Risk-averse two-stage stochastic programs in furniture plants

OR SpectrumDOI 10.1007/s00291-012-0312-5

REGULAR ARTICLE

Risk-averse two-stage stochastic programs in furnitureplants

Douglas Alem · Reinaldo Morabito

© Springer-Verlag Berlin Heidelberg 2012

Abstract We present two-stage stochastic mixed 0–1 optimization models to hedgeagainst uncertainty in production planning of typical small-scale Brazilian furnitureplants under stochastic demands and setup times. The proposed models consider cut-ting and drilling operations as the most limiting production activities, and synchro-nize them to avoid intermediate work-in-process. To design solutions less sensitive tochanges in scenarios, we propose four models that perceive the risk reductions overthe scenarios differently. The first model is based on the minimax regret criteria andoptimizes a worst-case scenario perspective without needing the probability of thescenarios. The second formulation uses the conditional value-at-risk as the risk mea-sure to avoid solutions influenced by a bad scenario with a low probability. The thirdstrategy is a mean-risk model based on the upper partial mean that aggregates a riskterm in the objective function. The last approach is a restricted recourse approach, inwhich the risk preferences are directly considered in the constraints. Numerical resultsindicate that it is possible to achieve significant risk reductions using the risk-aversestrategies, without overly sacrificing average costs.

Keywords Furniture industry · Production planning · Risk aversion ·Minimax with regret · Mean-risk · Restricted recourse · Conditional value-at-risk

The research was partially supported by grants from FAPESP and CNPq.

D. Alem (B)Department of Production Engineering, Federal University of São Carlos,Sorocaba, Brazile-mail: [email protected]

R. MorabitoDepartment of Production Engineering, Federal University of São Carlos,São Carlos, Brazile-mail: [email protected]

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D. Alem, R. Morabito

1 Introduction

The economic relevance of the Brazilian furniture industry has become increasinglysignificant and currently it is among the most important sectors of the transformationindustry in Brazil, mainly due to: (i) the sector consists of 14,400 industries that areresponsible for almost 2 % of the Brazilian commercial surplus; (ii) the sector producesaround 374 million products and 11.6 billion Euros in sales; (iii) the investments inthis sector totalled over 358 million Euros; (iv) its social impact is noteworthy: thesector has provided employment to more than 239,000 workers, i.e., approximately2.7 % of the total number of jobs in the industrial production of the country (Abimóvel2008).

In spite of this, due to the increase in recent international competition, the Brazilianfurniture industry is attempting to reengineer itself, e.g., by investing large capitalin more sophisticated machines and equipment, and acquiring new technology inprocesses and materials. Nevertheless, particularly in micro and small-scale plantsthat are not able to invest much in their facilities, focusing on reducing managerialand manufacturing problems could be the answer to surviving in the furniture market.One such problem is poor production planning and control, leading to waste in work-in-process, a high volume of backlog orders, improper storage policies, unnecessarycutting adjustments (setups), the inability to deal with demand fluctuation and varyingprocessing/setup times due to errors in handling operations, as well as difficulties ofestimating and managing risks associated to imprecise data.

Commonly, production planning in small-scale furniture companies involves twomain decisions concerning lot-sizing and cutting-stock. First, lot size informationprovides the quantity of furniture to be produced in each period of the planning timehorizon. Second, the cutting process generates the best possible cutting patterns toobtain the pieces of furniture. When these decisions are taken separately, undesirableeffects can emerge, e.g., a large amount of trim-loss and an excessive number of setupsof the cutting machine and high production costs are a consequence of the previousdecisions. On the other hand, when lot-sizing and cutting-stock problems are solvedsimultaneously, there is a tendency to anticipate future orders to have better cuttingpatterns (as it is easier to find good combinations for a larger variety of pieces), whichcan decrease trim-losses and setups of the cutting machine over the planning timehorizon. At the same time, there is an opposite tendency in postponing production toreduce holding costs. This tradeoff between anticipating and postponing productioncan be even more complex if overtime and backlog are permitted. The integrationbetween these decisions are refereed in the literature as the combined lot-sizing andcutting-stock problem.

