply chain cost or maximizing the total supply chain revenue (profit)

does not properly model and resolve the inherent conflicts.

Within the context of DEA, there are a number of methods that have the

potential to be used in supply chain efficiency evaluation. Seiford and Zhu

(1999) and Chen and Zhu (2004) provide two approaches in modeling

efficiency as a two-stage process. Färe and Grosskopf (2000) develop the

network DEA approach to model general multi-stage processes with

intermediate inputs and outputs (see also chapter 12). Golany, Hackman and

Passy (2006) provide an efficiency measurement framework for systems

Chapter 11

191

composed of two subsystems arranged in series that simultaneously compute

the efficiency of the aggregate system and each subsystem. Zhu (2003), on

the other hand, presents a DEA-based supply chain model to both define and

measure the efficiency of a supply chain and that of its members, and yield a

set of optimal values of the (intermediate) performance measures that

establish an efficient supply chain.

We point out that there are number of DEA studies on supply chain

efficiency. Yet, all of them tend to focus on a single supply chain member,

not the entire supply chain, where at least two members are present. This can

be partly due to the lack of DEA models for supply chain or multi-stage

structures.

Evaluation of supply chain efficiency, using DEA, has its advantages. In

particular, it eliminates the need for unrealistic assumptions inherent in

typical supply chain optimization models and probabilistic models; e.g., a

typical EOQ model assumes constant and known demand rate and lead-time

for delivery. These conventional approaches typically fail, however, to

consider the cooperation within the supply chain system.

Recently, Liang et al. (2006) developed two classes of DEA-based models

for supply chain efficiency evaluation, using a seller-buyer supply chain as

an example. First, the relationship between the buyer and the seller is

modeled as a non-cooperative two-stage game, and second, the relationship

is assumed to be cooperative. In the non-cooperative two-stage game, they

use the concept of a leader-follower structure. In the cooperative game, it is

assumed that the members of the supply chain cooperate on the intermediate

measures. The resulting DEA models are non-linear and can be solved using

a parametric linear programming technique.

The current chapter presents the approaches of Zhu (2003) and Liang et

la. (2006).

2. NOTIONS AND STANDARD DEA MODELS

Suppose there are N similar supply chains or N observations on one supply

chain. Consider a buyer-seller supply chain as described in Figure 11-1, where

for j = 1, …, N, X

A

=(

A

ij

x , i = 1, …, I ) is the input vector of the seller, and Y

A

=(

A

rj

y , r=1,…., R ) is the seller’s output vector. Y

A

is also an input vector of

the buyer. The buyer also has an input vector X

B

=(

B

s

j

x

, s = 1,…., S ) and the

output vector for the buyer is Y

B

(=

B

tj

y , t=1, …, T ).

We can use the following DEA model to measure the efficiency of the

supply chain shown in Figure 11-1 (Charnes, Cooper and Rhodes, 1978;

CCR)

Cook et al, DEA Models for Supply Chain or Multi-Stage Structure

192

Maximize

0

P

V =

0

1

0s 0

11

T

BB

tt

t

IS

AA BB

ii s

is

uy

vx vx

=

==

+

∑

∑∑

subject to

1

s

11

T

BB

ttj

t

IS

AA BB

iij sj

is

uy

vx vx

=

==

+

∑

∑∑

1

≤

j = 1, …, N

B

t

u ,

A

i

v ,

s

B

v 0≥ , t=1, …, T, i = 1, …, I, s = 1,…., S

(1)

Figure 11-1. Seller-Buyer Supply Chain

Zhu (2003) shows that DEA model (1) fails to correctly characterize the

performance of supply chains, because it only considers the inputs and

outputs of the supply chain system and ignores measures Y

A

associated with

supply chain members. Zhu (2003) also shows that if Y

A

are treated as both

input and output measures in model (1), all supply chains become efficient.

Zhu (2003) further shows that an efficient performance indicated by model

(1) does not necessarily indicate efficient performance in individual supply

chain members. For example, consider the example shown in Table 11-1.

