Description
Batch processes are widely adopted in many manufacturing systems with raw materials from mining or agricultural industries. Due to variations in both raw material quality and market conditions, variations in the recipes are used in production. Such recipe flexibility is not on design but on the operation that allows adjustments of recipe items aiming to achieve better performance than traditionally fixed recipes.
Proceedings of the 2013 Winter Simulation Conference
R. Pasupathy, S.-H. Kim, A. Tolk, R. Hill, and M. E. Kuhl, eds.
A SIMULATION-BASED APPROACH TO INVENTORY MANAGEMENT IN BATCH PROCESS
WITH FLEXIBLE RECIPES
Long He
Industrial Engineering and Operations Research
University of California, Berkeley
Berkeley, CA 94704, USA
Simin Huang
Industrial Engineering
Tsinghua University
Beijing 100084, CHINA
Zuo-Jun Max Shen
Industrial Engineering and Operations Research
University of California, Berkeley
Berkeley, CA 94704, USA
ABSTRACT
Batch processes are widely adopted in many manufacturing systems with raw materials from mining or
agricultural industries. Due to variations in both raw material quality and market conditions, variations in
the recipes are used in production. Such recipe ?exibility is not on design but on the operation that allows
adjustments of recipe items aiming to achieve better performance than traditionally ?xed recipes. In this
paper, we study the inventory investment, recipe selection and resource allocation decisions in batch process
systems with ?exible recipes. A two-stage stochastic mixed integer program formulation is developed for
each period. Moreover, the system updates its inventory investment decisions based on new demand data
from previous periods by a simulation-based approach. Bene?ts of implementing ?exible recipes over
traditional ?xed recipes are investigated in the numerical studies.
1 INTRODUCTION
Oil consumption has been escalating in the past decades, especially in emerging economies regions, such
as Asia. Meanwhile, the dramatically increasing oil price is impeding the growth of the world economy.
Despite its increasing trend, oil price also exhibits high volatility. After it reached the record peak US$
145 in July 2008, it fell signi?cantly to US$ 30.28 a barrel on December 23, 2008. Such increasing trend
together with jumps of prices also prevails in other commodities over the past decades as shown in Figure
1. This phenomenon leads to higher manufacturing costs as well as more dif?culties in supply chain
management under price uncertainty among many industries.
Facing such challenges, joint inventory investment and allocation decision making becomes an important
tool that makes the manufacturing systems robust. Consider an oil re?nery that converts crude oil into
pro?table petroleum products such as gasoline, diesel, kerosene, heating oil and asphalt. Those products are
actually inputs for further manufacturing processes. Generally, it operates in 3 phases: crude oil unloading
and blending, fractionation and reaction processes and product blending and shipping. In the ?rst phase,
crude oil of different grades is transported by crude oil marine vessels from different regions. Since the
properties of crude oil highly depend on its origins, there are usually dedicated storage tanks for crude
oil of different grades. In many situations, before crude oil enters distillation, the ?rst step of production,
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He, Huang, and Shen
Figure 1: Selected Commodity Prices in Past 20 Years.
different grades are blended to achieve certain properties, such as viscosity and density, in order to meet
the production requirements.
The manufacturing process presented above belongs to batch process that primarily schedules short
production runs of products (Connor 1986). Batch process industries often obtain their raw materials from
mining or from agricultural industries. These raw materials have natural variations in quality (Rutten and
Bertrand 1998). Some common batch processes can be found in ?elds such as oil re?ning, agricultural,
chemicals and fertilizers. (Schuster and Allen 1998) illustrates how Welch’s Inc manages grape-processing
among plants using linear program models. In that case, grapes are usually processed in plants located
near growing areas. To maintain national consistency, Welch’s often transfers juice for blending between
plants. The selection of recipes is a key decision that affects the pro?tability via both operational costs and
production capacity. The nature of variations in both raw material quality and market conditions often lead
to the variations in the recipes. Such recipe ?exibility is not on design but on the operation that allows
adjustments of recipe items aiming to achieve better performance than traditionally ?xed recipes. Here,
?exible recipe refers mainly to the adjustments of recipe items as input of batch process in response to
market conditions, i.e. demand arrivals.
In this paper, we simplify the system by considering three types of goods: raw materials, ingredients
and ?nal products. In the grape-processing case, we regard grapes from different growing areas as raw
materials, intermediate juice of different concentration as ingredients and packaged juice on market as ?nal
products. The batch process is simpli?ed into two phases: separation and blending. Since different raw
material grades have various concentration of desired ingredients, in the separation stage, those ingredients
are separated ?rst from raw materials. Then a combination of ingredients are blended into ?nal products
that meet certain speci?cations.
The remainder of this paper is organized as follows: We begin in §2 by reviewing related literature.
In §3, an application of ?exible recipes is illustrated by a simple case. A two-stage stochastic mixed
integer program for the basic model is formulated in §4. We then discuss a simulation-based approach
to the proposed model in §5. In §6, we conduct a numerical study to assess the performance of the
simulation-based approach and bene?ts of ?exible recipes over ?xed recipes. Finally, concluding remarks
are presented in §7.
2 LITERATURE
There are mainly two streams of literature related to our work. In the ?rst stream, given the structure of the
system, optimal investment decisions are analyzed under demand uncertainty and/or inventory procurement
cost variability. (Fine and Freund 1990) investigate the optimal capacity investment problem in single
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period by two-stage stochastic programming with discrete demand distribution. Their study focuses on
the case with two products, two dedicated resources, one ?exible resource. While in our study, we allow
the demand distribution to be unknown and the system handles multiple raw materials and ?nal products.
Following the lead of (Fine and Freund 1990), under a two-product ?rm, (Van Mieghem 1998) analyze the
optimal investment in ?exible resources as a function of margins, costs and multivariate demand uncertainty.
Contrary to the previous work, they show that it can be advantageous to invest in ?exible resource even
with perfectly positively correlated product demands. (Harrison and Van Mieghem 1999) study the optimal
investment strategy with a multi-dimensional newsvendor model and conclude a critical fractile property for
the optimal investment levels. Given the structure of an assemble-to-order system, (Akc¸ay and Xu 2004)
formulate the join inventory replenishment and component allocation problem into a two-stage stochastic
program and propose an order-based component allocation rule for the second stage problem.
In the second stream, the applications of ?exible recipes in batch processes are mostly studied. (Rutten
and Bertrand 1998) study the balancing of safety stock costs and recipe ?exibility costs for batch industries
with high customer service requirements. They conclude that under certain circumstances the use of recipe
?exibility can lead to lower costs when compared to using ?xed recipes. (Keesman 1993) investigates the
application of ?exible recipes for batch process optimization and applies an adaptive feedforward control
strategy for a priori known disturbances in the process inputs. Furthermore, a new framework that fully
exploits the inner ?exibility of batch processes at the plant level is developed by (Romero et al. 2003). Their
framework considers a batch recipe model that interacts with a plant-wide model to constitute the ?exible
recipe model. The most related work to ours, under batch process manufacturing, is done by (Karmarkar
and Rajaram 2001). They formulate the grade selection and blending problem as a nonlinear mixed-integer
program with ?xed cost for grade selection and inventory holding cost. However, they assume the annual
demand is known and constant for each ?nal products.
Our work is different from the literature mainly in two ways. First, in our model, the recipe ?exibility
is embedded in the operations of batch processes, rather than the system design as seen in literature on
process ?exibility. Second, we study the decisions of inventory investment, recipe selection and resource
allocation in an integrated model.
