Fuzzy Sets and Systems 159 (2008) 193 – 214
www.elsevier.com/locate/fss
An interactive possibilistic programming approach for multiple
objective supply chain master planning
S.A. Torabia,∗,1 , E. Hassinib
a Department of Industrial Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran
b DeGroote School of Business, McMaster University, 1280 Main St. W., Hamilton, ON, Canada L8S 4M4
Received 20 March 2007; received in revised form 27 June 2007; accepted 25 August 2007
Available online 31 August 2007
Abstract
Providing an efficient production plan that integrates the procurement and distribution plans into a unified framework is critical to
achieving competitive advantage. In this paper, we consider a supply chain master planning model consisting of multiple suppliers, one
manufacturer and multiple distribution centers. We first propose a new multi-objective possibilistic mixed integer linear programming
model (MOPMILP) for integrating procurement, production and distribution planning considering various conflicting objectives
simultaneously as well as the imprecise nature of some critical parameters such as market demands, cost/time coefficients and capacity
levels. Then, after applying appropriate strategies for converting this possibilistic model into an auxiliary crisp multi-objective linear
model (MOLP), we propose a novel interactive fuzzy approach to solve this MOLP and finding a preferred compromise solution.
The proposed model and solution method are validated through numerical tests. Computational results indicate that the proposed
fuzzy method outperforms other fuzzy approaches and is very promising not only for solving our problem, but also for any MOLP
model especially multi-objective mixed integer models.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Possibilistic programming; Supply chain master planning; Mixed-integer linear programs; Compromise solution
1. Introduction
The main focus of supply chain management (SCM) is the control of material flow among suppliers, plants, warehouses and customers efficiently such that the total cost in the supply chain can be minimized [38]. A major thrust of
recent research in this area is the development of optimization models that integrate different functions (e.g. purchasing,
production and distribution) in the supply chain. The basic idea behind this approach is to simultaneously optimize
decision variables of different functions that have traditionally been optimized sequentially [26]. In this regard, one of
the main issues facing with supply chain managers is the supply chain master planning (SCMP) problem. The major
task of SCMP is the determination of procurement, production and distribution quantities for facilities in different
echelons of a supply chain on a medium term basis [27]. Traditionally, these activities were conducted either independently or sequentially resulting in large inventories and very poor overall performance. But in the presence of emerging
∗ Corresponding author. Tel.: +9821 88021067; fax: +9821 88013102.
E-mail addresses: satorabi@ut.ac.ir (S.A. Torabi), hassini@mcmaster.ca (E. Hassini).
1 Currently a Post Doctoral Fellow at DeGroote School of Business, McMaster University, Hamilton, ON, Canada.
0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2007.08.010
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S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
computing power and increasing global competitive pressures, coordinating the procurement, production and distribution plans in an integrated and centralized framework is critical to the success of supply chains.
In such environments, where decisions involve resources and data that are owned by different entities within the
supply chain, there are two paramount characteristics of the problems that a decision maker will be faced with: (1)
Conflicting objectives that may arise from the nature of operations (e.g., minimize cost and at the same time increase
customer service) and the structure of the supply chain where it is often difficult to align the goals of the different
parties within the supply chain. (2) Lack of precise data (e.g., cost and lead time data) and/or presence of uncertainty
with imprecise parameters (e.g., demand fuzziness). Thus, it is important that models addressing problems in this area
should be designed to handle the foregoing two complexities.
In this paper we present a novel SCMP model consisting of multiple suppliers, one manufacturer and multiple
distribution centers which integrates the procurement, production and distribution plans considering various conflicting objectives simultaneously as well as the imprecise nature of some critical parameters such as market demands,
cost/time coefficients and capacity levels. We consider two important objective functions: the total cost of logistics
and the total value of purchasing. For constructing the former objective, we have used the modern concepts from the
costing literature known as the total cost of ownership (TCO) and activity-based costing (ABC) in order to have a
comprehensive total cost function. The paper has two important applied and theoretical contributions. First, it presents
a comprehensive and practical, but tractable, optimization model for supply chain master planning. The need for such
model by practitioners, for example for incorporation in advanced planning systems (APS), has been highlighted by
Tempelmeier [37]. And second, it introduces a novel solution procedure for finding an efficient compromise solution
to a fuzzy multi-objective mixed-integer program. In our literature survey we have felt a lack of studies in this field,
something that is understandable, given that mixed integer programming is known to be complex even when all data
is certain and precise.
The remainder of this paper is organized as follows. The relevant literature is reported in Section 2. In Section 3 we
define our notation, state our assumptions and propose a new multi-objective possibilistic mixed integer linear program
(MOPMILP) for the proposed SCMP problem. After applying appropriate strategies for converting the possibilistic
model into an auxiliary crisp multi-objective linear model (MOLP), we propose a novel interactive fuzzy approach to
solve this MOLP and find an efficient compromise solution in Section 4. The proposed model and solution method
are validated through numerical tests in Section 5. The data for these numerical computations have been inspired by a
real life industrial case as well as randomly generated data. Conclusion remarks about our computational results and
further research directions are the subject of Section 6.
2. Literature review
The considered SCMP problem deals with a medium term procurement, production and distribution planning problem in a three-echelon supply chain involving multiple suppliers, one manufacturer and multiple distribution centers. Based on characteristics of the problem which are explained in more details in the next section, we review the
most relevant and recent literature in three different but somewhat close streams: supplier selection and order lotsizing models, dynamic supply chain planning models, and application of fuzzy modeling in supply chain planning
problems.
2.1. Supplier selection and order lot-sizing
Ghodsypour and O’Brien [9] developed an integrated analytical hierarchy process (AHP) and linear programming
approach for solving a multi-objective, single-item, single-period capacitated supplier selection and order lot-sizing
problem. Xia and Wu [41] introduced a new model that integrates an improved AHP by rough sets theory with
a multi-objective mixed integer program to support supplier selection decisions in total business volume discounts
environments. Hassini [10] presented a supplier selection and order lot-sizing model for a single item, multiple period,
multiple capacitated suppliers offering lead time-dependent capacity reservation and unit price discounts. Sirias and
Mehra [33] studied quantity-dependent discounts versus lead time-dependent discounts in supply chains through a
simulation study. They concluded that the lead time-dependent discount systems could be more promising for supply
chains, especially for the manufacturing side. For additional studies in this area, the interested reader is referred to a
comprehensive review provided by Aissaoui et al. [2].
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195
2.2. Dynamic supply chain planning
Literature review of the SCMP area reveals that there are considerable amount of papers integrating production and
distribution decisions (see, for example, [3,8,21,26]). But, there are very limited papers integrating procurement and
distribution stages with production. Kanyalkar and Adil [14] considered a single supplier, multi-plant, multi-selling location and multi-dock problem and presented a comprehensive monolithic crisp mixed integer linear goal programming
model to obtain a time and capacity aggregated production plan, a detailed production plan, a detailed procurement
plan and a detailed distribution plan simultaneously to overcome the drawbacks of the hierarchical/sequential planning
approaches of not yielding a feasible and/or an optimal plan. Noorul Haq and Kannan [25] developed an integrated
supplier selection and multi-echelon distribution inventory model in a built-to-order supply chain involving single
selected supplier, multiple plants, multiple distributors, multiple wholesalers and multiple retailers. However, in the
supply side of the above integrated models, authors only considered the single supplier and/or single-item cases without
any discount structure. Sucky [34] stated that the coordination of order and production policies between buyers and
suppliers in supply chains is of particular interest and a coordinated order and production policy can reduce total cost
significantly.
2.3. Applications of fuzzy modeling in supply chain planning
Hsu and Wang [11] provided a possibilistic linear programming model to determine appropriate strategies regarding
the safety stock levels for assembly materials, regulating dealers’ forecast demands and numbers of key machines.
