Article pubs.acs.org/IECR
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 Production Engineering Department, Universidade Federal de São Carlos, Rodovia João Leme dos Santos (SP-264), Km 110, Sorocaba, São Paulo 18052-780, Brazil
‡
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 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.
1. 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 facesthe risk of spoilagehas to be mitigated in the production process at the operational decision level. In fact, the European Commission estimates that 39% 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 of 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 shelf-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 © 2013 American Chemical Society
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 work 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 the 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 two-stage 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 work is aligned with the gap pointed out by Seshadri and Subrahmanyam,9 namely, the fact that models Received: Revised: Accepted: Published: 17538
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Figure 1. Conceptual framework of this work from risk sources to risk mitigating strategies.
in Pahl and Voß12 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, the discrete lotsizing and scheduling problem, the continuous setup lot-sizing problem, the 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 multiobjective 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 toward practice, Kopanos et al.14,15 developed efficient models for production planning problems in the icecream industry. These models incorporate key elements in the food production planning, such as a multistage 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 toward 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 Among the various risk-averse methods, mean-risk models have been widely used to deal with risk mitigation.22−24 Basically, mean-risk models optimize simultaneously the expected outcome (·) and the dispersion of the outcomes (·), à la Markovitz:25,26
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 3 develops the risk-neutral 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 4. Afterward, section 5 describes the computational experiments and discusses the results. Section 6 indicates the main conclusions and future research directions.
2. 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. 2.1. 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 realworld 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 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. On the basis of the block planning approach, three mixed-integer linear programming models that integrate shelf-life issues into the planning of the packaging stage were proposed. More recently,
max (8ξ) − ϕ(8ξ)
(1)
where 8 ξ is the random outcome. The “user-controlled” parameter ϕ ∈ + 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. 17539
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Figure 2. Impact of varying customer related parameters p0 (a) and α (b) for products with low PQR.
Figure 3. Impact of p0 (a) and α (b) for products with high PQR.
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 and 3 describe the WTP functions that are empirically studied in ref 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.
Typically, to mitigate the variability of the second-stage or recourse costs, the dispersion (·) 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 a 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 2.2. Consumer Purchasing Behavior of Perishable Goods. Tsiros and Heilman29 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 work), it 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
WTP for products with a low PQR = p0 −
αp0 a u−1
(2)
WTP for products with a high PQR = p0 −
αp0 a ⎛ a ⎞⎟ ⎜2 − u − 1⎝ u − 1⎠
(3)
In the left panels of Figures 2 and 3, 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 = 5 is considered. Note that in each function the price is represented up to age 4, since at age 5 the products spoil and 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 ϵ (see (4) and (5)). The readers are referred to Amorim et al.30 for the complete proof. 17540
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Table 1. Illustrative Example of the Domain and Relation between Second-Stage Decision Variables inventory for given a = 0, 1, or 2 ψaktω
waktω t=1 t=2 t=3
θaktω
daktω
qlkt
0
1
2
0
1
2
0
1
2
0
1
2
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
ε ⎛ αp0 a ⎞ ̂ + p ⎜ ⎟ u−1 demand for products with a low PQR = d 0⎜ ⎟⎟ ⎜ p̂ ⎝ ⎠
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: (4)
PP-P-RN:
demand for products with a high PQR ⎛ p̂ + 0⎜ =d ⎜ ⎜ ⎝
αp0 a u−1
(2 − p̂
a u−1
)
⎞ε ⎟ ⎟⎟ ⎠
g (Z , ξ) = max − ∑ ( slk̅ ylkt + clkqlkt ) + {8[q, y, ξ(ω)]} l ,k ,t
(6)
subject to:
(5)
qlkt ≤
3. 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. 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 (> , - , Π), where > is a set of discrete outcomes or scenarios with corresponding probabilities of occurrence πω, such that πω > 0 and ∑ωπω = 1, equipped with a σ-algebra of events - and a probability measure Π. According to the twostage stochastic program methodology, we define the production lots and the setup schedule as first-stage decisions. Inventory and demand satisfaction policies are then the secondstage decisions. Consider the indices, parameters, and decision variables in the Nomenclature that are used in the stochastic formulation. In Table 1 the domain and relation between second-stage decision variables are 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 (ψ2ktω = 0). The value of θaktω is strictly linked to ψaktω, which used the production (qlkt) and the inventory (waktω) to fulfill demand (daktω). For example, in period t = 1 the demand (4) is completely fulfilled with fresh products (ψ0k1ω = 1). As the production output is 10 in the first period (qlk1 = 10), the 1 inventory in period 2 with age 1 turns out to be 6 (wk2ω = 6). Note that, in this solution, in period 3 we already have 5 spoiled products (w2k3ω = 5).
