DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633638
A biobjective model for collaborative planning in dyadic supply chain
Hamza Ben Abdallah
Université de Tunis El Manar, Faculté des Sciences de Tunis, LIP2LR99ES18, 2092, Tunis, Tunisia Hamzabenabdallah88@yahoo.fr
Zied Bahroun
ESM Graduate Program, College of Engineering American University of Sharjah, P.O. Box 26666, Sharjah, United Aarab Emirates zbahroun@aus.edu
Naoufel Cheikhrouhou
Laboratory for Production Management and Processes Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland naoufel.cheikhrouhou@epfl.ch
Mansour Rached
Université de Tunis El Manar, Faculté des Sciences de Tunis, LIP2LR99ES18, 2092 Tunis, Tunisia mansour.rached@hotmail.com
Abstract
—
The collaborative planning and the management of production and storage processes are important components in supply chain management. The goal of this paper is to
present the reliability of genetic algorithms on solving biobjective models compared to monoobjective models. To do this we will be based initially on the mono
objective Dudek’s model and then we
propose a division of the objective function in two objective functions. Finally we compare the results given by the genetic algorithms with the optimality result obtained using the LINGO solver on the mono
objective Dudek’s model.
. This model aims at simultaneously minimizing the total production cost and the total holding cost. To solve the proposed model, we use a genetic algorithm NSGAII. The proposed several test provide results that demonstrate and validate the effectiveness of the multiobjective approach and elitists genetic algorithms in solving this type of problem, compared to the literature in the proposed test.
The validation of our approach will allow us later to use this algorithm in solving complex multiobjective models approaching the real context.
Keywords
—
Supply Chain Management, Collaborative Planning, Mathematical Programming, MultiObjective Optimization Model, Elitist Genetic Algorithm NSGA II
I.
I
NTRODUCTION
Tactical planning is the determination of the quantities of products to be manufactured per period in order to meet as well as possible the demand at lower costs. The issues differ mainly according to two criteria: monolevel planning (Master Production Scheduling of finished products) or multilevel planning (finished products and components planning) and monosite or multisite planning. Commonly, the planning problems are formalized by mathematical models known as "lot sizing" problems. Among them, the "Capacitated Lot Sizing Problem" (CLSP) is considered as a reference model to treat the problems of generation of master production scheduling in a monosite context. For the multilevel planning, "Multi Level Capacitated Lot Sizing Problem" (MLCLSP) is recognized as the reference model. If the monosite problems were largely studied in the literature, the absence of a reference model for the multisite issues is highlighted in [19]. This can be explained by the diversity of supply chains and treated problems. Nevertheless, from the multilevel nature of the multisite production, the models of the literature (see for instance [21] and [5]) are derived from the MLCLSP model. Integral part of the multisite planning problems, collaboration of the production plans of the various production sites has represented for several years a major stake of the supply chain optimization. The theoretical contribution of this work is to validate the elitist
s’ genetic algorithms
to solving multiobjective problems in supply chain management, especially for the collaborative planning problem
.
Given that the proposed model is that the
Dudek’s model
developed in 2007 with the duplication of the objective function so as to obtain a biobjective model. So, our experimental study is based, firstly, on the comparison of results obtained from the resolution of the monoobjective
Dudek’s model and the bi
objective Du
dek’s model (which is
the proposed model) by the NSGAII algorithm to validate the
DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633638
performance of the genetic approach in solving the multiobjective problems rather than monoobjective problems. Second, we tend to solve the mono
objective Dudek’s model
by LINGO solver to obtain an optimal solution and compare it with the solution obtained when solving the biobjective model by the NSGAII in order to judge the performance of adopting a multiobjective approach to this type of problem and assess the effectiveness of NSGAII in solving this type of problems. Thus, the purpose of this paper is to develop and to solve a multiobjective model for optimal collaborative multi period planning in dyadic supply chain. The idea is to provide customer orders to the SC partners in order to coordinate and generate a global optimal production plan. In solving the multiobjective model, we use an elitist genetic algorithm based on NSGAII algorithm (Nondominated Sorting Algorithm II) and compare it to a benchmark solution from the literature. The paper is further organized as follows. Related literature is reviewed in Section 2. Characteristics of the case problem are presented in Section 3. The proposed model is detailed in Section 4. The resolution methodology is presented in section 4, followed by the computational results in section 5. A comparative study is done in section 6 to evaluate the performance of the proposed model. Finally, summary and potential research directions are drawn.
