International Journal of Mechanical and Production Engineering, ISSN: 2320-2092,
Volume- 2, Issue- 2, Feb.-2014
AN INTEGRATED STOCHASTIC MULTI-OBJECTIVE SUPPLIER
SELECTION PROBLEM WITH INCREMENTAL VOLUME
DISCOUNTS
1
REMICA AGGARWAL, 2S.P SINGH
1
Department of Management, Birla Institute of Technology & Science, Pilani, Rajasthan, India
2
Department of Management Studies, Indian Institute of Technology, Delhi, India
Email: [email protected]; [email protected]
Abstract— A stochastic multi-objective supplier selection model with lot size volume discounts is proposed in the paper.
The stochastic parameters involved in the model are the uncertainties related to supplier’s capacity, product demand, unit
cost of purchasing and lead time. These stochastic parameters are handled and converted to their deterministic equivalents
using chance constraint based approach. To validate the proposed model, data is generated randomly and solved in LINGO
10.
Keywords— Supplier selection; Chance constrained approach; Incremental quantity discount model
uncertainty. Section III presents model for
deterministic version of the Integrated Multi-objective
Supplier Selection Problem (IDMoSSP) while section
IV extends the IDMoSSP into Integrated Stochastic
Multi-objective
Supplier
Selection
Problem
(ISMoSSP). Section V presents an illustrative
example to demonstrate the proposed model.
Conclusions and directions for future research are
given in Section VI.
I. INTRODUCTION
Nowadays businesses largely depend on longer-term
working relationship with the suppliers. Supplier
selection is one of the tricky decision making process
as right choice of suppliers reduces costs, increases
profit margins, improves component quality and
ensures timely delivery. Selection of suppliers
becomes more challenging when suppliers offer
interesting deals such as better price and quality,
incremental or all units quantity discounts etc.
Selecting an optimal supplier under multiple
conflicting criteria such as purchasing cost, quantity
discounts, order size restrictions, product quality and
service levels, supplier capacity and lead time, often
gets complicated even for an experienced purchase
manager because competing suppliers have different
levels of achievement under these criteria. Also in the
event of uncertainty such as uncertain or fluctuating
demand, changing supplier’s capacity, supplier’s
unreliable lead time and varying quality, selection of
suppliers becomes even more complex. Although the
researches have been made in the field of selecting
suppliers under uncertainty or under discount pricing
and lot size restrictions, no research has integrated
quantity discounts model with uncertainty and lot size
restrictions at a time. In this paper, an attempt has
been made to present a mathematical model for
supplier selection supplying multiple products to a
buyer under uncertain scenario incorporating
incremental quantity discounts on lot sizes of
multiple products to be supplied by multiple
suppliers. Uncertainty in model parameters such as
capacity, lead time and demand uncertainty is
handled through the Chance constraints approach [13]. These uncertainties are captured by probability
distribution of capacity, demand, cost and lead time.
II.
LITERATURE REVIEW
Supplier Selection Problem with Quantity Discounts
Although the process of supplier selection has been
studied extensively, the problem of supplier selection
under multi-supplier with quantity discounts has
received attention quite recently.
A weighted fuzzy multi-objective model for the
supplier selection under price breaks or quantity
discounts environment in a supply chain is proposed
by [4], [5]. A multi-objective optimization problem is
proposed by [6], where one or more buyers order
multiple products from different vendors in a multiple
sourcing network using a pricing model under
quantity discounts. Similar kind of a problem is
considered by [7] using business volume discounts.
Reference [8] uses a scatter search algorithm for
supplier selection and order lot sizing under multiple
price discount environment.
A problem of allocation of orders for custom parts
among suppliers offering business volume discounts
in make to order manufacturing as a single or multiobjective mixed integer program is considered by [9].
All the above mentioned researches consider supplier
selection under all units, incremental or business
volume discounts in an exact or certain scenario. They
have not incorporated uncertainties related with each
of the capacity, demand or lead time parameters.
The paper is structured as follows. Section II presents
a review of the relevant literature on supplier selection
under volume discounts, lot size restrictions and
An Integrated Stochastic Multi-Objective Supplier Selection Problem With Incremental Volume Discounts
7
International Journal of Mechanical and Production Engineering, ISSN: 2320-2092,
A. Supplier selection problem under uncertainties
A two-phase stochastic program is proposed by [10]
where plant selection, product allocation and supplier
selection under uncertain costs, product prices and
demand is considered. Tang [11] observed four
drivers of operational risks, uncertain demand,
uncertain supply, uncertain lead times and uncertain
costs. Reference [12] also developed a supplier
selection model with demand uncertainty and
unreliable suppliers. Reference [13]consider both
operational risks and disruption risks in an inventory
problem by considering a reliable and an unreliable
supplier and derived expressions of optimal order
quantities under different conditions. A supplier
selection model under uncertain demand was
developed by [14] where suppliers have limits on
minimum and maximum order sizes. These researches
has not explored uncertainties related to demand,
capacity and lead time at a time and also lack the
concept of quantities discounts on the part of supplier.
Present paper integrates the concept of incremental
quantities discounts and lot size restrictions offered by
multiple suppliers in the event of uncertain demand,
supplier’s capacity and lead time.
III.
PROBLEM
DEFINITION
FORMULATION
m jk
Middle order quantity to be supplied from
supplier j of product k
u ' jk
u '' jk
kth product unit cost incurred on buyer from jth
 m jk