Clt y elk lkt
∑ (τlk̅ ylkt
∀ l ∈ [L], k ∈ [K ], t ∈ [T ]
+ elkqlkt ) ≤ Clt
(7)
∀ l ∈ [L], t ∈ [T ]
k
(8)
qlkt ≥ mlk ylkt qlkt ≥ 0;
∀ l ∈ [L], k ∈ [K ], t ∈ [T ]
(9)
ylkt ∈ {0, 1}
∀ l ∈ [L], k ∈ [K ], t ∈ [T ]
(10)
where the expectation {·} is evaluated as ∑ωπω8 [q,y,ξ(ω)] under a finite number of scenarios, ξ(ω) = [dω,αω,p0ω,βω] are the data of the second stage problem, and 8 [q,y,ξ(ω)] is the optimal value of the following problem: max
∑ [pk̂ ψktaωdkt0 ω − pk̅ βkω(wktaω − ψktaωdkt0 ω)] k ,t ,a
−
∑ pk̅ wktuω k
(11)
k ,t
∑
ψktaω ≤ 1
∀ k ∈ [K ], t ∈ [T ], ω ∈ [>]
a ∈ [A]
(12)
ψktaωdkt0 ω ≤ dktaω ∀ k ∈ [K ], t ∈ [T ], a ∈ [A], ω ∈ [>]
(13)
ψktaω ≤ θktaω ∀ k ∈ [K ], t ∈ [T ], a ∈ [A], ω ∈ [>]
(14)
wkta−ω1 − ψktaω− 1dkt0 ω ≤ (1 − θktaω)M ∀ k ∈ [K ], t ∈ [T ], a ∈ [A]\{0}, ω ∈ [>]
(15)
wktaω = (wka,−t −11, ω − ψka,−t −11, ωdk0, t − 1, ω)(1 − βkω) ∀ k ∈ [K ], t ∈ [T + 1], a ∈ [A]\{0}, ω ∈ [>] (16) 17541
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∑ qlkt = wkt0 ω
Article
Property 1. Problem (6)−(18) has a relatively complete recourse; i.e., for every feasible f irst-stage decision there exists a feasible second-stage decision. Let X be the vector of first-stage variables and ? denote the set of first-stage constraints, by taking into consideration the structure exhibited by model (6)−(18). Then, for X ∈ ? , the feasible set of the second-stage problem is non-empty; i.e., for every X ∈ ? the inequality 8 [q,y,ξ(ω)] < +∞ holds true for all ω ∈ >. 3.1. 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 ∈ > . Then, the wait-and-see (WS) solution is determined as follows:
∀ k ∈ [K ], t ∈ [T ], ω ∈ [>]
l
(17)
ψktaω , wktaω ≥ 0;
θktaω ∈ {0, 1}
k ∈ [K ], t ∈ [T ], a ∈ [A], ω ∈ [>]
(18)
The 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 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 shelf life, uk which is given by w ktω ). Equations 7−10 grasp the manufacturing environment 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 for capacity per period. Minimum lot sizes are imposed by (9) and (10) which represent the nonnegativity 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 before using an older inventory. Note that constraints (15) are only active in the case of θaktω = 1; 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 (usually expressed by a stamped best-before date, as occurs with milk) and, therefore, βkω = 0; (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.
WS = {max g[Z(ω), ξ(ω)]} = {g[Z(̂ ω), ξ(ω)]} Z
(19)
where Ẑ (ω) represents the optimal solution of each scenario ω ∈ > . 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 for 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] < ϵ, with ϵ 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 [ξ(ω)] = ξ(ω̅ ) and determine the solution of using this expectation as Z̅ = [X̅ ,Y̅], where X̅ and Y̅ are the corresponding first- and second-stage solutions, respectively. Consider also an approximation for the recourse problem that uses X̅ as solution. This problem is known as EEV or the expected value of using the EV solution: EEV = {max g[X̅ , Y (ω), ξ(ω)]} = {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 ∈ > . 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 < ϵ, with ϵ as small as possible, we can adopt the expected value solution with the corresponding EEV profit. In this case, 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 17542
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for the PP-P-RN, the resulting EEV is always feasible for any reference scenario, as shown in the next property. Property 2. Let (q*lkt,y*lkt) be an optimal f irst-stage solution of the EV problem for any reference scenario ξ(ω̃ ). If qlkt:= q*lkt 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)−(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 feasible, which is trivial, as w0ktω is unbounded for all k ∈ [K], t ∈ [T], ω ∈ > .
