International Journal of Scientific & Engineering Research Volume 2, Issue 5, May-2011 1

ISSN 2229-5518

Receding Horizon Control on Large Scale

Supply Chain

Mohammad Miranbeigi, Aliakbar Jalali

Abstract— Supply chain management system is a network of facilities and distribution entities: suppliers, manufacturers, distributors, retailers. The control system aims at operating the supply chain at the optimal point despite the influence of demand changes. In this paper, a centralized constrained receding horizon controller applying to a supply chain management system consist of two product, one plant, two distribution centers and three retailers.

Index Terms— supply chain, supply chain management system, suppliers, manufacturers, distributors, retailers , control system, demand, receding horizon controller.

1 INTRODUCTION

—————————— • ——————————
HE network of suppliers, manufacturers, distributors and retailers constitutes a supply chain management system. Between interconnected entities, there are
two types of process flows: information flows, e.g., an order requesting goods, and material flows, i.e., the actual shipment of goods. Key elements to an efficient supply chain are accurate pinpointing of process flows and tim- ing of supply needs at each entity, both of which enable entities to request items as they are needed, thereby re- ducing safety stock levels to free space and capital. The operational planning and direct control of the network can in principle be addressed by a variety of methods, including deterministic analytical models and stochastic analytical models, and simulation models, coupled with the desired optimization objectives and network perfor- mance measures [1].
The significance of the basic idea implicit in the reced- ing horizon control (RHC) or RHC has been recognized a long time ago in the operations management literature as a tractable scheme for solving stochastic multi period op- timization problems, such as production planning and supply chain management, under the term receding hori- zon [2]. In a recent paper [3], a RHC strategy was em- ployed for the optimization of production/distribution systems, including a simplified scheduling model for the manufacturing function. The suggested control strategy considers only deterministic type of demand, which re- duces the need for an inventory control mechanism [4,5].
For the purposes of our study and the time scales of in- terest, a discrete time difference model is developed [6].

————————————————

Mohammad Miranbeigi is currently pursuing Phd degree program in control engineering in University of Tehran, Iran, E-mail: m.miran@ut.ac.ir

A. A. Jalali, is associate professor in the Department of Electrical Engi-

neering, Iran University of Science and Technology, Narmak ,Tehran, Iran, E-mail : drjalali@iust.ac.ir.

The model is applicable to multi echelon supply chain networks of arbitrary structure. To treat process uncer- tainty within the deterministic supply chain network model, a RHC approach is suggested [7,8].
Typically, RHC is implemented in a centralized fashion [9]. The algorithm uses a receding horizon, to allow the incorporation of past and present control actions to future predictions [10,11,12,13].
In this paper, a centralized receding horizon controller applying to a supply chain management system consist of one plant (supplier), two distribution centers and three retailers.

2 DISCRETE TIME DIFFERENCE MODEL

In this work, a discrete time difference model is devel- oped[4]. The model is applicable to multi echelon supply chain networks of arbitrary structure, that DP denote the set of desired products in the supply Chain and these can be manufactured at plants, P, by utilizing various re- sources, RS. The manufacturing function considers inde- pendent production lines for the distributed products. The products are subsequently transported to and stored at warehouses, W. Products from warehouses are trans- ported upon customer demand, either to distribution cen- ters, D, or directly to retailers, R. Retailers receive time varying orders from different customers for different products. Satisfaction of customer demand is the primary target in the supply chain management mechanism. Un- satisfied demand is recorded as backorders for the next time period. A discrete time difference model is used for description of the supply chain network dynamics. It is assumed that decisions are taken within equally spaced time periods (e.g. hours, days, or weeks). The duration of the base time period depends on the dynamic characteris- tics of the network. As a result, dynamics of higher fre- quency than that of the selected time scale are considered negligible and completely attenuated by the network

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International Journal of Scientific & Engineering Research Volume 2, Issue 5, May-2011 2

ISSN 2229-5518

[4,14].
Plants P, warehouses W, distribution centers D, and re-
tailers R constitute the nodes of the system. For each
node, k, there is a set of upstream nodes and a set of

downstream nodes, indexed by (k ', k '') . Upstream nodes can supply node k and downstream nodes can be sup-

not require a separate balance for customer orders at nodes other than the final retailer nodes [4,15].

3 RHC DESIGNATION

RHC originated in the late seventies and has devel-

plied by k. All valid (k ', k )

and/or

(k , k ' ) pairs constitute

oped considerably since then. The term RHC does not
permissible routes within the network. All variables in the supply chain network (e.g. inventory, transportation loads) valid for bulk commodities and products. For unit products, continuous variables can still be utilized, with the addition of a post processing rounding step to identi- fy neighbouring integer solutions. This approach, though clearly not formally optimal, may be necessary to retain computational tractability in systems of industrial relev- ance.
A product balance around any network node involves the inventory level in the node at time instances t and t -
1, as well as the total inflow of products from upstream
nodes and total outflow to downstream nodes. The fol-
lowing balance equation is valid for nodes that are either
warehouses or distribution centers:

yi,k (t) = yi,k (t -1) + xi,k',k (t - Lk ',k ) - xi,k,k'' (t),

designate a specific control strategy but rather an ample
range of control methods which make explicit use of a
model of the process to obtain the control signal by mi-
nimizing an objective function. The ideas, appearing in
greater or lesser degree in the predictive control family,
are basically the explicit use of a model to predict the
process output at future time instants (horizon), the calcu-
lation of a control sequence minimizing an objective func- tion and the use of a receding strategy, so that at each instant the horizon is displaced towards the future, which involves the application of the first control signal of the
sequence calculated at each step. The success of RHC is due to the fact that it is perhaps the most general way of posing the control problem in the time domain. The use a finite horizon strategy allows the explicit handling of process and operational constraints by the RHC. The con- trol system aims at operating the supply chain at the op- timal point despite the influence of demand changes

\7'k E{W, D},

k'

t ET,

k '

i E DP

(1)

[12,13].
The control system is required to possess built in ca- pabilities to recognize the optimal operating policy
where

yi, k is the inventory of product i stored in node

through meaningful and descriptive cost performance

k; xi,k ',k denotes the amount of the i-th product trans-

indicators and mechanisms to successfully alleviate the

ported through route (k ', k ) ;

Lk ',k denotes the transporta-

detrimental effects of demand uncertainty and variability.

tion lag (delay time) for route (k ', k ) , i.e. the required time

periods for the transfer of material from the supplying node to the current node. The transportation lag is as- sumed to be an integer multiple of the base time period.
For retailer nodes, the inventory balance is slightly
modified to account for the actual delivery of the i-th product attained, denoted by di,k (t) .

yi,k (t ) = yi, k (t - 1) + xi, k ' ,k (t - Lk ' ,k ) - di,k (t ),

The main objectives of the control strategy for the supply chain network can be summarized as follows: (i) maxim- ize customer satisfaction, and (ii) minimize supply chain operating costs.
The first target can be attained by the minimization of back orders (i.e. unsatisfied demand) over a time period because unsatisfied demand would have a strong impact on company reputation and subsequently on future de- mand and total revenues. The second goal can be achieved by the minimization of the operating costs that

\7'k E {R},

k '

t ET ,

i E DP.

(2)

include transportation and inventory costs that can be
further divided into storage costs and inventory assets in
the supply chain network. Based on the fact that past and
The amount of unsatisfied demand is recorded as back- orders for each product and time period. Hence, the bal- ance equation for back orders takes the following form:

BOi,k (t) = BOi,k (t -1) + Ri,k (t) - di,k (t) - LOi,k (t),

present control actions affect the future response of the system, a receding time horizon is selected. Over the spe- cified time horizon the future behavior of the supply chain is predicted using the described difference model (Eqs. (1)–(3)).
In this model, the state variables are the product in-

\7'k E{R},

t ET,

i E DP.

(3)

ventory levels at the storage nodes, y, and the back or-
where Ri,k denotes the demand for the i-th product at the
ders, BO, at the order receiving nodes. The manipulated

k-th retailer node and time period t.

LOi,k denotes the

(control or decision) variables are the product quantities
amount of cancelled back orders (lost orders) because the network failed to satisfy them within a reasonable time limit. Lost orders are usually expressed as a percentage of unsatisfied demand at time t. Note that the model does
transferred through the network’s permissible routes, x,
and the delivered amounts to customers, d. Finally, the
product back orders, BO, are also matched to the output
variables. The inventory target levels (e.g. inventory set-

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International Journal of Scientific & Engineering Research Volume 2, Issue 5, May-2011 3

ISSN 2229-5518

points) are time invariant parameters. The control actions that minimize a performance index associated with the outlined control objectives are then calculated over the

RHC process complete (Fig. 1).
receding time horizon. At each time period the first con-

Order

Measurable

Measurable

trol action in the calculated sequence is implemented.
The effect of unmeasured demand disturbances and model mismatch is computed through comparison of the actual current demand value and the prediction from a stochastic disturbance model for the demand variability.
The difference that describes the overall demand uncer- tainty and system variability is fed back into the RHC scheme at each time period facilitating the corrective ac- tion that is required.
The centralized mathematical formulation of the per- formance index considering simultaneously back orders, transportation and inventory costs takes the following form[4]:

J total =

P W D R

RHC

Fig. 1 Centralized RHC on supply chain management system

4 SIMULATIONS

A three echelon supply chain system is used in the si- mulated examples. The supply chain network consists of

t + P

t

k E{W , D , R } iEDP

y ,i, k

( y i, k

(t ) - y

s ,i ,k

(t )) 2 }

two product, one production nodes, two distribution cen-
ters, and three retailer nodes.

t + M

+

t

t + P

kE{W , D , R } iEDP

w

x, i,k ' ,k

( x i, k ' ,k

(t )) 2 }

(4)

All possible connections between immediately succes-
sive echelons are permitted. Two product is being distri-
buted through the network. Inventory setpoints, maxi-

+

t kE{R } iEDP

.