Table 11-1. Efficient supply chains with inefficient supply chain members

DMU X

A

Y

A

Y

B

Seller

Efficiency

Buyer

Efficiency

Supply Chain

Efficiency

1 1 4 10 1 0.5 1

2 1 2 10 0.5 1 1

3 1 2 5 0.5 0.5 0.5

The above example shows that improvement to the best-practice can be

distorted. i.e., the performance improvement of one supply chain member

affects the efficiency status of the other, because of the presence of

intermediate measures.

Alternatively, we may consider the average efficiency of the buyer and

seller as in the following DEA model

Seller

Buyer

X

A

Y

A

X

B

Y

B

Chapter 11

193

Maximize

1

2

[

0

1

0

1

R

AA

rr

r

I

AA

ii

i

uy

vx

=

∑

+

0

1

0 0

11

T

BB

tt

t

RS

AA BB

rr ss

rs

uy

cy vx

=

==

+

∑

∑∑

]

subject to

1

R

AA

rrj

r

I

AA

iij

i

uy

vx

=

∑

1≤ j = 1, …, N

1

11

T

BB

ttj

t

RS

AA BB

rrj ssj

rs

uy

cy vx

=

==

+

∑

∑∑

1≤ j = 1, …, N

A

r

c ,

A

r

u ,

B

t

u ,

A

i

v ,

s

B

v

0≥

r=1, …, R, t=1, …, T, i = 1, …, I, s = 1,…., S

(2)

Although model (2) considers Y

A

, it does not reflect the relationship

between the buyer and the seller. Model (2) treats the seller and the buyer as

two independent units. This does not reflect an ideal supply chain operation.

3. ZHU (2003) APPROACH

1

Using the notation in the above section, the Zhu (2003) model for

situations when there are only two members in the supply chain can be

expressed as

)min(

buyerseller

θθ

+

subject to

Rr y

Nj

Rr yy

Iixx

r

j

r

N

j

rjj

i

seller

N

j

ijj

,...,1 0

~

,...,1 ,0

,...,1

~

,...,1

)seller(

A

0

A

0

1

A

0

1

A

=≥

=≤

∑

=

λ

θλ

1

Chapter 8 of Zhu, J., 2003, Quantitative Models for Performance Evaluation and

Benchmarking: Data Envelopment Analysis with Spreadsheets. Kluwer Academic

Publishers, Boston

Cook et al, DEA Models for Supply Chain or Multi-Stage Structure

194

Nj

t yy

r yy

ixx

(buyer)

j

t

N

j

tjj

r

N

j

rjj

s

buyer

N

j

sjj

,...,1 ,0

T1,...,

R1,...,

~

S1,...,

B

0

1

B

A

0

1

A

B

0

1

B

=≥

=≤

∑

=

β

θβ

where

A

0

~

r

y are decision variables.

Additional constraints can be added into the above model. For example,

if

A

r

y represents the number of units of a product shipped from the

manufacturer to the distributor, and if the capacity of this manufacturer in

producing the product is

r

C

, then we may add

A

0

~

r

y <

r

C

.

4. COOPERATIVE AND NON-COOPERATIVE

APPROACHES

In this section, we present several models due to Liang et al. (2006) that

directly evaluate the performance of the supply chain as well as its members,

while considering the relationship between the buyer and the seller. The

modeling processes are based upon the concept of non-cooperative and

cooperative games (see, e.g., Simaan and Cruz, 1973; Li, Huang and Ashley,

1995; Huang 2000).

4.1 The Non-cooperative Model

We propose the seller-buyer interaction be viewed as a two-stage non-

cooperative game with the seller as the leader and the buyer as the follower.

First, we use the CCR model to evaluate the efficiency of the seller as the

leader:

Maximize E

AA

=

0

1

0

1

R

AA

rr

r

I

AA

ii

i

uy

vx

=

∑

subject to

(3)

Chapter 11

195

1

R

AA

rrj

r

I

AA

iij

i

uy

vx

=

∑

1≤

j = 1, …, N

A

r

u ,

A

i

v 0≥ r=1, …, R, i = 1, …, I

This model is equivalent to the following standard DEA multiplier model:

Maximize

AA

E

=

0

1

R

AA

rr

r

y

μ

=

∑

subject to

11

0

IR

AA AA

iij rrj

ir

xy

ωμ

==

−≥

∑∑

j = 1, …, N

0

1

I

AA

ii

i

x

ω

=

∑

A

r

μ

,

A

i

ω

0≥ r=1, …, R , i = 1, …, I

(4)

Suppose we have an optimal solution of model (4)

*A

r

μ

,

*

A

i

ω

,

*

AA

E (r=1,

…, R , i = 1, …, I), and denote the seller’s efficiency as

*

AA

E . We then use

the following model to evaluate the buyer’s efficiency:

Maximize

AB

E =

0

1

0 0

11

T

BB

tt

t

RS

AA BB

rr ss

rs

uy

Dyvx

μ

=

==

×+

∑

∑∑

subject to

1

11

T

BB

ttj

t

RS

AA BB

rrj ssj

rs

uy

Dyvx

μ

=

==

×+

∑

∑∑

1≤ j = 1, …, N

0

1

R

AA

rr

r

y

μ

=

∑

=

*

AA

E

11

0

IR

AA AA

iij rrj

ir

xy

ωμ

==

−≥

∑∑

j = 1, …, N

0

1

I

AA

ii

i

x

ω

=

∑

A

r

μ

,

A

i

ω

,

B

t

u ,

s

B

v , 0D ≥

r=1, …, R, t=1, …, T, i = 1, …, I, s = 1,…., S

(5)

Note that in model (5), we try to determine the buyer’s efficiency given

that the seller’s efficiency remains at

*

AA

E . Model (5) is equivalent to the

following non-linear model:

Maximize

AB

E =

0

1

T

BB

tt

t

y

μ

=

∑

Cook et al, DEA Models for Supply Chain or Multi-Stage Structure

196

subject to

111

0

RST

AA BB BB

rrj ssj ttj

rst

dy x y

μωμ

===

×+−≥

∑∑∑

j = 1, …, N

0 0

11

1

RS

AA BB

rr ss

rs

dy x

μω

==

×+=

∑∑

0

1

R

AA

rr

r

y

μ

=

∑

=

*

AA

E

11

0

IR

AA AA

iij rrj

ir

xy

ωμ

==

−≥

∑∑

j = 1, …, N

0

1

I

AA

ii

i

x

ω

=

∑

A

r

μ

,

A

i

ω

,

B

t

μ

,

s

B

ω

,

0d ≥

r=1, …, R, t=1, …, T, i = 1, …, I, s = 1,…., S

(6)

Note that

0 0

11

1

RS

AA BB

rr ss

rs

dy x

μω

==

×+=

∑∑

and

0

1

R

AA

rr

r

y

μ

=

∑

=

*

AA

E .

Thus, we have 0 <

0

1

R

AA

rr

r

dy

μ

=

<

∑

=

*

1

AA

E

. i.e, we have the upper and

lower bounds on d . Therefore, d can be treated as a parameter, and model

(6) can be solved as a parametric linear program.

In computation, we set the initial d value as the upper bound, namely,

0

d

=

*

1

AA

E , and solve the resulting linear program. We then start to decrease

d

according to

*

1

tAA

dE t

ε

=−× for each step t, where

ε

is a small

positive number

2

. We solve each linear program of model (6) corresponding

to

t

d and denote the optimal objective value as

*

BA

E (

t

d ).

Let

)(

**

tBA

t

BA

dEMaxE =

. Then we obtain a best heuristic search solution

*

BA

E to model (6)

3

. This

*

AB

E represents the buyer’s efficiency when the

seller is given the pre-emptive priority to achieve its best performance. The

efficiency of the supply chain can then be defined as

AB

e = )(

2

1

∗∗

+

ABAA

EE

2

In the current study, we set

ε

= 0.01. If we use a smaller

ε

, the difference only shows in

the fourth decimal place in the current study.

3

The proposed procedure is a global solution approximation using a heuristic technique, as it

searches through the entire feasible region of d when d is decreased from its upper bound

to lower bound of zero. It is likely that estimation error exists. The smaller the decreased

step, the better the heuristic search solution will be.

