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DYNAMIC PROGRAMMING APPROACH FOR MINIMIZING THE TRANSPORTATION COSTS IN A SUPPLY CHAIN
IFAC MCPL 2007 The 4th International Federation of Automatic Control Conference on Management and Control of Production and Logistics September 27-30, Sibiu - Romania
DYNAMIC PROGRAMMING APPROACH FOR MINIMIZING THE TRANSPORTATION COSTS IN A SUPPLY CHAIN Zerouk Mouloua , Ammar Oulamara Equipe MACSI LORIA-INRIA Lorraine, antenne Ecole des Mines de Nancy, Parc de Saurupt, 54042,Nancy, {mouloua, oulamara }@loria.fr
Abstract: In this paper we study the transportation issues in supply chain given a fixed production schedule. We consider a supply chain with one manufacturer and one or several products. We distinguish the case where we produce one type of products and the case where there are several types of products. The objective is to minimize the transportation costs, i.e. making the optimal delivery schedule that minimize the number of travels from the manufacturer and the customers. We give dynamic programming based approaches to handle the two cases. Copyright © 2007 IFAC Keywords: supply chain, scheduling, dynamic programming, algorithms, graphs.
1.
Mouloua and Oulamara (2006) study the scheduling and cooperation issues in a supply chain where several suppliers collaborate to reduce the overall inventory holding costs by determining the quantity that each supplier has to manufacture. Agnetis et al (2006) study a number of operational issues that are important to the scheduling of supply chains organized to achieve just-in-time (JIT) goals. They consider two consecutive stages of the supply chain, consisting of one supplier and several manufacturers. They define and optimal schedule for each stage that minimizes the overall costs subject to resource constraints at that stage.
INTRODUCTION
The literature of supply chain is very rich when considering the planning or strategic aspects, but it is relatively poor when considering the operational (scheduling) level. Supply chain scheduling is often considered with the coordination and cooperation issues. Hall and Potts (2003) study the benefits and the challenges of coordinated decision making within supply chain scheduling models and considering the batching and delivery problems. Their objective is to minimize the overall scheduling and delivery costs. This is done by scheduling the jobs and forming them into batches, each of which is delivered to the next downstream stage as a single shipment.
Several papers deal with the coordination between the transportation, inventory holding and production planning functions of the supply chain. Kreipl and Pinedo (2004) focus on models and solution approaches for planning ad scheduling in a supply chain. They describe how planning and scheduling models can be used for the design of decision support systems in a supply chain. Lee et .al (2006) treat an integrated model considering both the cross-docking and vehicle routing scheduling. The cross-docking is defined as the continuous process to the final destination through the cross-dock, without storing products and materials in a distribution center (Apte and Viswanathan, 2002). In the vehicle routing problem they consider both pickup and delivery
Selvarajah and Steiner (2005) study the suppliers problem of minimizing the sum of inventory holding costs and delivery costs in a supply chain. The jobs are grouped and delivered to the costumer in batches. Chen and Hall (Chen and Hall, 2001) consider an assembly system in which several suppliers provide components parts to a manufacturer and study the issue of cooperation in such manufacturing system. They demonstrate that cooperation between decisions makers in an assembly system can greatly reduce costs comparing to a system in which there is no cooperation.
753
processes. Since this problem is NP-hard, they propose a tabu search based heuristic algorithm. Earliness-tardiness criteria usually induce NPcomplete problems. Chang and lee (2004) study a class of the two stage scheduling problem in which the first stage is job production and the second stage is job delivery. The focus is on the integration of production scheduling and transportations scheduling. They address the situation in which jobs require different amounts of storage space during delivery. The objective is to minimize the delivery time to customers. They propose heuristics approach with worst analysis. Wang and Lee (2005) consider a new type of scheduling problems arising in logistics systems in which the manufacturing and the delivery of the products are considered simultaneously. In the delivery stage there are two different transportation modes with different times and different costs. The approach they propose integrates the two stages to achieve the maximum benefit with the objective of minimizing the sum of the total transportation costs and the total weighted tardiness cost. They provide a branch and bound algorithm and a mathematical model that is solvable by CPLEX.