Although lot-sizing and cutting-stock problems are both well-known and have beenstudied in-depth in production and operations research literature, only a few papershave focused on their integration to better coordinate production activities and savecosts, as pointed out in Thomas and Griffin (1996), Drexl and Kimms (1997), Pochetand Wolsey (2006) and Jans and Degraeve (2008). In general, the papers on this subjectrefer to a specific industrial process and propose strategies to deal with the combinedproblem in a tractable way. Farley (1988) was one of the first authors to take intoaccount planning decisions embedded in a cutting-stock problem based on the clothing

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Risk-averse two-stage stochastic programs in furniture plants

industry. Farley’s model considers characteristics of the production process such as thecutting and sewing operations, while maximizing the total revenue. Reinders (1992)addressed a combined cutting-stock and production management in a centralized woodprocessing industry. The cutting-stock decisions are determined beforehand usingknapsack problems and the cutting patterns are used in a tactical planning problemto provide a multiperiod production plan for tree trucks. Hendry et al. (1996) studieda coupled cutting-stock and production scheduling problem in the cooper industry,focusing on minimizing the costs of the foundry operation. Several papers proposedcombined approaches in the paper industry, e.g., Krichagina et al. (1998), Respícioand Captivo (2002), Menon and Schrage (2002), Correia et al. (2004) and Poltroniereet al. (2008).

It is worth commenting that to the best of our knowledge, only Krichagina et al.(1998) studied a coupled lot-sizing and cutting-stock problem in an uncertain environ-ment. The study emphasized the close relationship between cutting-stock decisionsand the amount of paper produced in the paper industry. To deal with both deci-sions together, it is proposed a suboptimal two-step procedure which involves an LPmodel in the first step and a Brownian motion analysis in the second. Nonas andThorstenson (2000, 2008) presented a coupled lot-sizing and cutting-stock problemin a Norwegian company that manufactures special kinds of off-road trucks. Arbiband Marinelli (2005) proposed integrating operational and mid-term planning deci-sions in the production of gear belts. Basically, cutting-stock decisions were combinedwith in-process inventory to decrease trim-loss and/or to reuse leftovers. Aktin andZdemir (2009) proposed a two-phase approach for pattern generation and cutting plandetermination of the one-dimensional cutting stock problem in coronary stent man-ufacturing. In the first phase, the objective is to generate a set of cutting patterns tominimize the trim-loss. With these pre-determined patterns, the second phase seeks tominimize total production cost, as material inputs, number of setups, overtime usage,among others. Karelahti et al. (2011) presented a production planning model in thestainless steel industry that combines the issue of determining proper dimensions forlarger objects and how to cut small items from larger objects to minimize the trim-loss.Gramani and França (2006), Gramani et al. (2009) and Alem and Morabito (2012)dealt with combined lot-sizing and cutting-stock problems in furniture settings. Thefirst two papers focused on deterministic models, whereas the second explored robustoptimization techniques.

Our work in this paper strives to extend and improve upon previous studies byexploring risk-averse stochastic programming approaches in production planning oftypical small-scale Brazilian furniture plants. Our contribution is in twofold. First,we present a deterministic mathematical formulation that takes into account the mostcritical production processes in real small-scale furniture plants, which are the cuttingand drilling operations. This assumption permits identifying the production bottleneckprocess, by analyzing the capacity usage and/or overtime, such that it is possibleto have distinct bottlenecks for different instances. The model also considers setuptimes of both operations in a synchronized fashion to avoid intermediate stock ofworkpieces between both phases. In contrast, the previous models studied in Gramaniand França (2006), Gramani et al. (2009) and Alem and Morabito (2012) arise infurniture plants where the cutting process is definitively the production bottleneck,

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setup times are negligible, and the assembly line is rather limiting. Second, in this paper,we propose different two-stage stochastic programming models to deal with risk-neutral and risk-averse issues, discussing their performance in terms of optimal values,service levels, recourse solutions, and computational effort. A potential disadvantageof the robust optimization approach invoked in Alem and Morabito (2012) is that alldecision variables must be determined before the actual realization of the uncertain data(here-and-now decisions), while our application naturally yields a group of wait-and-see decisions that can be adjusted to the realization of the uncertain data to producemore reliable solutions (slacks and surplus). The two-stage stochastic models withrecourse paradigm was a natural choice for determining the decisions in differentstages, without leading to a much extra computational burden for some risk-aversestrategies.