3 EXAMPLE OF FLEXIBLE RECIPE APPLICATION
In this section, we brie?y illustrate the application of ?exible recipes in batch process. Consider a
manufacturer, i.e. re?nery or food processing factory, whose operations can be categorized as separation
and blending stages. There are 3 ?nal products made from 3 raw materials. The raw material inventory
is given as Z = (z
1
= 200, z
2
= 300, z
3
= 400) units, where z
i
is the inventory of raw material i. The raw
material cost is C = (c
1
= 6, c
2
= 4, c
3
= 3) dollars per unit, where c
i
is the purchase cost of raw material
i per unit. In the batch process, raw materials are separated ?rst into 3 ingredients, depending on their
concentration in raw materials. The ingredient concentration matrix for raw materials is
P =
0.6 0.3 0.1
0.4 0.4 0.2
0.3 0.4 0.3
where row i represents raw material i and column j represents ingredient j. The element on row i and
column j, denoted by p
i j
, is the proportion of ingredient j contained in a unit of raw material i. Then in
the blending stage, ?nal products are blended from those ingredients. The ingredient requirement matrix
for ?nal products is
A =
0.8 0.2 0
0.7 0.2 0.1
0.6 0.3 0.1
where row k represents ?nal product k and column j represents ingredient j. The element on row k and
column j, denoted by ?
k j
, is the proportion of ingredient j required in a unit of ?nal product k.
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He, Huang, and Shen
In a system that implements ?xed recipes, it determines the optimal ?xed recipe before any demand
arrivals. The resulting optimal ?xed recipe in this case is:
B =
0.53 0.44 1.03
0.40 0.18 1.29
0.32 0.11 1.22
where row k represents ?nal product k and column i represents raw material i. This matrix is similar to
BOM. That is, the element on raw k and column i, denoted by b
ki
, is the amount of raw material i required
in a unit of ?nal product k.
For simplicity, we assume that the demand for each ?nal product follows Bernoulli distribution with 0.5
probability equals 100 units and 0.5 probability equals 200 units. When the system sees demand arrivals, it
determines the optimal ?exible recipes that maximize its revenue with R = (r
1
= 10, r
2
= 8, r
3
= 6) dollars,
where r
k
represents the revenue of a unit ?nal product k.
Table 1 summarizes the computational results of expected pro?t and shows that the ?exible recipes enable
Table 1: Expected Pro?t Summary (in dollars)
Flexible Recipes (R1) Fixed Recipes (R2) Improvements=
R1?R2
R2
Current Setting 3525 2904.6 21.36%
Demand ×2 4350 3199.8 35.95%
Demand ×0.8 2880 2643.7 8.94%
the system to achieve higher pro?t via better resource utilization, especially under the presence of large
demand variance.
4 THE BASIC MODEL
As shown in Figure 2, the batch process consists of two stages: in the separation stage, raw materials are
processed into a set of ingredients; in the blending stage, a selection of ingredients are blended into ?nal
products to ful?ll the demands. In the oil re?nery industry, for instance, the raw materials are different crude
oil grades. The three most quoted oil grades are North America’s West Texas Intermediate crude (WTI),
North Sea Brent Crude, and the UAE Dubai Crude. Depending on the mixture of hydrocarbon molecules,
crude oil varies in color, composition and consistency. Different oil-producing areas yield signi?cantly
different varieties of crude oil. The ingredients are the intermediate products such as light ends, naphtha,
kerosene, distillate, atmospheric residua, vacuum gas oil and vacuum residua. Final products are various
gasoline types, lubricants, petrochemicals, diesel, asphalt, etc,.
Figure 2: Simpli?ed Batch Process
Our model is to determine the inventory level at the beginning of each period before any demand
arrivals. We assume all leftover inventory by the end of each period is disposed because of time sensitivity
of raw materials, i.e. perishability in grape-processing. As a result, at the beginning of each period t, we
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He, Huang, and Shen
only need to solve a subproblem of single period planning with available historical data up to period t.
Therefore, we focus on solving a single period problem.
There are basically three types of costs: raw material procurement cost, grade selection cost and ?nal
product revenue. Similar to the de?nition by (Karmarkar and Rajaram 2001), the grade selection cost incurs
when a grade is selected by a recipe into separation stage and the cost varies by grades. Raw material
procurement cost can be the real option or spot market price. The model we propose is able to optimize
investment on raw materials together with ?exible recipe selection for production when the system sees
demands. It can be easily extended to the systems with many separation and blending stages in serial
structure.
The event sequence is illustrated in Figure 3. At the beginning of period t, the system ?rst invests in
raw material inventory with total procurement cost ?
i
c
i
z
i
, where c
i
is the unit cost of raw material i and
z
i
is the amount of raw material i purchased. After the system sees the demand arrivals D = (d
k
), where
d
k
is the demand of ?nal product k, it then selects the optimal recipe to satisfy the demands. Under the
presence of grade selection cost, it is not always pro?table to satisfy as much demand as possible. As
discussed above, the recipe selection cost is the sum of grade selection costs, given by ?
i
f
i
y
i
, where f
i
is the grade selection cost of raw material i and y
i
is the selection decision of raw material i. Lastly, the
ful?llment of demands generates total revenue ?
k
r
k
x
k
, where r
k
is the unit revenue of ?nal product k and
x
k
is the ful?lled demand of ?nal product k. We assume that material transformation processes are linear.
Another assumption is that the material loss in separation and blending stages are negligible. Indeed, if the
loss is consistent and fractional in the process, we can introduce some discount factors in the formulation
so that the structure of the model is still preserved.
Figure 3: Event Sequence Diagram in Period t
The decision making process for the system with ?exible recipes differentiates itself from the newsven-
dor model with the postponement in ?nal product blending. In the newsvendor model, the ?nal products
are produced ahead and the total manufacturing cost can be assessed before demand arrivals. As a result,
in the newsvendor model, the critical fractiles and subsequently the raw material inventory levels can be
determined in advance. While in a batch process with ?exible recipes, the system makes tradeoffs between
revenue and recipe selection cost after demand arrivals.
We formulate the basic model as a two-stage stochastic mixed-integer program:
?= maxE
D
?(Z, D) ?
?
i?I
c
i
z
i
(1)
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He, Huang, and Shen
where
?(Z, D) =max
?
k?K
r
k
x
k
?
?
i?I
f
i
y
i
(2)
s.t.
?
k?K
?
k j
x
k
?
?
i?I
p
i j
z
i
y
i
, ?j ? J (3)
x
k
? d
k
, ?k ? K (4)
y
i
? {0, 1}, ?i ? I (5)
Uppercase is for vectors while lowercase is for scalars. The ?rst stage of the formulation, equation (1),
maximizes the expected total pro?t of the system. Given rawmaterial inventory level vector Z = (z
1
, z
2
, ..., z
I
)
and demand arrival vector D = (d
1
, d
2
, ..., d
K
), the second stage with recourse maximizes as in equation (2)
the total revenue minus total grade inclusion cost. The constraint (3) states that the supply of each ingredient
?
i?I
p
i j
z
i
y
i
must be no less than the ingredient requirement for producing X amount ?nal products which is
?
k?K
?
k j
x
k
. Here ?
k j
is the amount of ingredient j required to produce a unit of ?nal product k and p
i j
is the
amount of ingredient j contained in a unit of raw material i. Because the initial inventory investment is sunk
cost, (z
1
y
1
, z
2
y
2
, ..., z
I
y
I
) shows that the system chooses to use up all selected raw materials. Constraint (4)
represents that demand ful?llment can not exceed the demand arrivals. The recipe selection is determined
by the binary decision vector Y expressed in constraint (5).