Chapter 11

197

Similarly, one can develop a procedure for the situation when the buyer

is the leader and the seller the follower. For example, in the October 6, 2003

issue of the Business Week, its cover story reports that Walmart dominates

its suppliers and not only dictates delivery schedules and inventory levels,

but also heavily influences product specifications.

We first evaluate the efficiency of the buyer using the standard CCR ratio

model:

Maximize

BB

E =

0

1

0 0

11

T

BB

tt

t

RS

AA BB

rr ss

rs

uy

uy vx

=

==

+

∑

∑∑

subject to

1

11

T

BB

ttj

t

RS

AA BB

rrj ssj

rs

uy

uy vx

=

==

+

∑

∑∑

1≤ j = 1, …, N

A

r

u ,

B

t

u ,

s

B

v 0≥

r=1, …, R, t=1, …, T , s = 1,…., S

(7)

Model (7) is equivalent to the following standard CCR multiplier model:

Maximize

BB

E =

0

1

T

BB

tt

t

y

μ

=

∑

subject to

111

0

RST

AA BB BB

rrj ssj ttj

rst

yxy

μωμ

===

+−≥

∑∑∑

j = 1, …, N

0 0

11

1

RS

AA BB

rr ss

rs

yx

μω

==

+=

∑∑

A

r

μ

,

B

t

μ

,

s

B

ω

0≥

r=1, …, R, t=1, …, T , s = 1,…., S

(8)

Let

*

A

r

μ

,

*

B

t

μ

,

*

s

B

ω

,

*

BB

E ( r=1, …, R, t=1, …, T , s = 1,…., S) be an

optimal solution from model (8), where

*

BB

E represents the buyer’s

efficiency score. To obtain the seller’s efficiency given that the buyer’s

efficiency is equal to

*

BB

E , we solve the following model:

Maximize

BA

E =

0

1

0

1

R

AA

rr

r

I

AA

ii

i

Uy

vx

μ

=

×

∑

subject to

Cook et al, DEA Models for Supply Chain or Multi-Stage Structure

198

1

R

AA

rrj

r

I

AA

iij

i

Uy

vx

μ

=

×

≤

∑

j = 1, …, N

*

0

1

T

BB

tt BB

t

yE

μ

=

∑

111

0

RST

AA BB BB

rrj ssj ttj

rst

yxy

μωμ

===

+−≥

∑∑∑

j = 1, …, N

0 0

11

1

RS

AA BB

rr ss

rs

yx

μω

==

+=

∑∑

A

i

v ,

A

r

μ

,

B

t

μ

,

s

B

ω

, 0U ≥

r=1, …, R, t=1, …, T, i = 1, …, I, s = 1,…., S

(9)

Model (9) is equivalent to the following non-linear program:

Maximize

BA

E

=

0

1

R

AA

rr

r

uy

μ

=

×

∑

subject to

11

0

IR

AA A A

iij rrj

ir

xu y

ωμ

==

−× ≥

∑∑

j = 1, …, N

0

1

I

AA

ii

i

x

ω

=

∑

*

0

1

T

BB

tt BB

t

yE

μ

=

∑

111

0

RST

AA BB BB

rrj ssj ttj

rst

yxy

μωμ

===

+−≥

∑∑∑

j = 1, …, N

0 0

11

1

RS

AA BB

rr ss

rs

yx

μω

==

+=

∑∑

A

i

ω

,

A

r

μ

,

B

t

μ

,

s

B

ω

, 0u ≥

r=1, …, R, t=1, …, T, i = 1, …, I, s = 1,…., S

(10)

This model (10) is similar to model (6) and can be treated as a linear

program with

u

as the parameter. We next show how to select the initial

value of this parameter.

We first solve the following model:

Maximize

BA

EF =

*

0

1

0

1

R

AA

rr

r

I

AA

ii

i

Uy

vx

μ

=

×

∑

subject to

*

1

R

AA

rrj

r

I

AA

iij

i

Uy

vx

μ

=

×

≤

∑

j = 1, …, N

,0

A

i

vU≥

i = 1, …, I

(11)

Chapter 11

Zhu J., Cook W.D. (Eds.) Modeling Data Irregularities and Structural Complexities in Data Envelopment Analysis

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