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Proceedings of the 2012 Industrial and Systems Engineering
Research Conference
G. Lim and J.W. Herrmann, eds.
Robust Integrated Production Planning and Order Acceptance
Abstract ID: 197
T. Aouam
School of Business Administration, Al Akhawayn University,
Ifrane, Morocco.
Corresponding Author
[email protected]
N. Brahimi
Department of Industrial Engineering and Management, University
of Sharjah, United Arab
Emirates
[email protected]
Abstract
The aim of this work is to formulate a model that integrates
production planning and order acceptance decisions
while taking into account demand uncertainty and capturing the
effects of congestion. Orders are classified into
classes based on their marginal revenue and their level of
variability in order quantity. The proposed integrated
model provides the flexibility to decide on the fraction of
demand to be satisfied from each customer class giving the
planner the choice of selecting among the highly profitable yet
risky orders or less profitable but possibly more
stable orders. Furthermore, when the production stage exceeds a
critical utilization level, it suffers the consequences
of congestion via elongated lead-times which results in
backorders and erodes the firm's revenue. Through order
acceptance decisions, the planner can maintain a reasonable
level of utilization and avoid increasing delays. A robust
optimization approach is adapted to model demand uncertainty and
non-linear clearing functions characterize the
relationship between throughput and workload to reflect the
effects of congestion on production lead times.
Illustrative simulation and numerical experiments show the
integrated model characteristics, the effects of congestion
and variability, and the value of integrating production
planning and order acceptance decisions.
Keywords Production Planning; Order Acceptance; Clearing
Functions; Congestion; Robust Optimization;
1. Introduction Models of production and inventory systems have
been developed since the early days of the Operations Research
and Management Science field. A major concern in the area has
been to formulate models that can be solved
efficiently, yet these models should not be based on over
simplifying assumptions. Classical production planning
models determine the minimum cost or maximum profit production
plans in order to meet pre-specified demands. In
classical production planning models, three common simplistic
assumptions are usually made: (1) The demand is
deterministic and known in advance. (2) The production rate or
throughput, and consequently the lead time, does not
depend on the utilization of the resources. That is, even if the
utilization is very high congestion effects that result in
increasing lead times is not taken into consideration. (3) The
demand is an aggregation of several customer orders
without distinction. This means that even if backlogging or
shortages are accepted, the customers are not
distinguished.
With respect to the first assumption, it is well known that
orders are usually subject to great uncertainty in
terms of order size and due date which can be critical
especially for manufacturers with long production lead times.
In the case of semiconductor manufacturing for example, well in
advance of the ultimate due date, customers provide
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an indication, a demand signal, of what their orders will
ultimately be. As time evolves and after assessment of their needs
customers adjust their orders (quantities and due dates) until a
firm order is obtained. Despite the manner in which orders may
change after being signaled, customers still require that orders be
met within a short
period after their eventual due date, even though this date may
only be known with limited advance notice [1,2]. The
uncertainty inherent in orders should affect both production
planning and order acceptance decisions. Demand
uncertainty is modeled following the robust optimization (RO)
approach developed by Bertsimas and Sim in [3]. A
recent attempt to model inventory systems using the RO framework
was done by Bertsimas and Thiele [4].
The dependency between resource utilization and lead times (or
equivalently available capacity) has been
addressed to some degree by some authors. As a result of using
queuing models in production planning and
scheduling, Hopp and Spearman in [5] show that lead times
increase non-linearly as system utilization increases and
approaches 100%. Several authors use clearing functions (CFs) to
model the dependency between workload and lead
times, [6-9]. Aouam and Uzsoy in [10] compare the performance of
various production planning models with
workload-dependent lead times under demand uncertainty. In this
paper, the proposed formulations use CFs to model
the relationship between throughput and WIP levels. Two
production modes are distinguished based on a pre-
specified critical utilization level: low utilization mode and
high utilization mode. In the latter mode, congestion
effects are taken into consideration, i.e., when utilization
approaches 100% lead times become increasingly higher.
Grouping orders of customers in a single demand (per time
period, for example) is part of aggregation
decisions made on data in order to simplify the planning models,
or for managerial purposes. However, in practice
customer orders need to be distinguished for several reasons.