Risk-averse stochastic programming has gained a lot of attention in literaturerecently. Roughly speaking, the idea of a risk measure is to reduce the variability ofspecific outcome realizations over the scenarios. The concept of risk was introducedin the classical mean-variance model of Markowitz (1952) that used the variance ofrandom returns as a risk measure. Years later, Mulvey et al. (1995) proposed a moregeneral methodology, called robust optimization, to generate solutions less sensitiveto scenario realizations. Since then, there has been considerable research effort thatdeals with various aspects on this theme, see Artzner et al. (1999), Ogryczak andRuszczynski (1999, 2001), Takriti and Ahmed (2004), Schultz and Tiedemann (2006)and Gollmer et al. (2008), amongst many others. The field of risk-averse stochasticprogramming also contains a vast collection of application problems, e.g., capac-ity expansion (Laguna 1998), international sourcing (González Velarde and Laguna2004), aggregate and tactical planning (Leung and Wu 2004; Aghezzaf et al. 2010;Al-e Hashem et al. 2012), revenue management (Lai and Ng 2005; Lai et al. 2007;Escudero et al. 2012), multisite production planning problems in lingerie companies(Leung et al. 2007), production planning in sawmills (Zanjani et al. 2009), mediumrange planning in copper extraction (Alonso-Ayuso et al. 2011), supply chain in agilemanufacturing (Pan and Nagi 2010), production loading problems in global supplychain management (Wu 2006), operational planning of multisite production and distri-bution networks (Verderame and Floudas 2011), production planning and schedulingin chemical industries and oil refineries (Suh and Lee 2001; Jia and Ierapetritou 2004;Khor et al. 2008; Li and Ierapetritou 2008), bus scheduling (Yan and Tang 2009),natural gas procurement (Aouam et al. 2010), sign processing (Ukkusuri et al. 2010),disaster management (Noyan 2012; Bozorgi-Amiri et al. 2011), financial planning(Pinar 2007; Mansini et al. 2007; Consiglio and Staino 2012), power/hydro-powersystems and energy (Kuhn and Schultz 2009; Pousinho et al. 2011, 2012; Guigues andSagastizábal 2012), etc.

In this paper, we present a risk-neutral model and explore four strategies to handlerisk aversion, namely: (i) minimax with regret; (ii) conditional value-at-risk; (iii) mean-risk with upper partial mean; and (iv) restricted recourse. Their choice is justified by thefollowing. Minimax is well-known and used in the literature, and various authors claimthat it provides robust (or risk-averse) solutions from a worst-case scenario perspective.Mean-risk models have become increasingly important in risk management, as theyprovide convenient tradeoff analysis between variability and cost. Also, the upper

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partial mean (UPM) is an asymmetric and tractable risk measure that resembles theso-called mean-absolute deviation measure (MAD), but MAD is not suitable in thiscase, as it penalizes positive and negative deviations from the expected second-stagecosts. The restricted recourse (RR) model has received less attention in the literature,but it is equivalent to the proposed mean-risk model and it is computationally tractableas well (Ahmed and Sahinidis 1998). From a practical viewpoint, RR can be usefulbecause it incorporates explicitly the maximum risk that the manager is willing toaccept. The motivation for using a CVaR-based model is to avoid solutions influencedby a bad scenario with a low probability, as it can happen in the minimax approach.The risk-neutral and risk-averse approaches were solved using real-life instances, andtheir performance was compared mostly in terms of average costs and risk reduction.

The remainder of this paper is organized as follows: Section 2 describes the flowprocess in typical small-scale furniture plants and Sect. 3 presents the correspondingdeterministic production planning model. Section 4 develops a two-stage scenario-based stochastic program to hedge against uncertainty in demands and setup times.Section 5 proposes the four risk-averse approaches to deal with risk considerations.Section 6 analyzes a real-life case by comparing the effectiveness of the risk-neutralstrategy and the four risk-averse ones. In Sect. 7 we report our concluding remarksand suggest extensions of this study.