Wang and Liang [39] presented an interactive possibilistic linear programming approach for solving the multi-product
aggregate production planning problem with imprecise forecast demand, related operating costs and capacity. Kumar
et al. [15] proposed a fuzzy programming model with flexibility in some constraints for a single-item, single period
vendor selection problem. Mula et al. [23] provided a new linear programming model for the medium term production
planning in a capacitated multi-product, multi-level and multi-period MRP system, and transformed it into the three
different auxiliary crisp models with flexibility in the objective function, market demand and available capacity resources
using different fuzzy aggregation operators. Liang [20] developed an interactive multi-objective linear programming
model for solving the fuzzy multi-objective transportation problems with piecewise linear membership function. In
another work, Liang [21] proposed an interactive fuzzy multi-objective linear programming model for solving an
integrated production-transportation planning problem in supply chains. Selim and Ozkarahan [32] developed an
interactive fuzzy goal programming for the supply chain distribution network design. Chen and Chang [5] developed
an approach for deriving the membership function of the fuzzy minimum total cost in a multi-product, multi-echelon,
multi-period supply chain with fuzzy parameters using -cut representation and the fuzzy extension principle. However,
in most of the aforementioned works [11,15,20,21,39], the authors have applied the max–min approach of Zimmermann
[42] to solve the auxiliary single-objective model. But, it is well-known that the solution yielded by max–min operator
might not be unique nor efficient [17–19].
To the best of our knowledge, there is no research work in the supply chain master planning literature integrating
procurement, production and distribution planning activities in a fuzzy environment. Hence, in this study we develop
a comprehensive multi-objective SCMP framework for a three-echelon supply chain involving multiple suppliers,
one manufacturer and multiple distribution centers in a fuzzy environment to determine the coordinated purchasing,
production and distribution quantities over a given multi-period decision horizon. The main contributions of this paper
can be summarized as follows:
• Introducing a novel SCMP model in a fuzzy environment for integrating different activities in a multi-echelon, multiproduct and multi-period supply chain network, where we consider a simultaneous quantity and lead time-dependent
discount policy for the multi-item, multi-period supplier selection and order lot-sizing sub-problem.
• Proposing a new fuzzy programming approach for solving the auxiliary crisp multi-objective linear programming
model to provide a preferred compromise solution for the decision maker.
It should be noted that unlike previous studies that assume dynamic deterministic demand (e.g., [6,10]), critical
parameters (such as market demands and capacity levels) are imprecise (fuzzy) in nature due to incompleteness and/or
unavailability of required data over the mid-term decision horizon. In such conditions, the retailer/distributor knows its
demand requirements almost certainly, but quotes it in an imprecise manner, e.g., 100 ± 10 units. Therefore, we have to
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estimate the problem parameters subjectively based on current insufficient data and the decision maker’s experience.
That is why in this paper we chose to apply a fuzzy modeling approach. For a recent review of different approaches
for dealing with uncertainty in production planning problems the reader is advised to consult Mula et al. [24].
The proposed SCMP model could be used by every manufacturer dealing with different suppliers offering lead timedependent discounts as well as multiple customer zones dispersed in different areas for distributing its final products.
This seems to be especially relevant for global chemical processing companies, as has been mentioned to one of the
authors by a chemical engineer that attended a conference presentation of an earlier version of this study. Furthermore,
the proposed multi-objective framework enables the manufacturer to consider the various criteria in developing an
integrated procurement, production and distribution plan in order to reach a compromise solution considering different
supply parties’ concerns (i.e., implementing the win–win strategy).
3. Problem description and formulation
A manufacturer produces different types of products using a common set of input items (i.e., raw materials and/or subassembly parts) which are supplied from a pre-determined set of qualified suppliers. The final products are then delivered
to different distribution centers (warehouses/wholesalers) in order to satisfy their associated dynamic demands. This
problem in fact integrates three planning sub-problems: (1) supplier selection and order lot-sizing in the first echelon,
(2) production planning at the manufacturer in the second echelon, and (3) distribution planning in the last echelon
of the supply chain. Our objective is to find the best planning decisions over a multi-period mid-term horizon, in an
integrated and coordinated manner, for the following issues:
• Procurement plan: The purchase quantity for each item from each supplier in each period.
• Production plan: The production quantity for each final product in each period.
• Distribution plan: The number of each final product to be delivered to each distribution center in each period.
In order to develop the supply chain master plan, we consider two quantitative objectives: minimizing the total cost
of logistics and maximizing the total value of purchasing which are the most important objectives in SCMP problem
(e.g., [14,25,41]).
Decision making in such a complex supply chain network requires considering conflicting objectives as well as
different constraints imposed by the suppliers, manufacturers and distributors. Moreover, in practical situations, most
of the parameters embedded in SCMP problem are frequently fuzzy in nature because of incompleteness and/or
unavailability of required data over the mid-term horizon, and can be just obtained subjectively [5,39]. For example,
in a real decision problem, market demands, cost/time coefficients and amount of available resources are usually
imprecise over the planning horizon, and therefore assigning a set of crisp values for such ambiguous parameters is not
appropriate. We rely on possibility theory to model this fuzziness. This theory uses possibility distributions to handle
this inherent ambiguous phenomenon in the problem parameters [11,16,20,39].
3.1. Problem assumptions and notation
Below are the main characteristics and assumptions used in the problem formulation:
• The final products have deterministic dynamic demand at each distribution center over a given finite planning horizon
(which is usually between 3 and 6 months).
• A pool of pre-determined qualified suppliers is given.
• The production system at the manufacturer and each supplier is aggregated into a capacitated single stage system
(i.e., by focusing on the bottleneck stage).
• Production capacities at the manufacturer and suppliers are estimated by considering a rough estimate of various
contingencies (such as set-ups and machine break downs) and possible capacity expansions (using overtime and/or
subcontracting).
• The customers do not wait to receive their orders in a future period. Therefore, inventory shortages (stockouts) are
not allowed at each echelon.
• It is assumed that the different supply parties have been located in close proximity, so the procurement and distribution
lead times are negligible (in contrast to the length of each planning period).
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197
Fig. 1. The triangular possibility distribution of fuzzy parameter n.
• Due to competitive environments in bidding, there is some flexibility in the replenishment scheme of each supplier. In
particular, they offer lead time-dependent capacity reservation and unit price discounts in their discount schedule by
considering possible capacity expansions in the future periods. The practical situations of such lead time-dependent
discount schemes can be found in several industries such as the biologics and semiconductor industries [10,28] where
the suppliers and their customers engage in flexible quantity contracts that envision different pricing and capacity
availabilities depending on lead times.
• Each supplier may indicate a minimum acceptable utilized capacity (MAUC) in each period based on a corresponding
minimum acceptable utilization rate (MAUR) of its reserved capacity. That is, each supplier only accepts orders
which for the utilized capacity would be equal or greater than an economic pre-defined amount. Considering such
capacity constraints in the model can improve the win–win strategy between supply chain partners and enforces the
manufacturer to select fewer suppliers in each period.
• Inventory holding costs are linear and non-decreasing in succeeding echelons.
• Due to incompleteness and/or unavailability of required data over the mid-term decision horizon, critical parameters
(such as market demands and capacity levels) are assumed to be imprecise (fuzzy) in nature. Furthermore, the pattern
of triangular fuzzy number is adopted to represent each fuzzy parameter. The triangular possibility distribution is
the most common tool for modeling the imprecise nature of the ambiguous parameters due to its computational
efficiency and simplicity in data acquisition [20,42]. Generally, a possibility distribution can be stated as the degree
of occurrence of an event with imprecise data. Fig. 1 presents the triangular possibility distribution of fuzzy number
n = (np , nm , no ), where np , nm and no are the most pessimistic value, the most possible value, and the most optimistic
value of n estimated by a decision maker.
The indices, parameters and variables used to formulate the problem mathematically are described below. We use the
superscripts 1, 2 and 3 to denote the different echelons of the supply chain, i.e., the suppliers’ level, the manufacturer’s
and the distributors’ level, respectively.