Aggregate unit cost incurred when order
quantity  m jk
u
jk
F
j
Q
jk
Fixed cost of operating with supplier j
Percentage of good quality items of product k
procured from supplier j
L
Lead time of product k from supplier j
jk
q
j
Minimum order quantity to be supplied from
supplier j
ljk
Mean lead time of product k from supplier j
 k Level of probability that units supplied satisfies
the demand of kth product
&
 jk
Level of probability that
h jk  c a p j k x j k
l Level of risk for the calculated value of lead time
to be greater than aspired level
F Dk1
Constant inverse probability distribution function
for random demand
1
Fcap
jk xjk
Constant inverse probability distribution function
for random capacity
FA11 l 
Constant inverse probability distribution
function for random lead time
B. Deterministic model
Integrated Deterministic multi-objective supplier
selection problem (IDMoSSP) integrating incremental
quantity discounts and lot size restrictions can be
written as:
J
Minimize Z 1 
K
u
J
J
M ax im ize Z 2 
K
x jk +   F j x jk (1)
jk
j 1 k 1
j 1 k 1
K
Q
jk
h jk x jk
(2)
j 1 k 1
J
M inim ize Z 3 
K
L
jk
h jk x jk
j 1 k 1
(3)
Subject to
jk
jk
 m jk
supplier if order quantity
1 if supplier j is assigned as supplier of
product k, 0 otherwise
h
kth product unit cost incurred on buyer from jth
supplier if order quantity
A supply chain management problem with single
buyer and multiple suppliers is considered. Suppliers
offers incremental quantities discounts on lots to be
purchased by the buyer provided their lot size
aspirations for each product is fulfilled by the buyer.
There is quality level associated with each supplier for
every product which needs to be maximized. Another
objective before a buyer is to minimize the total cost
which may include variable as well as fixed costs.
There is lead time associated with each product. Each
product has an associated demand which need to be
met to the fullest extent. Hence, this supplier selection
problem have multiple conflicting objectives such as
total cost , quality and lead time subject to the
constraints such as demand and limited capacity. The
problem is stochastic in nature since some of the
parameters such as total cost, lead time, demand and
capacity are not certain or deterministic rather
stochastic or probabilistic in nature. This section
formulates an Integrated Stochastic Multi-objective
Supplier Selection Problem (ISMoSSP) with quantity
discounts and lot size restrictions. Following notations
are used to formulate the model.
j
1,2,...,J suppliers
k
1,2,...K products
x
Volume- 2, Issue- 2, Feb.-2014
J

h jk  D k ,  k  K
(4)
j1
The amount of product k shipped from
h jk  cap jk x jk ,  j  J ,  k  K
supplier j
D k
Demand for product k
C a p jk
Capacity at supplier j for product k
K

h jk x jk  q j ,  j  J ,  k  K
k 1
An Integrated Stochastic Multi-Objective Supplier Selection Problem With Incremental Volume Discounts
8
(5)
(6)
International Journal of Mechanical and Production Engineering, ISSN: 2320-2092,
x jk  [0 ,1]  j  J ,  k  K
u
jk
 u ' jk h j k
if h jk  m
independently. Using equation (17) this situation can
be modeled with the separate chance constraint as
given below:
(7)
h jk  0 ,  j  J ,  k  K
(8)