5 SIMULATION-BASED OPTIMIZATION
Our solution approach to the proposed two-stage stochastic mixed-integer program consists of two modules:
demand simulator and SAA optimizer, as shown in Figure 4. Given the available historical demand data,
the demand simulator generates simulated demand arrivals for period t. There are several ways to simulate
or forecast demands based on previous information. Here, we use Bootstrap sampling in the demand
simulator.
Figure 4: Solution Approach
With the simulated demand arrivals, the second module ?nds the optimal inventory levels by Sample
Average Approximation (SAA), an simulation-based approach. The algorithm is modi?ed from the SAA
method provided in (Akc¸ay and Xu 2004). A detailed introduction of the SAA method can be found
in (Shapiro and Homem-de Mello 1998). Let ?(Z
?
) be the optimal solution to the two-stage stochastic
program in the basic model (1)-(5) and Z
?
be the associated optimal inventory levels. We start with
generating M independent samples of random vector D from the demand simulator, each of size N. That
is, D
l
= (D
l,1
, D
l,2
, ..., D
l,N
) is the realization of the lth sample, where D
l,h
= (d
l,h
1
, d
l,h
2
, ..., d
l,h
K
) with d
l,h
k
as the realization of demand for ?nal product k in the hth vector of the lth sample realization. We solve
the SAA problem referring to each sample, as below:
max
Z
l
?
N
(Z
l
) =
1
N
N
?
h=1
?(Z
l
, D
l,h
) ?
?
i?I
c
i
z
l
i
(6)
s.t.
Constraints (3) ?(5) for each D
l,h
, h = 1, ..., N (7)
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He, Huang, and Shen
where Z
l
= (z
l
1
, z
l
2
, ..., z
l
I
)
For l = 1, ..., M, let ?
N
(
ˆ
Z
l
) be the corresponding optimal solution to the above SAA problem and
ˆ
Z
l
be the associated optimal inventory investment. Since Z
?
is always a feasible solution to (6)-(7), we have
?
N
(
ˆ
Z
l
) ??(Z
?
) for all l = 1, ..., M. We then have,
E[
¯
?
N
] ??(Z
?
), where
¯
?
N
=
1
M
M
?
l=1
?
N
(
ˆ
Z
l
)
Therefore, E[
¯
?
N
] is used as the estimate of an upper bound of ?(Z
?
).
In order to have an unbiased estimator of ?(
ˆ
Z
l
), we again generate one large number of independent
sample from the demand simulator, D
N
= (D
1
, D
2
, ..., D
N
). Then, for each inventory level vector
ˆ
Z
l
,
compute the estimate of ?(
ˆ
Z
l
) by
ˆ
?
N
(
ˆ
Z
l
) =
1
N
N
?
h=1
?(
ˆ
Z
l
, D
h
) ?
?
i?I
c
i
ˆ z
l
i
where ?(
ˆ
Z
l
, D
h
) is the optimal solution to the second stage allocation optimization with inventory vector
ˆ
Z
l
and demand realization D
h
. The estimated optimal inventory vector
ˆ
Z
?
is then determined by choosing
the one that gives the largest
ˆ
?
N
(
ˆ
Z
l
) among all candidate inventory vectors
ˆ
Z
l
with sampled demand
realizations, as follows:
ˆ
Z
?
? argmax{
ˆ
?
N
(
ˆ
Z
l
), l = 1, ..., M}
Since
ˆ
Z
?
is a feasible solution to the basic model (1)-(5), we further have
E[
ˆ
?
N
(
ˆ
Z
?
)] ??(Z
?
)
As a result, E[
ˆ
?
N
(
ˆ
Z
?
)] can serve as a lower bound of ?(Z
?
). Thus, the difference between E[
¯
?
N
] and
E[
ˆ
?
N
(
ˆ
Z
?
)] is an estimate of the optimality gap of SAA solution. In brief, the SAA method works in the
following procedure:
Step 0 Determine appropriate values for N, M and N
, and initialize l = 0;
Step 1 Set l = l +1 and generate an independent sample D
l
= (D
l,1
, D
l,2
, ..., D
l,N
); Solve the SAA
problem (6)-(7) for
ˆ
Z
l
and ?
N
(
ˆ
Z
l
); If l < M, go to Step 1; otherwise, go to Step 2;
Step 2 Generate an independent sample D
N
= (D
1
, D
2
, ..., D
N
); Initialize l = 0;
Step 3 Set l = l +1 and solve the SAA problem with D
N
and
ˆ
Z
l
for
ˆ
?
N
(
ˆ
Z
l
); If l < M, go to Step 2;
otherwise go to Step 4;
Step 4 Choose
ˆ
Z
?
? argmax{
ˆ
?
N
(
ˆ
Z
l
), l = 1, ..., M};
The quality of the solution, measured by the optimality gap, improves as the sample sizes N and N
grow. However, larger sample sizes require higher computational capacity. Therefore, tradeoff between
sample sizes and computational effort need to be considered.
6 NUMERICAL STUDY
In this section, we apply the proposed approach to a real-world ?our manufacturing system with one-year
demand data and some scaled cost values. The system produces 18 kinds of ?our “A” to “R” for different
uses from 3 grades of wheat numbered “1” to “3” from different origins. The ingredients are mainly
starch, protein and ?ber. The wheat with higher protein concentration costs more. The costs, ingredient
concentration, and requirement matrices are summarized in Table 2.
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He, Huang, and Shen
Table 2: Parameters in Numerical Study
Raw material cost
($/kg)
Grade selection cost
(10
3
$ )
Price
($/kg)
Starch
(100%)
Protein
(100%)
Fiber
(100%)
Wheat 1 2.10 2.0 0.80 0.10 0.10
Wheat 2 1.83 2.5 N/A 0.60 0.15 0.25
Wheat 3 1.78 3.0 0.50 0.30 0.20
Product A 2.63 0.98 0 0.02
Product B 2.43 0.95 0 0.05
Product C 2.21 0.72 0.1 0.18
Product D 2.38 0.88 0 0.12
Product E 2.40 0.88 0.02 0.1
Product F 2.19 0.8 0 0.2
Product G 2.15 0.68 0.15 0.17
Product H 2.41 0.93 0 0.07
Product I 2.06 0.75 0 0.25
Product J N/A N/A 1.79 0.65 0 0.35
Product K 1.77 0.2 0.7 0.1
Product L 2.33 0.76 0.1 0.14
Product M 2.13 0.57 0.2 0.23
Product N 2.37 0.57 0.33 0.1
Product O 2.23 0.33 0.52 0.15
Product P 2.41 0.3 0.65 0.05
Product Q 2.36 0.09 0.87 0.04
Product R 1.48 0 1 0
The sample demand arrivals are generated by the demand simulator module, which implements Bootstrap
sampling in the current setting. The SAA optimizer module is realized via CPLEX solver with N = 100,
M = 30 and N
= 500. We compute the average pro?t, inventory investment in dollar value and gaps for
various parameter settings listed in Table 3. As mentioned in the SAA algorithm, the gap is de?ned as the
difference between the upper and lower bounds. The upper bound is estimated by
¯
?
N
=
1
M
?
M
l=1
?
N
(
ˆ
Z
l
) and
the lower bound is estimated by
1
N
?
N
h=1
?(
ˆ
Z
?
, D
h
) ??
i?I
c
i
ˆ z
?
i
. For current parameter setting, the optimal
solution given by our approach is $2721.74×10
6
with inventory levels [5718.194;0;4041.851] for wheat
1, 2 and 3 respectively. We summarize the computational results of run 1 to 5 for ?exible recipe system
in Table 4.