delivery orders in known quantities for known delivery dates. Several customers can make orders simultaneously. The customers are geographically dispersed and thus transportation times are different from a customer to another. For transportation of finished products to final customers destination we have a fleet of capacitated vehicles, this fleet is homogeneous, then all the vehicles have the same capacity, which is a number of finished product items. We assume that the transportation activity is carried out by a third party, thus we have an unlimited number of vehicles. Given that the production planning and scheduling is already fixed and known, our objective is to find the schedule of transportation that minimizes the total transportation costs, i.e. find the minimum number of travels between the manufacturer and the customers. Actually, as we don’t consider warehousing constraints, the optimal production policy (from the manufacturer point of view) is to produce as soon as possible (left shifted production). Then, in this first problem, we only consider the transportation issues by minimizing the number of travels between the manufacturer and its clients. Table1 shows an example where we have three periods, for each period we have customers demands and the production capacity.
In this paper we consider a supply chain with one manufacturer and one or several customers. We study the problem of minimizing the transportations costs by minimizing the number of travels between the manufacturer and the customers. We assume that the production schedule is already made. At first, in section (2) we study the case where the manufacturer produces a single type of products, the case of multiple types of products is considered in section (3). For each problem we give a problem description, the notations and the proposed approach. Finally, section 4 provides some concluding remarks and directions for future works. 2.
Table 1 example of demands and production with two customers. demand1 demand2
1 3 4
2 5 6
production
8
12
3 6 7 11
2.1. Notations The following notations will be used throughout this section : n : the number of scheduling horizon periods which is the number of times where a delivery demand is made m : number of customers. C : vehicles capacity.
SINGLE PRODUCT SUPPLY CHAIN PROBLEM
We consider a supply chain with one or several suppliers, one manufacturer and one or several customers. The suppliers replenish the manufacturer with the raw materials needed for the manufacturing of one type of products. The raw materials arrive at defined dates and with known quantities. The manufacturing system could be modeled as a single machine system, all the raw materials pass by this machine for processing. The manufacturer has to satisfy the demands of the customers. The customers make
d ij : quantity asked by the j th client at the i th period.
dd i : delivery date associated to the i th period. CQDi : cumulative asked quantity until the i th delivery date dd i (included).
754
CQDij : cumulative asked quantity by client j from the first period until period i (period i
e
condition that there exists an arc from X i −1 to
X il .
included).
We add two virtual vertexes, one at the origin called S and one at the arrival called F.
ΔPQi : produced quantity between dd i −1 and dd i .
l
only under the following conditions :
X ij : total transported quantity for the client j during the period i .
X ie+1 j ≥ d i +1 j − S ijl
S i : quantity in stock at the beginning of period i.
m
∑X j =1
We model our problem as a problem of searching the shortest path in an oriented and acyclic graph. The shortest path problem is a very well known problem for which optimal solutions exist. We define a set of vertices, a set of arcs, how we generate the arcs and the cost associated to each arc.
S : quantity in stock associated to the l of the period i .
Cost function on arcs: As our objective is to minimize transportation costs by finding the minimum cost delivery schedule (with the minimum number of travels), then the cost
state
l
function of transition from the state X i of period
S ijl : cumulative quantities that were transported in addition of the orders to client j . Then the l
th
X ijl .
can be reached only by one arc (except for the states of the last period), and many arcs can have origin from a single state. There are no arcs between the states of the same period. An arc from period i can reach only a state of period i + 1 . From each state of the last period goes an arc to the arrival state F with a cost equal to zero.
state. th
(2)
Then, the number of the possible permutations is bounded, which means that the number of states at each period is bounded, and maybe equal to zero. If the number of states is equal to zero in at least one state then the problem is infeasible (i.e. ei = 0 then the problem is infeasible). A state
X ijl : transported quantity during period i for l i
≤ ΔPQi +1 + S il
(1)
bound and upper bound to the values of
Set of vertices: Each period i is represented by a set of finished number vertices (that might be empty). A state is represented by a vector which contains the transported quantity during period i for each customer, and the cumulative quantities that were transported in addition of the orders. We define the following variables:
th
e i +1 j
∀j
Condition (1) ensures that the demands of each client are satisfied. Condition (2) guarantees that we cannot transport more than the produced quantity. These two constraints give a lower
2.2. Proposed Approach
client j in the l
e
The arcs: There is an arc from state X i to X i +1 ,
i to state X ie+1 of period i + 1 is defined by the number of travels incurred by such a transition,
state of period i is represented by
e
thus it depends on the values of the vector X i +1 .
l i
vector X as :
The cost function is given by
X il = (X il1 , X il2 ,..., X iml , S il1 , S il2 ,..., S iml , S il )
m ⎡Xe ⎤ i +1 j fct X il , X ie+1 = fct X ie+1 = ∑ ⎢ ⎥ (3) j =1 ⎢ C ⎥
(
ei defines the number of the states of period i .
