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Collaborative Planning in Supply Chains by Lagrangian Relaxation

Lanshun Nie, Xiaofei Xu, and Dechen Zhan School of Computer Science and Technology, Harbin Institute of Technology, Harbin, China

[email protected]; [email protected]; [email protected]

Abstract

A collaborative planning framework based on the Lagrangian Relaxation method is developed to coordinate and optimize the production planning of the independent partners linked by material flows in multiple tier supply chains. Linking constraints and dependent demand constraints were added to the monolithic Multi-Level, multi-item Capacitated Lot Sizing Problem (MLCLSP) for supply chains. Model MLCLSP was Lagrangian relaxed and decomposed into facility-separable subproblems based on the separability of it. Surrogate gradient algorithm was used to update Lagrangian multipliers, which coordinated decentralized decisions of the facilities in supply chains. Production planning of independent partners could be appropriately coordinated and optimized by this framework without intruding their decision authorities and private information. This collaborative planning scheme was applied to a large set problem in supply chain production planning. Experimental results show that the proposed coordination mechanism and procedure come close to optimal results as obtained by central coordination.

1. Introduction

Supply chain management (SCM) deals with the management of the multiple relationships across the supply chain, i.e. the network of organizations involved in creating final customer products and services [1]. Supply Chain Operations Planning (SCOP) forms a key aspect of SCM. Its objective is to coordinate the release of materials and resources in the supply network under consideration such that customer service constraints are met at minimal cost [2]. It is often proposed that operations planning in supply chains can be organized in terms of a hierarchical planning system. This approach assumes a single decision maker who grasps “total visibility” of system details and makes centralized decisions for the entire supply chain. It might be suitable in an intra-

organizational supply chain or a focal inter-organizational supply chain. However, if partners are reluctant to reveal all of their information or it is too costly to keep the information of the entire supply chain up-to-date, the hierarchical approach is unsuitable or infeasible [3]. As a result, the question arises of how to link, coordinate and optimize production planning of independent partners in the supply chain without intruding their decision authorities and private information.

Supply chain coordination has attracted many researchers in recent years. First, there is a large and growing stream of literature by contracts. Several different contract types have been studied to coordinating the newsvendor problem where there is one selling reason with stochastic demand and one retailer has single opportunity to order inventory from the single supplier before the selling season begins, e.g. buyback contracts [4], revenue-sharing contracts [5], quantity-flexibility contracts [6], sales-rebate contracts [7] and quantity-discount contracts [8]. Another class of papers is concerned with problems of coordination in the single-location base-stock model and two-location base-stock model [9], [10].

A second area of research deals with multi-agent systems for the coordination of supply chains. Swaminathan et al. develop a software library which contains functional agents such as plants, suppliers, etc. and control agents for inventory management, transportation flow and demand planning [11]. Azevedo et al. describe an advanced agent-based order planning system for dynamic networked enterprises [12]. They propose a multi-agent system architecture for real-time customer-order planning in distributed manufacturing enterprises, addressing the requirement of a make-to-order environment. Caridi et al. give an overview of multi-agent systems in production planning and control [13]. They point out that there is still no clear understanding where and how multi-agent systems can provide better results than ‘traditional’ models and provide some indication of in what fields multi-agent approach turns out to be effective.

Proceedings of the First International Multi-Symposiums on Computer and Computational Sciences (IMSCCS'06) 0-7695-2581-4/06 $20.00 © 2006 IEEE

Finally, a field of research is concerned with mathematical programming models for supply chain planning. The majority of this kind of literature assume central decision making. Erenguc et al. present sub-models for production, distribution and inventory planning in supply chains and give a review of mathematical programming planning models within the supply chains [14].

A few contributions combine mathematical programming approaches with distributed decision making. The simplest coordination scheme is “upstream planning” [15]. Beginning with the most downstream facility, the plan is prepared and defines supply requirements for its suppliers. These requirements are passed to suppliers, and the procedure continues in upstream direction. This approach is easy to implement but results in sub-optimization or even capacity infeasibility. Dudek and Stadlter propose a negotiation-based coordination scheme for a buyer and a supplier [16]. This scheme can obtain near optimal solutions, but it is only suitable to the supply chains composed of two partners. Ertogral and Wu develop a coordination mechanism based on auction theory [17]. The objective of this approach is to achieve maximum fairness instead of optimization and effectiveness of the production planning in supply chains. Zimmer proposes a coordination scheme by extending the upstream planning by anticipation at the buyer domain and gets improved results, but this scheme need extensive information about partners [18].