P 

(9)
jk

(10)
u jk  u jk  u ' jk * mjk  (hjk  m jk )u '' jk , if hjk  m jk
J
K
j 1
Q
j
x
jk
jk
h
jk
x
jk
j1

k 1
l
jk
h
jk
x
jk
k
j1

  

k
,
 k  K
(18)
J

h
jk
 F Dk1   k

k  K
(19)
j 1
Following the similar steps in the preceding section,
the capacity constraint given in inequality (15) can be
expanded as follows:
h1 1  cap11 x11 ; h12  ca p1 2 x12 ;......h1 k  cap1 k x1 k
h 21  cap 21 x 21 ; h 22  cap 2 2 x 22 ;......h 2 k  cap 2 k x 2 k
.
.
.
.
.
.
h j 1  cap j1 x j1 ; h j 2  cap j 2 x j 2 ;......h jk  ca p jk x jk
cap
(20)
x
jk
jk
Assuming
follows probability distribution
with
cumulative
distribution
function
F c a p j k x jk
satisfying capacity of each supplier for each
product independently. Using inequality (20) this
situation can be modeled with the separate chance
constraint as given below:
(11)
P
(12)
K
 

F
1
 D
Deriving Capacity uncertainty [inequality (15)]
K
 
+
jk
k 1
J
3
x
jk
The deterministic equivalent in (19) preserves the
linearity of inequality (18).
K
j  1
M in im iz e Z
jk
j1 k
 

2
u
k 1
J
M a x im iz e Z
J
 

h
 J

F D k   h jk    k
j

1


C. Stochastic Multi-Objective Supplier Selection
Problem (ISMoSSP)
Applying chance constraint approach given by [1-3],
the deterministic model given in (1) to (10) is
converted to ISMoSSP described from (11) to (16) as
follows:
1
J

Deterministic equivalent to the chance constraint in
(18) can be written as:
Objective function Z1 minimizes the total cost
incurred by the buyer.
Objective function Z2
maximizes the total quality of purchased products.
Objective function Z3 minimizes the total lead time.
Constraint given by (4) ensures that shipments from
the suppliers cover the entire demand for each
product. Equation (5) restricts the amount shipped
from the suppliers to their capacity. Constraint (6)
provides the lot size restriction on different products.
Constraints in (7) and (8) enforce binary and non
negativity condition in the decision variables
respectively. Constraints (9) and (10) indicate the
incremental quantities discount provided by the jth
supplier for kth product based on the number of
components ordered.
M in im iz e Z
Volume- 2, Issue- 2, Feb.-2014
h
jk
 cap
jk
x
jk


jk
, j J ,k  K
(21)
Deterministic equivalent to the chance constraint in
inequality (21) is simply
(13)
 P  hjk  cap jk x jk   1   jk , j  J , k  K
Subject to
1
 Fcap jk x jk  h jk   1   jk  hjk  Fcap
1   jk  j  J , k  K
jk x jk
J
Deriving demand uncertainty [equation (14)]
(22)
The deterministic equivalent (22) preserves the
linearity of inequality (21).
Deriving Lead time uncertainty [equation (16)]
Let A1 be the aspired lead time of the lead time
objective function. This aspiration level may be
obtained by getting the ideal solution of solving lead
time objective separately. Applying chance constraint
approach the probabilistic equation for lead time
objective function can be written as :
The demand constraint given in (14) can be expanded
as follows:

P

h j k  F Dk1  k


k  K
(14)
j 1
h j k  F c a 1p jk x j k 1  
J

jk
j  J ,k  K
(15)
K
 
j1

l
k 1
h
jk
jk
x
 F
jk
1
A1
 l 
(16)
Constraints (6)-(10)
h
h
h
.
.
.
h
1 2


h
h
1 3

h
1 1
1
K

h
2 2


h
h
2 3

h
2 1
2
K

3 2


. . . . . .h
. . . . . .h
3 3

. . . . . .h
3 1
h
3
K

J

1
J
J
. . . . . .h
2
3
D

D
D
J K

K

 l j k h jk x jk  A 1   

l
(23)
Deterministic equivalent to the chance constraint
in (23) is simply written as
1