Table 3: Experiment Setting of Selected Runs
Wheat costs Grade selection costs
Run 1 [1.680;1.464;1.424] [2;2.5;3]
Run 2 [1.890;1.647;1.602] [2;2.5;3]
Run 3 [2.100;1.830;1.780] [2;2.5;3]
Run 4 [2.310;2.013;1.958] [2;2.5;3]
Run 5 [2.520;2.196;2.136] [2;2.5;3]
Table 4: Results for Flexible Recipes
Avg. pro?t (10
6
$) Inv. (10
6
$ ) Gap
Run 1 6750.69 17631.84 1.82%
Run 2 4666.52 18948.49 0.91%
Run 3 2721.74 19275.04 0.05%
Run 4 938.55 18000.56 0.41%
Run 5 30.22 3310.80 0.15%
For the selected run 1 to 5, all gaps are less than 2%. This suggests the good performance of the
simulation-based approach with our choice of N, M and N’. If the gap is big, the number of samples N
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He, Huang, and Shen
and N’ should be increased accordingly. Besides, both the average pro?t and total inventory investment
decrease convexly as the raw material cost increases. This implies that the system tends to stock less
inventory when raw materials cost is high. Recall that in newsvendor model, the critical fractile decreases
linearly with production cost and thus the inventory level also decreases convexly if the demand distribution
is concave, i.e. Normal distribution.
The impact of large grade selection cost is illustrated in Figure 5. The increasing grade selection costs
decrease the average pro?t at a mild rate. Meanwhile, grade selection cost increase does not change the
inventory investment signi?cantly. Since the grade selection costs are only scaled by multipliers, their
relative ranking among different rawmaterials are not changed. As a result, the preference among wheats are
not much affected. This makes the inventory investment decision and average revenue almost unchanged.
Therefore, the average pro?t, which equals average revenue less inventory investment and grade selection
costs, decreases linearly in the multipliers of grade selection costs.
Figure 5: Average Pro?t and Inventory Investment for Different Grade Selection Costs
We consider three simple ?xed recipes. That is, ?xed recipes with single source: wheat 1, 2 or 3. Table
5 summarizes the computational results of the average pro?t, inventory investment as well as the ratio
between average pro?t of the ?exible recipes and those of the ?xed recipes. For all experiments, ?exible
recipes can always achieve larger average pro?t than ?xed recipes. For example, in run 1, the “best” ?xed
recipe can at most generate 86.43% of the average pro?t of ?exible recipe. Meanwhile, in run 5, which is
an extreme case, wheat 1 and 2 are too costly to be used as raw materials. This leaves wheat 3 as the only
choice as raw material for the ?exible recipe. Therefore, in run 5, the ?exible recipe is equivalent to the
?xed recipe with wheat 3. In addition, as we see from run 1 to 3, the optimal solution of the ?exible recipes
requires not much more or even signi?cantly less inventory investment than the “best” ?xed recipes. That
is, ?exible recipes achieve higher average pro?t with lower inventory investment than ?xed recipes.
Table 5: Comparison of Flexible Recipes and Fixed Recipes
Flexible recipe Fixed recipe: wheat 1 Fixed recipe:wheat 2 Fixed recipe: wheat 3
Avg.
pro?t
(10
6
$)
Inv.
(10
6
$ )
Avg.
pro?t
(10
6
$)
Ratio Inv.
(10
6
$ )
Avg.
pro?t
(10
6
$)
Ratio Inv.
(10
6
$ )
Avg.
pro?t
(10
6
$)
Ratio Inv.
(10
6
$ )
Run 1 6750.69 17631.84 4869.70 72.14% 14188.77 5834.94 86.43% 18140.89 5039.26 74.65% 16165.77
Run 2 4666.52 18948.49 3244.55 69.53% 14384.60 3551.48 76.11% 18677.88 3151.62 67.54% 14579.06
Run 3 2721.74 19275.04 1695.32 62.29% 14035.47 1595.87 58.63% 19062.30 1786.61 65.64% 12421.80
Run 4 938.55 18000.56 448.34 47.77% 11636.23 85.50 9.11% 7330.68 695.27 74.08% 10315.34
Run 5 30.22 3310.80 0 0% 0 0 0% 0 29.62 98.02% 3256.07
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He, Huang, and Shen
7 CONCLUSION AND FUTURE WORK
In this paper, we propose a two-stage stochastic mixed-integer program to an inventory management
problem in batch process with ?exible recipes. In the ?rst stage, the model determines inventory levels
for each period based on past demand data. After demand arrivals are realized, the second stage recourse
makes recipe selection and allocation decisions in production. With available historical demand data, a
simulation-based approach based on SAA algorithm is developed to solve the stochastic program. As
the historical demand data updates along the time, the inventory levels are set iteratively in each period
using the most updated demand information. The results of numerical study show the performance of the
approach on various cost settings as well as the bene?ts of ?exible recipes over ?xed recipes.
In the proposed approach, we focus on the application of the SAA algorithm and use Bootstrap
sampling as the default in demand simulation. A direction of future improvement is to incorporate better
techniques in the simulation of future demand arrivals based on historical demand data. Those techniques
may consider some properties of the demand, such as seasonality and autocorrelation. Also, with limited
demand information, a robust optimization model might be developed that considers the worst cases.
Moreover, since our model assumes any inventory leftover at the end of each period is disposed, the
extension that relaxes this assumption and introduces inventory holding cost in multi-period setting should
also be investigated.
ACKNOWLEDGEMENTS
This research was partially supported by the National Science Foundation Grants CMMI 1068862, CM-
MI1031637, CMMI1265671 and the National Science Foundation of China Grants 71071084, 71128001,
71210002.
REFERENCES
Akc¸ay, Y., and S. Xu. 2004. “Joint inventory replenishment and component allocation optimization in an
assemble-to-order system”. Management Science 50 (1): 99–116.
Connor, S. J. 1986. “Process Industry Thesaurus”. American Production & Inventory Control Society.
Fine, C., andR. Freund. 1990. “Optimal investment inproduct-?exible manufacturingcapacity”. Management
Science 36 (4): 449–466.
Harrison, J., and J. Van Mieghem. 1999. “Multi-resource investment strategies: Operational hedging under
demand uncertainty”. European Journal of Operational Research 113 (1): 17–29.
Karmarkar, U., and K. Rajaram. 2001. “Grade selection and blending to optimize cost and quality”.
Operations Research 49 (2): 271–280.
Keesman, K. J. 1993. “Application of ?exible recipes for model building, batch process optimization and
control”. AIChE journal 39 (4): 581–588.
Romero, J., A. Espu˜ na, F. Friedler, and L. Puigjaner. 2003. “Anewframework for batch process optimization
using the ?exible recipe”. Industrial & Engineering Chemistry Research 42 (2): 370–379.
Rutten, W., and J. Bertrand. 1998. “Balancing stocks, ?exible recipe costs and high service level requirements
in a batch process industry: A study of a small scale model”. European Journal of Operational
Research 110 (3): 626–642.
Schuster, E., and S. Allen. 1998. “Raw Material Management at Welchs, Inc.”. INTERFACES 28 (5): 13–24.
Shapiro, A., and T. Homem-de Mello. 1998. “A simulation-based approach to two-stage stochastic pro-
gramming with recourse”. Mathematical Programming 81 (3): 301–325.
Van Mieghem, J. 1998. “Investment strategies for ?exible resources”. Management Science 44 (8): 1071–
1078.
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AUTHOR BIOGRAPHIES
LONG HE received the B.Eng degree in Logistics Management and Engineering from The Hong Kong
University of Science and Technology, Hong Kong in 2010. He is currently pursuing the Ph.D. degree in
Industrial Engineering and Operations Research from the University of California at Berkeley, Berkeley CA,
United States. His research interests includes supply chain management, capacity planning and inventory
control. His email address is [email protected].
