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Computers
&Chemical
Engineering
Computers and Chemical Engineering 24 (2000) 329-335
www.elsevier.com/locate/compchemeng

A model predictive framework for planning and scheduling
problems: a case study of consumer goods supply chain
S. Bose, J.F. Pekny *
School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-1283, USA

Abstract

Model Predictive Control is a well established technique for the control of processes and plants. We present a similar concept
for planning and scheduling problems. There have mainly been two approaches to solve the planning and scheduling problems.
The first approach is to model the planning and scheduling as one monolithic problem and solve it for the entire horizon. Needless
to say, this approach requires an extensive computational effort and becomes impossible to solve in the case of large-scale
scheduling problems. The other approach is to hierarchically-decompose the problem into a planning level problem and a
scheduling level problem. This approach leads to tractable problems. Neither of these approaches provide the framework for
incorporating uncertainties in the processing time of batches, or random equipment breakdowns, or demand uncertainties in the
future. Furthermore, these approaches only provide ‘one snapshot’ of the planning problem and not a ‘walk through the timeline’.
Model predictive planning and scheduling provides a framework for studying dynamics. Model predictive planning and scheduling
requires a forecasting model and an optimization model. Both these models work in tandem in a simulation environment that
incorporates uncertainty. The similarity with the model predictive approach which is widely used in process-control is that in each
period, the forecasting model calculates the target inventory (controlled variable) in the future periods. These inventory levels
ensure desired customer service level while minimizing average inventory. The scheduling model then tries to achieve these target
inventory levels in the future periods by scheduling tasks (manipulated variables). 0 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Supply chain; Consumer; Planning and scheduling

1. Introduction instead depend critically on the relationships and inter-
dependencies among different organizational units.
Operation of a supply chain for fast moving con- This, in turn, translates into establishing efficient mate-
sumer goods is a challenging and difficult task. Fast rial and information flows across the supply chain.
moving consumer goods are characterized by promo- The ongoing competitiveness of an organization is
tional demands, which take a toll on the target cus- linked with the dynamics of the supply chain(s) in
tomer service level as well as result in higher inventory which it participates, and recognition of this fact is
costs. The operation of a supply chain, in which the leading to considerable change in the way organizations
demands are known accurately, is tractable. So a interact with their supply chain partners. The interac-
difficulty in controlling a supply chain for fast moving tion among the parts of supply chain can be captured
consumer goods arises from the highly uncertain nature
by studying the coordination structure of the supply
of the promotional part in the demand signal. To cope
chain.
with the challenges of improving customer service level
For the purposes of this study, we shall remain
and reducing inventory costs, organizations cannot rely
focused on the coordination between the production
solely on isolated changes to specific parts of the supply
units and demand units of the supply chain. The key
chain, like suppliers or distributors or retailers, but
issues that will be addressed are:
1. What is the best organizational structure for the
* Corresponding author. Tel.: + l-317-4947901; fax: + 1-317-
4940805.
effective coordination of activities between the sup-
E-mail addresses.. shantanu@ecn.purdue.edu (S. Bose), ply and demand units (centralized control, decen-
pekny@ecn.purdue.edu (J.F. Pekny). tralized control or an intermediate form of control)?

0098-1354/00/$- see front matter 0 2000 Elsevier Science Ltd. All rights reserved.
PII: SOO98-1354(00)00469-5

330 S. Bose, J.F. Pekny /Computers and Chemical Engineering 24 (2000) 329-335

2. What are the appropriate ways of measuring fore- ulated variables). This is repeated till the end of the
cast accuracy, specifically, the accuracy of forecast- planning period. Refer to Fig. 1.
ing promotional demand?