2 Problem definition

This section presents a brief description of a typical small-scale Brazilian furnitureplant in the northwest of São Paulo State. The company produces home (modu-lar, upholstered, bedroom, dining-room and living-room) and office furniture, butthe largest production is due to the first type, mainly bedroom furniture, such aswardrobes, bedside tables, dressers, etc. The main raw material is hardboard of var-ious types, as MDF (medium density fibreboard) and MDP (medium density par-ticleboard), which are basically manufactured wood that combines wax and resinbinders. These hardboards are also provided in standard sizes (2, 750 × 1, 830 mm2

and 2, 750 × 1, 850 mm2) of different thicknesses (3, 9, 12, 15 and 20 mm), butdepending on the retailer, the widths and/or lengths can also vary. In general, furnitureplants buy large quantities of hardboard to negotiate better prices and to have moreproduction flexibility in a short/medium time horizon.

Although the furniture manufacturing involves several production processes, inthis paper we are particularly interested in two major operations, cutting and drilling,mainly because both present preparation steps basically composed by manual opera-tions that can be very time-consuming depending on the task and skills of the machineoperators. In the cutting process, most of the hardboards are cut in pre-determinedcutting patterns using an automatic saw machine. Some small pieces or pieces ofmore sophisticated shapes are manufactured in manual saw machines. Figure 1 showsa three-door wardrobe that consists of nine types of pieces, which are required in dif-ferent quantities as well. Each type of piece (A−I) can have a different length, widthand thickness, and refers to a distinct part of the final product, such as the top of thewardrobe (H), the doors (B), the bottom of the wardrobe (E), the front of the drawers

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B

A

C

D

EF

G

H

I

Fig. 1 An illustrative example of nine pieces (A–I ) required to assemble a three-door wardrobe. Each typeof piece usually has different dimensions and refers to a distinct part of the final product, such as the top ofthe wardrobe (H ), the doors (B), the bottom of the wardrobe (E), the front of the drawers (C), etc.

Fig. 2 Two-staged guillotine cutting pattern (left), and three-staged guillotine cutting pattern (right)

(C), etc. Before a new cutting pattern is processed, setup operations are needed to allo-cate the hardboards in the machine, and to adjust the position of the knives and saws.Other cutting operations include rotating the hardboard to obtain more sophisticatedcutting patterns and changing the worn-out saws. Due to these manual operations andmachine constraints, small-scale furniture plants usually prefer using two-stage andthree-stage guillotine cutting patterns (Fig. 2), as they are easier to cut and demand afewer number of setups. Homogeneous cutting patterns are also often used due to thesame reasons (a pattern is homogeneous if it contains only pieces of the same shapeand size). Other types of cutting patterns might be used to improve the productivityof the cutting machine, such as the n-group patterns (Morabito and Arenales 2000).Roughly speaking, a k-staged guillotine pattern is determined by a sequence of at mostk stages of guillotine cuts applied to the original board and to the subsequent smallerrectangles obtained after the previous cuts (dos Santos et al. 2011).

After the cutting process, workpieces are processed in an automatic drilling machineto receive holes (circular, countersinking, spot facing, etc.) into or through the facesand sides, in order to further screw and assemble the furniture parts. Most of the

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Fig. 3 Process flow for manufacturing furniture in typical small-scale plants

pieces are drilled separately, as some holes cannot go through them and have only oneopening. An important aspect of this operation is the speed of the drilling machine,which must be relatively low to avoid injuries in the workpieces. The setup of thedrilling machine includes choosing and/or changing type, position and speed of thebit drill, and then checking the size of the hole to make additional adjustments beforestarting a new process. Downstream operations involve edge gluing and painting. Thepainting phase involves another set of operations, such as sanding, ultraviolet (UV)painting and drying. At the end, the final pieces are wrapped with their respectiveaccessories, packaged and shipped to warehouses. The packaged products are loadedonto vehicles for shipment to distribution centers or customers. A simplified flowprocess is shown in Fig. 3.