This work draws from and combines aspects of all three research domains: contract design, multi-agent system, and mathematical programming. A novel collaborative planning framework is presented for coordinating the production planning of the independent partners with cooperative behavior in a multiple tiers supply chain. The coordination mechanism for aligning individual planning is based on the Lagrangian Relaxation. We formulate the production planning of the supply chain as a monolithic Multi-Level, multi-item Capacitated Lot Sizing Problem (MLCLSP) and decompose it into facility-separable subproblems using the Lagrangian Relaxation method base on the separability of it. A feasibility restoring method based on the upstream planning mechanism is proposed. Then, we propose a coordination structure and procedure among the coordination center and the facilities and adopt surrogate gradient algorithm to update the Lagrangian multipliers, which coordinate decentralized decisions of the facilities. As collaborative planning affects the cost outcomes of individual partners, contract terms need to be adapted based on the planning result in order to achieve mutual benefit and win-win. Finally,

the coordination scheme can be implemented as a multi-agent system for supply chain planning.

2. Monolithic production planning model and model decomposition

2.1. Monolithic model for production planning in supply chains

Operations planning in a multiple tier supply chain can be formulated as the Multi-Level, multi-item Capacitated Lot Sizing Problem(MLCLSP). It can be defined as follows: given external demand for end items over a time horizon, a bill-of-material structure for each end item where the production of components may be spread across multiple facilities, find a production plan over multiple facilities which minimizes total inventory holding cost and setup cost. The main restrictions are: (1) items can only be produced after all their predecessor components are available; (2) resources within each facility have limited capacity; and (3) no backlogging is allowed for the end items. We give a formulation of the MLCLSP problem similar to that presented in [19].

The following notations are adopted: Indices t planning period Tj resource Jk item KIndex sets T set of planning periods J set of resources K set of items Databjt available capacity of resource j in period ttbk production time per unit of item ktrk setup time of item k

akinumber of units of item k required to produce one unit of item i

Kj set of items that are produced by resource j

Nkset of items that are immediate successors ofitem k

z(k) deterministic minimal lead time for item kdkt external demand for item k in period thk inventory holding cost for item ksk setup cost for item kM a large number Variables qkt lot size for item k in period tykt inventory of item k at the end of period t

kt binary setup variable for item k in period tThe monolithic model for production planning in

supply chains is as follows:


Min 1 1

(T K

k kt k ktt k

C h y s ) (1)

s.t.

, 1 , ( ) ,

;k

k t k t z k ki it kt kti N

y q a q y

k K t T

d (2)

( ) , ;j

k kt k kt jtk K

tr tb q b j J t T (3)

0, ;kt ktq M k K t T (4)

0, 0, {0,1}, ;kt kt ktq y k K t T (5) The objective (1) is to minimize total inventory holding costs and setup costs. Constraints (2) for each item and period describe the balance relationships between the inventory level at the beginning and the end of the period, the external demand, the demand derived from production orders for successor items and the production quantity of an item. Constraints (3) enforce resource (capacity) limitations. The production and setup relationship is represented by constraints (4). Without loss of generality, we exclude production costs from our formulation and assume that all lead times z(k) are zero.

2.2. Problem Reformulations

MLCLSP is a monolithic model as it implicitly assumes that each facility in the supply chain is willing to reveal its local constraints and cost parameters, and is ready to implement a centrally imposed solution. To explore the coordination of multi-facilities’ production planning, we reformulate model MLCLSP based on the following rules.

The first rule is that sharing demand information of end items across the supply chain. The facilities in supply chains develop long-term partnership based on mutual trust. Those facilities facing customers are willing to sharing the demand information of end items to others. This will reduce the information distortion and increase the responsiveness of the supply chain. The second rule is that just-in-time (JIT) delivering items among the facilities. The upstream facilities may hold inventory of products for economic of scale in manufacturing while the downstream facilities do not hold any inventory of materials.

We use the following notation:

dkitx

the quantity of item k delivered (by the facility which produces item k) in period t forproducing item i

rkitx

the quantity of item k received (by the facility which produces item i) in period t forproducing item i

We reformulate constraints (2) as the following three constraint sets according to the JIT delivering rule.

, 1 , ; k

dk t k kit kt kt

i Ny q x y d k K t T (2’)

, ; ;d rkit kit kx x k K i N t T (6)

, ; ;rkit ki it kx a q k K i N t T (7)

Constraints (2’) describe the mass-balance relationship for each item. Constraints (6) and (7) represent the JIT delivering policy where no component is sent before needed.