Indices:
i
j
k
t
r
index of items (i = 1, . . ., I )
index of suppliers (j = 1, . . ., J )
index of final products (k = 1, . . ., K)
index of distribution centers ( = 1, . . ., L)
index of time periods (t = 1, . . ., T )
index of discount intervals (which are supplier-dependent)
Parameters:
dkt
si
pj
DIj
SDj r
jr
ur
c
ap1j r
aij1
demand of final product k at the distribution center in period t
set of qualified suppliers offering item i (i.e., s i ⊆ {1, . . . , J })
set of items offered by supplier j (i.e., pj ⊆ {1, . . . , I })
number of discount intervals offered by supplier j, (DIj T )
{Lj r , . . ., Uj r }; the mutually exclusive sub-intervals from {1, . . ., T } representing the rth discount interval proposed by supplier
j · Lj r and Uj r are the shortest and longest periods within this interval, respectively
j t = ur
j r , ∀t ∈ SDj r )
minimum acceptable capacity utilization rate for supplier j at each period of rth discount interval (i.e., ur
reserved production capacity of supplier j at each period of rth discount interval (i.e., c
ap1j t = c
ap1j r , ∀t ∈ SDj r )
unit capacity requirement at supplier j for item i
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S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
c
ap2t
ak2
bik
vri
Wr2
vfk
Wf2
W3
production capacity of manufacturer at period t
unit capacity requirement at manufacturer for final product k
quantity of item i required to produce one unit of final product k
unit storage volume required for item i
storage capacity (in volume) of receiving warehouse at the manufacturer
unit storage volume required for final product k
storage capacity (in volume) of shipping warehouse at the manufacturer
storage capacity (in volume) at the distribution center TCO
TCP
TCD
TC
slc
j
slc
olc
jt
olc
ulc
cij r
total cost of ownership (i.e., the total purchasing costs)
total production costs
total distribution costs
total costs of logistics system
total supplier level costs over the planning horizon
total costs associated with supplier j over the planning horizon
total order level costs over the planning horizon
total costs incurred by placing an order to supplier j in period t (such as ordering, transportation and receiving costs per order)
total unit level costs over the planning horizon
unit price of item i charged by supplier j at each period of rth discount interval of this supplier’s discount schedule (i.e., cij t = cij r ,
∀t ∈ SDj r )
additional unit level costs of item i bought from supplier j at period t
total unit level costs of item i bought from supplier j at period t
unit variable production cost of final product k at period t (except material cost)
unit shipping cost of final product k to distribution center at period t
unit holding cost of item i at period t (based on the average price in period t)
unit holding cost of final product k at the manufacturer in period t
unit holding cost of final product k at the distribution center in period t
safety stock level of final product k at the distribution center in period t
average defective rate of item i supplied by supplier j
manufacturer’s acceptable defective rate for incoming shipments of item i
average service level (the percentage of on-time deliveries) of supplier j
manufacturer’s acceptable service level per period
overall score (weight) of supplier j considering qualitative performance factors
total value of purchasing
an upper bound on xij t value
ij t
aulc
ij t
ulc
ckt
p
tckt
2it
hr
2kt
hf
3kt
hf
ss kt
qij
i
Q
j
sl
SL
Rj
TVP
ubxij t
Decision variables:
xij t
pkt
skt
Ir 2it
If 2kt
If 3kt
yj t
zj
purchasing quantity of item i from supplier j in period t
production quantity of final product k in period t
shipping quantity of final product k to distribution center in period t
ending inventory level of item i at the manufacturer in period t
ending inventory level of final product k at the manufacturer in period t
ending inventory level of final product k at the distribution center in period t
1, if an order is placed with supplier j in period t, 0, otherwise
1, if an order is placed with supplier j over the decision horizon, 0, otherwise
Note that the lead time-dependent capacity reservations and unit price discounts offered by suppliers indicate that
ap1j,r+1 > c
ap1j,r .
we would have: cij,r+1 < cij r , and c
3.2. Problem formulation
Generally, fuzzy mathematical programming is classified into two following major classes [13]:
• Fuzzy mathematical programming with vagueness when there is flexibility in the given target values of objective
functions and the elasticity of constraints. This class is referred to as flexible programming [15,21,23,32].
• Fuzzy mathematical programming with ambiguous coefficients in objective functions and constraints which is called
possibilistic programming [11,16,17,20,39].
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
199
In flexible programming models, the membership functions of fuzzy objectives and constraints are generally
preference-based and determined by the decision maker subjectively. Possibilistic programming is based on the objective degree of event occurrence for each imprecise data and hence the related possibility distributions are determined
objectively relying on some available historical data with an analogous to the probability distributions. Our proposed
fuzzy programming model lies in the second group because some parameters in the total cost objective function, and
some technological coefficients and right-hand sides in some constraints are ambiguous in nature.
3.2.1. Objective functions
and the toWe consider two important and conflicting objectives in our SCMP problem: the total cost of logistics (TC)
tal value of purchasing (TVP). Several other recent studies have also considered similar objectives [e.g., 14,21,25,32,41].
3.2.1.1. Objective 1: minimizing the total cost of logistics. The total costs of logistics is the most practical decision
objective which is usually used in the supply chain master planning models [14,21,25,27]. The total costs of logistics
include the total costs of purchasing, production and distribution activities and could be calculated by the following
equation:
= TCO
+ TCP
+ TCD.
Min TC
In this work, in order to construct a comprehensive total cost function, we have adopted the activity-based costing
(ABC) and the total cost of ownership (TCO) concepts. The ABC is an effective costing tool which is being increasingly
applied in different industries [30]. The ABC enables the cost analyzer to categorize and analyze all the related activities
and the associated costs of a specific process (e.g. purchasing process) on the basis of their hierarchical structure. It
should be noted that although this concept provides a more descriptive rather than a normative tool for cost management,
it provides a framework to understand cost behavior and translate the hierarchical cost structure of a given process into
a mathematical programming model to make optimal decisions [7].
On the other hand, the TCO is a comprehensive financial estimate approach which reflects all the resources consumed
in performing the purchasing-related activities and measures all the costs and benefits associated with these activities
within the company’s value chain (i.e. the life cycle costs of the purchased items/services). In other words, TCO could
help a company to assess their direct and indirect costs related to the purchase of any capital investment (including
initial purchase price, repairs, maintenance, and personnel training, among other expenses).
Degraeve et al. [6,7] presented a typical hierarchical structure of purchasing activities consisting of three levels: (1)
the supplier level activities, (2) the order level activities and (3) the unit level activities. The first hierarchical level
describes costs incurred and conditions imposed whenever the purchasing company actually uses the supplier over the
decision horizon. Examples of costs on the supplier level include a quality audit cost incurred by the buyer for the
evaluation of a supplier, and additional research and development costs due to using a particular supplier. The order
level parameters indicate costs incurred and conditions imposed each time an order is placed with a particular supplier
and include, amongst others, costs associated with reception, invoicing, transportation, ordering and receiving credit
notes. At the unit level we find costs incurred and conditions imposed related to the units of the products for which
the procurement decision has to be made, for example, price, internal failure (e.g. due to quality problems), external
failure, and inventory holding costs. It is important to make this classification of activities into separate levels since the
overall cost driver (i.e., number of suppliers, number of orders, number of units procured) for each level of activity is
independent of the activities in other levels.
Applying the ABC and TCO concepts, the total purchasing cost consists of the total supplier level costs, the total
order level costs and the total unit level costs which could be estimated as follows:
+ olc
+ ulc,
= slc
TCO
such that:
=
slc
J
j =1
j · zj ,
slc
=
olc
J
T t=1 j =1
j t · yj t ,
olc
=
ulc
J T t=1 j =1 i∈pj
ij t ) · xij t +
(cij t +aulc
I
T 2it · Ir 2 .
hr
it
t=1 i=1
The unit prices quoted by suppliers are assumed to be crisp, and hence the total unit level costs of item i bought from
ij t .
ij t = cij t + aulc
supplier j at period t is estimated by ulc
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The total production cost is equal to the sum of variable production costs (except material costs) and inventory
holding costs of final products at the manufacturer:
=
TCP
T K
kt · If 2 ).