)
(
)
l
The computing of the quantities S i in stock is
Dynamic programming : We adapt the dynamic programming based algorithm for the research of the shortest path problem in an acyclic oriented graph. The basic idea of the algorithm is that the shortest path until period i is the sum of the
given as follows m
S il = ΔPQi −1 − ∑ X ie−1 j + S ie−1 with the j =1
755
shortest path until period i − 1 , and the minimum cost arc between periods i and i − 1 .
by:
The dynamic programming formulation is given
⎧ F (0 ) = 0 ⎪ ⎨ F (i ) = ( F (i − 1) + fct ( X il )) (4) Min ⎪⎩ l =1,ei
Example : Figure1 shows a graph modeling the problem given by table 1.
Fig. I. the graph associated to table 1.
d ijk : asked quantity by the j th client at the i th
3. MULTI PRODUCT SUPPLY CHAIN PROBLEM
period.
In this section we generalize the last problem to the case where the manufacturer produces several types of finished products. As for the last problem, we keep the same assumptions about the organization of the supply chain. We want to find the delivery schedule that minimizes the transportations costs from the manufacturer to the customers (number of travels). We assume that the different products have the same volume, and thus they could be transported in the same vehicle.
Pk : processing time of an item of product k . R : number of products.
ΔPQik : Quantity of product k manufactured in period i between dd i −1 et dd i .
St k :
setup
time
before
starting
manufacturing of an item of product
the
k.
CQTijk : Cumulative transported quantity of product k until period i for client j .
3.1. Notations The following notation will be used throughout this section, we keep also the notations of the precedent section :
756
(
X ijk : Quantity of product k transported during the period i for client j .
l l X ikl = X il1k , X il2k ,..., X imk , Sil1k , Sil2k ,...,Simk , Sikl
the vector that defines the l state of period i for the product k . Here is an example of the set of states of period i.
Yij : number of travels done for client j during period i . res ik : Quantity in stock of product k at the end of period i .
1 1 1 1 1 ⎛ X i111 , X i121 ,..., X im ⎞ 1 , S i11 , S i 21 ,..., S im1 , S i1 ⎜ ⎟ 1 1 1 1 1 ⎜ X i112 , X i122 ,..., X im ⎟ , S , S ,..., S , S 2 i12 i 22 im 2 i2 ⎜ ⎟ ⎜ ......... ⎟ ⎜ ........ ⎟ ⎜ 1 ⎟ 1 1 1 1 1 ⎟ ⎜ X i1k , X i12 k ,..., X imk , S , S ,..., S , S i1k i 2k imk ik ⎠ ⎝ ....... ........ ........
Constraints 1.
Transportations constraint : the transported quantity must be greater or equal to the asked quantity of each product, for every customer and for each period.
2.
Production constraint : the cumulative produced quantity must be greater or equal to the cumulative asked quantity at each period and for every type of products.
Objective: minimize the transportation costs, i.e. minimizing the number of travels.
⎛ X iei11 , X iei21 ,..., X imei 1 , S iei11 , S iei21 ,..., S imei 1 , S iei1 ⎞ ⎜ ⎟ ⎜ X iei12 , X iei22 ,..., X imei 2 , S iei12 , S iei22 ,..., S imei 2 , S iei2 ⎟ ⎜ ⎟ ⎜ ......... ⎟ ⎜ ........ ⎟ ⎜ ei ⎟ ei ei ei ei ei ⎟ ⎜ X i1k , X iei2 k ,..., X imk , S , S ,..., S , S 1 2 i k i k imk ik ⎝ ⎠
3.2. Proposed Approach The dynamic programming approach that we propose here is a generalization of the proposed approach for the first problem. In fact, we model the multi product problem as the shortest path problem in oriented acyclic graph.