We define the dependent demand of item k in period t as:

is not an end item

is an end item

, .

,k

ki iti N

kt

kt

a dd kdd

d k . We add a dependent demand constraint for each item in each supplier-customer pair based on the sharing demand information rule. The total amount that a facility sends out by period tm has to be greater than or equal to the total dependent demand by period tm.

1 1, ;

m m

k

t tdkit kt m

t i N tx dd k K t T (8)

Although constraints (8) are redundant for the monolithic model, they are effective for the facility sub-models below.

2.3. Lagrangian decomposition

The Lagrangian Relaxation (LR) method solves complex optimization problems by “relaxing” or temporarily ignoring the coupling constraints and solving the problem as if they did not exist. The Lagrangian decomposition procedure, based on the dual optimization theory, generates a separable problem by integrating some coupling constraints into the objective function, through “penalty factors,” which are functions of the constraint violation [20].

The dependent relationships among the facilities in supply chains are represented only in constraints (6) after reformulating model MLCLSP. We then Lagrangian relax these linking constraints (6) between facilities, thus MLCLSP becomes facility separable. The following notations are adopted. e facility EE set of facilities in supply chain

reK set of items that facility e receives from

upstream facilities p

eK set of items that facility e produces deK set of items that facility e deliveries to

downstream facilities


efthe objective function of facility e in the facility sub models

eC the subset of constraints (2’), (3), (4), (5), (7) and (8) associated with facility e

Then the Lagrangian decomposed model of the problem is expressed as follows: (LD): { | ,e e

e},Min f C e E (9)

where

1

, ( ) ,

[ ( )

( )

pe

r p de k e e k

T

e k kt k ktt k K

rkit kit kit kit

k K i N K k K i N

f h y s

x ].dx

Constraints (8) represent that demand of items is bounded by its dependent demand. This guarantees the rationality of the facility sub-models. Because the linking constraints (6) are equations, the Lagrangian multiplier can be any real number. The Lagrangian dual problem is as follows: (LDD): { { | , }e e

e}Max Min f C e E (10)

3. Collaborative planning scheme

3.1. Coordination based on surrogate gradient algorithm

Lagrangian Relaxation provides a coordination mechanism for separable mathematical programming problems. In a nutshell, the key idea of the approach is decomposition and coordination, where decomposition is based on the separability of models, and coordination is based on the pricing concept of a market economy [21]. The standard method to solve the Lagrangian dual problem (LDD) is subgradient searching. However, subgradient searching method requires solving all the subproblems to obtain the subgradient direction, and this can be time consuming. It also usually results in oscillation and slow convergence [17], [21]. The surrogate subgradient method is a better alternative for solving Lagrangian dual problem. A proper surrogate direction to update the multipliers can be obtained without solving all the subproblems. In fact, only an approximate solution of one subproblem is needed to obtain a surrogate subgradient direction. Also, the surrogate subgradient directions are smooth and can avoid the notorious zigzagging difficulties associated with the subgradient method [21].

We introduce the definition of surrogate dual:

( , ) | , .e ee

LS x f C e E (11)

The corresponding surrogate subgradient is defined as ( ) , ; ; .r d

kit kit kg x x x k K i N t T (12) The key idea of the surrogate subgradient algorithm

is that when the surrogate dual, which can be obtained by approximate optimization, is less than the optimal dual LDD, the surrogate subgradient is in acute angle

with the direction towards *, therefore it is a proper

direction. Although the number of binary variables and

constraints of the problems decreases after decomposition of the monolithic model MLCLSP, solving each of the subproblems is still time consuming. It is appropriate to update the Lagragian multipliers by using the surrogate subgradient algorithm since it can obtain a proper direction without solving all the subproblems.

3.2. Restoring feasibility of the solution

Restoring feasibility of the solution is a key component when Lagrangian Relaxation method is adopted. Monolithic model such as MLCLSP is usually necessary to support this. However, this is not suitable when facilities are not willing to reveal all their local cost parameters and constraints. This study proposes a distributed feasibility restoring method based on the upstream planning mechanism, which does not need monolithic model and private information of the facilities.

All facilities cooperate to restore a global feasible solution using the upstream planning mechanism given some Lagrangian multiplier . In this procedure, d

kitx sthe in facility submodels are treated as parameters instead of decision variables. Their input values are equal to those of the corresponding r

kitx s decided by downstream facilities. We denote the objective facility e obtains in this procedure as '

ef . Beginning with the most downstream facility, the subproblem is solved and the decision result of r

kitx s defines supply requirements for its suppliers. These demands on the items are passed to suppliers, and the procedure continues in upstream direction. If some facility’s subproblem has no feasible solution, the procedure terminates with failure.