SIMIN HUANG is a Professor and Associate Head of the Department of Industrial Engineering, Tsinghua
University. He received his Ph.D. from SUNY at Buffalo in 2004. He has served as associate editor
of IIE Transactions since 2005, the executive council member of the China Society of Logistics since
2006, editorial board member of Industrial Engineering Journal (Chinese) since 2011. His current research
interests include supply chain risk management, scheduling and network design. His email address is
[email protected].
ZUO-JUN MAX SHEN is the Chancellor’s Professor in the department of Industrial Engineering and
Operations Research at UC Berkeley. He is also af?liated with Tsinghua University. He received his Ph.D.
from Northwestern University in 2000. He has been active in the following research areas: integrated supply
chain design and management, market mechanism design, applied optimization, and decision making with
limited information. He is currently on the editorial/advisory board for several leading journals. He received
the CAREER award from National Science Foundation in 2003. His e-mail is [email protected].
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doc_560208662.pdf
Batch processes are widely adopted in many manufacturing systems with raw materials from mining or agricultural industries. Due to variations in both raw material quality and market conditions, variations in the recipes are used in production. Such recipe flexibility is not on design but on the operation that allows adjustments of recipe items aiming to achieve better performance than traditionally fixed recipes.
Proceedings of the 2013 Winter Simulation Conference
R. Pasupathy, S.-H. Kim, A. Tolk, R. Hill, and M. E. Kuhl, eds.
A SIMULATION-BASED APPROACH TO INVENTORY MANAGEMENT IN BATCH PROCESS
WITH FLEXIBLE RECIPES
Long He
Industrial Engineering and Operations Research
University of California, Berkeley
Berkeley, CA 94704, USA
Simin Huang
Industrial Engineering
Tsinghua University
Beijing 100084, CHINA
Zuo-Jun Max Shen
Industrial Engineering and Operations Research
University of California, Berkeley
Berkeley, CA 94704, USA
ABSTRACT
Batch processes are widely adopted in many manufacturing systems with raw materials from mining or
agricultural industries. Due to variations in both raw material quality and market conditions, variations in
the recipes are used in production. Such recipe ?exibility is not on design but on the operation that allows
adjustments of recipe items aiming to achieve better performance than traditionally ?xed recipes. In this
paper, we study the inventory investment, recipe selection and resource allocation decisions in batch process
systems with ?exible recipes. A two-stage stochastic mixed integer program formulation is developed for
each period. Moreover, the system updates its inventory investment decisions based on new demand data
from previous periods by a simulation-based approach. Bene?ts of implementing ?exible recipes over
traditional ?xed recipes are investigated in the numerical studies.
1 INTRODUCTION
Oil consumption has been escalating in the past decades, especially in emerging economies regions, such
as Asia. Meanwhile, the dramatically increasing oil price is impeding the growth of the world economy.
Despite its increasing trend, oil price also exhibits high volatility. After it reached the record peak US$
145 in July 2008, it fell signi?cantly to US$ 30.28 a barrel on December 23, 2008. Such increasing trend
together with jumps of prices also prevails in other commodities over the past decades as shown in Figure
1. This phenomenon leads to higher manufacturing costs as well as more dif?culties in supply chain
management under price uncertainty among many industries.
Facing such challenges, joint inventory investment and allocation decision making becomes an important
tool that makes the manufacturing systems robust. Consider an oil re?nery that converts crude oil into
pro?table petroleum products such as gasoline, diesel, kerosene, heating oil and asphalt. Those products are
actually inputs for further manufacturing processes. Generally, it operates in 3 phases: crude oil unloading
and blending, fractionation and reaction processes and product blending and shipping. In the ?rst phase,
crude oil of different grades is transported by crude oil marine vessels from different regions. Since the
properties of crude oil highly depend on its origins, there are usually dedicated storage tanks for crude
oil of different grades. In many situations, before crude oil enters distillation, the ?rst step of production,
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He, Huang, and Shen
Figure 1: Selected Commodity Prices in Past 20 Years.
different grades are blended to achieve certain properties, such as viscosity and density, in order to meet
the production requirements.
The manufacturing process presented above belongs to batch process that primarily schedules short
production runs of products (Connor 1986). Batch process industries often obtain their raw materials from
mining or from agricultural industries. These raw materials have natural variations in quality (Rutten and
Bertrand 1998). Some common batch processes can be found in ?elds such as oil re?ning, agricultural,
chemicals and fertilizers. (Schuster and Allen 1998) illustrates how Welch’s Inc manages grape-processing
among plants using linear program models. In that case, grapes are usually processed in plants located
near growing areas. To maintain national consistency, Welch’s often transfers juice for blending between
plants. The selection of recipes is a key decision that affects the pro?tability via both operational costs and
production capacity. The nature of variations in both raw material quality and market conditions often lead
to the variations in the recipes. Such recipe ?exibility is not on design but on the operation that allows
adjustments of recipe items aiming to achieve better performance than traditionally ?xed recipes. Here,
?exible recipe refers mainly to the adjustments of recipe items as input of batch process in response to
market conditions, i.e. demand arrivals.
In this paper, we simplify the system by considering three types of goods: raw materials, ingredients
and ?nal products. In the grape-processing case, we regard grapes from different growing areas as raw
materials, intermediate juice of different concentration as ingredients and packaged juice on market as ?nal
products. The batch process is simpli?ed into two phases: separation and blending. Since different raw
material grades have various concentration of desired ingredients, in the separation stage, those ingredients
are separated ?rst from raw materials. Then a combination of ingredients are blended into ?nal products
that meet certain speci?cations.
The remainder of this paper is organized as follows: We begin in §2 by reviewing related literature.
In §3, an application of ?exible recipes is illustrated by a simple case. A two-stage stochastic mixed
integer program for the basic model is formulated in §4. We then discuss a simulation-based approach
to the proposed model in §5. In §6, we conduct a numerical study to assess the performance of the
simulation-based approach and bene?ts of ?exible recipes over ?xed recipes. Finally, concluding remarks
are presented in §7.
2 LITERATURE
There are mainly two streams of literature related to our work. In the ?rst stream, given the structure of the
system, optimal investment decisions are analyzed under demand uncertainty and/or inventory procurement
cost variability. (Fine and Freund 1990) investigate the optimal capacity investment problem in single
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He, Huang, and Shen
period by two-stage stochastic programming with discrete demand distribution. Their study focuses on
the case with two products, two dedicated resources, one ?exible resource. While in our study, we allow
the demand distribution to be unknown and the system handles multiple raw materials and ?nal products.
Following the lead of (Fine and Freund 1990), under a two-product ?rm, (Van Mieghem 1998) analyze the
optimal investment in ?exible resources as a function of margins, costs and multivariate demand uncertainty.
Contrary to the previous work, they show that it can be advantageous to invest in ?exible resource even
with perfectly positively correlated product demands. (Harrison and Van Mieghem 1999) study the optimal
investment strategy with a multi-dimensional newsvendor model and conclude a critical fractile property for
the optimal investment levels. Given the structure of an assemble-to-order system, (Akc¸ay and Xu 2004)
formulate the join inventory replenishment and component allocation problem into a two-stage stochastic
program and propose an order-based component allocation rule for the second stage problem.