BO ,i, k

(BO

i, k

(t ))2 }

mum storage capacities at every node, and transportation cost data for each supplying route are reported in Table 1.
The performance index, J, in compliance with the out- lined control objectives consists of four quadratic terms.
A prediction horizon of 20 time periods and a control
horizon of 10 time periods were selected and was consi-
Two terms account for inventory and transportation costs
dered

LO = 0

for every times. So each delay was re-
throughout the supply chain over the specified prediction and control horizons (P , M). A term penalizes back or- ders for all products at all order receiving nodes (e.g. re- tailers) over the receding horizon P.
placed by its 4th order Pade approximation (after system
model transformed to continuous time model and then
returned to discrete time model).

Table 1. Supply chain data

The weighting factors,

w y,i ,k , reflect the inventory sto-

rage costs and inventory assets per unit product,

wx,i,k',k ,

account for the transportation cost per unit product for

route (k ', k ) .Weights

wBO,i,k

correspond to the penalty
imposed on unsatisfied demand and are estimated based
on the impact service level has on the company reputa-
tion and future demand. Factors

w y,i ,k ,

wx,i,k',k and

wBO,i,k

are cost related that can be estimated with a relatively
good accuracy.
The weighting factors in cost function also reflect the
relative importance between the controlled (back orders
and inventories) and manipulated (transported products)
variables. Note that the performance index of cost func-
tion reflects the implicit assumption of a constant profit
margin for each product or product family. As a result,
production costs and revenues are not included in the
index.
In this centralized implementation, RHC will opti-
mized for whole policy and then will sent downstream optimal inputs to upstream joint nodes to those nodes which it is coupled, as measurable disturbances.
Each node completely by a centralized RHC optimizes for whole policy. At each time period, the first control action in the calculated sequence is implemented until
The simulated scenarios lasted for 50 time periods with Eq. 5. Response to constant demand is presented in Fig. 2. In this part, centralized RHC method is applied to the supply chain network with a constant customer de-

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International Journal of Scientific & Engineering Research Volume 2, Issue 5, May-2011 4

ISSN 2229-5518

mand that is seeing in figures 2 and 3.

200

100

0

40

20


0 5 10 15 20 25 30 35 40 45 50

T y1, D1

I y2, D1

I y

T D1

ID2

I D

T x1, P, D1

I x1, D1, R1

I x1, D1, R 2

I

I x2 , P , D1

I x

I 2 , D1, R1

I x2, D1, R 2

I x

T I1

I I 2

I I 3

I

I I 4

I I 5

I

I I 6

I

0

100

50

0

200

100

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

1, D 2 3

I

I 1, P, D 2

I I 7

0

0 5 10 15 20 25 30 35 40 45 50

I y2 , D 2

I y

I 1, R1

I y2, R1

I y

I 1, R 2

I y2 , R 2

Y = I

I 1, R 3

I y

ID4

I R1

I I R2

I R3

I

= I R4

I R5

I

I R

I x1, D 2 , R1

I x

I 1, D 2 , R 3

I x2, P, D 2

I

I x2, D 2, R1

I x

, U = I 2, D 2, R3

I d1, R1

I d

I I 8

I I 9

I

II10

I I I11

II12

= I

I d1

I

100

50

0

100

50

0

100

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

2, R3 6

I I

2, R1 d 2

50

I BO1, R1

I BO

I 2 , R1

I BO1, R 2

IBO

I 2, R 2

I BO1, R3

L BO2 , R 3 J16X1

I B1

I B2

I

I B3

I B4

I

I B5

L B6 J16 X1

I d1, R 2

I d

I 2, R 2

I d1, R3

I

I d2 , R 3

I R1, R1

I

I R2 , R1

I R

I 1, R 2

I R2, R 2

I R

I 1, R3

I d 3

I d 4

I

I d 5

I

I d 6

I r1

I

I r 2

I r3

I

I r 4

I r5

I

(5)

0

100

50

0

50

0

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

Time (days)

IL R2 , R 3 J24 X1

IL r 6 J24 X1

Fig. 3 RHC inputs of supply chain management system

400

200

0

400

200

0

400

200

0

400

200

0

200

100

0

200

100

0

200

0

-200

200

100

0

100

50

0

100

50

0

100

0

-100

100

0

-100

100

0

-100

100

0

-100

100

0

-100

100

0

-100

Plant Outputs
















0 5 10 15 20 25 30 35 40 45 50

Time (days)

CONCLUSION

The large majority of successful RHC applications ad- dress the case of multivariable control in the presence of constraints, motivating its extensive distribution for ap- plications where traditional control usually comes close to its limits. The success of RHC is due to the fact that it is perhaps the most general way of posing the control prob- lem in the time domain. The use a finite horizon strategy allows the explicit handling of process and operational constraints by the RHC. Typically, RHC is implemented in a centralized fashion. In this paper, a centralized re- ceding horizon controller applying to a supply chain management system consist of one plant (supplier), two distribution centers and three retailers.

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