(
p ckt · pkt + hf
kt
2
t=1 k=1
The total distribution cost is equal to the sum of transportation costs and inventory holding costs of final products at
the distributors:
=
TCD
T L K
kt · If 3 ).
(tckt · skt + hf
kt
3
t=1 =1 k=1
Thus, the objective of minimizing the total cost of logistics can be stated as follows:
=
min TC
J
j · zj +
slc
j =1
+
T J
j t · yj t +
olc
t=1 j =1
K
T T J ij t ) · xij t +
(cij t + aulc
t=1 j =1 i∈pj
kt · If 2 ) +
(
p ckt · pkt + hf
kt
2
t=1 k=1
T
L K
T I
2it ·Ir 2
hr
it
t=1 i=1
kt · If 3 ).
(tckt · skt + hf
kt
3
(1)
t=1 =1 k=1
3.2.1.2. Objective 2: maximizing the total value of purchasing. The total value of purchasing [9,41] considers the
impact of qualitative (intangible) performance criteria in purchasing decisions (such as after sale services, business
structure and technical capabilities of the suppliers), and can be estimated as follows:
max TVP =
J
j =1
Rj
T ·xij t ,
(2)
t=1 i∈pj
where Rj is the overall score (weight) of supplier j. These global weights could be generated using, for example, the
fuzzy analytical hierarchy process (FAHP) [4,25]. Other important quantitative factors affecting the supplier selection
and order lot-sizing decision (i.e., the quality and on-time delivery criteria) will be incorporated in the model as soft
constraints.
3.2.2. Model constraints
3.2.2.1. Inventory level constraints. All relevant inventory balancing constraints at the manufacturer and distribution
centers are summarized as follows:
Ir 2i,t−1 +
xij t − Ir 2it =
bik · pkt ∀i, t,
(3)
j ∈s i
If 2k,t−1 + pkt − If 2kt =
k
skt
∀k, t,
(4)
∀k, , t,
(5)
If 3k,t−1 + skt − If 3kt = dkt
If 3kt ss kt
∀k, , t.
(6)
Constraints (3) and (4) are inventory balancing equations for components and final products at the manufacturer’s
warehouses, respectively. The right-hand sides in constraints (3) represent the dependent demands of components based
on the production plan of final products. Moreover, constraints (5) and (6) indicate the inventory equations for final
products and the safety stock levels at the distribution centers, respectively. Also, ss kt = k dk,t+1 , where k is the
forward inventory coverage policy factor for product k [14]. It is noted that the safety stock level for the last period T
is calculated based on the first period’s demand.
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201
3.2.2.2. Capacity constraints. The following constraints are related to the production capacity levels at the suppliers
and the manufacturer:
ap1j t ∀j, t,
(7)
aij1 · xij t c
i∈pj
j t · c
ap1j t · yj t
aij1 · xij t ur
∀j, t,
(8)
i∈pj
ap2t
ak2 · pkt c
∀t.
(9)
k
Constraints (7) and (8) represent the maximum and minimum level of utilized capacities at the suppliers in each period,
respectively. Constraints (8) ensure that the manufacturer to take into account the suppliers’ MAUC requirements. These
types of constraints would be active whenever a supplier j is selected in period t. Constraints (9) state the manufacturer’s
production capacity in each period. Moreover, the warehouses’ space limitations at the manufacturer and the distributors
are as follows:
vri · Ir 2it Wr2 ∀t,
(10)
i
vfk · If 2kt Wf2
∀t,
(11)
k
∀, t.
vfk · If 3kt W3
(12)
k
Constraints (10) and (11) indicate the limited storage space at the receiving and shipping warehouses at the manufacturer, respectively. The storage capacities at the distributors are imposed by constraints (12).
3.2.2.3. Quality and on-time delivery constraints. Two other important quantitative factors imposed by the manufacturer on the supplier selection and order lot-sizing sub-problem are the minimum acceptable levels of quality and
on time delivery (or service level) provided by each supplier, respectively [9,10,15,41]. These requirements could be
imposed by the following constraints:
i
xij t ∀i, t,
(13)
qij · xij t Q
j ∈s i
i
j ∈s i
j ∈s i
j xij t SL
sl
i
xij t
∀t.
(14)
j ∈s i
3.2.2.4. Constraints on variables. The integrality and non-negativity constraints are as follows:
xij t ubxij t · yj t ∀i, j ∈ s i , t,
zj yj t ∀j,
(15)
(16)
t
yj t zj ∀j, t,
yj t , zj ∈ {0, 1} ∀j, t,
xij t , pkt , skt , If 3kt , If 2kt , Ir 2it 0
(17)
(18)
∀i, j ∈ s i , k, , t.
(19)
Constraints (15) ensure that if there will be an order from supplier j at period t (i.e., yj t = 1), then the amount of order
for each item say i (xij t ) will be limited to its upper bound which can be calculated by the following equation:
⎞
⎛
K L
T capj1ot o ⎠
ubxij t = min ⎝ 1p ,
∀i, j ∈ s i , t,
bik dkh
aij
h=t k=1 =1
202
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
1p
where capj1ot and aij are the most optimistic and the most pessimistic values of capacity level and consumption rate,
o denotes the most optimistic value of demand. When y = 1, the corresponding order level
respectively. Also, dkh
jt
cost (olcj t ) will be charged by the total cost function (1). Constraints (16) and (17) are integrality constraints. Using
constraint (16), the decision variable zj will be equal to zero, if the model suggests not to buy from the supplier
j over the planning horizon (i.e., t yj t = 0), while constraint (17) forces zj to be equal to 1, if during some
j ) will be charged
time period, an order is placed with supplier j. In this case, an appropriate supplier level cost (slc
through the total cost function (1). Finally, constraints (18) and (19) are non-negativity constraints of continuous
variables.
4. Solution methodology
Recalling above-mentioned objective functions and constraints, we are dealing with a multiple objective possibilistic
mixed integer linear programming model (MOPMILP). To solve this problem, we apply a two-phase approach. In the
first phase, the original problem is converted into an equivalent auxiliary crisp multiple objective mixed integer linear
model. Then, in the second phase, a novel interactive fuzzy programming approach is proposed for finding a preferred
compromise solution through an interaction between the decision maker and model analyzer.
4.1. An auxiliary multi-objective mixed integer linear model
We apply an extended version of a well-known approach proposed by Lai and Hwang [16,18] to transform the
MOPMILP model into an auxiliary crisp multiple objective mixed integer linear programming model. To do so, we
should apply appropriate strategies for converting the fuzzy total cost objective function as well as some soft constraints
into the equivalent crisp equations.
4.1.1. Treating the imprecise total cost objective function
Given the imprecise coefficients in the objective function, in general, one cannot guarantee an ideal solution to problem
(1)–(19). There have been several approaches for obtaining compromise solutions in the literature [16,22,31,35,36].
