Risk Management in Production Planning of
Perishable Goods
Pedro Amorim,∗,† Douglas Alem,‡ and Bernardo Almada-Lobo†
INESC TEC, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias,
s/n, 4600-001 Porto, Portugal, and Production Engineering Department, Universidade
Federal de São Carlos, Rodovia João Leme dos Santos (SP-264), Km 110, Sorocaba, SP
18052-780, Brazil
E-mail: [email protected]
Abstract
In food supply chain planning, the trade-off between expected profit and risk is emphasized by the perishable nature of the goods that it has to handle. In particular, the
risk of spoilage and the risk of revenue loss are substantial when stochastic parameters
related to the demand, the consumer behavior and the spoilage effect are considered.
This paper aims to expose and handle this trade-off by developing risk-averse production planning models that incorporate financial risk-measures. In particular, the
performance of a risk-neutral attitude is compared to the performance of models taking
into account the upper partial mean and the conditional value-at-risk. Insights from an
illustrative example show the positive impact of the risk-averse models in operational
performance indicators, such as the amount of expired products. Furthermore, through
an extensive computational experiment, the advantage of the conditional value-at-risk
∗
To whom correspondence should be addressed
INESC TEC
‡
Universidade Federal de São Carlos
†
1
model is evidenced, as it is able to dominate the solutions from the upper partial
mean for the spoilage performance indicator. These advantages are tightly related to
a sustainable view of production planning and they can be achieved at the expense of
controlled losses in the expected profit.
Introduction
Supply chains of perishable food goods are becoming more global and complex than ever.
Customers of such goods demand an increasing variety of products with high freshness standards as well as all year round supply of exotic goods. Furthermore, customers have become
more aware and concerned about product quality, safety and overall supply sustainability 1 .
Companies competing and cooperating in these supply chains have to deal with several risk
sources that have to be properly managed when planning their activities. Ignoring the impact
of these risk sources may yield disastrous supply chain disruptions, such as the interdiction
of selling a certain product or a considerable amount of spoiled inventory.
Most of the research on supply chain risk management has focused primarily, on a decision
level perspective, either at the strategic (long term) or at the tactical (medium term) level 2 .
Regarding the supply chain processes, it is in the distribution process that more work has
been developed. However, empirical data suggest that one key risk that the supply chain of
perishable goods faces – the risk of spoilage – has to be mitigated in the production process at
the operational decision level. In fact, the European Commission estimates that 39 percent
of total food spoilage, excluding loss at the farm level, is generated at the processing stage 3 .
According to Pfohl et al. 4 , supply chain risk management consists in a collaborative and
structured approach to risk management, embedded in the planning and control processes of
the supply chain, to handle the risks that might adversely affect the achievement of supply
chain goals. The food manufacturer goal is usually to maximize its profit in a sustainable
manner. To do so it is important to account for the multiple uncertainty sources that may
affect his operation 5 , such as traveling times, processing times, demand, decay rates or shelf2
lives. These uncertainties result directly in a multitude of risks that need to be mitigated
in order to build a sustainable competitive advantage. To list some of the specific risks of
food supply chains, it is worth mentioning the risk of contamination, spoilage, stock out and
the money tied up in inventory. Note that these risks can be correlated among themselves.
For example, in order to decouple the production and distribution processes, stock has to
be built and there is a risk of having too much capital in inventory that is emphasized by
the risk of spoilage that these products naturally yield. One strategy for mitigating these
risks is to modify the traditional mathematical models for supply chain planning in order
to account for them explicitly in the corresponding formulations. This may be achieved by
modifying the objective functions to minimize batch dispersion and decrease the impact of
the risk of contamination 6,7 , or to maximize goods’ quality and tackle the risk of spoilage 8 .
This paper intends to assess the suitability of financial risk-measures for mitigating crucial
risks in the production planning of perishable food goods. Figure 1 frames the scope of this
research in light of the previous discussion. From several risk sources, we consider uncertainty
in the demand level, decay rates and consumer purchasing behavior in face of products with
different ages. The randomness of these factors has a direct impact on the risk of spoilage and
revenue loss. These risks are to be managed by the producers of perishable food goods and we
analyze the differences in the effectiveness of risk mitigation of two financial risk-measures:
upper partial mean and conditional value-at-risk. To this end, we start by proposing a twostage stochastic model that incorporates the mentioned stochastic parameters / risk sources.
This model is further extended to integrate the risk-averse perspectives. The contribution of
this paper is aligned with the gap pointed out by Seshadri and Subrahmanyam 9 , namely the
fact that models that are able to quantify and concretize the amount of conceptual research
in supply chain risk management are promised to be of great use.
The remainder of this paper is organized as follows. The next section reviews the work
related to production planning of perishable goods, uncertainty in production planning and
consumer purchasing behavior of perishable products. Section develops the risk-neutral
3
Risk Mitigating
Strategy
...
Risk
...
Uncertainty /
Risk Source
...
Incorporate
Financial Risk
Measures
Force
Traceability
Contamination
Raw Material
Availability/
Quality
Spoilage
Revenue Loss
Decay Rates
Demand Level
Minimize Batch
Dispersion
Stock Out
Consumer
Behaviour
(towards
perishability)
Processing Times
...
...
...
Figure 1: Conceptual framework of this paper from risk sources to risk mitigating strategies.
production planning model for perishable goods and clarifies some stochastic programming
concepts. This model serves as the basis for the risk-averse models developed in Section
. Afterwards, Section describes the computational experiments and discusses the results.
Section indicates the main conclusions and future research directions.
Literature Review
The literature review is divided in two topics: (i ) production planning of perishable goods
and risk management; (ii ) mathematical expressions that describe the consumer purchasing
behavior of perishable food goods.
Production Planning with Perishability and Risk
In order to account explicitly for the perishability of food products, the formulation of the
production planning problem has to keep track of the age of inventories and/or products
sold. An example of a work that deals with perishability is found in Marinelli et al. 10 . In
this work a solution approach for a real-world capacitated lot-sizing and scheduling problem
with parallel machines is proposed. The underlying industry produces yogurt and the model
accounts for perishability by using a make-to-order production strategy. Obviously, this
4
production strategy will ensure a high freshness standard of the products delivered, however,
in the fast moving consumer goods, this policy can be very hard to implement due to the
large variety of products.
Still in the yogurt packaging industry, Lütke Entrup et al. 11 were probably the first to
include perishability in a capacitated production planning model with dynamic demand.
Based on the block planning approach, three mixed-integer linear programming models that
integrate shelf-life issues into the planning of packaging stage were proposed. More recently,
in Pahl and Voß 12 and Pahl et al. 13 , well-known lot-sizing and scheduling models were
extended by including deterioration and perishability constraints. These extended formulations have included the capacitated lot-sizing problem, discrete lot-sizing and scheduling
problem, continuous setup lot-sizing problem, proportional lot-sizing and scheduling problem
and the general lot-sizing and scheduling problem. One of the key insights of these works
is the importance of the minimum batch size constraint in the amount of spoiled inventory.
Amorim et al. 8 partially conserved the constraints developed in the previous works and proposed a multi-objective framework to differentiate between cost minimization and freshness
maximization. Therefore, the result of the lot-sizing and scheduling problem is a Pareto
front trading off these two dimensions. More oriented towards practice, Kopanos et al. 14
and Kopanos et al. 15 developed efficient models for production planning problems in the
ice-cream industry. These models incorporate key elements in the food production planning,
such as a multi-stage setting.
Although the risk of spoilage is especially imminent in the food supply chain, the importance of risk management tools has not been assessed in the corresponding production
planning literature. Roughly speaking, quantitative risk management approaches consist of
developing mathematical expressions to reflect risk-aversion, i.e., decision maker’s preferences towards risk. The so-called risk-averse models generate low-variability solutions that
avoid not meeting a certain target profit from a financial perspective 16–21 .
Amongst the various risk-averse methods, mean-risk models have been widely used to deal
5
with risk mitigation 22–24 . Basically, mean-risk models optimize simultaneously the expected
outcome E(·) and the dispersion of the outcomes D(·), à lá Markovitz 25,26 :
max
E(Qξ ) − φ · D(Qξ ),
(1)
where Q(ξ) is the random outcome. The “user-controlled” parameter φ ∈ R+ provides
risk preferences by trading-off profit and risk. High-variability solutions with high expected
profits are obtained when φ → 0. As φ → ∞, low-variability solutions are achieved at the
expense of profit losses.