However, this is an attemptable procedure and not all the Lagrangian multipliers will result in global feasible solutions. So, we incorporate this feasibility restoring method into the procedure of updating multipliers in order to succeed in obtaining solutions.


3.3. Coordination structure

The coordination structure of the collaborative planning scheme in supply chains based on the Lagrangian decomposition, surrogate subgradient algorithm and the feasibility restoring method is shown in Fig. 1.

Coordinationcenter

Facility 2Facility m Facility 1rkitxr

kitx

'ef

ef rkitx d

kitxTs

'ef

'ef

ef rkitx d

kitxef rkitx d

kitx Ts Ts

Figure 1. Coordination structure of the collaborative planning scheme

The coordination center (acted by the leader or committee of strategic alliance in the supply chain) initializes Lagrangian multiplier and broadcasts it to the facilities. Since not all the subproblems are required to solve, coordination center send solving instruction (“Ts” in Fig. 1.) to part of the facilities based on some strategy. On receiving the multipliers, facilities may make two kinds of decisions using the facility submodel. Firstly, the facility which receives the solving instruction finds its facility-best solution and computes the objective ef , the demands r

kitx and

the deliveries dkitx given the multiplier and local

constraints. Secondly, all facilities cooperate to restore a global feasible solution using the feasibility restoring method mentioned above for the multiplier . If some facility’s subproblem has no feasible solution, the procedure terminates with failure. Then, facilities report their decision results for all the multipliers, including ef s, r

kitx s , dkitx s and '

ef s (or failure to restore a global feasible solution). The coordination center updates the Lagrangian multipliers based on the decision results reported by the facilities. The coordination centre sends new multiplier and solving instruction to the facilities and the facilities make decisions with this new multiplier. This procedure iterates until some satisfying solution has been obtained.

The collaborative planning scheme will result in savings of the total cost for supply chains. It usually bears savings to the upstream facilities but cost increases to the downstream facilities. The upstream facilities should contract to share the savings to the

downstream facilities in order to offer an incentive to them, such as “quantity commitment with bonus reward” contract [22]. This will also enhance partnerships and cooperation among the facilities in supply chains.

3.4. Coordination procedure

The algorithm (entitled LRSGSCP) of the collaborative planning scheme in supply chains is outlined as follows: Step 0. Initialize the parameters such as the upper

bound of the total cost of the supply chain TCup,the maximum number of iteration MaxK, the thresholds 1 and 2 . Coordination center

initializes the multiplier 0 and broadcasts it to all the facilities. The facilities do the step 4 and step 5, then we get the initial point 0x .

Step 1. The coordination center updates the multipliers. Given the current point ( , )k kx at the k th iteration: (a) Compute the surrogate subgradient kgusing (12). (b) Compute the stepsize. If all the facilities succeed in restoring the feasibility,

2'2( ) / ,k ke e

e es f f g

else2

2( ) / .k kup e

es TC f g

(c) The Lagrangian multiplier is updated according to

1k k ks g k

up

.Step 2. If '

ee

f TC , set .'up e

eTC f

Step 3. The coordination centre sends the multiplier to all the facilities, selects some of the facilities according to some strategy and sending the solving instruction to them.

Step 4. The facility finds its facility-best solution given the multiplier, computes ef , r

kitx s and dkitx s and

reports the results to the coordination centre. Step 5. Each facility computes '

ef for the multipliers according to the feasibility restoring procedure and reports the results to the coordination centre.

Step 6. Given 1k , ef s, rkitx s and d

kitx s, the

coordination center compares kx and 1kx ,which is obtained by updating kx using new


rkitx s and d

kitx s. If1 1 1( , ) ( ,k k k kLS x LS x ),

reset 1k kx x .Step 7. Check the stopping criteria. If one of the

criteria’s given by 1

1k kx x

12

k k

k MaxKis met, output the optimal solution and stop. Otherwise, set , go to Step 1. 1k k

4. Simulation experiments

We use a subset of standard test problems in [19]. Each problem involves 10 items and three facilities. The product structure and the assignment of items to facilities are depicted in Fig. 2. The 75 test problems are generated using three end-item demand structures, five combinations of time-between-orders (a factor for the determination of setup costs), and five capacity utilization profiles.

1

2

3

5

6

74

8

9

10

Facility 1Facility 2Facility 3

Figure 2. Problem structure The coordination procedure LRSGSCP was

implemented in MS Visual Basic. The facilities’ optimization sub-models were solved using the LINGO 8 standard mathematical programming solver. The control parameter setting of the procedure is maximum number of iteration MaxK=50.