Firstly, even if the finished good is the same, different
customers might impose particular conditions on the source of
the raw material (this was partially addressed in [11])
or on the quality control tests made during the manufacturing
process of their orders. Secondly, there are situations
where the planners need to satisfy the demands partially. This
happens in case of shortages or for profitability
reasons. If shortages happen under the form of backlogs or lost
sales, the planners have to decide which order they
will not satisfy properly. Furthermore, even if there is enough
capacity to avoid shortage, it is not always clear that
all orders should be accepted even if the unit price the
customer will pay exceeds the variable production cost. Kefili
et al. in [12] show that the marginal prices of capacitated
resources are not necessarily equal to zero when the
utilization is less than one. This means that even in the case
where capacity is available, the revenue from an
additional order should at least offset the variable production
cost plus the dual of the capacity constraints that take
into account workload. Therefore, models that integrate
production planning decisions and order acceptance
decisions have a great potential to improve the overall
profitability of the firm.
In this paper, we provide a robust model that integrates
production planning and order acceptance decisions.
To the best of our knowledge, our model (which is a production
planning model under uncertainty) is the first model
to incorporate: (i) integration of production planning and order
acceptance decisions, (ii) a robust optimization
approach to model demand uncertainty with demand signals and
firm orders, (iii) two production modes based on
utilization, reflecting the effects of congestion, (iv) and
multiple customer classes.
The rest of the paper is organized as follows. In section 2, we
present a production planning model with
congestion based on CFs. In section 3, we formulate a robust
production planning model where demands are
aggregated. We formulate the robust integrated production
planning and order acceptance model in section 4. In
section 5, we present numerical experiment and a simulation
study to compare the models. We conclude in section 6.
2. Production Planning with Congestion Consider a single
capacitated production stage. Raw materials are released into the
stage at the beginning of each
time period . The units remain in work-in-process (WIP) for a
certain production lead time which depends on the WIP level and
once a finished item is produced it is kept in stock. The unit
costs of releasing raw material,
holding WIP, and holding finished goods are given by , , and
respectively. Furthermore, if shortage occurs, a unit penalty cost,
, is incurred. The demand for period t is denoted by and the
cumulative demand up to time t by assumed to be deterministic in
the current section. We will use a similar notation for cumulative
quantities all throughout the paper. We denote the quantity
released by , the quantity produced (throughput) by , the WIP level
at end of period t by , the finished goods inventory level by , and
the backlog level by . The maximum throughput of the production
stage (capacity) over one period is denoted by .
The proposed production planning model with congestion is based
on the utilization defined for each time t
by
. Two operation modes are distinguished based on a critical
utilization . We say that the production
stage is under low utilization mode when , and the production
stage is under high utilization mode when . In the latter mode,
congestion effects are taken into consideration, i.e., when
utilization approaches 100%, lead-time increases non-linearly.
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Under low utilization mode, items are assumed to spend on
average L units of time, where L is the
production lead time which includes processing time and delay
time. We use the linear control rule in the form of a
CF based on Little's law, [6]. The throughput of the production
stage is expressed as follows,
(1)
where represents the resource load for period t, or the total
amount of work that becomes available
for processing during the period. Given that the same
proportion,
for example 25%, of the WIP is always
produced (cleared from the stage), the last unit to enter the
stage and hence added to the WIP should wait L = 4
periods in the stage before it is produced.
Now, in order to model the high utilization mode, a CF, denoted
by , that is increasing and concave with to relate the throughput
to the WIP as follows,
(2) We followed [7] in writing our CF as a function of the
resource load for period t, or, the total amount of work that
becomes available for processing during the period. It is also
assumed that under low utilization, i.e.,
, we have
. This assumption will make sure that the congestion reflected
by the clearing
function constraints can be binding only for the high
utilization mode. This assumption is not really restrictive
because the CF does not play any role under low utilization.
Following [8,9] and for tractability reasons, we
approximate the CF using an outer linearization. In fact, can be
approximated by the convex hull of a set of affine functions of the
form,
{ } (3)
with is a strictly decreasing series and . Given initial
inventories and and assuming no initial backlogs, the production
planning model with congestion is formulated as
(PPC):
Minimize [ ]
(4)
Subject to:
(5) (6)
(7)
(8) (9) (10)
The objective function in equation (4) minimizes total cost over
the planning horizon. Constraints (5) and (6) define
WIP and finished goods inventory balances respectively for each
period. Constraints (7) represent the linear control
rule and make sure that units spend L units of time as WIP
before being cleared from the production stage and will
be binding in periods under low utilization. Constraints (8)
represent capacity constraints on the throughput based on
the CF, and will be active for the case of high utilization.
Constraints (9) define the measure of WIP as the total
amount of work that becomes available for processing for a given
period.
3. Robust Production Planning with Uncertainties on Aggregate
Demand In this section, we propose a robust model for the
production planning model under demand uncertainty based on the
RO approach developed by [3]. Demand in each period is regarded
as the aggregate orders that the firm has already
committed to deliver at the end of the same period. This model
does not consider order acceptance decisions and will
serve as a benchmark to evaluate the benefits from integrating
production and order acceptance decisions.