3 Deterministic mixed 0–1 model for production planning in furniture plants

In this section, we propose a deterministic mixed 0–1 program to optimize short-termproduction plans in small-scale furniture settings. The proposed mathematical model,named CLC (combined lot-sizing and cutting-stock problem), aims to determine pro-duction plans to meet customers’ demands over the planning horizon (e.g., a fewweeks) at a high service level and low cost. To address the optimization model, let usconsider the notation as follows:

Indices/sets

I = {1, . . . , I } Set of productsJ = {1, . . . , J } Set of cutting patternsP = {1, . . . , P} Set of piecesT = {1, . . . , T } Set of periods

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Input data

cit Production cost of product i in period th+

i t Holding cost of product i in period th−

i t Shortage cost of product i in period tw j t Trim-loss cost of cutting pattern j in period t

ot Overtime cost in period tdit Demand for product i in period tν I

j Cutting time of pattern j

ν I Ip Drilling time of piece p

apj Number of times that piece p appears in cutting pattern jr pi Number of pieces p needed to assemble product iφ I

j t Setup time of cutting pattern j in period t

φ I Ipt Setup time of drilling piece p in period tCt Regular capacity in period t (in terms of work-shifts)

C Et Extra machine capacity in period t (in terms of extra work-shifts)

δpj Indicator function equal to 1, if apj > 0, and 0, otherwiseM Large number

Decision variables

Xit Amount of product i produced in period tY jt Production frequency of cutting pattern j in period tZ jt Binary variable to indicate if the machine is setup to produce pattern j

in period t (Z jt = 1) or not (Z jt = 1)I +i t Inventory level for product i at the end of period t

I −i t Backlogging level for product i at the end of period t

Ot Overtime in period t

The deterministic mixed 0–1 model for production planning in furniture plants canbe posed as follows:

(CLC) min∑

i∈I

∑

t∈T

(cit Xit + h+

i t I +i t + h−

i t I −i t

) +∑

j∈J

∑

t∈Tw j t Y jt +

∑

t∈Tot Ot , (1)

s.t.: Xit + I +i,t−1 − I +

i t + I −i t − I −

i,t−1 = dit , i ∈ I, t ∈ T (2)∑

j∈Japj Y jt ≥

∑

i∈Irpi Xit , p ∈ P, t ∈ T (3)

∑

j∈J

(ν I

j Y jt + φ Ij t Z jt

)≤ Ct + Ot , t ∈ T (4)

∑

p∈P

∑

j∈J

(ν I I

p apj Y jt + φ I Ipt δpj Z jt

)≤ Ct + Ot , t ∈ T (5)

Y jt ≤ M Z jt , j ∈ J , t ∈ T (6)

Xit ≥ 0, i ∈ I, t ∈ T (7)

I +i t , I −

i t ≥ 0, i ∈ I, t ∈ T (8)

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0 ≤ Ot ≤ C Et , t ∈ T (9)

Y jt ≥ 0, Z jt ∈ {0, 1} , j ∈ J , t ∈ T . (10)

The objective function consists of minimizing the total cost incurred in pro-duction, inventory, backlogging, trim-loss and overtime utilization. The inventorybalance constraints (2) define the relation between demand, production, inventoryand backlogging levels for each product i and time period t. Without loss of gener-ality, we assume I +

i0 = I −i0 = 0. The coupled constraints (3) enforce the balance

between the pieces produced from the cutting phase (left-hand side) and the piecesneeded to assemble final products (right-hand side), for each piece p and period t.We remind that the furnishings are composed by diverse parts or pieces required indifferent quantities, e.g., one unit of the three-door wardrobe requires 26 pieces of typeG (Fig. 1). Consequently, the production of product i causes an internal demand forpieces p, which is expressed by

∑i r pi Xit , where rpi is given by the bill of materials.

These constraints link lot-sizing (Xit ) and cutting-stock (Y jt ) decisions.The capacity constraints (4) and (5) ensure capacity limit in cutting and drilling