Typically, to mitigate the variability of the second-stage or recourse costs, the dispersion
D(·) is modeled via variance, standard deviation and/or semideviations 27 . More recently,
there has been a great effort in studying mean-risk models based on financial measures, as the
value-at-risk (VaR) and the conditional value-at-risk (CVaR). The motivation for using VaR
and CVaR is to avoid solutions influenced by a very pessimistic scenario, as both measures
rely on specific percentile of the worst-case realizations of the random variables. However,
there is no risk approach that is unrestrictedly recommended for general problems 28 .
Consumer Purchasing Behavior of Perishable Goods
Tsiros and Heilman 29 performed empirical research in order to analyze the effects of perishability on the purchasing pattern of customers across different perishable products. The
conclusions of this study indicate that customer willingness to pay (WTP) decreases throughout the course of the products’ shelf-life; this decrease follows a linear function for products
with a low product quality risk (PQR) and an exponential negative function for products
with a high PQR. PQR is defined as the expected negative utility associated with a given
product as it reaches its expiry date, and WTP is the maximum price a customer is willing
to pay for a given product.
For operational production planning problems (as the one under study in this paper), it
6
is assumed that customers’ WTP is subjected to a fixed price and, therefore, the parameter
that reflects the consumer behavior is the demand. In Amorim et al. 30 the deduction to
convert WTP in function of the age to demand in function of the age is described. Two
functions (one for low PQR and one for high PQR) were used. They have a similar behavior
as they are monotonically decreasing, having its maximum value for the product with a
maximum freshness (p0 ) and a value of 0 at the end of shelf-life u. The closeness of the
WTP to 0 monetary units as the product reaches its shelf-life is controlled by parameter
α. This parameter, which varies between 0 and 1, represents the customer sensitivity to the
decaying freshness of the product.
Equations (2)-(3) describe the WTP functions that are empirically studied in Tsiros and
Heilman 29 . Figures 2 and 3 show the impact of the customer related parameters (p0 and α)
in the WTP curve throughout the age of the product.
αp0 a
u−1
(2)
αp0 a
a
(2 −
)
u−1
u−1
(3)
WTP for products with a Low PQR = p0 −
WTP for products with a High PQR = p0 −
3
3
2.5
2.5
2
p0 (3)
p0 (2)
1
p0 (1)
0.5
WTP
WTP
2
1.5
α (1)
1.5
α (0.5)
1
α (0)
0.5
0
0
0
1
2
3
4
5
0
Age (a)
1
2
3
4
5
Age (a)
(a)
(b)
Figure 2: Impact of varying customer related parameters p0 (a) and α (b) for products with
Low PQR.
In the left figures p0 is varied in the set {1, 2, 3} and in the right ones there is a variation
of α in the set {0, 0.5, 1}. For all of them, a shelf-life u equal to 5 is considered. Note that
in each function the price is represented up to age 4, since at age 5 the products spoil and
7
3
3
2.5
2.5
2
WTP
WTP
2
p0 (3)
1.5
p0 (2)
1
α (1)
1.5
α (0.5)
1
p0 (1)
α (0)
0.5
0.5
0
0
0
1
2
3
4
5
0
1
2
Age (a)
3
4
5
Age (a)
(a)
(b)
Figure 3: Impact of varying customer related parameters p0 (a) and α (b) for products with
High PQR.
can no longer be sold. For the same parameters setting the products with a High PQR have
a WTP always below the one related to Low PQR, since the WTP drops very fast as soon
as the product is produced.
Using the WTP functions (2) and (3), it was then possible to derive the corresponding
demand functions for a fixed price p̂ and a price elasticity of demand (see (4) and (5)).
The readers are referred to Amorim et al. 30 for the complete proof.
0
a
p̂ + αp
u−1 )
Demand for products with a Low PQR = d (
p̂
0
0
Demand for products with a High PQR = d (
p̂ +
αp0 a
(2
u−1
p̂
−
(4)
a
)
u−1 )
(5)
A Risk-Neutral Production Planning Model for Perishable Goods
This section presents the risk-neutral two-stage stochastic production planning model to deal
with perishable food goods that considers uncertainty in consumers’ purchasing behavior,
demand levels and spoilage rates simultaneously. Let k = 1, ..., K be the products that
are produced. Products are scheduled on parallel production lines l = 1, ..., L over a finite
planning horizon that is divided in periods t = 1, ..., T . These periods correspond to days,
weeks or months. In food production planning the sequence of products is usually defined.
8
Therefore, only sequence independent setup times and costs are considered here. Each
product has a given shelf-life (uk ), after which it cannot be sold. The demand for a product
depends on its age and products may spoil, decreasing the respective stock availability.
The stochastic data are modeled on some probability space (W, F, Π), where W is a set
of discrete outcomes or scenarios with corresponding probabilities of occurrence πω , such that
P
πω > 0 and ω πω = 1, equipped with a σ-algebra of events F and a probability measure Π.
According to the two-stage stochastic program methodology, we define the production lots
and the setup schedule as first-stage decisions. Inventory and demand satisfaction policies
are then the second-stage decisions. Consider the following indices, parameters, and decision
variables that are used in the stochastic formulation.
Indices
l ∈ [L]
parallel production lines
k ∈ [K]
products
t ∈ [T ]
periods
a ∈ [A]
ages (in periods)
ω ∈ [W ]
scenarios
Deterministic Parameters
9
Clt
capacity (time) of production line l available in period t
elk
capacity consumption (time) needed to produce one unit of
product k on line l
clk
production costs of product k (per unit) on line l
p¯k
opportunity cost of producing product k as it gets spoiled
uk
shelf-life duration of product k right after being produced
(time)
mlk
minimum lot size (units) of product k when produced on
line l
s̄lk (τ̄lk )
setup cost (time) of a changeover to product k on line l
pˆk
price of each product k sold
k
price elasticity of demand for product k
Stochastic Parameters
daktω demand for product k with age a (> 0) in period t in scenario
ω (calculated using (4) and (5))
d0ktω
demand for product k with age 0 in period t in scenario ω
αkω
customer’s sensitivity to the ageing of product k in scenario ω
p0kω
willingness to pay for product k in its fresher state in scenario
ω
βkω
spoilage rate for product k in scenario ω
First-Stage Decision Variables
qlkt
quantity of product k produced in period t on line l
ylkt
equals 1, if line l is set up for product k in period t (0 otherwise)
Second-Stage Decision Variables
10
a
wktω
initial inventory of product k with age a available at period t
in scenario ω, a = 0, ..., min{uk , t − 1}
a
ψktω
fraction of the demand for product k delivered with age a at
period t in scenario ω, a = 0, ..., min{uk − 1, t − 1}
a
θktω
equals 1, if inventory of product k with age a is used to satisfy demand in period t in scenario ω (0 otherwise), a =
0, ..., min{uk − 1, t − 1}
In Table 1 the domain and relation between second-stage decision variables is illustrated
with an example for a given product k with a shelf-life of 2 in scenario ω. Regarding the
domains of the decision variables, notice that they are dynamic with the advancement of
the planning periods. Since we have uk = 2, it is not possible to sell products with this age
2
a
a
(ψktω
= 0). The value of θktω
is strictly linked to ψktω
, which used the production (qlkt ) and
a
the inventory (wktω
) to fulfill demand (daktω ). For example, in period t = 1 the demand (4)
0
is completely fulfilled with fresh products (ψk1ω
= 1). As the production output is 10 in the
1
first period (qlk1 = 10), the inventory in period 2 with age 1 turns out to be 6 (wk2ω
= 6).
2
Remark that in this solution, in period 3 we already have 5 spoiled products (wk3ω
= 5).