As stated in Hsu and Hwang [11], the last four approaches [22,31,35,36] have restrictive assumptions and are often
difficult to implement in practice, thus, we chose to implement that of Lai and Hwang [16].
have triangular possibility distributions, the
Since some of the parameters in the fuzzy total cost of logistics TC
objective function would have a triangular possibility distribution as well. Geometrically, this fuzzy objective can
TC
be fully defined by the three prominent points (TC p , 0), (TC m , 1) and (TC o , 0). So, this imprecise objective can be
minimized by pushing the three points towards the left. Consequently, minimizing the imprecise objective function
requires minimizing TC p , TC m and TC o simultaneously. However, there may exist a conflict in the simultaneous
TC
minimization of these crisp objectives. So, using the Lai and Hwang’s approach [16] which is also adopted by other
researchers [11,39], we minimize TC m , maximize (TC m − TC p ), and minimize (TC o − TC m ) instead of minimizing
TC p , TC m and TC o simultaneously. These three objectives still serve the purpose of pushing the three objective points
to the left. In this manner, the original fuzzy total cost of logistics (1) is replaced by the following three crisp objectives
to obtain a compromise solution:
min Z1 = TC m =
J
j =1
+
+
T
I
slcjm · zj +
hrit2m · Ir 2it +
t=1 i=1
L K
T J
T t=1 j =1
K
T olcjmt · yj t +
J T t=1 j =1 i∈pj
m
(cij t + aulcij
t ) · xij t
m
(pckt
· pkt + hfkt2m · If 2kt )
t=1 k=1
m
3m
(tckt
· skt + hfkt
· If 3kt ),
(20)
t=1 =1 k=1
max Z2 = (TC m − TC p ) =
J
j =1
p
(slcjm − slcj ) · zj +
J
T t=1 j =1
p
(olcjmt − olcj t ) · yj t
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
+
J T t=1 j =1 i∈pj
+
T K
p
m
(aulcij
t − aulcij t ) · xij t +
2p
t=1 k=1
min Z3 = (TC o − TC m ) =
J
j =1
+
t=1 j =1 i∈pj
+
K
T 2p
(hrit2m − hrit ) · Ir 2it +
t=1 i=1
(hfkt2m − hfkt ) · If 2kt +
T J I
T T L K
o
m
(aulcij
t − aulcij t ) · xij t +
3p
t=1 j =1
(21)
(olcjot − olcjmt ) · yj t
(hrit2o − hrit2m ) · Ir 2it +
t=1 i=1
(hfkt2o − hfkt2m ) · If 2kt +
p
m
(pckt
− pckt ) · pkt
t=1 k=1
p
I
T L K
T K
T m
3m
[(tckt
− tckt ) · skt + (hfkt
− hfkt ) · If 3kt ],
t=1 =1 k=1
J
T (slcjo − slcjm ) · zj +
203
T K
o
m
(pckt
− pckt
) · pkt
t=1 k=1
o
m
3o
3m
[(tckt
− tckt
) · skt + (hfkt
− hfkt
) · If 3kt ].
(22)
t=1 =1 k=1
t=1 k=1
4.1.2. Treating the soft constraints
To resolve the imprecise demands in the right-hand sides of constraints (5) and (6), the weighted average method
[16,20,39] is used for the defuzzification process and converting the dkt parameter into a crisp number. So, if the
minimum acceptable degree of feasibility (i.e., minimum acceptable possibility), , is given, then the equivalent
auxiliary crisp constraints can be represented as follows:
p
m
o
If 3k,t−1 + skt − If 3kt = w1 dkt, + w2 dkt,
+ w3 dkt,
p
If 3kt k [w1 dk,t+1,
m
o
+ w2 dk,t+1,
+ w3 dk,t+1, ]
∀k, , t,
∀k, and t = 1, . . . , T − 1,
(23)
(24)
where w1 + w2 + w3 = 1, and w1 ,w2 and w3 denote the weights of the most pessimistic, the most possible and the
most optimistic value of the fuzzy demand, respectively. The suitable values for these weights as well as usually are
determined subjectively by the experience and knowledge of the decision maker. However, based on the concept of
the most likely values proposed by Lai and Hwang [16] and considering several relevant works [20,39], we set these
parameters as: w2 = 4/6, w1 = w3 = 1/6 and = 0.5. Furthermore, as it is mentioned earlier, the safety stock level
for the last period T is calculated based on the first period’s demand.
Moreover, regarding the other soft constraints (i.e., capacity levels, quality and service level constraints) which have
imprecise parameters both in the left-hand side and right-hand side, we can use the fuzzy ranking concept [16,29,39],
and replace each imprecise constraint with three equivalent auxiliary inequality constraints. In this manner, we can
obtain the following auxiliary capacity constraints:
1p
1p
aij, · xij t capj t, ∀j, t,
(25)
i∈pj
1m
1m
aij,
· xij t capj t,
∀j, t,
(26)
1o
1o
aij,
· xij t capj t,
∀j, t,
(27)
i∈pj
i∈pj
1p
p
1p
aij, · xij t urj t, · capj t, · yj t
∀j, t,
(28)
1m
m
1m
aij,
· xij t urj t, · capj t, · yj t
∀j, t,
(29)
1o
o
1o
aij,
· xij t urj t, · capj t, · yj t
∀j, t,
(30)
i∈pj
i∈pj
i∈pj
k
2p
2p
ak, · pkt capt,
∀t,
(31)
204
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
2m
2m
ak,
· pkt capt,
∀t,
(32)
2o
2o
ak,
· pkt capt,
∀t.
(33)
k
k
In a similar way we can have the following auxiliary constraints for the fuzzy quality and service level constraints:
p
p
qij, · xij t Qi,
xij t ∀i, t,
(34)
j ∈s i
j ∈s i
m
m
qij,
· xij t Qi,
j ∈s i
j ∈s i
o
o
qij,
· xij t Qi,
j ∈s i
i
p
∀i, t,
(35)
xij t
∀i, t,
(36)
j ∈s i
p
slj, xij t SL
m
m
slj,
xij t SL
j ∈s i
i
xij t
j ∈s i
i
j ∈s i
o
o
slj,
xij t SL
i
j ∈s i
i
j ∈s i
i
j ∈s i
xij t
∀t,
(37)
xij t
∀t,
(38)
xij t
∀t.
(39)
Consequently, we would have an auxiliary crisp multi-objective mixed integer linear programming model (MOMILP)
as follows:
MOMILP:
min
s.t.
Z = [Z1 , −Z2 , Z3 , −Z4 ],
Z1 = TC m , Z2 = TC m − TC p ,
Z3 = TC 0 − TC m , Z4 = TVP
v ∈ F (v),
(40)
where v denotes a feasible solution vector involving all of the continuous and binary variables in the original problem.
Also, F (v) denotes the feasible region involving crisp constraints (3)–(4), (10)–(12), (15)–(19) and (23) up to (39).
4.2. Proposed interactive fuzzy programming solution approach
There are several methods in the literature for solving multi-objective linear programming (MOLP) models, among
them; the fuzzy programming approaches are being increasingly applied. The main advantage of fuzzy approaches is
that they are capable to measure the satisfaction degree of each objective function explicitly. This issue can help the
decision maker to make her/his final decision by choosing a preferred efficient solution according to the satisfaction
degree and preference (relative importance) of each objective function.
Zimmermann developed the first fuzzy approach for solving a MOLP called max–min approach [42], but it is wellknown that the solution yielded by max–min operator might not be unique nor efficient [17–19]. Therefore, after that
several methods were proposed to remove this deficiency. Of particular interest, Lai and Hwang [17] developed the
augmented max–min approach (hereafter the LH method), Selim and Ozkarahan [32] presented a modified version of
Werners’ approach [40] (hereafter the MW method), and Li et al. [19] proposed a two-phase fuzzy approach (hereafter
the LZL method). A brief discussion of these three approaches has been presented in Appendix A.
In our initial numerical tests for applying these approaches to solve the problem, we have observed some deficiencies.
Among the single-phase methods (i.e. the LH and MW) which solve the original model directly by just one auxiliary
crisp model, the LH method sometimes generates inefficient solutions dominated by the solution of LZL method, and the
MW method usually yields an efficient but unbalanced and poorly compromised solution so that the satisfaction degrees
of objectives have considerable differences, which is often not acceptable by the decision maker. On the other hand,
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
205
although LZL always produces an efficient solution, it applies a two-phase method which needs more computational
efforts than the single-phase methods, especially for solving the multi-objective mixed integer linear models.
Consequently, we tried to develop a new single-phase approach removing above deficiencies which led to a new
fuzzy approach. The proposed approach (hereafter the TH method) is actually a hybridization of the LH and MW
methods. Of particular interest, we were able to prove the efficiency of this method using a similar technique to that
used by Li et al. [19]; which has been represented in Appendix B.
In summary, our proposed interactive solution procedure to solve the original MOPMILP model is as follows:
Step 1: Determine appropriate triangular possibility distributions for the imprecise parameters and formulate the
original MOPMILP model for the SCMP problem.
into the three equivalent crisp objectives (20)–(22).