In the following, we assume that demand in every period t, is
random and can be expressed as follows, (11)
where [ ] is the demand scaled deviation from the mean. are the
mean and standard deviation of and k>0 is the variability
factor. Demand uncertainty only affects the FGI balance constraints
(6). For each period t,
the inventory balance will reduce to one of two constraints that
will be binding at optimality one in the case of excess
inventory and the other for the case of shortage,
(12) (13)
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Let us first model uncertainty in constraint (11) that can be
rewritten for each period t as follows,
(14)
The worst case in terms of violating the constraint corresponds
to . This is not surprising because this case would correspond to
the case of minimum demand and hence we would expect that excess
inventory will
remain at the end of the period, leading to high holding costs.
However, this scenario is very unlikely to happen in a
real situation and corresponds to an extreme case that would
lead to production plans with very high inventory.
Following the RO approach, detailed in the previous section,
large deviations are eliminated by allocating
uncertainty budgets for each period t,
| |
(15)
The inventory balance for the case of holding can be written
as,
(16)
where is the optimal solution of the following problem,
Maximize
(17)
Subject to:
(18)
(19) and are the dual variables corresponding to constraints 18
and 19, respectively. The quantity
is
the maximum deviation that is admissible, i.e. within the budget
limit forced by constraint (18). This deviation serves
as a protection for the inventory constraint not to be violated.
Following similar arguments, one can write the
inventory balance in the case of shortage for each period as
follows,
(20)
Now, if we consider problem (16 - 18) as a primal problem that
is a feasible and bounded linear program, by strong
duality the primals optimal value is equal to the optimal value
of its dual. Therefore, the maximum admissible
deviation
can be replaced by
. Furthermore, only one of the two constraints (12) or (13)
should be active, i.e., in each period t either a shortage cost or
an inventory cost will be incurred. To tackle this
modeling issue we introduce the new variable . The robust
counterpart of the CPP problem, i.e., the robust production
planning model with congestion RPPC is formulated as follows,
(RPPC):
Minimize [ ]
(21)
Subject to: Constraints (7)-(9), and
(22)
(
) (23)
(
) (24)
(25) (26)
The robust counterpart of the PP problem, i.e., the robust
production planning model without congestion (RPP) is
formulated in the same way as the RPPC except that constraints
(7) (9) are replaced by constraints a regular capacity constraint
on the throughput. The uncertainty budgets are increasing in time
since uncertainty increases with
the number of future time periods considered. Also, the
uncertainty budgets cannot increase by more than 1 at each
time period, i.e., , . This means that the increase should not
exceed the number of new parameters added at each time period. We
suggest the use of , where is referred to as the budget factor. The
case of corresponds to the worst case where all demands before
period t are set either to their maximum or minimum values and the
case of corresponds to the nominal case.
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4. The Integrated Production Planning and Order Acceptance
Problem In this section, we formulate an integrated model that
determines jointly production planning and order acceptance
decisions. The production planner has the flexibility to decide
on which orders to satisfy, considering the fact that
each order has a unique marginal revenue (reservation price) and
a different level of uncertainty (measured by the
variance). The order acceptance flexibility allows the planner
to decide among the highly profitable, yet risky, orders
or less profitable, but possibly more stable, orders. In what
follows, we refer to a customer class to denote all
customer orders that have the same reservation price and
variance. These classes can represent single orders,
customers, or markets. Notice that there is no restriction on
the number of orders per class. If the planner would like
to consider each order separately, in such case a class would
correspond to a single order.