Table 1: Illustrative example of the domain and relation between second-stage decision
variables.
a
wktω
qlkt
t
a
ψktω
a
θktω
daktω
a
−
0
1
2
0
1
2
0
1
2
0
1
2
1
2
3
10
10
5
10
10
5
−
6
6
−
−
5
1
0.8
0.5
−
0.2
0.5
−
−
−
1
1
1
−
1
1
−
−
−
4
5
6
2
1
3
0
0
0
Consider the vector Z that contains all decision variables. The risk-neutral two-stage
stochastic program for production planning of perishable food under influence of consumer
purchasing behavior (PP-P-RN) reads as follows:
11
(PP-P-RN)
g(Z, ξ) = max −
X
(s̄lk · ylkt + clk · qlkt ) + E{Q[q, y, ξ(ω)]}
(6)
∀l ∈ [L], k ∈ [K], t ∈ [T ]
(7)
l,k,t
subject to:
qlkt ≤
X
Clt
· ylkt
elk
(τ̄lk · ylkt + elk · qlkt ) ≤ Clt
∀l ∈ [L], t ∈ [T ]
(8)
k
qlkt ≥ mlk · ylkt
∀l ∈ [L], k ∈ [K], t ∈ [T ]
qlkt ≥ 0; ylkt ∈ {0, 1} ∀l ∈ [L], k ∈ [K], t ∈ [T ]
where the expectation E{·} is evaluated as
P
ω
(9)
(10)
πω · Q[q, y, ξ(ω)] under finite number of
scenarios, ξ(ω) = [dω , αω , p0ω , β ω ] are the data of the second stage problem and Q[q, y, ξ(ω)]
is the optimal value of the following problem:
max
X
a
a
a
pˆk · ψktω
· d0ktω − p¯k · βkω · (wktω
− ψktω
· d0ktω )
k,t,a
−
X
p¯k ·
uk
wktω
(11)
k,t
X
a
ψktω
≤ 1 ∀k ∈ [K], t ∈ [T ], ω ∈ [W]
a∈[A]
12
(12)
a
· d0ktω ≤ daktω
ψktω
a
a
≤ θktω
ψktω
∀k ∈ [K], t ∈ [T ], a ∈ [A], ω ∈ [W]
∀k ∈ [K], t ∈ [T ], a ∈ [A], ω ∈ [W]
(13)
(14)
a−1
a−1
a
wktω
− ψktω
· d0ktω ≤ (1 − θktω
)·M
∀k ∈ [K], t ∈ [T ], a ∈ [A] \ {0}, ω ∈ [W]
(15)
a−1
a−1
a
wktω
= (wk,t−1,ω
− ψk,t−1,ω
· d0k,t−1,ω ) · (1 − βkω )
∀k ∈ [K], t ∈ [T + 1], a ∈ [A] \ {0}, ω ∈ [W]
(16)
X
(17)
0
qlkt = wktω
∀k ∈ [K], t ∈ [T ], ω ∈ [W]
l
a
a
a
, wktω
≥ 0; θktω
∈ {0, 1}
ψktω
k ∈ [K], t ∈ [T ], a ∈ [A], ω ∈ [W]
(18)
Model (6)−(18) is a two-stage program. The first-stage decides the production lots of
the perishable food and the setup schedule. The second-stage settles both the inventory
13
and demand fulfillment policy based on the first-stage plan and the materialized scenario.
The objective is to maximize the expected total profit over the planning horizon, which is
determined by the income from the revenue of each unit sold, reduced by the costs due
to production, setup, uncontrolled spoiled products and unused stocks (at the end of the
uk
). Equations (7)–(10) grasp the manufacturing environment
shelf-life, which is given by wktω
requirements. Constraints (7) force a given line to be correctly set up for a product before its
production starts. Constraints (8) ensure that production and setups do not exceed each line
available capacity per period. Minimum lot-sizes are imposed by (9) and (10) represent the
non-negativity and integrality constraints of the first-stage production and setup variables.
The management of the available stock along the planning horizon is given by constraints (12)–(18). Naturally, this stock depends on the produced quantities, met demand
and spoilage rates of each product. For food goods, the customer demand for a product
peaks at its fresher state. Nevertheless, he still has some remaining demand for older products. Constraints (12) guarantee that the total fulfilled demand does not exceed the demand
at the fresher state. Moreover, constraints (13) limit the sales of a product with a given age
to the demand for that age (this complete demand parameter is derived using expressions
(4) and (5)). It is well known that customers pick from the retailers’ shelves perishable
products with the highest degree of freshness. Such kind of “last-expired-first-out policy”
is not respected by requirements (12) and (13). The sets of constraints (14) and (15) bring
into the model this instinctive customer purchasing behavior, by ensuring that stock of a
given product cannot be used to satisfy demand in case a respective less fresher-state stock
has been used beforehand. In other words, fresher inventory has to be completely depleted
a
before using an older inventory. Note that constraints (15) are only active in case θktω
equals
to one, i.e., the inventory of a product k of age a in period t is picked up. These variables
are properly defined in (14). The inventory balancing constraints (16) ensure the correctness
of the quantity and age of the available stock along the planning horizon. Two situations
have to be differentiated: 1) the planner only accounts for product k expected shelf-life
14
(usually expressed by a stamped best-before-date, as occurs with milk) and, therefore, βkω
equals to zero; 2) the planner wishes also to account for a more unpredictable pattern of
spoilage due to, for example, varying temperature of storage or handling of products and,
therefore, βkω > 0. The amount of uncontrolled spoilage over a period increases with the
magnitude of this parameter (βkω ∈ [0, 1]). The initial stock of a product at its fresher
state is determined by the produced quantity, as ensured by (17). This family of constraints
connects the first-stage and second-stage decision variables, bridging the production and
logistics environments. Finally, constraints (18) define the second-stage variables domain.
Property 0.1 Problem (6)−(18) has a relatively complete recourse, i.e., for every feasible
first-stage decision there exists a feasible second-stage decision.
Let X be the vector of first-stage variables and X denote the set of first-stage constraints,
by taking into consideration the structure exhibited by model (6)−(18). Then, for X ∈ X ,
the feasible set of the second-stage problem is non-empty i.e., for every X ∈ X the inequality
Q[q, y, ξ(ω)] < +∞ holds true for all ω ∈ W.
Perfect Information and Stochastic Solution
The Expected Value of Perfect Information (EVPI) and the Value of Stochastic Solution
(VSS) are two quantities widely used not only to evaluate the potential “gain” by using
stochastic solutions over deterministic approximations but also to provide bounds on the
optimal value of the two-stage problem.
Let g[Z? , ξ] be the optimal value of the recourse problem and consider g[Z(ω), ξ(ω)] the
objective function for scenario ω ∈ W. Then, the wait-and-see (WS) solution is determined
as follows:
n
o
n
o
WS = E max g[Z(ω), ξ(ω)] = E g[Ẑ(ω), ξ(ω)] ,
Z
15
(19)
where Ẑ(ω) represents the optimal solution of each scenario ω ∈ W. EVPI is evaluated as
the difference between WS and the optimal recourse value:
EVPI = WS − g[Z? , ξ].
(20)
EVPI is commonly interpreted as the cost of uncertainty or the maximum amount to pay
in exchange of knowing precisely future outcomes. Higher EVPIs mean that randomness
plays an important role in the problem 31 . In this case, it is expected that the stochastic
solution performs better than the deterministic one. On the other hand, if WS−g[Z? , ξ] < ,
with a sufficiently small positive number, then Ẑ(ω) is a good approximation for the optimal
recourse solution Z? , and WS is (possibly) a tight upper bound for g[Z? , ξ].
Now, let the expected value over the scenarios be defined as E[ξ(ω)] = ξ(ω̄) and determine
the solution of using this expectation as Z̄ = [X̄, Ȳ ], where X̄ and Ȳ are the corresponding
first and second-stage solutions, respectively. Consider also an approximation for the recourse
problem that uses X̄ as solution. This problem is known as EEV or the expected value of
using the EV solution:
n
o
EEV = E max g[X̄, Y (ω), ξ(ω)] = E {g[Y (ω), ξ(ω)]} .
Y
(21)
Notice that EEV is only a second-stage problem, since the first-stage is fixed according to
the EV solution. Moreover, EEV is a decoupled problem for each scenario ω ∈ W. Finally,
VSS is evaluated as follows:
VSS = g[Z? , ξ] − EEV.