J

2
j 1 k 1
3
D
K
 J K
FA1     l h jk x jk
jk
 j 1 k 1
(17)

  l

J

K

j 1 k 1
l jk
h jk x jk  F A11  l 
(24)
Assuming the Dk follows probability distribution with
F
cumulative distribution function Dk and we are
concerned with satisfying demand for each product
The deterministic equivalent in inequality (24)
preserves the linearity of inequality (23).
An Integrated Stochastic Multi-Objective Supplier Selection Problem With Incremental Volume Discounts
9
International Journal of Mechanical and Production Engineering, ISSN: 2320-2092,
IV.
Volume- 2, Issue- 2, Feb.-2014
 be the corresponding
problem and 0 
objective function value then final problem can be
formulated using this optimal solution as follows:
0
GOAL PROGRAMMING APPROACH
0
g  , ,X
*
Each of the objectives Z1, Z2, Z3 has ideal value Z 1
0
*
*
, Z 2 and Z 3 associated with them. These are the
optimal values for a given objective in the multiobjective model ignoring other objectives. Since
multiple objectives conflict with each other, the ideal
value can never be reached for all objectives
simultaneously in a multi-objective optimization
problem. Therefore some target values which are fixed
percentages of ideal values can be assigned to each of
these objectives in order to get a compromised
solution. For example, formulation of problem from
(11) to (16) incorporating the ideal values for each of
the objective becomes
J
K
J
Stage 2:
Minimize
g ( ,  , x )   1  1   2 
J
J
M a x im ize Z 2 

J
jk
j 1 k 1
J
x jk + 
K
F j x jk   1   1  Z 1*

(35)
j 1 k 1
Q
jk
h
jk
x
jk

2
 2  Z
2
*
j  1 k 1
J
J

j 1 k 1
 l h jk x jk   3   3  F A1 1  l   Z 3*
jk
0
0
j  1 k 1
l jk
(37)
0
g ( ,  , x )  g 0 ( ,  , x )
(38)
Constraints (30)-(31) ; Constraints (6)-(10)
(26)
Where  i ,  i and  i are the weights, positive and
negative
deviational
variables
respectively
corresponding to ith objective function ( i=1,2,3).
K

(36)
K
K
 Q jk h jk x jk
(34)
K
 
(25)
j  1 k 1
M inim ize Z 3 
K
u
K
j 1 k 1
 33
Subject to
Minimize Z1    u jk x jk +   F j x jk
j 1 k 1
2
h jk x jk
(27)
Subject to
J
K
J
u
jk
j 1 k 1
J
V.
K
x jk +   F j x jk  Z 1*
j 1 k 1
jk
h jk x jk  Z 2*
(29)
j 1 k 1
J
Consider Five suppliers (S1, S2, S3, S4, S5) supplying
three different products (P1, P2, P3) to the buyer.
Stochastic data corresponding to the capacity,
minimum order, demand, unit cost, fixed cost, quality
levels and stochastic lead time are given from Table I
to Table VI. All data are randomly generated. The
reliability level for the capacity is set at  jk = 0.95
j  1, 2,...5; k  1, 2,3 meaning that at least 95% of
demand should be met. The risk level is set
(28)
K
Q
K

j 1 k 1
l jk
NUMERICAL ILLUSTRATION
h jk x jk  F A1 1  l   Z 3*
(30)
Constraints (6)-(10), (14), (15)
This multi-objective problem when solved gives an
infeasible solution. In order to obtain compromise
solution, goal programming technique is used. The
above formulation of the problem is solved in two
stages as described below.
at
k ,l  0.05.
Stage 1:
M in im iz e g 0 ( ,  , x )  
jk

'
jk
(31)
Subject to
J
h
  jk   jk  F Dk1  k
jk

k  K
j 1
1
Fcap
1   jk   h jk   'jk   'jk  0 j  J , k  K
jk x jk
(32)
(33)
Constraints (6)-(10)