II.
M
ULTI
O
BJECTIVE
P
ROBLEMS IN
S
UPPLY
C
HAINS
M
ANAGEMENT
:
A
L
ITERATURE
R
EVIEW
In the last decade, a large number of multiobjective optimization problems and resolution methods have been presented in the literature on supply chain management. (Jayaraman, 1999) [29] developed a weighted multiobjective model for the equipment locating service problem to evaluate the compromise between the demand cover and the number of equipment. (Ertogral and Wu, 2000) [17] developed a coordination mechanism for the production planning in the various stages of the supply chain based on the Lagrangian relaxation of a global model of the chain. This global model is decomposed into specific submodels with respect to the different actors. (Sabri and Beamon, 2000) [7] developed an integrated multiobjective model, for simultaneous, strategic and operational supply chain planning by taking account the supply chain cost, the customer service level and the flexibility of delivery as objectives. In [15], the authors developed a mathematical programming model in order to reduce the operating cost while maintaining the execution of customer orders at a high level for a supply chain network. In [4], the authors formulated production and distribution planning model using multiobjective nonlinear mixed integer programming for an equitable distribution of benefits through the supply chain. (Hugo et al., 2005) [1] proposed a based multiobjective optimization model for strategic investment planning and supply chains design of hydrogen, considering both investment and environmental criteria. In [18], the authors developed a multiobjective optimization method based on simulation for the optimization problem of storage policies in supply chains, taking into account the total holding cost and the service level. In [6], the authors solved a multiobjective linear programming problem at two levels for supply chains planning, using a technique adapted from fuzzy logic. (Chern and Hsieh, 2007) [2] proposed a heuristic algorithm for solving the central planning of supply chain problem, with three goals, including penalties, capacity utilization and the total cost. (Pokharel, 2008) [25] optimized the operation cost and reliability of delivery for supply chain design problem. (Raj and Lakshminarayanan, 2008) [28] examined three performance criteria: customer satisfaction, breaking orders and excess inventories. [14] developed a multiobjective linear programming model for production and distribution collaborative planning problem using the fuzzy goal programming approach. (Liang, 2008) [26] developed a fuzzy multiobjective linear programming model to simultaneously minimize total costs and total delivery time in a supply chain by adopting the fuzzy objective programming method. (You and Grossmann, 2008) [9] proposed a multiperiod planning model for the design and the planning of supply chains under economic criteria with uncertainties in demand. (Torabi and Hassini, 2009) [23] considered the multiobjective optimization problem for the multilevel supply chain planning. Franca et al. (2010) [22] proposed a multiobjective stochastic model considering both the benefits and defects of raw materials obtained from the supply chain. (PintoVarela et al. 2011) [27] used an optimization approach adapted from the symmetric fuzzy linear programming in order to solve a biobjective model for the planning and design of supply chains by considering the economic and environmental aspects. (Liu and Papageorgiou, 2012) [24] focused on the production planning, the distribution, and global supply chains capacities, considering simultaneously the costs and the customer service level. They developed a multiobjective mixed linear programming mixed approach with the total cost, the delivery time and the total lost sales as key objectives. (Ben Yahia et al., 2012) [30] studied the problem of a multiobjective optimization for multiitems cooperative supply chain multi period planning. They have developed a biobjective model for cooperative planning between different plants belonging to the same supply chain; this model aims at simultaneously minimizing the total production cost and the average inventory level considering multiitems production. We note that (Dudek and Stadtler, 2005) [10] proposed a coordination mechanism of a supply chain involving a supplier and a manufacturer, they have developed a monoobjective model for the MultiLevel Capacited Lot Sizing Problem. This model aims to minimize the total cost of supply chain which is equal to the sum of production cost and holding cost which are considered as objectives to our model that is described in the next sections. III.