To address the issues given above, one must capture 3. Coordination structure of the supply chain:
the dynamics of the supply chain by modeling each definitions
element of the supply chain (the supply unit and the
individual markets) and the interaction among these The coordination structure of a supply chain is
elements. The framework chosen for this study captures jointly determined by its decision rights structure
the forecasting, optimization and simulation aspects of (whether the plant or the distributors decide what to
the supply chain. One must also classify the coordina- produce and when) and its information structure (local
tion structure by some quantifiable parameters. The knowledge of promotions, distributor’s stock visibility,
rest of the paper explains these in detail and also etc.). The coordination structures can be classified in
presents the results obtained from the study so far. the following categories.
Industrial data has been used to obtain results and
draw conclusions to be sure that realistic assumptions 3.1. Centralized control
have been made about supply chain behavior.
The quantities to be supplied to each market are
centrally determined. The forecasts for the demands are
2. The model predictive approach based on past sales and recorded data. The forecasts of
the promotional part of the demand incorporate infor-
Here we present a model predictive approach for mation that is centrally available, and do not include
solving this problem. Model predictive planning and local knowledge of promotions. Thus, in a centralized
scheduling requires a forecasting model and an opti- control, even though most of the promotional events
mization model. Both these models work in tandem in are known, the actual quantity of the promotional
a simulation environment that incorporates uncertainty. demand and the exact week of occurrence have a lot of
The similarity with the model predictive approach uncertainty.
which is widely used in process-control is that in each
period, the forecasting model calculates the target in- 3.2. Decentralized control
ventory (controlled variable) in the future periods.
These inventory levels ensure desired customer service The individual markets place orders with the supply
level while minimizing average inventory. The schedul- unit based on local promotional knowledge. Each indi-
ing model then tries to achieve these target inventory vidual market is unaware of the capacity constraints in
levels in the future periods by scheduling tasks (manip- the factory, as well as, demands placed by the other

Output Vorlrble flaqet Inventory)

mvu?

S. Bose, J.F. Pekny / Computers and Chemical Engineering 24 (2000) 329-335 331

markets. Thus, even though the forecast quantity l Uncertainty in the actual quantity of the promo-
of promotional demands is relatively accurate, the tional demand.
markets do not provide enough lead-time for l Uncertainty in the actual week of occurrence of the
SUPPlY. promotional demand.

3.3. Distributed Control For various values of the above, different forecasts
can be generated. With each forecast, a parameter c1
The quantities are determined using both central and can be associated which is defined as the fraction of the
local promotional knowledge. There is information total promotional demand that is expected (known).
sharing and negotiation among the supply units and
Known Prom. Demand
individual markets.
’ = Total Prom Demand
The parameter CImeasures the overall efficiency of the
4. Forecasting-optimization-simulation framework business processes in the supply chain.

This framework is developed to study the supply
chain dynamics under uncertainty. Each module of the 5. De-convolution of the demand signal
framework is explained below. Please refer to Fig. 2 for
the schematic diagram of ‘forecasting-optimization-sim- The following method is used to de-convolute the
ulation architecture’ which shows the interaction demand signal into a base demand and promotional
among each module of the framework. demand. Since it is not known a priori what the promo-
tional demand is and in which of the weeks it appears
4.1. Forecasting module we have to use an iterative (trial and error) method.
This involves estimating a base demand. The estimation
The forecasting module de-convolutes the demand is done by assuming the base demand to be of the
data into a base demand and a promotional demand. following functional form:Base demand = A + Bt +
The de-convolution technique is explained in the next C sin(w x t)where A, B, C and w are the estimated
sub-section section. The promotional demand consists parameters and t denotes the time (measured in week in
of an expected promotional demand (with some uncer- this case).
tainty) and an unexpected promotional demand. As- Promotional demand is defined as any demand that
suming a reasonable market growth, a weekly demand is in excess of the base demand. Initially, all the sales
forecast is generated. This also takes into account the data is assumed to be base demand (i.e. no promotional
coordination structure of the supply chain. The coordi- demand). As a first step of the iteration, the base
nation structure is captured by the following para- demand parameters are fitted to the total sales. The sale
meters: in any given week is then compared with the predicted
l How far ahead are these promotional demands base demand. If the actual sales is in excess of the
known to the factory. predicted base demand by a certain value (specified by