In the second stream, the applications of ?exible recipes in batch processes are mostly studied. (Rutten
and Bertrand 1998) study the balancing of safety stock costs and recipe ?exibility costs for batch industries
with high customer service requirements. They conclude that under certain circumstances the use of recipe
?exibility can lead to lower costs when compared to using ?xed recipes. (Keesman 1993) investigates the
application of ?exible recipes for batch process optimization and applies an adaptive feedforward control
strategy for a priori known disturbances in the process inputs. Furthermore, a new framework that fully
exploits the inner ?exibility of batch processes at the plant level is developed by (Romero et al. 2003). Their
framework considers a batch recipe model that interacts with a plant-wide model to constitute the ?exible
recipe model. The most related work to ours, under batch process manufacturing, is done by (Karmarkar
and Rajaram 2001). They formulate the grade selection and blending problem as a nonlinear mixed-integer
program with ?xed cost for grade selection and inventory holding cost. However, they assume the annual
demand is known and constant for each ?nal products.
Our work is different from the literature mainly in two ways. First, in our model, the recipe ?exibility
is embedded in the operations of batch processes, rather than the system design as seen in literature on
process ?exibility. Second, we study the decisions of inventory investment, recipe selection and resource
allocation in an integrated model.
3 EXAMPLE OF FLEXIBLE RECIPE APPLICATION
In this section, we brie?y illustrate the application of ?exible recipes in batch process. Consider a
manufacturer, i.e. re?nery or food processing factory, whose operations can be categorized as separation
and blending stages. There are 3 ?nal products made from 3 raw materials. The raw material inventory
is given as Z = (z
1
= 200, z
2
= 300, z
3
= 400) units, where z
i
is the inventory of raw material i. The raw
material cost is C = (c
1
= 6, c
2
= 4, c
3
= 3) dollars per unit, where c
i
is the purchase cost of raw material
i per unit. In the batch process, raw materials are separated ?rst into 3 ingredients, depending on their
concentration in raw materials. The ingredient concentration matrix for raw materials is
P =
0.6 0.3 0.1
0.4 0.4 0.2
0.3 0.4 0.3
where row i represents raw material i and column j represents ingredient j. The element on row i and
column j, denoted by p
i j
, is the proportion of ingredient j contained in a unit of raw material i. Then in
the blending stage, ?nal products are blended from those ingredients. The ingredient requirement matrix
for ?nal products is
A =
0.8 0.2 0
0.7 0.2 0.1
0.6 0.3 0.1
where row k represents ?nal product k and column j represents ingredient j. The element on row k and
column j, denoted by ?
k j
, is the proportion of ingredient j required in a unit of ?nal product k.
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He, Huang, and Shen
In a system that implements ?xed recipes, it determines the optimal ?xed recipe before any demand
arrivals. The resulting optimal ?xed recipe in this case is:
B =
0.53 0.44 1.03
0.40 0.18 1.29
0.32 0.11 1.22
where row k represents ?nal product k and column i represents raw material i. This matrix is similar to
BOM. That is, the element on raw k and column i, denoted by b
ki
, is the amount of raw material i required
in a unit of ?nal product k.
For simplicity, we assume that the demand for each ?nal product follows Bernoulli distribution with 0.5
probability equals 100 units and 0.5 probability equals 200 units. When the system sees demand arrivals, it
determines the optimal ?exible recipes that maximize its revenue with R = (r
1
= 10, r
2
= 8, r
3
= 6) dollars,
where r
k
represents the revenue of a unit ?nal product k.
Table 1 summarizes the computational results of expected pro?t and shows that the ?exible recipes enable
Table 1: Expected Pro?t Summary (in dollars)
Flexible Recipes (R1) Fixed Recipes (R2) Improvements=
R1?R2
R2
Current Setting 3525 2904.6 21.36%
Demand ×2 4350 3199.8 35.95%
Demand ×0.8 2880 2643.7 8.94%
the system to achieve higher pro?t via better resource utilization, especially under the presence of large
demand variance.
4 THE BASIC MODEL
As shown in Figure 2, the batch process consists of two stages: in the separation stage, raw materials are
processed into a set of ingredients; in the blending stage, a selection of ingredients are blended into ?nal
products to ful?ll the demands. In the oil re?nery industry, for instance, the raw materials are different crude
oil grades. The three most quoted oil grades are North America’s West Texas Intermediate crude (WTI),
North Sea Brent Crude, and the UAE Dubai Crude. Depending on the mixture of hydrocarbon molecules,
crude oil varies in color, composition and consistency. Different oil-producing areas yield signi?cantly
different varieties of crude oil. The ingredients are the intermediate products such as light ends, naphtha,
kerosene, distillate, atmospheric residua, vacuum gas oil and vacuum residua. Final products are various
gasoline types, lubricants, petrochemicals, diesel, asphalt, etc,.
Figure 2: Simpli?ed Batch Process
Our model is to determine the inventory level at the beginning of each period before any demand
arrivals. We assume all leftover inventory by the end of each period is disposed because of time sensitivity
of raw materials, i.e. perishability in grape-processing. As a result, at the beginning of each period t, we
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He, Huang, and Shen
only need to solve a subproblem of single period planning with available historical data up to period t.
Therefore, we focus on solving a single period problem.
There are basically three types of costs: raw material procurement cost, grade selection cost and ?nal
product revenue. Similar to the de?nition by (Karmarkar and Rajaram 2001), the grade selection cost incurs
when a grade is selected by a recipe into separation stage and the cost varies by grades. Raw material
procurement cost can be the real option or spot market price. The model we propose is able to optimize
investment on raw materials together with ?exible recipe selection for production when the system sees
demands. It can be easily extended to the systems with many separation and blending stages in serial
structure.
The event sequence is illustrated in Figure 3. At the beginning of period t, the system ?rst invests in
raw material inventory with total procurement cost ?
i
c
i
z
i
, where c
i
is the unit cost of raw material i and
z
i
is the amount of raw material i purchased. After the system sees the demand arrivals D = (d
k
), where
d
k
is the demand of ?nal product k, it then selects the optimal recipe to satisfy the demands. Under the
presence of grade selection cost, it is not always pro?table to satisfy as much demand as possible. As
discussed above, the recipe selection cost is the sum of grade selection costs, given by ?
i
f
i
y
i
, where f
i
is the grade selection cost of raw material i and y
i
is the selection decision of raw material i. Lastly, the
ful?llment of demands generates total revenue ?
k
r
k
x
k
, where r
k
is the unit revenue of ?nal product k and
x
k
is the ful?lled demand of ?nal product k. We assume that material transformation processes are linear.
Another assumption is that the material loss in separation and blending stages are negligible. Indeed, if the
loss is consistent and fractional in the process, we can introduce some discount factors in the formulation
so that the structure of the model is still preserved.
Figure 3: Event Sequence Diagram in Period t
The decision making process for the system with ?exible recipes differentiates itself from the newsven-
dor model with the postponement in ?nal product blending. In the newsvendor model, the ?nal products
are produced ahead and the total manufacturing cost can be assessed before demand arrivals. As a result,
in the newsvendor model, the critical fractiles and subsequently the raw material inventory levels can be
determined in advance. While in a batch process with ?exible recipes, the system makes tradeoffs between
revenue and recipe selection cost after demand arrivals.
We formulate the basic model as a two-stage stochastic mixed-integer program:
?= maxE
D
?(Z, D) ?
?
i?I
c
i
z
i
(1)
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He, Huang, and Shen
where
?(Z, D) =max
?
k?K
r
k
x
k
?
?
i?I
f
i
y
i
(2)
s.t.
?
k?K
?
k j
x
k
?