processes, respectively. In both processes, processing and setup times cannot exceedregular plus overtime production capacity. Notice that we use the same setup variableZ jt for both cutting and drilling production phases in order to synchronize them, i.e.,the drilling machine has to be setup for processing workpieces of type p only if theupstream cutting operation generated workpieces of type p. This assumption considersthat all workpieces p obtained from the cutting machine in period t are drilled in thesame period t, i.e., there is no intermediate stock of workpieces between cutting anddrilling machines. Mathematically, we define an indicator function δpj ∈ {0, 1} (inputof the problem) to synchronize the setup, such that it values 1 if the cutting pattern jcontains at least one piece p, and 0, otherwise. If Z jt = 0, then pattern j (as well as itspieces p) is not produced in period t and the setup times due to pattern j and its pieces pare not taken into account in constraints (4) and (5). Conversely, if Z jt = 1, the cuttingmachine is prepared to produce pattern j and we have to check if pattern j generates apiece p or not. In the first case, δpj ·Z jt = 1, which activates the setup time in (5). In thelatter, δpj · Z jt = 0, and the setup time is not considered in (5). Based on observationsin practical settings, we consider that both cutting and drilling processes use an equalcapacity that is measured in work-shifts, and that there are common workers at eachprocess. Thus, as overtime is an extension of the work-shift for all employees andthe overtime usage is the same for both production processes. Although this modelassumption reflects the usual practice in typical small-scale Brazilian furniture plants,the extension to take into account different capacities and overtimes for these processesis straightforward.

The setup constraints (6) force that if the production of cutting pattern j takes placein period t, then the setup time is incurred in that period. Note that we can strengthenconstraint (6) replacing M on its right-hand side by M jt , which provides an upperbound for the number of possible hardboards j cut in period t. The domain of decisionvariables (7)−(10) states that all decision variables are continuous and non-negative,with the exception of the binary setup variable Z jt . Finally, constraint (9) limits theovertime usage in each period t.

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4 Risk-neutral stochastic approach for production planning in furniture plants

Based on the small-scale furniture plants, we consider the following assumptions: (i)costs are fixed or known in advance because there are contracts between the companyand the retailers that establish the prices for the hardboards during a short-term horizon,in general, 4–8 weeks; (ii) processing times are deterministic values, as the correspond-ing operations are automatized; (iii) cutting and drilling setup times are uncertain para-meters since both are much dependent on the ability of the operator to handle manualoperations in dedicated machines; (iv) demands are also uncertain, as they are totallydependent on the market factors, seasonality effects, changing customer preferences,product life cycles, etc; (v) demands and setup times are mutually independent randomvariables in the sense of statistical independence, i.e., the demand’s realization has noinfluence on the setup time’s realization. This assumption is reasonable in the proposedsituation because the realizations of the demand depend mainly on the market factors,whereas the realizations of the setup times depend on the skills of the workers.

Thus, in order to deal with the random data in furniture production planning, wepropose a two-stage stochastic mixed 0–1 version of the problem by using the recourseparadigm. This model relies on situations where a set of so-named first-stage decisionvariables must be determined before the random data are observed, and correctiveactions can be taken once the uncertain parameters become known under each scenariotaken into consideration. These corrected actions are called second-stage variables.It is usual to represent random data on some probability space (�,F ,�), where �

is a set of scenarios (where each specific realization is denoted by ξs for scenarios ∈ � ) equipped with a σ -algebra of events F with a probability measure �. Inthis application, we assume that the random data for scenario s are represented by the

triple ξs =(

dits, φIj ts, φ

I Ipts

), with a probability of occurrence πs that represents its

importance in the overall problem, so that πs > 0 and∑S

s=1 πs = 1. So, accordingto the two-stage paradigm, the decision variables are partitioned as follows:

– production level Xit must be determined here-and-now; otherwise, all demandis trivially postponed. As final products depend on the cutting phase, and it ismandatory to setup machines before producing, Y jt and Z jt are also first-stagedecision variables.

– inventory and backlogging levels may serve to adjust the first-stage productionto the known demand for each scenario. Moreover, overtime usage may serve toadjust the first-stage consumed capacity to the known setup time for each scenario.Therefore, I +

i ts, I +i ts and Ots are the second-stage decision variables.