Table 1
Range of values for control parameters

Parameters Regimes

Low Medium High

LT: lead-time for satisfying promotional demand 4 weeks 8 weeks 12 weeks
Q: accuracy in the quantity of promotional demands (denoted by mean Qm, and Mean = 1.0; Mean = 1.O; Mean = 1.O;
S.D. Qd) S.D. = 0.25 S.D. = 0.25 SD. = 0
X: accuracy in the exact week of occurrence (denoted by mean Xm, and S.D. Xd) +4 weeks +2 weeks *O weeks

the user as ‘Promotional Criteria’), then that week is 5.2. Simulation module
marked as a promotional week. This is done for all the
weeks. Now the base demand estimated again to fit the The simulation module simulates production in the
predicted base demand to on& those weeks, which are factory for one week, at a time during which the
not marked as promotional. The sale for each week is schedule is frozen. The inventory levels at the market
compared with the predicted base demand and the warehouses are updated accounting for the actual de-
weeks, which meet the promotional criterion, are mands and allowing time for transportation of stock
marked and added to the previous set of promotional keeping units.
weeks. This iteration is performed until no more weeks Each of the three modules is invoked for each period
can be marked as promotional weeks. of the plan.

5.1. Optimization module
6. Design of simulation runs
The optimization module solves a Mixed Integer
Linear Program (MILP) to decide on a production In order to study the supply chain dynamics, it is
schedule which ensures a target customer service level important to perform simulation runs, which are spe-
while minimizing inventory costs at the same time. The cifically designed to address the key issues. The simula-
optimization considers a time horizon over which it tion runs should bring out the differences in the various
solves. The horizon determines how many weeks of possible organizational and coordination structures of
demand information are available to the optimizer. the supply chain. Each run must be classified as either
This production schedule is frozen for a single period (a Centralized Control or Decentralized Control or Dis-
week, in this case). Since detailed scheduling is a hard tributed Control.
problem to solve and the difficulty increases exponen-
tially with the problem size, the optimization problem 6.1. Control parameters
does detailed scheduling only for the first week and a
rough cut capacity planning for the rest of the weeks in There are several control parameters that affect the
the horizon. Since only the first week is implemented, performance of the supply chain. They are:
the rough cut capacity planning does not pose any 1. How far ahead in the future are the promotional
unfeasibility. The optimization module enforces the demands known (i.e. how much lead-time is avail-
minimum-run-length- constraint for the tasks. The opti- able to meet the promotional demands).
mization module also ensures that the production fre- 2. The accuracy in the actual quantity of the promo-
quency of the stock keeping units does not exceed the tional demands.
stipulated production frequency as practiced by the 3. The accuracy in the actual week of occurrence of the
plant. promotions.

Table 2
Relation between control parameters and coordination structure

Parameters Type of coordination structure

Centralized De-centralized Distributed

LT: lead-time for satisfying promotional demand HIGH LOW MEDIUM
Q: accuracy in the quantity of promotional demands LOW HIGH MEDIUM
X: accuracy in the exact week of occurrence LOW HIGH MEDIUM


4.2. Centralized

All the promotions originate from the supply unit,
hence the information about the promotions is avail-
able to the supply unit (which decides the demand
forecasts in case of centralized coordination) well in
advance. This means that lead-time is also high for
centralized control. The uncertainty lies in knowing the
actual effect of these promotions that are offered by the
plant. Since knowledge from the local markets is not
available to the factory, the actual orders requested by
the markets is substantially different from what the
factory forecasts. Hence, the ‘low’ accuracy in the exact
quantity as well as the exact occurrence of the promo-
tional demands. Furthermore, the individual markets
often create unnecessary noise in the supply chain, by
excessive hoarding or stock clearance, which reduce the
accuracy of promotional demands even more. So, over-
all the centralized coordination structure promises
Fig. 3. Sensitivity of overall customer service level on the exact week
of occurrence of promotional demands. longer lead-times but loses out in the accuracy of
forecasting promotional demands.