?
i?I
p
i j
z
i
y
i
, ?j ? J (3)
x
k
? d
k
, ?k ? K (4)
y
i
? {0, 1}, ?i ? I (5)
Uppercase is for vectors while lowercase is for scalars. The ?rst stage of the formulation, equation (1),
maximizes the expected total pro?t of the system. Given rawmaterial inventory level vector Z = (z
1
, z
2
, ..., z
I
)
and demand arrival vector D = (d
1
, d
2
, ..., d
K
), the second stage with recourse maximizes as in equation (2)
the total revenue minus total grade inclusion cost. The constraint (3) states that the supply of each ingredient
?
i?I
p
i j
z
i
y
i
must be no less than the ingredient requirement for producing X amount ?nal products which is
?
k?K
?
k j
x
k
. Here ?
k j
is the amount of ingredient j required to produce a unit of ?nal product k and p
i j
is the
amount of ingredient j contained in a unit of raw material i. Because the initial inventory investment is sunk
cost, (z
1
y
1
, z
2
y
2
, ..., z
I
y
I
) shows that the system chooses to use up all selected raw materials. Constraint (4)
represents that demand ful?llment can not exceed the demand arrivals. The recipe selection is determined
by the binary decision vector Y expressed in constraint (5).
5 SIMULATION-BASED OPTIMIZATION
Our solution approach to the proposed two-stage stochastic mixed-integer program consists of two modules:
demand simulator and SAA optimizer, as shown in Figure 4. Given the available historical demand data,
the demand simulator generates simulated demand arrivals for period t. There are several ways to simulate
or forecast demands based on previous information. Here, we use Bootstrap sampling in the demand
simulator.
Figure 4: Solution Approach
With the simulated demand arrivals, the second module ?nds the optimal inventory levels by Sample
Average Approximation (SAA), an simulation-based approach. The algorithm is modi?ed from the SAA
method provided in (Akc¸ay and Xu 2004). A detailed introduction of the SAA method can be found
in (Shapiro and Homem-de Mello 1998). Let ?(Z
?
) be the optimal solution to the two-stage stochastic
program in the basic model (1)-(5) and Z
?
be the associated optimal inventory levels. We start with
generating M independent samples of random vector D from the demand simulator, each of size N. That
is, D
l
= (D
l,1
, D
l,2
, ..., D
l,N
) is the realization of the lth sample, where D
l,h
= (d
l,h
1
, d
l,h
2
, ..., d
l,h
K
) with d
l,h
k
as the realization of demand for ?nal product k in the hth vector of the lth sample realization. We solve
the SAA problem referring to each sample, as below:
max
Z
l
?
N
(Z
l
) =
1
N
N
?
h=1
?(Z
l
, D
l,h
) ?
?
i?I
c
i
z
l
i
(6)
s.t.
Constraints (3) ?(5) for each D
l,h
, h = 1, ..., N (7)
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He, Huang, and Shen
where Z
l
= (z
l
1
, z
l
2
, ..., z
l
I
)
For l = 1, ..., M, let ?
N
(
ˆ
Z
l
) be the corresponding optimal solution to the above SAA problem and
ˆ
Z
l
be the associated optimal inventory investment. Since Z
?
is always a feasible solution to (6)-(7), we have
?
N
(
ˆ
Z
l
) ??(Z
?
) for all l = 1, ..., M. We then have,
E[
¯
?
N
] ??(Z
?
), where
¯
?
N
=
1
M
M
?
l=1
?
N
(
ˆ
Z
l
)
Therefore, E[
¯
?
N
] is used as the estimate of an upper bound of ?(Z
?
).
In order to have an unbiased estimator of ?(
ˆ
Z
l
), we again generate one large number of independent
sample from the demand simulator, D
N
= (D
1
, D
2
, ..., D
N
). Then, for each inventory level vector
ˆ
Z
l
,
compute the estimate of ?(
ˆ
Z
l
) by
ˆ
?
N
(
ˆ
Z
l
) =
1
N
N
?
h=1
?(
ˆ
Z
l
, D
h
) ?
?
i?I
c
i
ˆ z
l
i
where ?(
ˆ
Z
l
, D
h
) is the optimal solution to the second stage allocation optimization with inventory vector
ˆ
Z
l
and demand realization D
h
. The estimated optimal inventory vector
ˆ
Z
?
is then determined by choosing
the one that gives the largest
ˆ
?
N
(
ˆ
Z
l
) among all candidate inventory vectors
ˆ
Z
l
with sampled demand
realizations, as follows:
ˆ
Z
?
? argmax{
ˆ
?
N
(
ˆ
Z
l
), l = 1, ..., M}
Since
ˆ
Z
?
is a feasible solution to the basic model (1)-(5), we further have
E[
ˆ
?
N
(
ˆ
Z
?
)] ??(Z
?
)
As a result, E[
ˆ
?
N
(
ˆ
Z
?
)] can serve as a lower bound of ?(Z
?
). Thus, the difference between E[
¯
?
N
] and
E[
ˆ
?
N
(
ˆ
Z
?
)] is an estimate of the optimality gap of SAA solution. In brief, the SAA method works in the
following procedure:
Step 0 Determine appropriate values for N, M and N
, and initialize l = 0;
Step 1 Set l = l +1 and generate an independent sample D
l
= (D
l,1
, D
l,2
, ..., D
l,N
); Solve the SAA
problem (6)-(7) for
ˆ
Z
l
and ?
N
(
ˆ
Z
l
); If l < M, go to Step 1; otherwise, go to Step 2;
Step 2 Generate an independent sample D
N
= (D
1
, D
2
, ..., D
N
); Initialize l = 0;
Step 3 Set l = l +1 and solve the SAA problem with D
N
and
ˆ
Z
l
for
ˆ
?
N
(
ˆ
Z
l
); If l < M, go to Step 2;
otherwise go to Step 4;
Step 4 Choose
ˆ
Z
?
? argmax{
ˆ
?
N
(
ˆ
Z
l
), l = 1, ..., M};
The quality of the solution, measured by the optimality gap, improves as the sample sizes N and N
grow. However, larger sample sizes require higher computational capacity. Therefore, tradeoff between
sample sizes and computational effort need to be considered.
6 NUMERICAL STUDY
In this section, we apply the proposed approach to a real-world ?our manufacturing system with one-year
demand data and some scaled cost values. The system produces 18 kinds of ?our “A” to “R” for different
uses from 3 grades of wheat numbered “1” to “3” from different origins. The ingredients are mainly
starch, protein and ?ber. The wheat with higher protein concentration costs more. The costs, ingredient
concentration, and requirement matrices are summarized in Table 2.
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He, Huang, and Shen
Table 2: Parameters in Numerical Study
Raw material cost
($/kg)
Grade selection cost
(10
3
$ )
Price
($/kg)
Starch
(100%)
Protein
(100%)
Fiber
(100%)
Wheat 1 2.10 2.0 0.80 0.10 0.10
Wheat 2 1.83 2.5 N/A 0.60 0.15 0.25
Wheat 3 1.78 3.0 0.50 0.30 0.20
Product A 2.63 0.98 0 0.02
Product B 2.43 0.95 0 0.05
Product C 2.21 0.72 0.1 0.18
Product D 2.38 0.88 0 0.12
Product E 2.40 0.88 0.02 0.1
Product F 2.19 0.8 0 0.2
Product G 2.15 0.68 0.15 0.17
Product H 2.41 0.93 0 0.07
Product I 2.06 0.75 0 0.25
Product J N/A N/A 1.79 0.65 0 0.35
Product K 1.77 0.2 0.7 0.1
Product L 2.33 0.76 0.1 0.14
Product M 2.13 0.57 0.2 0.23
Product N 2.37 0.57 0.33 0.1
Product O 2.23 0.33 0.52 0.15
Product P 2.41 0.3 0.65 0.05
Product Q 2.36 0.09 0.87 0.04
Product R 1.48 0 1 0
The sample demand arrivals are generated by the demand simulator module, which implements Bootstrap
sampling in the current setting. The SAA optimizer module is realized via CPLEX solver with N = 100,
M = 30 and N
= 500. We compute the average pro?t, inventory investment in dollar value and gaps for
various parameter settings listed in Table 3. As mentioned in the SAA algorithm, the gap is de?ned as the
difference between the upper and lower bounds. The upper bound is estimated by
¯
?