Step 2: Convert the original fuzzy total cost of logistics TC
Step 3: Given the minimum acceptable possibility level for imprecise parameters, , convert the fuzzy constraints
into the corresponding crisp ones, and formulate the auxiliary crisp MOMILP model.
Step 4: Determine the positive ideal solution (PIS) and negative ideal solution (NIS) for each objective function by
solving the corresponding MILP model as follows [1,12,15–21,32,39,42]:
Z1PIS = min TC m ,
Z1NIS = max TC m ,
Z2PIS = max(TC m − TC p ),
Z2NIS = min(TC m − TC p ),
Z3PIS = min(TC o − TC m ),
Z3NIS = max(TC o − TC m ),
Z4PIS = max TVP,
s.t. v ∈ F (v).
Z4NIS = min TVP.
It should be noted that determining the above ideal solutions requires solving eight mixed integer linear program
which could be computationally very cumbersome especially in large-sized problem instances. In order to alleviate the
computational complexity, we apply the following heuristic rules:
• Obtaining an approximate positive ideal solution for each objective function by solving the corresponding MILP
heuristically to obtain a satisfactory feasible integer solution. To do so, the MIP solver is run until reaching prespecified termination criteria, e.g., CPU time and/or optimality gap [14].
• Instead of solving a separate MILP for determining each NIS, we can estimate them using the positive ideal solutions.
Let vh∗ and Zh (vh∗ ) denote the decision vector associated with the PIS of hth objective function and the corresponding
value of hth objective function, respectively. So, the related NIS could be estimated as follows:
ZhNIS = max {Zh (vk∗ )}; h = 1, 3;
k=1,...,4
ZhNIS = min {Zh (vk∗ )}; h = 2, 4.
k=1,...,4
Step 5: Specify a linear membership function for each objective function as follows:
1 (v) =
⎧
1
⎪
⎪
⎨ Z NIS − Z
1
if Z1 < Z1PIS ,
1
NIS
PIS
⎪
⎪
⎩ Z1 − Z 1
0
⎧
1
⎪
⎪
⎨ Z − Z NIS
2
2
2 (v) =
PIS
NIS
⎪
Z
−
Z
⎪
2
2
⎩
0
⎧
⎪
1
⎪
⎪
⎨ Z NIS − Z
3
3
3 (v) =
PIS
NIS
⎪
Z
− Z3
⎪
⎪
⎩0 3
if Z1PIS Z1 Z1NIS ,
(41)
if Z1 > Z1NIS ,
if Z1 > Z2PIS ,
if Z2NIS Z2 Z2PIS ,
(42)
if Z2 < Z2NIS ,
if Z3 < Z3PIS ,
if Z3PIS Z3 Z3NIS ,
if Z3 > Z3NIS ,
(43)
206
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
Fig. 2. Linear membership function for Z1 (Z3 ).
Fig. 3. Linear membership function for Z2 (Z4 ).
⎧
1
⎪
⎪
⎨ Z − Z NIS
4
4
4 (v) =
PIS − Z NIS
⎪
Z
⎪ 4
4
⎩
0
if Z4 > Z4PIS ,
if Z4NIS Z4 Z4PIS ,
(44)
if Z4 < Z4NIS .
In fact, h (v) denote the satisfaction degree of hth objective function for the given solution vector v. Figs. 2 and 3
represent the graphs of these membership functions.
Step 6: Convert the auxiliary MOMILP model into an equivalent single-objective MILP using the following new
auxiliary crisp formulation (45).
Auxiliary MILP:
max (v) = 0 + (1 − )
h h (v)
h
s.t.
0 h (v), h = 1, . . . , 4,
v ∈ F (v), 0 and ∈ [0, 1],
(45)
where h (v) and 0 = minh {h (v)} denote the satisfaction degree of hth objective function and the minimum satisfaction
degree of objectives, respectively. This formulation has a new achievement function defined as a convex combination of
the lower bound for satisfaction degree of objectives (0 ), and the weighted sum of these achievement degrees (h (v))
to ensure yielding an adjustably balanced compromise solution. Moreover, h and indicate the relative importance of
the hth objective function and the coefficient of compensation, respectively. The h parameters are determined by the
decision maker based on her/his preferences such that h h = 1, h > 0. Also, controls the minimum satisfaction
level of objectives as well as the compromise degree among the objectives implicitly. That is, the proposed formulation
is capable of yielding both unbalanced and balanced compromised solutions for a given problem instance based on the
decision maker’s preferences through adjusting the value of parameter .
In this regard, a higher value for means more attention is paid to obtain a higher lower bound for the satisfaction
degree of objectives (0 ) and accordingly more balanced compromise solutions. On the contrary, the lower value for
means more attention is paid to obtain a solution with high satisfaction degree for some objectives with higher
relative importance without any attention paid to the satisfaction degree of other objectives (i.e., yielding unbalanced
compromise solutions).
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
207
It is noteworthy that there exists a correlation between and the range of h values (i.e. maxh {h } − minh {h }) so
that there will be a limited reasonable interval of in which it could be selected for a given vector. For example, for
the considerably large values of this range, corresponding should be selected as a small value (e.g. smaller than 0.3)
because of explicit preference of the decision maker for getting an unbalanced compromise solution in this case.
Step 7: Given the coefficient of compensation and relative importance of the fuzzy goals ( vector), solve the
proposed auxiliary crisp model (45) by the MIP solver. If the decision maker is satisfied with this current efficient
compromise solution, stop. Otherwise, provide another efficient solution by changing the value of some controllable
parameters say and , and then go back to Step 3.
5. Computational experiments
5.1. Experimental design
To demonstrate the validity and practicality of the proposed model and solution method, an industrial case scenario
inspired form a home appliances manufacturer is presented. This supply chain involves four suppliers, one manufacturer
and three distribution centers located in different customer zones. The factory produces three types of products using
ten common purchased items. The supplier-item matrix representing the pj and s i sets is given in Table 1 where pair
(j, i) is 1 if the supplier j offers item i, and it is zero, otherwise.
These eligible suppliers have been selected through an initial screening process performed by the quality control
department. Especially, the fuzzy AHP process [25] was used to determine the overall scores of suppliers, and finally
the above four suppliers having the highest overall scores Rj were selected. The overall global weights vector after
normalization was adjusted to Rj = (0.32, 0.26, 0.18, 0.24).
The planning horizon is 3 months consisting of 12 weekly periods. The longest material acquisition lead time is 2
days (less than one week) and indicates that the lead times can be ignored in contrast with the length of each period.
Because of confidentiality as well as the lack of some required data, we decided to generate most of the required
parameters randomly. However, the generation of random data was done in such a way that they will be close to the
real data available in the company. Without loss of generality and just to simplify the generation of fuzzy parameters,
we applied symmetrical triangular possibility distribution for our numerical test. So, the most possible value of each
imprecise parameter was first generated with an appropriate probability distribution, and then the corresponding most
pessimistic and optimistic values were determined by multiplying the most possible value with 0.8 and 1.2, respectively
[32]. Moreover, for lead time-dependent capacity reservation and unit price discounts, the number of discount intervals
offered by each supplier (DIj ) was first generated randomly between 1 and 3, and then the associated sub-intervals
(SDj r ) were considered with equal periods. Table 2 summarizes the information about the source of random data
generation.
Other input data have been shown in Tables 3 and 4. Table 3 represents the bill of material data (bik ) where the entry
of (k, i) denotes the number of items i required to produce one unit of product k. Also, the forward inventory coverage
policy factors for determination of safety stock levels were considered as k = 5% for all final products.
Regarding the manufacturer capacity, we could check the following necessary feasibility conditions:
t L K
m
ak2m dkh
h=1 =1 k=1
t
caph2m ,
t = 1, . . . , T .