P
ROBLEM
D
ESCRIPTION
The most significant classical problem in relation to the type of problem under interest is the MultiLevel Capacited Lotsizing Problem (MLCLSP).The objective is to determine, in a collaborative way, the quantities to be manufactured for
DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633638
the various products over the planning horizon, in dyadic supply chain, in order to satisfy the demand (without breaking stock), with respecting the capacity constraints and to minimize the total cost by generating a global optimal production plan. The supply chain considered is dyadic. Despite of the asymmetrical nature of information flows between the actors of the supply chain, they share some, such as production capacity and production costs. Moreover, we assume for simplicity reasons, that the planning horizon of T time periods is the same for all supply chain partners and each production site has eight work hours per day, but differs in the overtime available and the time of production of each product. IV.
P
LANNING
M
ODEL
Based on the structure of the studied supply chain, assuming that each partner uses the model MLCLSP (multilevel capacitated lotsizing problem) to generate its own local production, neglecting the time setting as well as generate the global optimal production plan. The planning is based on reaching two main objectives; the first is the minimization of production cost and the second is the minimization of holding cost.
Knowing that the objective function of the
Dudek’s
model is minimizing the total cost which is the sum of the production cost and the holding cost and that both models have the same constraints, explains the equivalence between the biobjective model detailed below and the monoobjective
Dudek’s model
. Our problem can be formulated as follows:
Set Indexes
T: Set of planning periods.
J: Set of operations.
M: Set of resources.
S
j
: Set of direct successor operations of j.
Indexes
t: Planning period, t= l, ...,T.
j: O
peration, j = 1, …, J.
m: R
esource, m = 1, …, M.
Parameters
ch
j
: Unit holding cost of operation j.
cv
j
: Unit cost of operation j.
cf
j
: Fixed setup cost of operation j.
co
m
: Unit cost of overtime (capacity expansion) at resource m.
D
j,t
: (external) demand for operation j in period t.
B
j,t
: Large constant.
C
m,t
: Capacity at resource m in period t.
I
j
: Starting inventory of operation j.
a
m,j
: Unit requirement of resource m by operation j.
r
j,k
: Unit requirement of operation j by successor operation k.
Variables
C
p
: Total production cost.
C
s
: Total holding cost.
x
j,t
: Output of operation j in period t.
i
j,t
: Inventory level of operation j at the end of period t.
y
j,t
: Setup variable of operation j in period t(y
j,t
=1 if product j is set up in period t ; y
j,t
=0 otherwise).
o
m,t
: Overtime at resource m in period t.
The model Min (C
p
, Cs) (1) S.t.
C
p
=
∑∑[(cf
j
∗y
j,t
)+(cv
j
∗x
j,t
)]+
∈∈
∑∑(co
m
∗o
m,t
)
∈∈
(1.1) C
s
=
∑∑(ch
j
∗i
j,t
)
∈∈
(1.2)
i
j,t−1
+x
j,t
=D
j,t
+∑r
j,k
∗x
k,t
+i
j,t ∈
∀
j
∈
J, t
∈
T
(2)
∑(a
m,j
∗x
j,t
)
∈
≤C
m,t
∗o
m,t
∀
m
∈
M, t
∈
T (3)
x
j,t
≤
,
∗y
j,t
∀
j
∈
J, t
∈
T (4)
x
j,t
≥0 ∀
j
∈
J, t
∈
T
(5)
i
j,t
≥0 ∀
j
∈
J, t
∈
T
(6)
i
j,
=I
j
∀
j
∈
J
(7)
o
m,t
≥0 ∀
m
∈
M, t
∈
T
(8)
y
j,t
∈
{0, 1}
∀
j
∈
J, t
∈
T
(9) This is a program which aims to minimize the total production cost (1.1), including manufacturing costs and execution costs, which results in a linear function of the changes of manufacturing (y
j,t
), the produced quantities (x
j,t
) and overtime on resources (o
m,t
), and the total holding cost (1.2) which results in a linear function of inventory levels (i
j,t
). Constraints are used to describe:
evolution of inventory levels between two successive periods (2): For each item, the sum of the stock at period t1 and the produced quantity at period t must satisfy the sum of internal (requirements of successor operations) and external demand (customer demand) of this article,
limit of production due to resources capacity (3),
coupling between decision variables of production and change series (assigning the value 1 to the variable execution when there is a production) (4),
DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633638
that the variables representing the productions are positive or null (5),
the fact that inventory levels are positive or null, it
doesn’t allow as
stock shortage (6),
The initialization of the opening inventories (7),
That overtime on resources are positive or null (8),
Whether there exists or not the production concerned over the period (9). V.