(22)
VSS provides the profit that one may obtain by adopting the recourse decision rather
than the approximated mean-value decision. Similarly, VSS shows the cost of ignoring the
uncertainty in choosing a decision 32 . When g[Z? , ξ] − EEV < , with as small as possible,
we can adopt the expected value solution with the corresponding EEV profit. In this case,
16
EEV is (possibly) a tight lower bound for the true recourse problem.
In practical problems, the first-stage fixation may result in infeasible EEV problems even
though the deterministic version is feasible. In this case, we assume that VSS −→ +∞.
Particularly for the PP-P-RN, the resulting EEV is always feasible for any reference scenario,
as shown in the next property.
?
?
Property 0.2 Let (qlkt
, ylkt
) be an optimal first-stage solution of the EV problem for any
?
?
reference scenario ξ(ω̃). If qlkt := qlkt
and ylkt := ylkt
in problem (6)−(18), ∀l ∈ [L], k ∈
[K], t ∈ [T ], then the approximation problem EEV is always feasible.
Since the first-stage decisions are fixed, then constraints (7)−(10) always hold true. Also,
constraints (12), (13), (14), (15) and (16) are independent of the first-stage decisions. Thus,
it suffices to show that constraint (17) that links first and second-stage variables is always
0
feasible, which is trivial, as wktω
is unbounded for all k ∈ [K], t ∈ [T ], ω ∈ W.
Downside Risk-Averse Production Planning Models for
Perishable Goods
In this section, we present two downside risk-averse models to deal with risk management
issues: the upper partial mean and the conditional value-at-risk. Both models minimize the
risk associated with scenarios whose revenues are below a certain threshold. Consequently,
the risk of spoilage and the risk of revenue loss are mitigated. Also, both risk approaches
are computationally tractable for modeling mixed-integer programs.
The Upper Partial Mean Model (UPM)
Given R? ∈ R+ , Fabian 33 defines the expected shortfall with respect to a target R? as
E[(R? − Qω )+ ], where (·)+ denotes the positive part of a real number. We propose to use a
risk-deviation model that considers the expected second-stage profit as the preselected target.
17
The proposed expected shortfall below the expected profit is also known as upper partial
mean 34 and it is a semideviation-based measure. Although this risk-deviation model has
been criticized as not being (always) consistent with the Pareto-Optimality condition 16,35 ,
it is an asymmetric and tractable risk deviation model, and it provides a very intuitive risk
analysis, as it resembles the standard-deviation over the scenarios. For simplicity, let us
consider the expected second-stage profit Q[q, y, ξ(ω)] = Qω . Then,
Qω =
X
a
a
a
· d0ktω ) −
− ψktω
· d0ktω − p¯k · βkω · (wktω
pˆk · ψktω
k,t,a
X
uk
.
p¯k · wktω
k,t
Mathematically, the upper partial mean is defined as δωE = [E(Qω ) − Qω ]+ , ∀ω ∈ [0, W].
The corresponding UPM model reads:
(UPM)
max −
X
X
a
(s̄lk ylkt + clk qlkt ) +
pˆk · ψktω
· d0ktω − p¯k · βkω ·
l,k,t
k,t,a
X
X
uk
a
a
πω · δωE
p¯k · wktω
−φ·
(wktω
− ψktω
· d0ktω ) −
k,t
ω
s.t.: deterministic constraints (7) − (10)
stochastic constraints (12) − (18)
X
δωE ≥
πω? Qω? − Qω ∀ω ∈ [W]
(23)
ω?
δωE ≥ 0 ∀ω ∈ [W].
Variable δωE is zero when scenario ω yields a profit higher than the expected profit
P
P
E
ω ? πω ? Qω ? . Otherwise, δω assumes the difference between the expected profit
ω ? πω ? Q ω ?
and the corresponding second-stage profit Qω , ∀ω ∈ [W].
18
The Conditional Value-at-Risk Model (CVaR)
Although the previous model considers the fluctuation of specific scenarios, the solution
can be influenced by scenarios with a low probability of resulting in “bad” revenues. To
overcome this issue, we use a financial measure called as conditional value-at-risk 36,37 . The
CVaR is defined as the expected profit of the (1−α)·100% scenarios exhibiting lowest profit.
CVaR accounts for the expected profit below a measure η called value-at-risk (VaR) at the
confidence level α. VaR is the maximum profit such that its probability of being lower than
or equal to this value is lower than or equal to (1 − α). Although CVaR and VaR are closely
related, CVaR is more appropriate to model risk-deviation in large-scale problems, as VaR
requires additional binary variables for its modeling.
The CVaR at the confidence level α (CVaRα ) is defined as follows:
CVaRα (Qω ) = max η −
η∈R
1
E[η − Qω ]+ ,
1−α
The corresponding CVaRα mean-risk model becomes:
19
(24)
(CVaRα )
max −
X
(s̄lk ylkt + clk qlkt ) +
l,k,t
X
a
pˆk · ψktω
· d0ktω − p¯k · βkω ·
k,t,a
a
a
· d0ktω ) −
− ψktω
(wktω
X
uk
+φ·
p¯k · wktω
k,t
η−
1
1−α
!
X
πω δωcvar
ω
s.t.: deterministic constraints (7) − (10)
stochastic constraints (12) − (18)
δωcvar ≥ η − Qω
(25)
∀ω ∈ [W]
δωcvar ≥ 0 ∀ω ∈ [W]
η ∈ R.
Variable δωcvar is zero if scenario ω yields a profit higher than η. Otherwise, δωcvar
assumes the difference between the value-at-risk η and the corresponding second-stage profit
Qω , ∀ω ∈ [W]. The value-at-risk belongs to the first-stage decision variables and δωcvar
belongs to the second-stage ones.
Computational Experiments
The objectives of the computational experiments are threefold. Firstly, investigate the impact of the random variables in the production planning of perishable goods. Secondly, understand how the solutions perceive risk-aversion, as well as analyze the advantages/disadvantages
of each risk-averse strategy. Thirdly, study the trade-off between expected profit and the
proposed risk measures from a financial planning perspective. To achieve these objectives,
this section is organized as follows. Section discusses the generation of the scenario tree.
Section presents the experimental setup. Section analyses a detailed example, and Section
presents a more comprehensive computational analysis. All the programs were implemented
20
and solved using IBM ILOG CPLEX Optimization Studio 12.5 on an Intel Core i7-3770-3.40
GHz processor under a Microsoft Windows 7 platform.
Scenario Tree Structure
In this paper, the scenario trees were generated by the combination of the four stochastic
parameters (d0ktω , αkω , p0kω and βkω ). We assume that αkω and p0kω are dependent random
variables positively correlated. The remaining parameters are mutually independent random
variables. Thus, the total number of scenarios |W| is given by n1 × n2 × n3 , where n1 , n2
and n3 are the number of realizations of d0ktω , αkω (p0kω ) and βkω , respectively. This suggests
a possible huge number of scenarios as the number of realizations of each random variable
increases. If, on the one hand, a large number of scenarios probably yields more accurate
solutions, on the other hand, the computational burden can be prohibitive.
In order to analyze the most suitable scenario tree size, we solved the risk-neutral model
by considering n1 = n2 = n3 = n, from n = 1 (one single scenario) to 7 (343 scenarios). In
all cases, the values that each random parameter may take were drawn in an equiprobable
manner from the respective distribution (Section ). Figure 4 presents the average results
over 20 simulations for each scenario tree. The squares represent the average results and the
vertical lines connect the maximum and minimum values among all instances. The largest
variation on the optimal value occurred when the number of scenarios increased from 1 to
27. In this case, the average objective function decreased from 22747 to 10691 (approx.
52%). From the 27 scenarios on, the expected profit tends to a locally asymptotically stable
behavior, which is more pronounced when |W| ≥ 125.
The results suggest that it is unnecessary to plan with a large number of outcomes, as
optimal values are stabilized with a relatively small number of scenarios. Therefore, we chose
to deeply analyze an illustrative instance with 27 scenarios (Section ) and perform extensive
computational results for scenario trees with 27 and 216 scenarios (Section ).
21
30000
Objective Function
25000
20000
15000
10000
5000
0
1
8
27
64
125
216
343
Number of Scenarios
Figure 4: Results for the expected profit over several scenario trees.