'
TABLE II. STOCHASTIC DEMAND DATA (IN UNITS)
P1
P2
P3
N(210,36) N(250,49) N(250,64)
'
Where jk , jk , jk and jk are negative and positive
deviational variables corresponding to k demand
constraints and j*k capacity constraints respectively.
 ,  , X  be the optimal solution to the above
0
Let
0
0
An Integrated Stochastic Multi-Objective Supplier Selection Problem With Incremental Volume Discounts
10
International Journal of Mechanical and Production Engineering, ISSN: 2320-2092,
TABLE III. FIXED COST ( IN DOLLARS ) FOR
SUPPLIERS
S1 S2 S3 S4 S5
100 200 150 150 120
Volume- 2, Issue- 2, Feb.-2014
ujk  u ' jk hjk ,if hjk  mjk ; j 1,2..5;k 1,2,3

ujk  ujk  u ' jk *mjk  (hjk  mjk )u'' jk , if hjk  mjk ; j  1,2..5; k  1,2,3
(44)
The above problem is solved in LINGO 10 and the
results are displayed below in table VII and table VIII.
Number of units supplied has been approximated to
the nearest decimal places.
TABLE VII . RESULTS IN TERMS OF COST, QUALITY
TABLE IV. UNIT COST DATA WITH PRICE BREAKS
(IN DOLLARS)
AND LEAD TIME
TABLE V. Q UALITY DATA (IN % OF GOOD ITEMS)
P1
0.95
0.95
0.9
0.9
0.9
S1
S2
S3
S4
S5
P2
0.95
0.97
0.9
0.93
0.92
P3
0.93
0.99
0.9
0.9
0.97
TABLE VIII. NUMBER OF UNITS SUPPLIED FROM
EACH SUPPLIER
TABLE VI. STOCHASTIC LEAD TIME DATA (IN DAYS)
P1
N(10,6)
N(5, 2)
N(8,2)
N(3,1)
N(8,2)
S1
S2
S3
S4
S5
P2
N(9, 5)
N(2,1)
N(3,1)
N(4,2)
N(2,1)
P3
N(1,0)
N(8,1)
N(9,2)
N(6,2)
N(4,1)
CONCLUSIONS AND FUTURE DIRECTIONS
3h32x32 9h33x33 3h41x41 4h42x42 6h43x43 8h51x51  2h52x52 4h53x53
The paper provides a multi-objective supplier
selection with incremental quantities discount and lot
size restrictions optimization model
under
uncertainties in demand, capacity and lead time
associated with supplier. Stochastic model with
deterministic equivalents is solved assuming a fixed
level of probability and risk which in practice can be
varied to test different scenarios. Goal programming
approach has been used to provide a compromise
feasible solution to the supplier selection problem.
The problem can be extended to include the cases of
multiple buyers, business volume discounts and all
unit discounts as well.
(39)
REFERENCES
On putting the various parameters, the SMoSSP
for the given example appears as:
Minimize u11x11 u12x12 u13x13 u21x21 u22x22 u23x23 u31x31 u32x32
u33x33 u41x41 u42x42 u43x43 u51x51 u52x52 u53x53 100(x11  x12  x13)
200 x21  x22  x23 150 x31  x32  x33 150 x41  x42  x43 120 x51  x52  x53
Maximize 0.95h11x11 0.95h12x12 0.93h13x13 0.95h21x21 0.97h22x22 0.99h23x23 0.9h31x31
0.9h32x32 0.9h33x33 0.9h41x41 0.93h42x42 0.9h43x43 0.9h51x51 0.92h52x52 0.97h53x53
Minimize 10h11x11 9h12x12 h13x13 5h21x21 2h22x22 8h23x23 8h31x31
Subject to
[1]
h11 h21 h31 h41 h51 219.87; h12 h22 h32 h42 h52  259.87;
(40)
h13 h23 h33 h43 h53  263.16;
[2]
h11  46x11; h12  41x12 ; h13  92x13 ; h21  83x21; h22  46x22 ; h23 18x23 ;
h31  64x31 ; h32  46x32 ; h33  138x33; h41  76x41; h42  46x42 ; h43  46x43 ;
[3]
h51  64x51; h52  92x52 ; h53  64x53
(41)
h11x11 h12x12 h13x13 100; h21x21 h22x22 h23x23 100; h31x31 h32x32 h33x33 100;
h41x41 h42x42 h43x43 100; h51x51 h52x52 h53x53 100;
(42)
[4]
[5]
xjk [0,1]  j 1,2,...5,k 1,2,3 hjk  0 , j 1,2,...5 ,k 1,2,3
(43)
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