N
=
1
M
?
M
l=1
?
N
(
ˆ
Z
l
) and
the lower bound is estimated by
1
N
?
N
h=1
?(
ˆ
Z
?
, D
h
) ??
i?I
c
i
ˆ z
?
i
. For current parameter setting, the optimal
solution given by our approach is $2721.74×10
6
with inventory levels [5718.194;0;4041.851] for wheat
1, 2 and 3 respectively. We summarize the computational results of run 1 to 5 for ?exible recipe system
in Table 4.
Table 3: Experiment Setting of Selected Runs
Wheat costs Grade selection costs
Run 1 [1.680;1.464;1.424] [2;2.5;3]
Run 2 [1.890;1.647;1.602] [2;2.5;3]
Run 3 [2.100;1.830;1.780] [2;2.5;3]
Run 4 [2.310;2.013;1.958] [2;2.5;3]
Run 5 [2.520;2.196;2.136] [2;2.5;3]
Table 4: Results for Flexible Recipes
Avg. pro?t (10
6
$) Inv. (10
6
$ ) Gap
Run 1 6750.69 17631.84 1.82%
Run 2 4666.52 18948.49 0.91%
Run 3 2721.74 19275.04 0.05%
Run 4 938.55 18000.56 0.41%
Run 5 30.22 3310.80 0.15%
For the selected run 1 to 5, all gaps are less than 2%. This suggests the good performance of the
simulation-based approach with our choice of N, M and N’. If the gap is big, the number of samples N
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He, Huang, and Shen
and N’ should be increased accordingly. Besides, both the average pro?t and total inventory investment
decrease convexly as the raw material cost increases. This implies that the system tends to stock less
inventory when raw materials cost is high. Recall that in newsvendor model, the critical fractile decreases
linearly with production cost and thus the inventory level also decreases convexly if the demand distribution
is concave, i.e. Normal distribution.
The impact of large grade selection cost is illustrated in Figure 5. The increasing grade selection costs
decrease the average pro?t at a mild rate. Meanwhile, grade selection cost increase does not change the
inventory investment signi?cantly. Since the grade selection costs are only scaled by multipliers, their
relative ranking among different rawmaterials are not changed. As a result, the preference among wheats are
not much affected. This makes the inventory investment decision and average revenue almost unchanged.
Therefore, the average pro?t, which equals average revenue less inventory investment and grade selection
costs, decreases linearly in the multipliers of grade selection costs.
Figure 5: Average Pro?t and Inventory Investment for Different Grade Selection Costs
We consider three simple ?xed recipes. That is, ?xed recipes with single source: wheat 1, 2 or 3. Table
5 summarizes the computational results of the average pro?t, inventory investment as well as the ratio
between average pro?t of the ?exible recipes and those of the ?xed recipes. For all experiments, ?exible
recipes can always achieve larger average pro?t than ?xed recipes. For example, in run 1, the “best” ?xed
recipe can at most generate 86.43% of the average pro?t of ?exible recipe. Meanwhile, in run 5, which is
an extreme case, wheat 1 and 2 are too costly to be used as raw materials. This leaves wheat 3 as the only
choice as raw material for the ?exible recipe. Therefore, in run 5, the ?exible recipe is equivalent to the
?xed recipe with wheat 3. In addition, as we see from run 1 to 3, the optimal solution of the ?exible recipes
requires not much more or even signi?cantly less inventory investment than the “best” ?xed recipes. That
is, ?exible recipes achieve higher average pro?t with lower inventory investment than ?xed recipes.
Table 5: Comparison of Flexible Recipes and Fixed Recipes
Flexible recipe Fixed recipe: wheat 1 Fixed recipe:wheat 2 Fixed recipe: wheat 3
Avg.
pro?t
(10
6
$)
Inv.
(10
6
$ )
Avg.
pro?t
(10
6
$)
Ratio Inv.
(10
6
$ )
Avg.
pro?t
(10
6
$)
Ratio Inv.
(10
6
$ )
Avg.
pro?t
(10
6
$)
Ratio Inv.
(10
6
$ )
Run 1 6750.69 17631.84 4869.70 72.14% 14188.77 5834.94 86.43% 18140.89 5039.26 74.65% 16165.77
Run 2 4666.52 18948.49 3244.55 69.53% 14384.60 3551.48 76.11% 18677.88 3151.62 67.54% 14579.06
Run 3 2721.74 19275.04 1695.32 62.29% 14035.47 1595.87 58.63% 19062.30 1786.61 65.64% 12421.80
Run 4 938.55 18000.56 448.34 47.77% 11636.23 85.50 9.11% 7330.68 695.27 74.08% 10315.34
Run 5 30.22 3310.80 0 0% 0 0 0% 0 29.62 98.02% 3256.07
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7 CONCLUSION AND FUTURE WORK
In this paper, we propose a two-stage stochastic mixed-integer program to an inventory management
problem in batch process with ?exible recipes. In the ?rst stage, the model determines inventory levels
for each period based on past demand data. After demand arrivals are realized, the second stage recourse
makes recipe selection and allocation decisions in production. With available historical demand data, a
simulation-based approach based on SAA algorithm is developed to solve the stochastic program. As
the historical demand data updates along the time, the inventory levels are set iteratively in each period
using the most updated demand information. The results of numerical study show the performance of the
approach on various cost settings as well as the bene?ts of ?exible recipes over ?xed recipes.
In the proposed approach, we focus on the application of the SAA algorithm and use Bootstrap
sampling as the default in demand simulation. A direction of future improvement is to incorporate better
techniques in the simulation of future demand arrivals based on historical demand data. Those techniques
may consider some properties of the demand, such as seasonality and autocorrelation. Also, with limited
demand information, a robust optimization model might be developed that considers the worst cases.
Moreover, since our model assumes any inventory leftover at the end of each period is disposed, the
extension that relaxes this assumption and introduces inventory holding cost in multi-period setting should
also be investigated.
ACKNOWLEDGEMENTS
This research was partially supported by the National Science Foundation Grants CMMI 1068862, CM-
MI1031637, CMMI1265671 and the National Science Foundation of China Grants 71071084, 71128001,
71210002.
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AUTHOR BIOGRAPHIES
LONG HE received the B.Eng degree in Logistics Management and Engineering from The Hong Kong
University of Science and Technology, Hong Kong in 2010. He is currently pursuing the Ph.D. degree in
Industrial Engineering and Operations Research from the University of California at Berkeley, Berkeley CA,
United States. His research interests includes supply chain management, capacity planning and inventory
control. His email address is [email protected].
SIMIN HUANG is a Professor and Associate Head of the Department of Industrial Engineering, Tsinghua
University. He received his Ph.D. from SUNY at Buffalo in 2004. He has served as associate editor
of IIE Transactions since 2005, the executive council member of the China Society of Logistics since
2006, editorial board member of Industrial Engineering Journal (Chinese) since 2011. His current research
interests include supply chain risk management, scheduling and network design. His email address is
[email protected].
ZUO-JUN MAX SHEN is the Chancellor’s Professor in the department of Industrial Engineering and
Operations Research at UC Berkeley. He is also af?liated with Tsinghua University. He received his Ph.D.
from Northwestern University in 2000. He has been active in the following research areas: integrated supply
chain design and management, market mechanism design, applied optimization, and decision making with
limited information. He is currently on the editorial/advisory board for several leading journals. He received
the CAREER award from National Science Foundation in 2003. His e-mail is [email protected].
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