6.3. De-centralized

In case of de-centralized control, the forecasting is
primarily based on demand information provided by
Theoretically, these parameters can vary over a con- the individual markets. The result is high accuracy in
tinuous range of values. To perform any meaningful the forecast quantity of the promotional demand as
simulations, we must first set some bounds on these well as the exact week of occurrence. Since the markets
parameters and then divide the range between the can see their own stock levels, they place fairly accurate
bounds into regimes, such as low, medium and high. orders based on their current stock levels and local
Depending upon the coordination in the supply chain, promotional knowledge. But, the down side is that they
the parameter values will lie in one of the regimes. do not give enough lead-time for production. When the
Table 1 defines the regimes for the parameters. lead-time given is too small, the factory may not be
The bounds on the parameters must be substantiated able to accommodate the demand because of capacity
by data and observations. From the forecast data avail- limitations. So, overall the de-centralized coordination
able, only the accuracy in the quantity of the promo- structure promises high accuracy in predicting promo-
tional demands could be verified. The corresponding tional demands but settles for lower lead-times.
values from the forecast data were calculated to be 0.98
(mean) and 0.45 (S.D.). The bounds for the other two 6.4. Distributed
parameters (LT and X) are based on reasonable esti-
mates. In the case of ‘lead-time’, the upper bound of 12 Distributed control is a mix of the centralized and
weeks is justified because it is highly unlikely to sched- de-centralized control structures. Obviously, there is
ule for more than 12 weeks at a time. Hence, any trade-off between the centralized and de-centralized
demands beyond 12 weeks will not be seen by the types of coordination. The distributed control can ei-
scheduling-model anyway. It was also observed during ther be highly efficient and have the benefits of both
the simulation runs that if the lead-time is reduced to centralized and de-centralized control, or it can repre-
less than 4 weeks, or, if the inaccuracy in the exact sent a highly unsynchronized supply chain. The Ideal
week of occurrence of promotional demand is more Coordination Structure would be achieved when the
than + / -4 weeks, the performance of the supply very best of both worlds (centralized and de-central-
chain deteriorates considerably. Hence, these values ized) is integrated.
were assumed to be bounds. The medium regime was
assumed to be the middle value of the range, since there 6.5. Performance measure
is no data to suggest otherwise.
The entries in Table 2 are explained below for each Each simulation run is measured by an overall cus-
coordination structure. tomer service level achieved and the overall inventory


level. Obviously, increasing the inventory safety stock- between 0 and 1. The u values were chosen to be
levels result in higher customer service level. The aim 0.2, 0.4, 0.6, 0.8 and 1.0, a total of five values. For
is to attain the desired customer service level of 99.7% each value of c1 the 27 combinations calculated above
with the minimum possible inventory. were simulated. That is a total of 27 times 5, i.e. 135
combinations. To determine the best possible customer
6.6. Computational effort service level in each run, six trials were made with
a different inventory level in each trial. That results
The number of simulations increases combinatori- in a total of 135 times 6, i.e. 810 combinations.
ally with the number of parameter and combinations Each run takes about 25 min for simulating 52 weeks
thereof. There are three control parameters (LT, Q on a Pentium II (400 MHz) processor. The total
and x), with three values each, hence, a total of 33 simulation time for all these runs is 20 250 min, which
times 3, i.e. 27 combinations. The parameter CI,which is about 14 days worth of computational time. Each
was introduced as a lumped parameter for measuring run is dependent on a random number seed, hence
the efficiency of business processes, can take any value for the same values of the parameters, several runs
should be simulated using a different random seed for
each run. It was found that the results of the runs do
not vary too much when a different random number
seed is used. This is because any peculiarities of a
random number are somewhat averaged over 52
weeks.