Data Generation
The data generator aims to be as realistic as possible. Hence, whenever it was possible, we
recurred to real-world information already published in the literature. The data generation
is divided in production data and market data.
Production Data. We considered six products (K = 6) to be produced on a single
production line (L = 1) over a horizon of 1 month, discretized in T = 30 time periods.
All products spend one unit time of capacity to be produced (elk = 1 and have negligible
minimum batch sizes mlk = 1. The setup costs and times between products (s̄lk and τ̄lk )
were randomly determined in the interval [1, 4]. We assume a constant capacity utilization
throughout the planning horizon of 0.6 in relation to the expected demand across all scenarios
for products in its fresher state. Therefore, the capacity per period Clt is determined as
P
Clt =
k
E(d0ktω )
,
0.6
∀l, t.
It is important to highlight that the utilization of capacity may correspond to an underestimate because besides the possibility of having scenarios with a demand above the average,
setup times do not influence the value of Clt .
Market Data. The number and type of products used in the computational experiments
is strictly linked to the amount of reliable and available information. Six different packaged
22
perishable food goods (lettuce, milk, chicken, carrots, yogurt and beef) were considered for
which there exists information of the related consumer purchasing behavior, demand patterns
and spoilage characteristics. The deterministic parameters related to these products are
organized in Table 2.
Table 2: Market and product deterministic parameters.
1
2
3
4
5
6
Product
clk (1)
pˆk and p̄k (1)
uk (1)
k (2)
PQR(1)
Lettuce
Milk
Chicken
Carrots
Yoghurt
Beef
0.249
0.27
0.299
0.169
0.062
0.268
2.49
2.7
2.99
1.69
0.62
2.68
10
14
7
21
21
7
−0.58
−0.59
−0.68
−0.58
−0.65
−0.75
Medium
Medium
High
Medium
Medium
High
Data extracted from:
(1)
Tsiros and Heilman 29 ,
(2)
Andreyeva et al. 38
As already mentioned, the uncertainty in the data is present in four parameters: d0ktω ,
αkω , p0kω and βkω . The first three are used to build the complete demand parameter for every
product across all product ages and periods (see Section for more details). Parameter d0ktω
was determined using the data available in van Donselaar et al. 39 for the demand of different
perishable products in a supermarket. This work uses a gamma distribution of the demand.
We further assumed that the production system has to serve 20 similar supermarkets. The
calculation of these parameters’ mean and variance (Table 3) follows the same methodology
of Broekmeulen and van Donselaar 40 .
Table 3: Statistics for the market and product stochastic parameters.
1
2
3
4
5
6
Product
E(d0ktω )(3)
σ(d0ktω )(3)
E(puk k −7 )(1)
E(puk k −1 )(1)
σ(puk k −7 )(1)
σ(puk k −1 )(1)
E(βkω )(4)
Lettuce
Milk
Chicken
Carrots
Yoghurt
Beef
37.71
54.86
96.57
37.71
54.86
96.57
41.91
84.18
97.09
41.91
84.18
97.09
2.15
2.14
2.58
1.39
0.51
2.33
1.08
0.71
1.38
0.70
0.26
1.20
0.63
0.82
0.73
0.53
0.19
1.63
0.79
0.78
0.93
0.56
0.22
0.90
0.013
0.005
0.011
0.006
0.004
0.011
Data extracted from:
(1)
Tsiros and Heilman 29 ,
(3)
van Donselaar et al. 39 ,
(4)
U. N. FAO 41
Concerning parameters αkω and p0kω , they are dependent on two other parameters puk k −7
23
and puk k −1 , for which there exists published data 29 . These two parameters reflect the customer
WTP when the product has an age of (uk − 7) and (uk − 1), respectively. Figure 5 illustrates
the process of generating the random parameters αkω and p0kω for a three-level discretization
of the respective distribution based on these two input parameters. First, it is assumed that
parameters puk k −7 and puk k −1 follow a truncated normal distribution (bounded to values > 0)
and having the moments presented in Table 3. For obtaining the scenarios with a high value
of p0kω we proceed as follows: (1) draw a number randomly from the interval [ 32 , 1] and get the
respective value from the cumulative normal distribution of parameter puk k −7 ; and (2) draw
a number randomly from the interval [0, 13 ] and get the respective value from the cumulative
normal distribution of parameter pkuk −1 . With these two values it is possible to extrapolate
u −1
p0kω with a linear regression and also obtain αkω , which is equal to
p0kω −pk k
u −1
pk k
. The opposite
intervals are used for the scenario with low p0kω and low αkω . For average values of p0kω the
same interval ] 13 , 23 [ is used for puk k −7 and puk k −1 .
𝑝𝑘0+
𝑝𝑘0
1 − α+
𝑝𝑘0−
1−α
𝑝𝑘𝑢−7
WTP
1 − α−
𝑝𝑘𝑢−1
0
𝑢𝑘 − 7
Age (a)
𝑢𝑘 − 1
𝑢𝑘
Figure 5: Estimating the parameters αkω and p0kω .
Having the values for d0ktω , αkω and p0kω , expressions (4) and (5) are used to obtain the
complete demand parameter (daktω ). Equation (4) for products with low PQR and equation
(5) for products with high PQR (Table 2).
24
The random parameter βkω is drawn using a similar methodology to the former stochastic
parameters and assuming an exponential distribution, which is the usual methodology for
mimicking the decay of perishable food goods 42 .
Figure 6 exemplifies and resumes the generation of the random parameters using the
different distributions for a scenario tree with 27 scenarios. In this case the levels of the
parameter values are classified according to low, medium and high. Notice that empirically
the least favorable scenario has low values for d0ktω and high values for αkω , p0kω and βkω .
Inversely, the most favorable scenario has high values for d0ktω and low values for αkω , p0kω
and βkω .
For generating parameters for scenarios trees with more than 27 scenarios, the methodology would be exactly the same, but the distribution of each parameter would be partitioned
in more equiprobable intervals (for example, 4 for a 64-scenario tree).
High
Medium
𝑃(𝑋 ≤ 𝑥)
Medium
High
(High for 𝑝𝑘𝑢−7 and Low for 𝑝𝑘𝑢−1 )
𝑃(𝑋 ≤ 𝑥)
𝑃(𝑋 ≤ 𝑥)
High
Medium
Low
Low
(High for 𝑝𝑘𝑢−7 and Low for 𝑝𝑘𝑢−1 )
Low
0
Gamma Distribution for 𝑑𝑘𝑡𝑤
0
Normal Distribution for α𝑘𝑤, 𝑝𝑘𝑤
Exponential Distribution for β𝑘𝑤
Figure 6: Illustrative example of the generation of the different stochastic parameters.
Illustrative Example Results
This section analyses an instance with a scenario tree composed by 27 scenarios, as shown
in Figure 7. The analysis focuses on the operational impact of including risk mitigation
procedures in the production planning formulation with perishable products.
The results are obtained for the three models developed in Sections and . In order
to compare the solutions of the three models, we allow the risk-averse results a maximum
25
d kt0 
Low
Medium
High
 k
p k0
Low
Low
Medium
Medium
High
High
Low
Low
Medium
Medium
High
High
Low
Low
Medium
Medium
High
High
 k
Low
Medium
High
Low
Medium
High
Low
Medium
High
Low
Medium
High
Low
Medium
High
Low
Medium
High
Low
Medium
High
Low
Medium
High
Low
Medium
High
d
0
kt ,
 k , pk0 ,  k 
Low, Low, Low, Low
Low, Low, Low, Medium
Low, Low, Low, High
Low, Medium, Medium, Low
Low, Medium, Medium, Medium
Low, Medium, Medium, High
Low, High, High, Low
Low, High, High, Medium
Low, High, High, High
Medium, Low, Low, Low
Medium, Low, Low, Medium
Medium, Low, Low, High
Medium, Medium, Medium, Low
Medium, Medium, Medium, Medium
Medium, Medium, Medium, High
Medium, High, High, Low
Medium, High, High, Medium
Medium, High, High, High
High, Low, Low, Low
High, Low, Low, Medium
High, Low, Low, High
High, Medium, Medium, Low
High, Medium, Medium, Medium
High, Medium, Medium, High
High, High, High, Low
High, High, High, Medium
High, High, High, High
Figure 7: Scenario tree composed by the combination of all levels of d0ktω , αkω , p0kω and βkω .