Two benchmark solutions are for evaluating the performance of the collaborative planning scheme. First, the central, global optimal of model MLCLSP is considered. Second, pure upstream planning results are considered. The comparison between the pure upstream planning and the LRSGSCP is given in Table 1. The 75 problems are classified into five categories according to the characteristic of setup cost. The number of capacity infeasible test problems is listed first. Secondly, the percentage gaps of the mechanism to corresponding central planning solution are given for the remaining test instances.

Table 1. Test results Upstream Planning LRSGSCP

Capacityinfeasible

Gap to central

Capacityinfeasible

Gap to central

Characteristic of setup cost

# Av. (%) # Av. (%)Balanced, all are low. 0 0.76% 0 0.69%

Balanced, all are medium. 0 11.07% 0 2.61%

Balanced, all are high. 9 9.23% 0 1.78%

Downstream low, upstream high. 0 9.55% 0 1.41%

Downstream high, upstream low. 4 9.98% 0 0.80%

Total 13 7.84% 0 1.46% As can be seen in Table 1, quality of solutions

obtained by upstream planning depends heavily on the characteristic of the test problems. The solutions obtained by upstream planning are near optimal when setup cost of facilities are all low and demand information distorts very weakly in the process of transferring from downstream to upstream. However, quality of solutions obtained by upstream planning is poor in other situation. Capacity overruns occur in upstream planning in 13 out of 75 instances when setup cost of all facilities or downstream facilities are high and no feasible solutions can be obtained. Upstream solutions of the remaining test problems deviate on average 7.84% from central planning. In contrast, capacity infeasible solutions do not occur in the collaborative planning scheme, indicating that capacity shortages present in upstream planning are successfully corrected during the coordination procedure. The results obtained with the collaborative planning scheme and algorithm LRSGSCP deviate on average by a mere 1.46% from central planning.

As average deviation is aggregate measure, the results are examined more closely in Fig. 3 which shows frequency distribution of the number of test instances as a function of the gap to central planning. The bars indicate that in upstream planning gaps to central planning between 5% and 15% are most frequent. It can also be seen that a noteworthy number of test problems exhibits gaps of 0 to 3%, indicating that upstream planning can yield good solutions in some problem structures, as discussed above. On the other hand, arguably weak results with gaps of 15% and beyond are too obtained in some 10% of test instances. This situation changes drastically when using the collaborative planning scheme. Here, the majority of test results (more than 50%) have gaps to central planning of less than 1%. Almost 97% of test problems have a gap to the optimal solution of less than 5 %. None of the test problems have gaps beyond 10%. In summary, the collaborative planning scheme


proposed in this study outperforms upstream planning mechanism in terms of both performance and robustness.

100%

80%

60%

40%

20%

0%

<1% <2% <3% <5% <10% <15% <20%Gap vs. central planning solution [%]

Perc

enta

ge o

f tes

t ins

tanc

es [%

]

LRSGSCP

Upstream Planning

>20%

Figure 3. Cumulated frequency distribution of optimality gaps.

5. Conclusion

A collaborative planning framework based on the Lagrangian Relaxation method for collaborative planning of independent partners with cooperative behaviors in supply chains is presented in this paper. We reformulate the monolithic Multi-Level, multi-item Capacitated Lot Sizing Problem for supply chains and Lagrangian decompose the model into facility-separable sub-problems. The coordination centre updates Lagrangian multipliers to coordination the decentralized decisions of the facilities. Surrogate gradient algorithm is used in the coordination procedure to update these multipliers for overcoming the oscillation and slow convergence caused by standard subgradient searching method. The collaborative planning scheme aims at reducing total supply chain costs. Since this brings savings to the upstream facilities but comes at a cost increase to the downstream facilities, payment terms among the partners need to be adapted accordingly in order to ensure that all the facilities realize benefits and the collaborative planning scheme can be successfully implemented. Simulation results show that the proposed scheme yield promising results. The results for all the problems deviate on average by a mere 1.46% from the central optimal. The proposed mechanism coordinates and optimizes production planning of independent partners with cooperative behavior in the supply chain without intruding their decision authorities and private information.

Also, the proposed framework and methodology can be extended to the problem of activity coordination in the make-to-order supply chains. Research about this is currently under way. We are developing a multi-

agent system implementing this coordination scheme, which will aid production planning in a supply chain composed of an automobile company and its main suppliers.

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