l
let E i be the matrix defining the l
state of the
⎛ X il11 , X il21 ,..., X iml 1 , S il11 , S il21 ,..., S iml 1 , S il1 ⎞ ⎜ ⎟ ⎜ X il12 , X il22 ,..., X iml 2 , S il12 , S il22 ,..., S iml 2 , S il2 ⎟ ⎜ ⎟ Eil = ⎜ ......... ⎟ ⎜ ........ ⎟ ⎜ l ⎟ l l l l l ⎟ ⎜ X i1k , X il2 k ,..., X imk , S , S ,..., S , S 1 2 i k i k imk ik ⎝ ⎠
constraints. In the first problem, a state was defined by a vector of the transported quantities for each client, in the multi product case we have a vector for each type of products. A state would be represented by a R × n matrix. We define the following variables :
X ijkl : quantity of product k transported during th
th
period i . We will have :
Set of states: For each period there is a set of finite number of states that maybe equal to zero. The set of states is built by all the permutations of X ijk that respect the transport and production
period i for the client j in the l
)
th
If the number of states is equal to zero in one state then the problem is infeasible.
state.
S ikl : the quantity in stock of product k at the
The arcs: An arc goes from a state X i to X i +1 ,
l th state of the period i . S ijkl : cumulative transported quantity of product
if and only if the following constraints are respected:
l
k until period i in addition to the demand of the client j .
X ie+1 jk ≥ d i +1 jk − S ijkl
757
∀ j , k , e, i
e
(5)
m
∑X j =1
e i +1 jk
≤ ΔPQik+1 + S ikl
of the components and the delivery of products to the final customer (Just-In-Time delivery).
∀j , k , e, i (6)
REFERENCES
As for the single product problem , we have an initial state and a terminal state with the same properties as for the single product problem. Our graph construction method assures us that we obtain only feasible solutions with a finite number of states (maybe equal to zero).
Agnetis A, Hall N.G and Pacciarelli D, (2006). Supply chain scheduling: Sequence coordination. Discrete Applied Mathematics, 154, Issue 15, 2044-2063 Apte, U. M., and Viswanathan, S. (2002). Strategic and technological innovations in supply chain management. International Journal of manufacturing Technology and management, 4, 264-282.
Cost function: The cost of a transition between a l
e
state E i of period i to state E i +1 of period
i + 1 is defined by the number of travels that such a transition implies which depends on the e
values of the matrix E i +1 . The cost function is
Chang Y., and Lee C.,(2004) Machine scheduling with job delivery coordination. European Journal of Operational Research.158, 470487.
given by :
(
)
( )
C Eil , Eie+1 = C Eie+1
⎡ R ⎤ X ie+1 jk ⎥ m ⎢∑ ⎥ (7) = ∑ ⎢ k =1 C ⎥ j =1 ⎢ ⎢⎢ ⎥⎥
Chen Z.L., and N.G. Hall (2001). Supply chain scheduling: Assembly systems. Working Paper, Department of Systems Engineering, University of Pennsylvania. Hall, N.G. and C.N. Potts (2003). Supply Chain Scheduling : Batching and Delivery. Operations Research, 51, 566-584. Kreipl S., and M. Pinedo (2004). Planning and scheduling in supply chains: An overview of issues in practice. Production and Operations management, 13, No 1, 77-92.
The dynamic programming formulation is given
⎧ F (0) = 0 ⎪ by: ⎨ F (i ) = ( F (i − 1) + C ( Eil )) (8) Min ⎪⎩ l =1,ei
Mouloua Z. and Oulamara A.(2006). Cooperation in supply chain scheduling : minimizing the inventory holding cost, . The International conference on Information systems, Logistics and Supply chain, Lyon.
4. CONCLUSION We have studied scheduling problems in a supply chain linked with the transportation issues. We developed an optimal approach to the problem of minimizing the number of travels between the manufacturer and the customers in both the cases the manufacturer makes a single or multiple products. We assumed that the production schedule is fixed before the transportation schedule. In further studies we will consider the problem of making the production and transportation schedules simultaneously. This would be an issue of the cooperation in supply chain. The cooperation between production and delivery functions is very challenging because the delivery costs are the greatest costs in supply chains. We can also add some constraints to the model, such as deadlines on both the availability
Selvarajah E. and G. Steiner (2005). Batch scheduling in a two-level supply chain—a focus on the supplier , European Journal of Operational Research, 173, Issue 1, 226-240. Wang H., Lee C.Y., (2005), Production and transport logistics scheduling with two Naval transport mode choices, Research Logistics. 52, Issue 8, 2005, Pages: 796-809. Young Hae Lee, Jung Woo Jung and Kyong Min Lee, Vehicle routing scheduling for crossdocking in the supply chain . Computers & Industrial Engineering, 51, Issue 2, October 2006, 247-256.