(46)
h=1
Table 1
The supplier–item matrix
Supplier (j )
1
2
3
4
Item (i)
1
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
0
0
0
1
0
1
0
1
0
1
1
0
1
0
1
0
1
0
208
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
Table 2
The sources of random generation of data set
Parameter
Corresponding random distribution
m
d1t
m
d2t
m
d3t
ak2m
cap2m
t
N(150, 10)
N(400, 20)
N(250, 15)
U (3, 5)
m )/T · U (1.1, 1.3)
(ak2m dkt
aij1m
cap1m
j1
U (1, 3)
cap1m
j r , (r > 1)
urjmr
cij 1
cij r (r > 1)
aulcm
ij r
m , (t = 1)
pckt
m , (t > 1)
pckt
m , (t = 1)
tckt
m , (t > 1)
tckt
hrit2m
U (1.1, 1.2) · capj ·r−1
U (0.2, 0.3)
U (4, 8)
U (0.8, 1) · cij ·r−1
U (0.1, 0.2) · cij r
U (1, 3)
m = pcm
pckt
k,t−1 · U (1, 1.1)
U (0.2, 0.4)
m = tcm
tckt
k,t−1 · U (1, 1.05)
0.01)
averagej ∈s i {cij t } · U (0.005,
hfkt2m
cij t /|s i |
i|bik =0 j ∈s i
hfkt2m · U (1.05, 1.10)
t
t
3m
hfkt
slcjm
olcjmt
Ir 2i0
qijm
Qm
i
sljm
SLm
k
i∈p j
k
m )/T · U (0.5, 0.8)
(aij1m bik dkt
m · U (0.005, 0.01)
· bik + pckt
U (1000, 1500)
U (100, 200)
P
k
bik dk1 · U (0.8, 1.0)
U (0.01, 0.03)
U (0.03, 0.05)
U (0.90, 0.95)
U (0.90, 0.95)
Table 3
Bill of material matrix (bik values)
Product (k)
1
2
3
Item (i)
1
2
3
4
5
6
7
8
9
10
2
1
1
1
3
0
0
1
2
1
0
1
0
1
1
2
2
1
1
0
3
0
1
0
1
2
1
1
2
1
These necessary feasibility conditions require the existence a sufficient cumulative capacity in each period based on
the most possible situation. Moreover, in the supply side there exists enough capacity provided by the suppliers and
therefore no need to check the capacity feasibility.
Furthermore, the decision maker provided the relative importance of objectives linguistically as: 1 > 4 > 2 = 3 ,
and based on this relationships we set the objectives weight vector as: = (0.5, 0.15, 0.15, 0.2). After some initial
experiments, the stopping criteria for solving each MIP as well as controllable parameters were set as: CPU time = 300
seconds, optimality gap = 0.05, = 0.01 and = 0.4, respectively. It is noted that the reason for selecting = 0.4
is that the Z1 is the most important objective and also Z2 and Z3 are actually relative measures from Z1 . Thus the
somewhat unbalanced compromise solution with highest satisfaction degree for Z1 is of particular interest. In this
respect, our initial experiments show that any value of between 0.3 and 0.8 could be appropriate for obtaining a
compromise solution with 1 > 4 > 2 = 3 . However, it seems that = 0.4 is more appropriate. It should be noted
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
209
Table 4
Storage capacity data
Unit storage volume required for item i (vri )
Unit storage volume required for product k (vfk )
Storage capacity of receiving warehouse at the manufacturer
Storage capacity of shipping warehouse at the manufacturer
Storage capacity (in volume) at the distribution center (W3 )
(3, 1, 2, 1, 1, 3, 2, 1, 2, 1)
(5, 8, 6)
(18500)
(1800)
(1300, 900, 1200)
that although the size of problem instances are considerable and acceptable in real-size scale (including about 1550
constraints, 800 continuous variables and 52 binary variables), the CPU time was not an issue in our experiments and
fortunately all of the above approaches lead to “good” feasible solution (with mean optimality gap less than 2%) within
just a few seconds.
5.2. Performance analysis
To evaluate the performance of the aforementioned fuzzy approaches (i.e. the LZL, LH, MW and proposed TH
methods), 50 problem instances were randomly generated. These approaches were coded in GAMS and the OSL solver
from IBM was used for solving the MIP models on a Pentium 4 with a 1.8 GHz CPU processor and 256 MB of RAM.
Due to space limitations, the details of the solutions found by the different approaches are not presented here, but can
be made available upon request. In summary, we make the following observations based on our numerical experiments:
• The solutions found by the Zimmermann max–min method are always inefficient and dominated by the solutions of
LZL and/or TH methods.
• In the 27 cases out of 50 problem instances (i.e., in about 54%), the solutions found by the LH method were equal
to the solution of LZL and/or TH method. Also, in the 10 cases out of 50 problem instances (i.e., in about 20%), its
solutions were inefficient and dominated by the solutions of LZL and/or TH method. However, it should be noted that
these solutions were close to the corresponding efficient solutions, and may be the main source of this inefficiency
is related to the computational errors.
• The solutions found by the MW method are usually efficient but at the same time unbalanced and poorly compromised
which would be often unacceptable by the decision maker. In other words, the minimum satisfaction level 0 is very
small and the most attention is just paid to objectives with higher weights without any attention paid to the satisfaction
degree of other objectives.
• The solutions found by the TH method are always efficient. Also in the 25 and 28 cases out of 50 problem instances
(i.e., in about 50% and 56%), the solutions found by the proposed TH method were equal to the solution of LH and
LZL methods, respectively.
In order to analyze and compare the performance of these fuzzy approaches, we have used two performance measures:
(1) the well-known distance measure, and (2) a new proposed balancing measure. The distance measure is used for
determining the degree of closeness of each solution to the corresponding ideal solution. In this regard, we define the
following family of distance functions [1,18]:
1/p
p
p
dp (v) =
h (1 − h (v))
; p 1 and integer.
(47)
h
Since the satisfaction degree of each objective is defined as the relative closeness of the solution to the ideal point or
the relative remoteness to the anti-ideal point, they are used explicitly in Eq. (47). The power p represents a distance
parameter and especially p = 1, 2 and ∞ are operationally important so that d1 (the Manhattan distance) and d2 (the
Euclidean distance) are the longest and shortest distances in the geometrical sense; and d∞ (the Tchebycheff distance)
is the shortest distance in the numerical sense. Generally speaking, when p increases, the amount of distance dp and
also the credibility of the distance function dp decreases [18]. It is noted that based on the definition of dp , the fuzzy
approach with minimum dp (especially for p = 1), would be preferred to the other methods.
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S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
Table 5
Performance comparison of fuzzy approaches for = (0.5, 0.15, 0.15, 0.2), = 0.01, and = 0.4
Fuzzy approaches
Distance measures
d1
0.357
0.360
0.310
0.353
LZL
LH
MW
TH
d2
0.204
0.203
0.181
0.199
Dispersion measure
RSD
0.317
0.304
0.771
0.311
d∞
0.168
0.165
0.133
0.159
Table 6
Sensitivity analysis on value in the MW and TH methods
-Value
0
0.1
0.2
0.3
0.4
0.5
0.6–0.9
1
Z1 (v)
TH method
Z2 (v)
Z3 (v)
Z4 (v)
MW method
Z1 (v)
Z2 (v)
Z3 (v)
Z4 (v)
0.925
0.944
0.714
0.685
0.685
0.685
0.685
0.678
0.061
0.116
0.467
0.5
0.5
0.5
0.5
0.5
0.939
0.884
0.533
0.5
0.5
0.5
0.5
0.5
0.292
0.219
0.546
0.573
0.573
0.573
0.573
0.557
0.925
0.909
0.929
0.917
0.925
0.943
0.685
0.678
0.061
0.074
0.056
0.062
0.061
0.042
0.5
0.5
0.939
0.926
0.944
0.938
0.939
0.958
0.5
0.5
0.292
0.308
0.289
0.312
0.292
0.271
0.573
0.557
Corresponding max–min solution = (0.5, 0.5, 0.5, 0.713).
Corresponding LZL solution = (0.619, 0.5, 0.5, 0.713).
Corresponding LH solution = (0.697, 0.5, 0.5, 0.557).