reduction of the expected profit in relation to the risk-neutral approach of 5%. We assume that this threshold is an admissible loss for the decision maker and, therefore, the
φ weight of both risk-averse formulations is automatically calibrated. The results per scenario are presented in Tables 4, 5 and 6. The headings of the tables are as follows. DeP
P
P
a
a
); Freshness (( k,t,a ((uk −
d0ktω ); Service Level ( k,t,a ψktω
mand Satisfaction ( k,t,a ψktω
P
P
a
a
a
a)/uk )ψktω
)/[d0ktω ]); Inventory ( k,t,a<uk wktω
); Controlled Spoilage ( k,t,a=uk wktω
); UnconP
a
a
trolled Spoilage ( k,t,a (wktω
− ψktω
d0ktω )βkω ). Demand Satisfaction indicates the amount
of sold products and is used to evaluate the Service Level that gives an indication of the
lost sales per scenario. The Freshness indicator 8 estimates the remaining shelf-life of the
delivered products. In operations management of food products, the freshness aspect is
very important to customer’s satisfaction. Inventory and Production indicators provide the
amount of products stocked and produced along the planning horizon, respectively. The last
two indicators are related to the amount of spoiled products, which is a reliable measure of
the operations’ “health” in supply chains of perishable products. Controlled Spoilage refers
to the amount of products that reach the end of their expected shelf-life without being sold
26
(expired products). Uncontrolled Spoilage is related to the chaotic nature of the perishability
phenomenon that affects products continuously (before reaching their expiry date).
27
28
L
L
L
L
L
L
L
L
L
M
M
M
M
M
M
M
M
M
H
H
H
H
H
H
H
H
H
Scenario
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Max
Min
Average
d0ktω
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
αkω
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
βkω
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
p0kω
25665
1658
11413
1658
1658
1658
1658
1658
1658
1658
1658
1658
6916
6916
6916
6916
6916
6916
6916
6916
6916
25665
25665
25665
25665
25665
25665
25665
25665
25665
13722
1656
7411
1658
1658
1658
1657
1657
1657
1656
1656
1656
6913
6910
6879
6908
6908
6873
6899
6898
6862
13722
13696
13663
13702
13695
13662
13678
13677
13647
k,t,a
d0ktω
k,t
X
Demand
Satisfaction
X
a
ψktω
d0ktω
Demand
100%
53.2%
84.3%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
99.5%
100%
100%
99.4%
100%
100%
99.2%
53.5%
53.4%
53.2%
53.4%
53.4%
53.2%
53.3%
53.3%
53.2%
k,t,a
Service
Level
X
a
ψktω
ktω
d0ktω
k,t,a
85.6%
55.3%
67.0%
59.2%
60.0%
57.1%
61.7%
60.2%
61.3%
62.3%
61.1%
61.8%
85.0%
85.6%
83.7%
84.4%
85.2%
84.7%
83.7%
84.4%
85.5%
55.9%
55.9%
55.9%
55.8%
55.6%
55.7%
55.3%
55.5%
55.3%
uk
X uk − a ψ a
Freshness
a
wktω
113557
27794
71760
113118
111361
108441
113557
111339
108573
113529
111164
108196
77410
76273
74651
77454
76316
74693
77559
76412
74778
28259
27941
27794
28329
28006
27857
28436
28106
27958
k,t,a<uk
X
Inventory
2958
0
907
2919
2736
2451
2939
2759
2470
2958
2776
2486
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
k,t,a=uk
Controlled
Spoilage
X
a
wktω
Table 4: Results for the risk-neutral model (PP-P-RN) for a single 27-scenario tree sample.
k,t,a
X
1308
24
378
178
622
1304
179
620
1305
178
622
1308
107
334
649
108
335
652
108
336
656
24
69
102
24
71
103
24
72
105
a
a
(wktω
− ψktω
d0ktω )βkω
Uncontrolled
Spoilage
29
L
L
L
L
L
L
L
L
L
M
M
M
M
M
M
M
M
M
H
H
H
H
H
H
H
H
H
Scenario
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Max
Min
Average
d0ktω
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
αkω
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
βkω
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
p0kω
25665
1658
11413
1658
1658
1658
1658
1658
1658
1658
1658
1658
6916
6916
6916
6916
6916
6916
6916
6916
6916
25665
25665
25665
25665
25665
25665
25665
25665
25665
12718
1652
6757
1658
1658
1658
1655
1655
1655
1652
1652
1652
5939
5934
5923
5936
5930
5920
5931
5925
5915
12718
12696
12669
12708
12695
12668
12691
12690
12667
k,t,a
d0ktω
k,t
X
Demand
Satisfaction
X
a
ψktω
d0ktω
Demand
100%
49.4%
78.3%
100%
100%
100%
100%
100%
100%
100%
100%
100%
85.9%
85.8%
85.6%
85.8%
85.7%
85.6%
85.8%
85.7%
85.5%
49.6%
49.5%
49.4%
49.5%
49.5%
49.4%
49.4%
49.4%
49.4%
k,t,a
Service
Level
X
a
ψktω
ktω
d0ktω
k,t,a
75.7%
51.1%
61.0%
55.4%
57.3%
55.5%
58.8%
58.9%
56.1%
57.9%
58.3%
57.7%
73.9%
73.3%
73.9%
74.7%
74.4%
74.8%
74.2%
75.7%
74.8%
51.4%
51.3%
51.2%
51.3%
51.3%
51.2%
51.2%
51.1%
51.1%
uk
X uk − a ψ a
Freshness
a
wktω
114443
27571
71845.3
114250
112108
109085
114087
112075
109155
114443
112339
109261
76897
75770
74152
76980
75850
74232
77129
75991
74369
27990
27707
27571
28166
27859
27717
28479
28155
28005
k,t,a<uk
X
Inventory
1950
0
592
1916
1781
1581
1934
1801
1595
1950
1813
1610
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
k,t,a=uk
Controlled
Spoilage
X
a
wktω
k,t,a
X
Table 5: Results for the upper partial mean model (UPM) for a single 27-scenario tree sample.
1147
20
335
158
550
1142
158
548
1146
159
551
1147
99
303
591
99
303
591
99
304
592
20
61
88
20
61
89
21
62
90
a
a
(wktω
− ψktω
d0ktω )βkω
Uncontrolled
Spoilage
30
L
L
L
L
L
L
L
L
L
M
M
M
M
M
M
M
M
M
H
H
H
H
H
H
H
H
H
Scenario
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Max
Min
Average
d0ktω
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
αkω
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
βkω
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
p0kω
25665
1658
11413
1658
1658
1658
1658
1658
1658
1658
1658
1658
6916
6916
6916
6916
6916
6916
6916
6916
6916
25665
25665
25665
25665
25665
25665
25665
25665
25665
12669
1651
6729
1658
1658
1658
1655
1655
1655
1651
1651
1651
5888
5882
5869
5887
5881
5868
5885
5879
5866
12669
12660
12634
12666
12660
12633
12660
12660
12633
k,t,a
d0ktω
k,t
X
Demand
Satisfaction
X
a
ψktω
d0ktω
Demand
100%
49.2%
78.0%
100%
100%
100%
100%
100%
100%
100%
100%
100%
85.1%
85.0%
84.9%
85.1%
85.0%
84.8%
85.1%
85.0%
84.8%
49.4%
49.3%
49.2%
49.4%
49.3%
49.2%
49.3%
49.3%
49.2%
k,t,a
Service
Level
X
a
ψktω
ktω
d0ktω
k,t,a
72.7%
50.4%
60.7%
59.3%
60.7%
55.6%
60.5%
60.0%
60.4%
61.6%
61.7%
61.6%
71.8%
71.0%
70.5%
72.3%
72.1%
71.9%
71.1%
70.6%
72.7%
50.5%
50.5%
50.5%
50.4%
50.5%
50.4%
50.5%
50.6%
50.5%
uk
X uk − a ψ a
Freshness
a
wktω
43699
10136
24521
43699
42880
41420
43536
42881
41398
43564
42810
41384
20723
20549
20416
20935
20757
20622
21192
21010
20872
10176
10152
10136
10176
10152
10136
10184
10161
10145
k,t,a<uk
X
Inventory
122
0
21
78
52
0
105
77
14
122
94
30
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
k,t,a=uk
Controlled
Spoilage
X
a
wktω
k,t,a
X
Uncontrolled
Spoilage
1152
20
336
157
546
1150
158
546
1152
158
544
1144
99
305
595
100
305
595
100
305
595
20
61
88
21
61
88
21
62
89
a
a
(wktω
− ψktω
d0ktω )βkω
Table 6: Results for the conditional value-at-risk model (CVaR) for a single 27-scenario tree sample.