758
DYNAMIC PROGRAMMING APPROACH FOR MINIMIZING THE TRANSPORTATION COSTS IN A SUPPLY CHAIN Zerouk Mouloua , Ammar Oulamara Equipe MACSI LORIA-INRIA Lorraine, antenne Ecole des Mines de Nancy, Parc de Saurupt, 54042,Nancy, {mouloua, oulamara }@loria.fr
Abstract: In this paper we study the transportation issues in supply chain given a fixed production schedule. We consider a supply chain with one manufacturer and one or several products. We distinguish the case where we produce one type of products and the case where there are several types of products. The objective is to minimize the transportation costs, i.e. making the optimal delivery schedule that minimize the number of travels from the manufacturer and the customers. We give dynamic programming based approaches to handle the two cases. Copyright © 2007 IFAC Keywords: supply chain, scheduling, dynamic programming, algorithms, graphs.
1.
Mouloua and Oulamara (2006) study the scheduling and cooperation issues in a supply chain where several suppliers collaborate to reduce the overall inventory holding costs by determining the quantity that each supplier has to manufacture. Agnetis et al (2006) study a number of operational issues that are important to the scheduling of supply chains organized to achieve just-in-time (JIT) goals. They consider two consecutive stages of the supply chain, consisting of one supplier and several manufacturers. They define and optimal schedule for each stage that minimizes the overall costs subject to resource constraints at that stage.
INTRODUCTION
The literature of supply chain is very rich when considering the planning or strategic aspects, but it is relatively poor when considering the operational (scheduling) level. Supply chain scheduling is often considered with the coordination and cooperation issues. Hall and Potts (2003) study the benefits and the challenges of coordinated decision making within supply chain scheduling models and considering the batching and delivery problems. Their objective is to minimize the overall scheduling and delivery costs. This is done by scheduling the jobs and forming them into batches, each of which is delivered to the next downstream stage as a single shipment.
Several papers deal with the coordination between the transportation, inventory holding and production planning functions of the supply chain. Kreipl and Pinedo (2004) focus on models and solution approaches for planning ad scheduling in a supply chain. They describe how planning and scheduling models can be used for the design of decision support systems in a supply chain. Lee et .al (2006) treat an integrated model considering both the cross-docking and vehicle routing scheduling. The cross-docking is defined as the continuous process to the final destination through the cross-dock, without storing products and materials in a distribution center (Apte and Viswanathan, 2002). In the vehicle routing problem they consider both pickup and delivery
Selvarajah and Steiner (2005) study the suppliers problem of minimizing the sum of inventory holding costs and delivery costs in a supply chain. The jobs are grouped and delivered to the costumer in batches. Chen and Hall (Chen and Hall, 2001) consider an assembly system in which several suppliers provide components parts to a manufacturer and study the issue of cooperation in such manufacturing system. They demonstrate that cooperation between decisions makers in an assembly system can greatly reduce costs comparing to a system in which there is no cooperation.
753
processes. Since this problem is NP-hard, they propose a tabu search based heuristic algorithm. Earliness-tardiness criteria usually induce NPcomplete problems. Chang and lee (2004) study a class of the two stage scheduling problem in which the first stage is job production and the second stage is job delivery. The focus is on the integration of production scheduling and transportations scheduling. They address the situation in which jobs require different amounts of storage space during delivery. The objective is to minimize the delivery time to customers. They propose heuristics approach with worst analysis. Wang and Lee (2005) consider a new type of scheduling problems arising in logistics systems in which the manufacturing and the delivery of the products are considered simultaneously. In the delivery stage there are two different transportation modes with different times and different costs. The approach they propose integrates the two stages to achieve the maximum benefit with the objective of minimizing the sum of the total transportation costs and the total weighted tardiness cost. They provide a branch and bound algorithm and a mathematical model that is solvable by CPLEX.
delivery orders in known quantities for known delivery dates. Several customers can make orders simultaneously. The customers are geographically dispersed and thus transportation times are different from a customer to another. For transportation of finished products to final customers destination we have a fleet of capacitated vehicles, this fleet is homogeneous, then all the vehicles have the same capacity, which is a number of finished product items. We assume that the transportation activity is carried out by a third party, thus we have an unlimited number of vehicles. Given that the production planning and scheduling is already fixed and known, our objective is to find the schedule of transportation that minimizes the total transportation costs, i.e. find the minimum number of travels between the manufacturer and the customers. Actually, as we don’t consider warehousing constraints, the optimal production policy (from the manufacturer point of view) is to produce as soon as possible (left shifted production). Then, in this first problem, we only consider the transportation issues by minimizing the number of travels between the manufacturer and its clients. Table1 shows an example where we have three periods, for each period we have customers demands and the production capacity.