We also propose a new measure, i.e., the range of satisfaction degrees (RSD) which is a dispersion index and is
computed as follows:
RSD(v) = max(h (v)) − min(h (v)).
h
h
(48)
In fact, this index measures the balancing amount of a compromise solution via calculating the maximum difference
between the satisfaction degrees of objectives. It also indicates the level of consistency between the priority vector quoted by the decision maker and the satisfaction degrees vector . For example, given strong differences in h values,
considerable differences in h values, i.e. the higher RSD will be more desirable. In this case, parameter should be
set as a small number (less than 0.3) indicating less attention is paid to the minimum satisfaction level 0 . Table 5
summarizes the numerical results of the four fuzzy approaches in terms of above-mentioned performance indices.
Furthermore, to analyze the impact of changing parameter on the final solution of MW and proposed TH method,
we solved some problem instances with different values of . As an example, for a specific instance (with seed number
27 in GAMS), the corresponding solutions have been provided in Table 6.
From the above comparison and sensitivity analysis, we can derive the following information:
• From Table 5, the MW method seems better than other methods in terms of distance measures, but at the same time
has the largest RSD value because the related solutions are usually unbalanced and poorly compromised. Generally,
such compromise solutions are often unacceptable by the decision maker.
• From Table 6 we can observe that MW method is sensitive to parameter so that it produces different unbalanced
solutions for values less than 0.6. On the other hand, the proposed TH method is not very sensitive to value so that
it produces appropriately unique balanced solution for values between 0.3 and 0.9. However, the MW and proposed
TH methods have very similar unbalanced solutions for small values of (especially for 0.1) and very similar
balanced solutions for large values of (especially for 0.6). Therefore, we can control the degree of compromise
among the objectives by the proposed TH method, and a spectrum of unbalanced and balanced solutions based on
the decision maker preferences, can be obtained for a given relative importance vector .
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
211
• Among LZL, LH and the proposed TH method which generally produce the balanced compromised solutions,
surprisingly the proposed TH method outperforms the others in terms of distance measure.
• Regarding the RSD measure, the TH and LZL methods have higher values than the LH method. Through more
investigation it can be realized that the main reason is related to the solutions characteristic so that the satisfaction
degree of objectives in the TH and LZL’s solutions are compatible with the decision maker’s preferences (i.e.
1 > 4 > 2 = 3 ), that is why these methods have higher RSD than the LH method.
Based on the above information, we can conclude that the proposed TH is the most appropriate method among the
considered fuzzy approaches because: (1) it is more robust and reliable than the LH and MW approach as it always
generates efficient solutions and is able to produce both unbalanced and balanced solutions based on the decision maker’s
preferences, (2) its solutions are consistent with the decision maker’s preferences (i.e., the consistency between weight
vector and satisfaction vector ), (3) it is more flexible than LZL and LH approaches because it is able to find different
efficient solutions for a specific problem instance with a given weight vector through changing the value, and finally
from the computational stand point (4) the proposed TH method, due to its single-phase characteristic, is more suitable
than the LZL method especially for solving multi-objective mixed integer linear models. In summary, the proposed TH
method contains all the advantages of existing methods and at the same time, it overcomes their shortcomings.
6. Conclusion remarks
This study proposes a novel multi-objective possibilistic programming model to formulate a supply chain master
planning problem integrating procurement, production and distribution planning in a multi-echelon, multi-product and
multi-period supply chain network. A two-phase interactive fuzzy programming procedure has been developed. In the
first phase, the possibilistic programming model is converted into an auxiliary crisp MOMILP by applying appropriate
strategies. Then, a novel fuzzy approach (called TH method) is applied to find an efficient compromise solution.
The numerical experiments indicate that the proposed TH method is very promising fuzzy approach which can produce both unbalanced and balanced efficient solutions based on the decision maker’s preferences along with offering
appropriate flexibility to provide different solutions to help the decision maker in selecting the final preferred compromise solution. This approach can also be used for solving other practical MOLP models due to it’s computationally
advantages.
Finally, there are some possible directions for further research. Among them, is to test the proposed approach on a
complete real life problem, such as a chemical processing supply chain where the use of lead-time dependent discounts
is common. In addition, extending the proposed model to more general supply chain networks with multiple plants
in parallel and also considering other important discount policies for example, business volume discounts [41] are of
particular interest. Moreover, as it mentioned earlier, the CPU time was not an issue in our numerical experiments. That
is why we used a limit of 300 s of CPU time in our numerical tests. However, in other large-scaled practical problems it
might be an issue. Therefore, developing an efficient metaheuristic algorithm to solve the corresponding MILP models
should be helpful in reaching efficient solutions.
Appendix A
In this appendix we provide an abstract version of three previously developed approaches (i.e., the LZL, LH and
MW methods).
A.1. Li et al. (LZL) two-phase method [19]
LZL model:
max
(v) =
h h (v)
h
s.t.
0h h (v),
v ∈ F (v),
h = 1, . . . , 4,
0h , h (v) ∈ [0, 1].
(A.1)
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S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
In the above formulation, 0h denotes the minimum satisfaction degree of hth objective function which is found by
solving the Zimmermann’s max–min approach [42] as follows:
max–min model:
max
s.t.
h (v), h = 1, . . . , 4,
v ∈ F (v), ∈ [0, 1].
(A.2)
A.2. Lai and Hwang (LH) augmented max–min method [17]
LH model:
max
(v) = 0 + h h (v)
h
s.t.
0 h (v), h = 1, . . . , 4,
v ∈ F (v), 0 ∈ [0, 1].
(A.3)
Here, 0 denotes the minimum satisfaction degree of objectives which is determined along with the variables h (v) via
solving the LH model directly in a single phase. Also, is a sufficiently small positive number which is usually set to
0.01 [17,18].
A.3. Selim and Ozkarahan extended Werners (MW) method [32]
MW model:
max
(v) = 0 + (1 − )
h h
h
s.t.
h (v)0 + h , h = 1, . . . , 4,
v ∈ F (v), , 0 and h ∈ [0, 1].
(A.4)
In this model, 0 and h (v) denote the minimum satisfaction degree of objectives and satisfaction degree of objective
h, respectively, which simultaneously are determined through solving the MW model. Moreover, is the coefficient of
compensation [32], and we have set it to 0.4 based on our initial tests.
Appendix B
In this appendix we proof a theorem to establish the efficiency of solutions produced by the proposed TH method.
We start by defining a fuzzy-efficient solution to (45).
Definition. A vector v ∗ is an optimal solution to auxiliary MILP model (45) or an efficient solution to MOMILP
model (40), iff there does not exist any v ∈ F (v) such that h (v) h (v ∗ ) for all h and s (v) > s (v ∗ ) for at least one
s ∈ {1, . . . , 4}.
Theorem. The optimal solution of auxiliary model (45) is an efficient solution to the MOMILP model (40).
Proof. Suppose that v ∗ is an optimal solution of (45) which is not an efficient solution to model (40). It means that
model (40) must have an efficient solution say v ∗∗ so that we have: h (v ∗∗ ) h (v ∗ ); ∀h and ∃s|s (v ∗∗ ) > s (v ∗ ).
Hence, for the minimum satisfaction level of objectives in v ∗ and v ∗∗ solutions, we would have 0 (v ∗∗ ) 0 (v ∗ ), and
S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193 – 214
213
regarding the related objective values we would have the following inequality:
⎡
⎤
(v ∗ ) = 0 (v ∗ ) + (1 − )
h h (v ∗ ) = 0 (v ∗ ) + (1 − ) ⎣
h h (v ∗ ) + s s (v ∗ )⎦
h
⎡
< 0 (v ∗∗ ) + (1 − ) ⎣
⎤
h=s
h h (v ∗∗ ) + s s (v ∗∗ )⎦ = (v ∗∗ ).
h=s
Thus, v ∗ is not the optimal solution of (45), a contradiction.
Acknowledgments
This research is supported by both the Natural Sciences and Engineering Research Council of Canada (NSERC)
and Tehran University, Iran. We are also grateful to the two anonymous reviewers for their valuable comments and
constructive criticism.
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