R
ESOLUTION OF
T
HE
P
ROPOSED
B
I
O
BJECTIVE
M
ODEL
During the last decade, there has been a growing interest in using genetic algorithms (GA) to solve a variety of single as well as multiobjective problems in production and operations management that are considered as combinatorial and NP hard [3],[8], [13] and [20]. To solve our problem, we use a genetic algorithm based on NSGAII (Nondominated Sorting Algorithm II), (Deb 2002 [16]), which retains optimal Pareto solutions found over generations: Each population is distributed in several fronts by a fast sorting procedure. If the number of individuals in a new generation is less than the population size, a crowding procedure is applied to the first next front F
i
not included in this generation. Thus, a new population is created by tournament selection, uniform crossover and mutation (binary and real).This GA has O (MN²) computational complexity (where M is the number of objectives and N is the population size) which is efficient from the computational point of view, compared to other Multiobjective Evaluation.
A.
Test Description
The proposed test was developed using NSGAII to compare the proposed model (which is the biobjective
Dudek’s model)
and the monoobjective
Dudek’s model
which are equivalent. This test is designed where the demand for products is given and has to be fulfilled while facing finite capacities of staff and resources. Our model needs only 1000 iterations to converge to the
optimal solution; however, Dudek’s model needs an important
number of generations to converge to the optimal solution. So, to work in the best condition for the comparison, the generation number used for the tests is equal to 2500 generations. Both of the resolution methods use the algorithm NSGAII.
Set of tests: 2P.2P.3P
In these tests 2P.2P.3P, the supply chain consists of two manufacturing Plants (2P) the planning is done over two periods (2P) to provide three types of Products (3P) in order to satisfy the customers demand, respectively. The storage of finished products is done in the second plant. The second product needs one item of the first operation to be produced. The third product needs one unit of operation 2 to be produced. The demand D
j,t
, is represented in Table 1, where j is the product index and t is the period index.
TABLE I. E
XTERNAL DEMAND OF DIFFERENT ITEMS
D1,1 D2,1 D3,1 D1,2 D2,2 D3,2
90 40 50 80 60 70
B.
Experimental Results and Sensitivity Analysis
It should be noted that the proposed model means the model biobjective Dudek
’s model
detailed previously. To evaluate the proposed model and the solution quality provided by the NSGAII, a benchmark is done with respect to the monoobjective model developed by (Dudek 2007 [11]). So, in order to evaluate the performance of the adopted approach which is the use of genetic algorithms in solving multiobjective problems of supply chains management, we use the NSGAII algorithm to solve the models studied which are the mono
objective Dudek’s model
and the biobjective proposed model equivalent to the
Dudek’s one.
The results are interpreted in order to validate that the genetic algorithm NSGAII is more effective in solving the collaborative planning problem by adopting a multiobjective approach than the resolution of the same problem by adopting a monoobjective approach. The generic solver LINGO 11.0 is used to provide the
global optimal solution for the Dudek’s model in order to
compare this solution to the obtained one for the proposed model using NSGA II: Since the optimal solution obtained by LINGO 11.0 provides the optimal total cost, which is the sum of the production cost and holding cost, then we can easily deduce the optimal values of two objective functions of the proposed model so as to evaluate its performance and its optimality. We assume that all inventories at the starting period are equal to zero. The constraints of the both models are the same.
The objective function of Dudek’s model is to minimize
the total cost (the sum total production cost C
p
and total holding cost C
s
represented in equation (10).