Let us assume that there are N customer classes and that at the
beginning of the planning horizon, customers
place their target orders (quantities and due dates) resulting
in demand forecasts, , for each customer class. These are the
demand signals as referred to by Kempf [1,2]. The actual demand for
each customer class n is random and can be expressed as
follows,
(27)
with [ ] . is the standard deviation of representing the
variability in orders and denote the variability factors. The order
acceptance problem under uncertainty consists of determining the
fraction of the
demand forecasts (initial orders) that the planner commits to
satisfy. The optimal acceptance fraction of demand for
class n in period t is defined by the following equation,
(28)
where, is the optimal quantity among announced orders that the
producer commits to satisfy. After the
realization of demand, i.e., firm orders are obtained, the
producer will supply if enough inventory is
available. In case inventory is not enough to satisfy all demand
penalty costs will occur. As stated in [2], despite the
manner in which orders may change after being signaled,
customers still require the order to be met. In such a case,
the inventory constraint for each period t can be written as
follows,
(29)
Following the RO approach in allocating uncertainty budgets for
each class n and in each period t, the inventory balance (29) can
be represented by one of the two constraints that will be binding
at optimality,
(30)
(
) (31)
with being the optimal solution of the following problem for
each (n,t)
Maximize
(32)
Subject to:
(33)
(34)
For each period t and for each customer class n, the
quantity
is the maximum deviation that is
admissible and represents the protection from violation of the
constraint, given the acceptance fraction
. The
linear program (32 - 34) is feasible and bounded, hence by
strong duality the optimal objective of this problem is
equal to the optimal objective of its dual formulated as
follows,
Minimize
(35)
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Subject to:
(36)
(37) Let us introduce the new variables . Following the RO
approach, the robust production planning and order acceptance
problem with congestion can be formulated as follows,
(RPPC-OA):
Minimize [
]
(38)
Subject to: Constraints (14)-(16) and
(39) (40)
( [
]
) (41)
( [
]
) (42)
(43)
(44)
Notice that the acceptance fraction
on the right hand side in equations (43) allows for the planner
to trade-off
between variability or risk and profit in the objective function
(38). When the variability of a customer class is too
high, the planner can choose to sacrifice profit for less risk
or inability to meet demand by decreasing . Similarly, the robust
production planning and order acceptance problem without congestion
(RPP-OA) can be formulated
simply by replacing constraints (7) (9) by regular capacity
constraints.
5. Experimental Results In this section, we present a numerical
example along with a simulation study to illustrate the proposed
models
characteristics, show the effects of various parameters on the
optimal acceptance fraction, and evaluate the added
value from integrating production and order acceptance
decisions. The optimization models have been implemented
in GAMS and solved using CPLEX 11.0.
5.1 A Numerical Example
Customer demand is forecasted over a horizon of three months,
each consisting of four working weeks (12 weeks
horizon). We assume that expected demand during months 1, 2, and
3 are equal to 80, 100, and 200 respectively with
a coefficient of variation equal to 0.2. Because of limited
production capacity, production takes place ahead of time
during early periods. The choice of values for and can be
arbitrary, but the values we use are those
recommended by Graves (1988), and . The capacity is (160% of the
average demand), and the unit costs are given by , , Based on [7],
the following functional form of the clearing function is
adopted:
where
, =0.8, and L = 2 weeks. We consider four customer classes
distinguished by
their marginal revenue (high/low profit margins) and their
demand coefficient of variation (high/low) as represented in Table
2. Specifically, , = = , = = 0.3, and = = 0.1. We also assume that
demand forecasts from the four classes are equal, that is
.
Variance
Marginal
Revenue
n1 High High
n2 Low Low
n3 High Low
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n4 Low High
Table 2: Customer Classes differentiated by Variance and
Reservation Price
In order to evaluate the different performance measures,
including, acceptance fractions, fill rate, profit,
revenue, and costs we develop a simulation procedure in which
two levels of randomization are considered: the
demand signal level and the replication level, corresponding to
firm orders. Multiple replications for a demand signal
are done to obtain statistically significant values of the
average performance corresponding to the demand signal.
Multiple demand signals are used to obtain the average
performance of the model to be evaluated. Four factors are
found to be most influential,
- average demand to capacity ratio. - : variability factor. We
assume that the variability factor of all customer classes is the
same. - : budget of uncertainty factor. We assume that the budget
of uncertainty for each customer class in
each period t is given by . - : shortage penalty factor. We
assume that the marginal penalty cost is given by
. The nominal case corresponds to the following values: . The
ranges for each parameter are given by: .
5.1 Summary of results
In this section, we study the effect of congestion on the
optimal acceptance fractions for the different customer
classes n, given by
, and for the entire system, . In order to reflect the effect of
congestion the two
integrated production planning and order acceptance models with
congestion, RPPC-OA, and without congestion,
RPP-OA, are compared . Figure 4 plots the optimal acceptance
fraction of each class, , and the average acceptance fraction, ,
while the internal parameters ( and ) are varied from low to high
in order to reflect the level of conservatism of the planner. As
expected, the average acceptance fraction is lower when congestion
is taken into account. Also, as the conservatism of the planner
increases a lower fraction of orders are accepted and less
order
from customer classes n1 and n3 are accepted as their orders
have higher variability. When congestion is modeled,
although orders from customer class n1 have higher marginal
revenue, fewer orders are accepted since they are
considered risky. On the contrary more orders from customer
class 2 are accepted as their variance is lower
Figure 1: Effect of congestion on the optimal acceptance
fractions
average acceptance fraction
5.3 The effect of order acceptance
We consider three states of the system: under-utilized ( = 0.6),
normal ( =0.9), and over-utilized ( =1.2). Then, we vary the
internal parameters ( and ) from low (both parameters are set to
their minimum) to high (both parameters are set to their maximum)
in order to reflect the level of conservatism of the planner. The
value of
integration is reflected through the following performance
measures estimated through simulation: the total cost
0
0,2
0,4
0,6
0,8
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Acceptance Fractions (with congestion)
n1 n2 n3 n4 Avg.