s.t.:
stochastic constraints (12) − (18) δω ≥
δω ≥ 0
η∈
1 [η − 8ω]+ 1−α
}
(24)
CVaRα: max − ∑ ( slk̅ ylkt + clkqlkt ) l ,k ,t
+
∑ [pk̂ ψktaωdkt0 ω − pk̅ βkω(wktaω − ψktaωdkt0 ω)] k ,t ,a
⎛
−
∑ pk̅ wktuω + ϕ⎜⎜η − k
⎝
k ,t
∑ [pk̂ ψktaωdkt0 ω − pk̅ βkω(wktaω − ψktaωdkt0 ω)]
s.t.:
1 1−α
⎞
∑ πωδωCVaR ⎟⎟ ω
⎠
(25)
deterministic constraints (7) − (10) stochastic constraints (12) − (18)
∑ pk̅ wktuω k
δωCVaR ≥ η − 8ω δωCVaR ≥ 0
Mathematically, the upper partial mean is defined as δω = [(8 ω) − 8 ω]+, ∀ ω ∈ [0, > ]. The corresponding UPM model reads as follows:
l ,k ,t
∑ [pk̂ ψktaωdkt0 ω − pk̅ βkω(wktaω − ψktaωdkt0 ω)]
5. COMPUTATIONAL EXPERIMENTS The objectives of the computational experiments are 3-fold: first, investigate the impact of the random variables in the production planning of perishable goods; second, understand
k ,t ,a k
ω
∀ ω ∈ [>]
Variable δCVaR is zero if scenario ω yields a profit higher than ω η. Otherwise, δCVaR assumes the difference between the valueω at-risk η and the corresponding second-stage profit 8 ω, ∀ ω ∈ [> ]. The value-at-risk belongs to the first-stage decision variables and δCVaR belongs to the second-stage ones. ω
max − ∑ ( slk̅ ylkt + clkqlkt )
∑ pk̅ wktuω − ϕ∑ πωδω
∀ ω ∈ [>]
η∈
UPM:
k ,t
{
The corresponding CVaRα mean-risk model becomes
k ,t
−
∀ ω ∈ [>]
CVaR α(8ω) = max η −
k ,t ,a
+
∀ ω ∈ [>]
Variable δω is zero when scenario ω yields a profit higher than the expected profit ∑ω*πω*8 ω*. Otherwise, δω assumes the difference between the expected profit ∑ω*πω*8 ω* and the corresponding second-stage profit 8 ω, ∀ ω ∈ [> ]. 4.2. Conditional Value-at-Risk Model. 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-atrisk.36,37 The conditional value-at-risk model (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. CVaR at the confidence level α (CVaRα) is defined as follows:
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. 4.1. Upper Partial Mean Model. Given R* ∈ +, Fabian33 defines the expected shortfall with respect to a target R* as [R* − 8 ω)+], 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. The proposed expected shortfall below the expected profit is also known as upper partial mean (UPM),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 8 [q,y,ξ(ω)] = 8 ω. Then,
−
∑ πω *Q ω * − Q ω ω*
4. DOWNSIDE RISK-AVERSE PRODUCTION PLANNING MODELS FOR PERISHABLE GOODS
8ω =
deterministic constraints (7) − (10)
(23) 17543
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Figure 4. Results for the expected profit over several scenario trees.