C=[cf
j
∗y
j,t
+cv
j
∗x
j,t
+ch
j
∗i
j,t
]
∈∈
+co
m
∗o
m,t∈∈
(10) Thus, for this model, the total holding cost is optimized indirectly. When the total production cost is minimized and consequently the produced quantity level, the inventory level is indirectly minimized. With re
spect to the Dudek’s model
(2007), the model developed in our work gives an emphasis on minimizing the total holding cost by considering it as a second objective function. The results are visualized in Table 2 which contains the produced quantities x
j,t
, the used overtime o
m,t
, the inventory levels i
j,t
, the objective functions (the total production cost C
p
and the total holding cost C
s
) and the total cost C, both for
Dudek’s model and the proposed model.
DOI: 10.1109/CSCWD.2013.6581034 Conference: Proceedings of the 2013 IEEE 17th International Conference on Computer Supported Cooperative Work in Design, CSCWD 2013, At Wistler, Volume: Article number 6581034, Pages 633638
C.
Optimality of the Proposed Model
It is clear that in the global optimal production plan provided by the generic solver LINGO 11.0, the quantities produced for the third item are equal to external demands. Moreover, the second resource does not need to use overtime in the two planning periods. It should be noted that in 75% of cases studied, the proposed model, solved with the NSGAII algorithm gives an optimal total production cost equal to that provided by the solver LINGO 11.0; this cost is equal to 13502.4 [UM]. One can deduce that starting from a population size greater than or equal to 150 and a number of generations equal to 1000, the proposed model provides an optimal production cost. The best total holding cost provided by the proposed model, in the studied cases, is equal to 12 [UM] with a population size equal to 300, while the optimal cost of holding is equal to 8 [UM]. So we can deduce that with population sizes greater than 300, the proposed model will converge to the optimal solution with regard to the total holding cost Fig.1.
TABLE II. T
EST RESULTS FOR THE BI

OBJECTIVE MODEL
,
MONO

OBJECTIVE MODEL AND THE LINEAR PROGRAMMING
LINGOPopulation size 100 150 200 300
Global optimal solution for monoobjective model provided by LINGO 11.0 solver
Model Monoobjectve model with NSGAII
Biobjective model with NSGAII
Monoobjectve model with NSGAII
Biobjective model with NSGAII
Monoobjectve model with NSGAII
Biobjective model with NSGAII
Monoobjectve model with NSGAII
Biobjective model with NSGAII
y
j,t
1 1 1 1 1 1 1 1 1
x
1,1
239 215 214 204 269 194 236 186 189
x
2,1
106 125 124 114 144 104 109 96 99
x
3,1
66 73 84 74 50 64 65 56 50
x
1,2
151 175 176 186 121 196 154 204 201
x
2,2
114 95 96 106 76 116 111 124 121
x
3,2
54 47 36 46 70 56 55 64 70
o
1,1
884 966 1080 780 1312 480 878 240 214
o
2,1
62 140 236 0 386 0 5 0 0
o
1,2
116 34 0 220 0 520 122 760 786
o
2,2
0 0 0 0 0 0 8 0 0
i
1,1
43 0 0 0 35 0 37 0 0
i
2,1
0 12 0 0 54 0 4 0 8
i
3,1
16 23 34 24 0 14 15 6 0
i
1,2
0 0 0 0 0 0 0 0 0
i
2,2
0 0 0 0 0 0 0 0 0
i
3,2
0 0 0 0 0 0 0 0 0
C
p
14432.4 15602.4 17842.4 13502.4 22412.4 13502.4 14372.4 13502.4 13502.4 C
s
96.5 58 68 48 106.5 28 89.5 12 8 C 14528.9 15660.4 17910.4 13550.4 22518.9 13530.4 14461.9 13514.4 13510.4
Figure 1. Convergence of the holding cost to the optimal solution
D.
Performance of the Adopted Approach
One can note from these results that the total holding cost of proposed model is lower than the total holding cost of
Dudek’s model by 100%. So, considering the total
holding cost as an objective function and not as a constraint clearly improved the results. The per
formance of Dudek’s model is
better than the performance of our proposed model (that with regard to the total cost and in particular the total production cost). However, for all other tests and for all population sizes greater than or equal to 150, the performances reached by the proposed model are bett
er than those of Dudek’s model.
For example, for a population size equal to 200, the total holding
cost (28) is lower than the Dudek’s model (106.5). Besides,