In this paper we consider a supply chain with one manufacturer and one or several customers. We study the problem of minimizing the transportations costs by minimizing the number of travels between the manufacturer and the customers. We assume that the production schedule is already made. At first, in section (2) we study the case where the manufacturer produces a single type of products, the case of multiple types of products is considered in section (3). For each problem we give a problem description, the notations and the proposed approach. Finally, section 4 provides some concluding remarks and directions for future works. 2.
Table 1 example of demands and production with two customers. demand1 demand2
1 3 4
2 5 6
production
8
12
3 6 7 11
2.1. Notations The following notations will be used throughout this section : n : the number of scheduling horizon periods which is the number of times where a delivery demand is made m : number of customers. C : vehicles capacity.
SINGLE PRODUCT SUPPLY CHAIN PROBLEM
We consider a supply chain with one or several suppliers, one manufacturer and one or several customers. The suppliers replenish the manufacturer with the raw materials needed for the manufacturing of one type of products. The raw materials arrive at defined dates and with known quantities. The manufacturing system could be modeled as a single machine system, all the raw materials pass by this machine for processing. The manufacturer has to satisfy the demands of the customers. The customers make
d ij : quantity asked by the j th client at the i th period.
dd i : delivery date associated to the i th period. CQDi : cumulative asked quantity until the i th delivery date dd i (included).
754
CQDij : cumulative asked quantity by client j from the first period until period i (period i
e
condition that there exists an arc from X i −1 to
X il .
included).
We add two virtual vertexes, one at the origin called S and one at the arrival called F.
ΔPQi : produced quantity between dd i −1 and dd i .
l
only under the following conditions :
X ij : total transported quantity for the client j during the period i .
X ie+1 j ≥ d i +1 j − S ijl
S i : quantity in stock at the beginning of period i.
m
∑X j =1
We model our problem as a problem of searching the shortest path in an oriented and acyclic graph. The shortest path problem is a very well known problem for which optimal solutions exist. We define a set of vertices, a set of arcs, how we generate the arcs and the cost associated to each arc.
S : quantity in stock associated to the l of the period i .
Cost function on arcs: As our objective is to minimize transportation costs by finding the minimum cost delivery schedule (with the minimum number of travels), then the cost
state
l
function of transition from the state X i of period
S ijl : cumulative quantities that were transported in addition of the orders to client j . Then the l
th
X ijl .
can be reached only by one arc (except for the states of the last period), and many arcs can have origin from a single state. There are no arcs between the states of the same period. An arc from period i can reach only a state of period i + 1 . From each state of the last period goes an arc to the arrival state F with a cost equal to zero.
state. th
(2)
Then, the number of the possible permutations is bounded, which means that the number of states at each period is bounded, and maybe equal to zero. If the number of states is equal to zero in at least one state then the problem is infeasible (i.e. ei = 0 then the problem is infeasible). A state
X ijl : transported quantity during period i for l i
≤ ΔPQi +1 + S il
(1)
bound and upper bound to the values of
Set of vertices: Each period i is represented by a set of finished number vertices (that might be empty). A state is represented by a vector which contains the transported quantity during period i for each customer, and the cumulative quantities that were transported in addition of the orders. We define the following variables:
th
e i +1 j
∀j
Condition (1) ensures that the demands of each client are satisfied. Condition (2) guarantees that we cannot transport more than the produced quantity. These two constraints give a lower
2.2. Proposed Approach
client j in the l
e
The arcs: There is an arc from state X i to X i +1 ,
i to state X ie+1 of period i + 1 is defined by the number of travels incurred by such a transition,
state of period i is represented by
e
thus it depends on the values of the vector X i +1 .
l i
vector X as :
The cost function is given by
X il = (X il1 , X il2 ,..., X iml , S il1 , S il2 ,..., S iml , S il )
m ⎡Xe ⎤ i +1 j fct X il , X ie+1 = fct X ie+1 = ∑ ⎢ ⎥ (3) j =1 ⎢ C ⎥
(
ei defines the number of the states of period i .