Several conclusions related to the stochastic parameters seem valid across the different
models. The stochastic parameter related to the demand level for products in their fresher
state (d0ktω ) has the greatest impact over the different scenarios. On one hand, for scenarios
with a high initial demand, service levels deteriorate considerably. This service level reduction translates into less demand satisfied and demand satisfied with a lower level of product
freshness. On the other hand, for scenarios with low initial demand, due to the high levels of
inventory, the amount of products that reach the end of their shelf-lives is high (controlled
spoilage). This amount of inventory is also responsible for generating more uncontrolled
spoilage. The results also suggest that more exigent customers (high values of αkω and high
values of p0kω ) together with products subject to intense perishability (high values of βkω ) lead
to a lower demand satisfaction and an increased number of uncontrolled spoiled products.
Consequently, the profit in these scenarios slightly decreases.
The reduction of 5% in the expected profit of the risk-averse models allow for a considerable reduction of expired products in the least favorable scenarios. In comparison with
the RN model, the CVaR model was able to reduce the controlled spoilage by almost 100%.
Also, the UPM approach provided a controlled spoilage 35% smaller than in the RN case.
Moreover, the uncontrolled spoilage diminished approximately 11% with both risk-averse
approaches. Therefore, risk-averse models are more in line with sustainable planning approaches in which low spoiled amounts of stocks are desired. However, these more conservative plans lead to reduction of about 6% in the service level. Both the fraction of demand
satisfied and the freshness of products delivered drops. This decrease is more evident for
scenarios with a medium demand level (scenarios 10 − 18). Notice that the same absolute
decrease in the production throughput has a higher impact for lower levels of demand.
Overall, comparing the risk-averse solutions for an allowed decrease of 5% of expected
profit in relation to the RN approach, the model maximizing the CVaR is able to achieve
lower inventory levels and less expired products (Controlled Spoilage), while not excessively
dropping the service levels both in terms of quantity and quality (Freshness). These are the
31
main differences as the remaining indicators are very similar between the two risk-averse
formulations.
Extensive Computational Results
In this computational study, we consider PP-P-RN, UPM and CVaR instances with 27 and
216 scenarios. For each scenario tree size, we generated a set of 100 instances introduced
also to evaluate the effectiveness of the risk aversion in the production planning of perishable
products. The risk-averse models were solved for φ ∈ {0.5, 1, 2, 3, 4}.
Table 7 presents both EVPI and VSS analysis for the risk-neutral model. The results
show that the cost of gathering perfect information about the future might be prohibitive.
Indeed, the expected value of perfect information ascends to around 119% (96%) of the
stochastic solution value for the 216- (27-) scenario tree sample. According to the stochastic
programming theory, it means that the randomness plays a major role in the production
planning model with perishability. Also, the relative VSS suggests that it is possible to obtain
significant savings by using the stochastic solution instead of determining the production
lots and the setup schedule based on average values, as the relative VSS is around 25% and
37% for the scenario trees composed by 216 and 27 scenarios, respectively. Apparently, as
the number of scenarios |W| increases, EVPI increases and VSS decreases. Perhaps this
phenomenon occurs because the impact of |W| on WS and EV is smaller than its impact on
RP (recourse problem), as one can observe in Table 7. Similar results in a different context
were pointed out in 43 .
Table 7: EVPI and VSS analysis over a sample of 100 instances.
No. of Scenarios
RP
WS
EVPI
27
216
10106
8959
19802
19657
9696
10697
EVPI × 100%
RP
96%
119%
EEV
VSS
6413
6698
3693
2261
VSS × 100%
RP
37%
25%
The results for the risk-averse and risk-neutral models are presented in Table 8. The
32
headings are as follows. The incumbent solution value (Zmip ); the expected profit (µ); the
normalized weighted expected profit over all scenarios, where 100 is the expected profit for
the risk neutral model (EP%); the standard deviation over all scenarios (σ); the reduction of
the standard deviation in comparison with the risk-neutral case, evaluated as 1 − σ(σ(·)
RN) ;
the upper partial mean value (UPM); the reduction of the upper partial mean in comparison
UPM(·) ; the conditional value-at-risk at
with the risk-neutral case, evaluated as 1 − UPM
(RN)
the 95% of confidence (CVaRα ); the sum of the weights of the scenarios with a negative
profit (P [< 0]); the conditional expectation of the scenarios with a negative profit (E[< 0]);
the percentage of spoiled products (B). Figures 8, 9, 10 and 11 depict the Pareto fronts
of the key performance indicators for the scenario trees composed by 27 and 216 scenarios,
respectively.
Overall, the risk-averse formulations are sensitive to the weight φ, as the expected profit
(µ) decreases substantially as risk-aversion is enforced with larger φ0 s. Along with profit
losses come a drastic reduction on the variability of the potential outcomes (σ and UPM)
and a reduction on the amount of spoilage (B). Moreover, the probability of having scenarios
with negative profit is very high for the risk-neutral solutions, but this drawback is further
mitigated by the risk-averse formulations, which reduce both the probability and the impact
of such scenarios in the expected profit (P < 0 and E < 0, respectively). Risk-averse
models are able to achieve such trade-offs by slightly lowering production outputs and by
scheduling production lots closer to the due dates and, therefore, reducing the chances of
spoiled inventory.
It is still possible to achieve much less risky solutions at the expense of minor decreases
in the expected profit, e.g. φ = 0.5 in the UPM model provides solutions up to 7% less risky,
but only 3% less profitable in both scenario-tree samples. Although in the CVaR model the
expected profit initially decreases faster than in the UPM model (see Figure 8 and 10), the
former yields better performance indicators for lower degrees of risk aversion. For example,
the solution obtained in the CVaR model for a φ = 0.5 and 27 scenarios is able to improve
33
the CVaR from −7712 to −745, the spoilage from 8% to 3%, the probability of having a
scenario with negative profit from 0.33 to 0.12, at the expense of a loss in the expected
profit of 14%. The corresponding results for the 216-scenario tree sample is similar, but less
pronounced. When comparing those indicators between RN and UPM approaches, we see
that they are also similar to each other, which suggests that UPM probably performs better
when risk aversion is larger.
Notice that, as the CVaR risk measure represents the expected profit of the (1−α)×100%
worst scenarios in our maximization problem, then both risk-averse models provide good
CVaR values for φ > 2 in the 27-scenario tree sample, and for φ > 3 in the 216-scenario
tree sample. However, the latter scenario tree provides worse values presumably because by
discretizing more the least favorable scenarios, it is harder to get solutions robust enough to
improve the worst scenarios consistently.
As expected, analyzing Figures 8, 9(a), 10 and 11(a), we reiterate that the UPM model
dominates the CVaR model for the UPM risk measure and that the CVaR model dominates
the CVaR model for the CVaR risk measure. Moreover, as expected, the UPM model is
able to more effectively reduce the the standard deviation of profit (σ). Figures 9(b) and
11(b) indicate that the CVaR model dominates the UPM model in the trade-off between the
amount of spoiled products and the expected profit loss. Again, this is a crucial insight when
dealing with the production planning of perishable goods. Furthermore, for small losses of
the expected profit with respect to the RN model, the CVaR formulation seems to approach
the Pareto front of the UPM model in the other indicators (UPM and σ).