0
0,2
0,4
0,6
0,8
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Acceptance Fractions (without congestion)
n1 n2 n3 n4 Avg.
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(TC), which includes the release cost (RC) and the holding costs
(HC) that includes WIP and FG inventory holding
costs, the total revenue (TR), the total profit (TP), the total
backordered quantity (TB), and the fill rate (FR).
The results show that when order acceptance is not considered
(RPPC) and as the level of conservatism of the
decision maker increases, higher quantities of raw material are
released into the production stage. In fact, when the
budget of uncertainty and shortage penalty factors are high, the
RPCC model needs to increase the FG inventory
levels (safety stocks). The FG inventory requirements increase
the releases that in turn increase the WIP level and
hence the amount cleared from the production stage (throughput)
is higher. However, as the level of conservatism
gets higher the WIP reaches its critical level, , and the
production system becomes in high utilization mode which is
characterized by the non-linear CF. Under high utilization mode, as
the releases increase the
production lead time increases and the throughput increases very
slowly leading to high levels of WIP inventory and
slow increases in FG inventory, especially in the cases of high
D/C ratio. When order acceptance decisions are
integrated with production decisions, only a reasonable number
of orders are accepted and hence release quantities
are kept relatively low for the sake of smooth production plans
with low levels of WIP inventory. As a consequence,
it is clear from Figure 2, that the total operating cost (TC)
for the traditional production planning case (RPPC) is
much higher than the operating costs of the integrated case
(RPPC-OA).
Figure 2: Comparison of total costs (TC) (excluding backlogging
cost)
The main performance measures for any production planner are the
total profit (TP) and the extent to which demands
are met, which is commonly measured by the fill rate (FR)
representing the fraction of total demand met from
inventory. In order not to bias the results we kept the penalty
factor equal to its nominal value (p=1.3) when
computing the total profit resulting from the optimal production
planning decisions in our simulation. Otherwise, the
total profit for the case of RPPC will be very small, especially
for very conservative decision makers. As one can
notice from Figure 13, RPPC-OA outperforms RPPC according to
total profits.
Figure 3: Comparison of total profits (TP)
0
100000
200000
300000
400000
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
TC for RPPC-OA
D/C = 0.6 D/C= 0.9 D/C= 1.2
0
100000
200000
300000
400000
500000
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
TC for RPPC
D/C = 0.6 D/C= 0.9 D/C= 1.2
-100000
0
100000
200000
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
TP for RPPC-OA
D/C = 0.6 D/C= 0.9 D/C= 1.2
-100000
0
100000
200000
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
TP for RPPC
D/C = 0.6 D/C= 0.9 D/C= 1.2
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This fact is mainly due to the fact that the RPPC model
increases the releases in order to increase the inventory
levels
as the level of conservatism resulting in very high operating
costs and unmet demand that lead to low profits and low
fill rates (Figure 14 - right), especially in the case of high
D/C ratio. The RPPC-OA commits to satisfy a reasonable
amount of orders taking into account utilization. Therefore, by
integrating the order acceptance and production
decisions, the production planner can increase the companys
total profits while maintaining a very high fill rate (Figure 14 -
left) independently of the D/C ratio.
Figure 4: Comparison of fill rates (FR)
6. Conclusion A capacitated production stage serving customer
orders from multiple demand classes characterized by different
marginal revenues and variability in their order quantities was
considered in this paper. The proposed integrated
model jointly determines production planning decisions and order
acceptance decisions while capturing uncertainty
in order quantities and workload dependent lead times. A robust
optimization approach is followed to capture
demand uncertainty while clearing functions are adopted to
capture the non-linear dependency between lead time and
utilization and reflect the effects of congestion.
Orders/customers are classified into classes based on their
marginal
revenue and their level of variability in order quantity (demand
variance). In this paper we show that the main value
of integrating the two decisions is that the planner has the
flexibility to select a reasonable number of orders that the
company commits to satisfy and hence release quantities and
utilization can be maintained at desirable levels. This
flexibility leads to high profits and high levels of customer
satisfaction, measured by the fill rate.
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0,00
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0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
FR for RPPC-OA
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