)
(
)
l
The computing of the quantities S i in stock is
Dynamic programming : We adapt the dynamic programming based algorithm for the research of the shortest path problem in an acyclic oriented graph. The basic idea of the algorithm is that the shortest path until period i is the sum of the
given as follows m
S il = ΔPQi −1 − ∑ X ie−1 j + S ie−1 with the j =1
755
shortest path until period i − 1 , and the minimum cost arc between periods i and i − 1 .
by:
The dynamic programming formulation is given
⎧ F (0 ) = 0 ⎪ ⎨ F (i ) = ( F (i − 1) + fct ( X il )) (4) Min ⎪⎩ l =1,ei
Example : Figure1 shows a graph modeling the problem given by table 1.
Fig. I. the graph associated to table 1.
d ijk : asked quantity by the j th client at the i th
3. MULTI PRODUCT SUPPLY CHAIN PROBLEM
period.
In this section we generalize the last problem to the case where the manufacturer produces several types of finished products. As for the last problem, we keep the same assumptions about the organization of the supply chain. We want to find the delivery schedule that minimizes the transportations costs from the manufacturer to the customers (number of travels). We assume that the different products have the same volume, and thus they could be transported in the same vehicle.
Pk : processing time of an item of product k . R : number of products.
ΔPQik : Quantity of product k manufactured in period i between dd i −1 et dd i .
St k :
setup
time
before
starting
manufacturing of an item of product
the
k.
CQTijk : Cumulative transported quantity of product k until period i for client j .
3.1. Notations The following notation will be used throughout this section, we keep also the notations of the precedent section :
756
(
X ijk : Quantity of product k transported during the period i for client j .
l l X ikl = X il1k , X il2k ,..., X imk , Sil1k , Sil2k ,...,Simk , Sikl
the vector that defines the l state of period i for the product k . Here is an example of the set of states of period i.
Yij : number of travels done for client j during period i . res ik : Quantity in stock of product k at the end of period i .
1 1 1 1 1 ⎛ X i111 , X i121 ,..., X im ⎞ 1 , S i11 , S i 21 ,..., S im1 , S i1 ⎜ ⎟ 1 1 1 1 1 ⎜ X i112 , X i122 ,..., X im ⎟ , S , S ,..., S , S 2 i12 i 22 im 2 i2 ⎜ ⎟ ⎜ ......... ⎟ ⎜ ........ ⎟ ⎜ 1 ⎟ 1 1 1 1 1 ⎟ ⎜ X i1k , X i12 k ,..., X imk , S , S ,..., S , S i1k i 2k imk ik ⎠ ⎝ ....... ........ ........
Constraints 1.
Transportations constraint : the transported quantity must be greater or equal to the asked quantity of each product, for every customer and for each period.
2.
Production constraint : the cumulative produced quantity must be greater or equal to the cumulative asked quantity at each period and for every type of products.
Objective: minimize the transportation costs, i.e. minimizing the number of travels.
⎛ X iei11 , X iei21 ,..., X imei 1 , S iei11 , S iei21 ,..., S imei 1 , S iei1 ⎞ ⎜ ⎟ ⎜ X iei12 , X iei22 ,..., X imei 2 , S iei12 , S iei22 ,..., S imei 2 , S iei2 ⎟ ⎜ ⎟ ⎜ ......... ⎟ ⎜ ........ ⎟ ⎜ ei ⎟ ei ei ei ei ei ⎟ ⎜ X i1k , X iei2 k ,..., X imk , S , S ,..., S , S 1 2 i k i k imk ik ⎝ ⎠
3.2. Proposed Approach The dynamic programming approach that we propose here is a generalization of the proposed approach for the first problem. In fact, we model the multi product problem as the shortest path problem in oriented acyclic graph.