34
35
216
RN
216
27
CVaR
CVaR
27
UPM
216
27
RN
UPM
Number of
Scenarios
Model
8959
−
6470
6667
7695
9016
10512
9517
10549
13070
15659
19174
0.5
1
2
3
4
0.5
1
2
3
4
7293
5356
3563
3105
3160
0.5
1
2
3
4
6601
5019
2949
2167
1874
10106
−
0.5
1
2
3
4
Zmip
φ
7481
5828
5330
4728
4087
8720
7815
6317
3097
3023
8959
8663
8426
7519
6489
5938
9858
7449
6180
3145
3184
10106
µ
83.5
65.0
59.5
52.8
45.6
97.3
87.2
70.5
34.6
33.7
−
86.0
83.2
74.4
64.2
59.0
97.8
73.5
61.1
31.1
31.6
−
EP (%)
6608
4739
3658
2872
2192
10041
6836
4260
854
793
13410
8232
7519
5545
4070
3325
12315
5099
2852
214
218
13264
σ
50.7
64.7
72.7
78.6
83.7
25.1
49.0
68.2
93.6
94.1
−
37.9
43.3
58.2
69.3
74.9
7.15
61.6
78.5
98.4
98.4
−
Reduction
of σ(%)
2746
2095
1652
1293
974
4238
2797
1684
310
287
5715
3625
3297
2363
1698
1392
5015
2094
1343
99.3
92.6
5482
UPM
52.0
61.8
69.9
76.4
82.2
25.8
49.0
69.3
94.3
94.8
−
33.9
39.9
56.9
69.0
74.6
8.51
61.8
75.5
98.2
98.3
−
Reduction
of UPM (%)
0.12
0.09
0.00
0.00
0.00
−745
−192
884
1506
1872
20.7
16.7
16.7
0.00
0.00
0.17
0.11
0.04
0.00
0.00
−9340
−5453
−2833
1020
1071
−4416
−1076
−377
139
534
0.33
0.33
0.00
0.00
0.00
0.00
−6492
835
1934
3058
3130
−13550
0.33
P [< 0]
−7712
CVaRα
(α = 0.95)
−3473
−467
−302
−44.4
0.00
−6808
−4648
−2348
0.00
0.00
−7789
−727
−296
0.00
0.00
0.00
−5501
0.00
0.00
0.00
0.00
−6727
E[< 0]
3.82
1.93
1.43
1.08
0.82
7.71
5.37
4.81
0.89
0.77
9.78
2.99
2.54
1.86
1.39
1.02
7.84
2.08
1.52
0.48
0.53
8.00
Spoilage
(%)
Table 8: Average results for the risk-neutral, UPM and CVaR models over a sample of 100 instances for each scenario tree.
6000
4000
2000
5000
0
-2000
3000
CVAR
UPM
4000
2000
0
2000
4000
6000
8000
10000
12000
-4000
-6000
-8000
1000
-10000
0
-12000
0
2000
4000
6000
8000
10000
12000
-14000
EXPECTED PROFIT
RN
UPM
EXPECTED PROFIT
CVaR
RN
(a)
UPM
CVaR
(b)
Figure 8: (a) Expected shortfall versus expected profit; (b) Conditional value at risk versus
expected profit. (27-scenario tree sample)
14000
12000
% SPOILAGE
SIGMA
10000
8000
6000
4000
2000
0
0
2000
4000
6000
8000
10000
12000
10%
9%
8%
7%
6%
5%
4%
3%
2%
1%
0%
0
2000
4000
EXPECTED PROFIT
RN
UPM
6000
8000
10000
12000
EXPECTED PROFIT
CVaR
RN
(a)
UPM
CVaR
(b)
Figure 9: (a) Standard deviation of profit versus expected profit; (b) Percentage spoilage of
total total production versus expected profit. (27-scenario tree sample)
4000
6000
2000
0
4000
-2000
3000
-4000
CVAR
UPM
5000
2000
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
-6000
-8000
1000
-10000
-12000
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
-14000
EXPECTED PROFIT
EXPECTED PROFIT
RN
UPM
RN
CVaR
(a)
UPM
CVaR
(b)
Figure 10: (a) Expected shortfall versus expected profit; (b) Conditional value at risk versus
expected profit. (216-scenario tree sample)
With Figures 12 and 13, we further investigate the distribution of profit for the three
models and for the two scenario trees, respectively. We compare the distribution of the
profits for φ = 0.5, which is the closest to a risk-neutral approach that seems to provide
a good trade-off between risk and profit according to the previous results. The insights
obtained in Section about the major impact of demand uncertainty are reiterated in these
36
14000
12000
% SPOILAGE
SIGMA
10000
8000
6000
4000
2000
0
0
1000
2000
3000
4000
5000
6000
7000
8000
10%
9%
8%
7%
6%
5%
4%
3%
2%
1%
0%
0
9000 10000
1000
2000
3000
4000
RN
UPM
5000
6000
7000
8000
9000
10000
EXPECTED PROFIT
EXPECTED PROFIT
RN
CVaR
(a)
UPM
CVaR
(b)
Figure 11: (a) Standard deviation of profit versus expected profit; (b) Percentage spoilage
of total total production versus expected profit. (216-scenario tree sample)
figures. It is clear to see across all models and scenario trees the levels of discretization of
the initial demand. By generating more frequently very low and very high values of demand
(216-scenario tree sample) is also reflected in the dispersion of the profit distribution. Notice
that the range spanned by the profit in the risk-averse approaches is much smaller than that
in the risk-neutral, which clearly shows that it is possible to mitigate the probability of more
pessimistic scenarios (with negative profits) by enforcing risk aversion. Moreover, the CVaR
model reacts quicker than the UPM by improving the least favorable profits and turning
the profit distribution smoother. For the 216-scenario tree, it seems that the CVaR model
is able to reduce risk, while not jeopardizing too much the probability of high profits. For
the risk-averse solutions with higher values of phi the described behaviors are even more
emphasized and, therefore, the dispersion of profit is smaller.
0.12
PROBABILITY
0.1
0.08
0.06
0.04
0.02
27500
25500
23500
21500
19500
17500
15500
13500
9500
11500
7500
5500
3500
-500
1500
-2500
-4500
-6500
-8500
-10500
-12500
-14500
0
PROFIT
UPM
CVaR
RN
Figure 12: Profit distribution over a sample of 2700 scenarios for the UPM (φ = 0.5), CVaR
(φ = 0.5) and risk-neutral models (27-scenario tree sample).
37
0.12
PROBABILITY
0.1
0.08
0.06
0.04
0.02
27500
25500
23500
21500
19500
17500
15500
13500
9500
11500
7500
5500
3500
-500
1500
-2500
-4500
-6500
-8500
-10500
-12500
-14500
0
PROFIT
UPM
CVaR
RN
Figure 13: Profit distribution over a sample of 21600 scenarios for the UPM (φ = 0.5), CVaR
(φ = 0.5) and risk-neutral models (216-scenario tree sample).
Conclusions and Future Research
Risk management plays an important role in production planning, especially in dealing with
perishable goods. In fact, the random nature of demands for fresh products, consumer
purchasing behavior and spoilage rates, pose an additional challenge for planners in the
food supply chain. In order to provide less risky decisions from operational and financial
viewpoints, we have proposed two tractable downside risk-averse approaches. The results
suggested that it is possible to reduce the percentage of expired products that reach the
end of their shelf-lives by using the risk-averse models. Moreover, although the risk-averse
solutions operate with worse profits, the probability of having scenarios with negative profits
is much improved, as risk aversion is enforced. When comparing the different risk-measures
implemented, it seems that the conditional value-at-risk is the most suitable to incorporate
in the production planning of perishable products. This risk approach is less sensitive to
variations in the risk-weighting factor, it dominates the UPM approach regarding the spoilage
indicator and it has similar solutions to the ones in the Pareto front of the UPM model for
small losses of the expected profit (with respect to the RN approach). Future research
should focus on understanding the impact of strategic decisions (e.g. facility location) in the
risk management of the food supply chain and studying other risk-averse approaches (e.g.
second-order dominance constraints).
38
Acknowledgement
The research was partially supported by CNPq and by FCT through the research grant
SFRH/BPD/89861/2012.
Supporting Information Available
Instances data is available.
This material is available free of charge via the Internet at http://pubs.acs.org/.
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