7. Results

There are several interesting results that can be ex-
tracted from the simulation runs. Although there are
many results and figures obtained from this study,
only the important ones are being discussed below.

7.1. Sensitivity analysis of control parameters

Sensitivity of overall customer service level on the
exact week of occurrence of promotional demands.
This is obtained by varying the parameter X and
Fig. 4. Sensitivity of overall customer service level on the exact keeping all the other parameters constant. Lead-time
quantity of promotional demands. is fixed at 8 weeks and CIis set to 0.8. Refer to Fig.
3.
Sensitivity of overall customer service level on the
exact quantity of promotional demands. This is ob-
tained by varying the parameter Q and keeping all
the other parameters constant. Lead-time is fixed at
8 weeks and a is set to 0.8. Refer to Fig. 4.
Sensitivity of overall customer service level on lead-
time. This is obtained by varying lead-time and
keeping all other parameters constant. The uncer-
tainties in the exact week of occurrence and the exact
quantity of promotional demands are both set to 0,
while CIis fixed at 0.8. Refer to Fig. 5.
Sensitivity of overall customer service level on tl.
This is obtained by varying the tl and keeping all the
other parameters constant. The other parameters are
I3
kept constant so that they represent a particular
1rlhW
coordination structure. Fig. 6 shows the case of
Fig. 5. Sensitivity of overall customer service level on lead-time. de-centralized coordination structure.


By comparing results for same tl and same inventory
level, it is observed that medium uncertainties in the
occurrence and quantity of promotional demands
have about the same effect. But, high uncertainty in
occurrence is more harmful to the performance than
high uncertainty in quantity of promotional de-
mands.
Although the sensitivity of performance on lead-
time cannot be compared with any other parameter,
it is observed that there is a steady increase in
customer service level as the lead-time increases.
This suggests more simulation runs with still higher
lead-times to observe the point at which the gain in
service level tapers off.
The dependence of customer service level on inven-
tory level in the system is very high. This result was
Fig. 6. Sensitivity of overall customer service level on CI. rather intuitive. The interesting fact, however, is the
gradient of the dependence. The gradient is very
high at lower inventory levels and changes to almost
zero gradient at higher inventory levels. This sug-
gests that keeping the average inventory level at
about 120- 150 tons gives the best performance.
This target inventory level is about 20% higher than
the current practice of inventory coverage. It is also
observed that the service level drops slightly when
target inventory levels are very high. This is not
immediately intuitive. The reason is that pushing the
plant to maintain higher inventory levels unneces-
sary bottlenecks in many of the weeks during the
52-week simulation, Hence, some products, which
have an immediate demand, are not produced in
Fig. 7. Effect of various coordination structures. those weeks, since the plant is busy producing other
products to maintain their high target inventories.
This may seem more like a problem with the
7.2. Effect of coordination structure on overall
scheduling, but most scheduling models minimize
customer service level
the deviation of inventory levels from the target
inventory levels.
This is the most important set of results derived from The dependence of customer service level on CI is
this study. They are obtained by varying the coordi- also very strong. A is a lumped parameter, which
nation structure and keeping CI constant. As de- reflects the efficiency of business processes in the
scribed before, the coordination structures fix the supply chain. A can be improved by better commu-
regimes of the control parameters (LT, Q and x). nications between the marketing and production
Fig. 7 shows the effect of various coordination struc- units, and, better knowledge of local information at
tures for c( = 0.8. the markets.
Perhaps the most important result can be drawn
from Fig. 5. This figure clearly demarcates the re-
Conclusions gions where different control structures are recom-
mended. For example, if we assume that the current
One of the objectives of this study was to determine status of the supply chain is CI 0.8 and average
=
which of the parameters is important. This can be inventory level = 115-125 tons, then we can con-
deduced from the sensitivity analysis shown in the clude that the centralized coordination structure is
results. We summarize the conclusions below. recommended.

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