l
let E i be the matrix defining the l
state of the
⎛ X il11 , X il21 ,..., X iml 1 , S il11 , S il21 ,..., S iml 1 , S il1 ⎞ ⎜ ⎟ ⎜ X il12 , X il22 ,..., X iml 2 , S il12 , S il22 ,..., S iml 2 , S il2 ⎟ ⎜ ⎟ Eil = ⎜ ......... ⎟ ⎜ ........ ⎟ ⎜ l ⎟ l l l l l ⎟ ⎜ X i1k , X il2 k ,..., X imk , S , S ,..., S , S 1 2 i k i k imk ik ⎝ ⎠
constraints. In the first problem, a state was defined by a vector of the transported quantities for each client, in the multi product case we have a vector for each type of products. A state would be represented by a R × n matrix. We define the following variables :
X ijkl : quantity of product k transported during th
th
period i . We will have :
Set of states: For each period there is a set of finite number of states that maybe equal to zero. The set of states is built by all the permutations of X ijk that respect the transport and production
period i for the client j in the l
)
th
If the number of states is equal to zero in one state then the problem is infeasible.
state.
S ikl : the quantity in stock of product k at the
The arcs: An arc goes from a state X i to X i +1 ,
l th state of the period i . S ijkl : cumulative transported quantity of product
if and only if the following constraints are respected:
l
k until period i in addition to the demand of the client j .
X ie+1 jk ≥ d i +1 jk − S ijkl
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∀ j , k , e, i
e
(5)
m
∑X j =1
e i +1 jk
≤ ΔPQik+1 + S ikl
of the components and the delivery of products to the final customer (Just-In-Time delivery).
∀j , k , e, i (6)
REFERENCES
As for the single product problem , we have an initial state and a terminal state with the same properties as for the single product problem. Our graph construction method assures us that we obtain only feasible solutions with a finite number of states (maybe equal to zero).
Agnetis A, Hall N.G and Pacciarelli D, (2006). Supply chain scheduling: Sequence coordination. Discrete Applied Mathematics, 154, Issue 15, 2044-2063 Apte, U. M., and Viswanathan, S. (2002). Strategic and technological innovations in supply chain management. International Journal of manufacturing Technology and management, 4, 264-282.
Cost function: The cost of a transition between a l
e
state E i of period i to state E i +1 of period
i + 1 is defined by the number of travels that such a transition implies which depends on the e
values of the matrix E i +1 . The cost function is
Chang Y., and Lee C.,(2004) Machine scheduling with job delivery coordination. European Journal of Operational Research.158, 470487.
given by :
(
)
( )
C Eil , Eie+1 = C Eie+1
⎡ R ⎤ X ie+1 jk ⎥ m ⎢∑ ⎥ (7) = ∑ ⎢ k =1 C ⎥ j =1 ⎢ ⎢⎢ ⎥⎥
Chen Z.L., and N.G. Hall (2001). Supply chain scheduling: Assembly systems. Working Paper, Department of Systems Engineering, University of Pennsylvania. Hall, N.G. and C.N. Potts (2003). Supply Chain Scheduling : Batching and Delivery. Operations Research, 51, 566-584. Kreipl S., and M. Pinedo (2004). Planning and scheduling in supply chains: An overview of issues in practice. Production and Operations management, 13, No 1, 77-92.
The dynamic programming formulation is given
⎧ F (0) = 0 ⎪ by: ⎨ F (i ) = ( F (i − 1) + C ( Eil )) (8) Min ⎪⎩ l =1,ei
Mouloua Z. and Oulamara A.(2006). Cooperation in supply chain scheduling : minimizing the inventory holding cost, . The International conference on Information systems, Logistics and Supply chain, Lyon.
4. CONCLUSION We have studied scheduling problems in a supply chain linked with the transportation issues. We developed an optimal approach to the problem of minimizing the number of travels between the manufacturer and the customers in both the cases the manufacturer makes a single or multiple products. We assumed that the production schedule is fixed before the transportation schedule. In further studies we will consider the problem of making the production and transportation schedules simultaneously. This would be an issue of the cooperation in supply chain. The cooperation between production and delivery functions is very challenging because the delivery costs are the greatest costs in supply chains. We can also add some constraints to the model, such as deadlines on both the availability
Selvarajah E. and G. Steiner (2005). Batch scheduling in a two-level supply chain—a focus on the supplier , European Journal of Operational Research, 173, Issue 1, 226-240. Wang H., Lee C.Y., (2005), Production and transport logistics scheduling with two Naval transport mode choices, Research Logistics. 52, Issue 8, 2005, Pages: 796-809. Young Hae Lee, Jung Woo Jung and Kyong Min Lee, Vehicle routing scheduling for crossdocking in the supply chain . Computers & Industrial Engineering, 51, Issue 2, October 2006, 247-256.
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