deeply analyze an illustrative instance with 27 scenarios (section 5.3) and perform extensive computational results for scenario trees with 27 and 216 scenarios (section 5.4). 5.2. 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. 5.2.1. 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 (sl̅ k and τl̅ k) 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
how the solutions perceive risk aversion, as well as analyze the advantages/disadvantages of each risk-averse strategy; third, 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 5.1 discusses the generation of the scenario tree. Section 5.2 presents the experimental setup. Section 5.3 analyses a detailed example, and section 5.4 presents a more comprehensive computational analysis. All of the programs were implemented 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. 5.1. Scenario Tree Structure. In this work, 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 |> | is given by n1 × n2 × n3, where n1, n2, and n3 are the number of 0 0 , αkω(pkω ), and βkω, respectively. This realizations of dktω 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 n = 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 5.2). 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 (approximately 52%). From the 27 scenarios on, the expected profit tends to a locally asymptotically stable behavior, which is more pronounced when |> | ≥ 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
Clt =
∑k (dkt0 ω) 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. 5.2.2. Market Data. The number and type of products used in the computational experiments are strictly linked to the amount of reliable and available information. Six different packaged 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. 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 2.2 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 γ distribution of the demand. We further assumed that the 17544
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Table 2. Market and Product Deterministic Parameters 1 2 3 4 5 6
product
clka
p̂k and p̅ka
uk a
εkb
PQRa
lettuce milk chicken carrots yogurt 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
a Data extracted from Tsiros and Heilman.29 bData extracted from Andreyeva et al.38
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 Concerning parameters αkω and p0kω, they are dependent on two other parameters pukk−7 and pukk−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 pukk−7 and pukk−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 [2/3, 1] and get the respective value from the cumulative normal distribution of parameter pukk−7, and (2) draw a number randomly from the interval [0, 1/3] and get the respective value from the cumulative normal distribution of parameter pukk−1. With these two values it is possible to extrapolate p0kω with a linear regression and also obtain αkω, which is equal to (p0kω − pukk−1)/pukk−1. The opposite intervals are used for the scenario with low p0kω and low αkω. For average values of p0kω the same interval ]1/3, 2/3[ is used for pukk−7 and pukk−1. Having the values for d0ktω, αkω, and p0kω, expressions (4) and (5) are used to obtain the complete demand parameter (daktω): eq 4 for products with low PQR and eq 5 for products with high PQR (Table 2). The random parameter βkω is drawn using a methodology similar 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.
Figure 5. Estimating the parameters αkω and p0kω.
Inversely, the most favorable scenario has high values for d0ktω and low values for αkω, p0kω, and βkω. For generating parameters for scenario 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). 5.3. 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 3 and 4. In order to compare the solutions of the three models, we allow the risk-averse results a maximum 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: demand satisfaction (∑k,t,aψaktω); service level (∑k,t,aψaktω); freshness ((∑k,t,a((uk − a)/uk)ψaktω)/ 0 a [d ktω ]); inventory (∑ k,t,a | increases, EVPI increases and VSS decreases. Perhaps this phenomenon occurs because the impact of |> | 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 ref 43. The results for the risk-averse and risk-neutral models are presented in Table 8. The headings are as follows: the
The reduction of 5% in the expected profit of the risk-averse models allows 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 drop. 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 main differences as the remaining indicators are very similar between the two risk-averse formulations. 5.4. Extensive Computational Results. In this computational study, we consider PP-P-RN, UPM, and CVaR instances 17547
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Table 5. Results for the Upper Partial Mean Model (UPM) for a Single 27-Scenario Tree Sample demand
Σ dkt0 ω
scenario
d0ktω
αkω
βkω
p0kω
k ,t
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 av
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
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
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
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
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 25665 1658 11413
demand satisfaction
Σ ψktaωdkt0 ω
k ,t ,a
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 12718 1652 6757
service level
Σ ψktaω
k ,t ,a
freshness
⎛u − Σ⎜ k k , t , a⎝ uk
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% 100% 49.4% 78.3%
inventory
a a ⎞ ψktω ⎟ 0 ⎠ dktω
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% 75.7% 51.1% 61.0%
Σ wktaω
k , t , a < uk
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 114443 27571 71845.3
controlled spoilage
Σ wktaω
k , t , a = uk
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 1950 0 592
uncontrolled spoilage
Σ (wktaω − ψktaωdkt0 ω)βkω
k ,t ,a
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 1147 20 335
respectively). Risk-averse models are able to achieve such tradeoffs 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 Figures 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 ϕ = 0.5 and 27 scenarios is able to improve 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 are 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, since 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
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 with the risk-neutral case, evaluated as [1 − (UPM(·)/ UPM(RN))]; the conditional value-at-risk at the 95% level of confidence (CVaRα); the sum of the weights of the scenarios with a negative profit (P[