In order to formulate the stochastic model, let the following notation:

Deterministic data

cit Production cost of product i in period th+

i t Holding cost of product i in period th−

i t Shortage cost of product i in period tw j t Trim-loss cost of cutting pattern j in period t

ot Overtime cost in period tν I

j Cutting time of pattern j

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ν I Ip Drilling time of piece p

apj Number of times that piece p appears in cutting pattern jr pi Number of pieces p needed to assemble product iCt Regular capacity in period t (in terms of work-shifts)

C Et Extra machine capacity in period t (in terms of extra work-shifts)

δpj Indicator function equal to 1, if apj > 0, and 0, otherwiseM Large number

Stochastic data

S Number of scenariosdits Demand for product i in period t in scenario sφ I

j ts Setup time of pattern j in period t in scenario s

φ I Ipts Setup time of piece p in period t in scenario sπs Probability of scenario s

First-stage variables

Xit Amount of product i produced in period tY jt Production frequency of cutting pattern j in period tZ jt Binary variable to indicate if the machine is setup to produce pattern j

in period t (Z jt = 1) or not (Z jt = 0)

Second-stage variables

I +i ts Inventory level for product i at the end of period t in scenario s

I −i ts Backlogging level for product i at the end of period t in scenario s

Ots Overtime in period t in scenario s

Model RN (risk-neutral stochastic CLC) consists of minimizing the first-stage costplus the expected second-stage cost for the set of feasible solutions. It can be expressedas follows:

(RN) min∑

i∈I

∑

t∈Tcit Xit +

∑

j∈J

∑

t∈Tw j t Y jt

+S∑

s=1

πs

[∑

i∈I

∑

t∈T

(h+

i t I +i ts + h−

i t I −i ts

) +∑

t∈Tot Ots

](11)

s.t. : Deterministic constraints (3), (6), (7) and (10) (12)

Xit + I +i,t−1,s − I +

i ts + I −i ts − I −

i,t−1,s = dits, i ∈ I, t ∈ T , s = 1, . . . , S (13)∑

j∈J

(ν I

j Y jt + φ Ij ts Z jt

)≤ Ct + Ots, t ∈ T , s = 1, . . . , S (14)

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∑

p∈P

∑

j∈J

(ν I I

p apj Y jt + φ I Iptsδpj Z jt

)≤ Ct + Ots, t ∈ T , s = 1, . . . , S (15)

0 ≤ Ots ≤ C Et , t ∈ T , s = 1, . . . , S (16)

I +i ts, I −

i ts ≥ 0, i ∈ I, t ∈ T , s = 1, . . . , S. (17)

The objective function (11) consists of two deterministic terms related to productionand trim-loss cost, and the expectation of the second-stage cost over the scenarios.The deterministic constraints present the coupled relationship between lot-sizing andcutting-stock problems and the setup production. The second stage (13)−(17) definesall scenario-dependent constraints. Note that for each feasible first-stage solution andeach scenario s = 1, . . . , S, the second-stage problem is well defined, i.e., the problem(11)−(17) has full recourse; let us name it RP. Therefore, for every (X, Y, Z) ∈ A,

the feasible set of the second-stage problem is non-empty, where:

A =⎧⎨

⎩(X, Y, Z) :∑

j∈Japj Y jt ≥

∑

i∈Irpi Xit ,

Y jt ≤ M Z jt , Xit , Y jt ≥ 0, Z jt ∈ {0, 1}, ∀i, j, t

⎫⎬

⎭ . (18)

4.1 Determining EVPI and VSS

In this study, we investigate the expected value of perfect information (EVPI) andthe value of stochastic solution (VSS) for the risk-neutral model. EVPI comparesstochastic with wait-and-see approaches; this measure can indicate whether or notrandomness has much impact on a given problem (Kall and Wallace 1994). It can bedefined as the expected price that the modeler should pay for having perfect informationabout the data and, then, it can be computed as the difference in the objective functionbetween the recourse model and the expected wait-and-see solution, as follows: (i)solve one deterministic problem for each scenario s. Such solutions W

s , s = 1, . . . , S,

represent the smallest total cost in the presence of perfect information; (ii) obtain theexpectation over the set of scenarios of the wait-and-see solutions, WS = ∑

s πs W s ;

(iii) evaluate this measure as EVPI = RP − WS.

VSS is a measure that compares stochastic with expected value approaches. Itquantifies the possible gain in solving the stochastic problem instead of using theexpected value problem EV. VSS is quantified as follows: (i) solve the expected valueproblem, EV, where the random variables are placed by their expectations; (ii) solveone problem for each scenario s, s = 1, . . . , S, with the first-stage variables fixedaccording to the EV problem, and obtain the expectation over the set of scenarios ofthese wait-and-see solutions (EEV); (iii) determine VSS = EEV − RP. The reader isreferred to Kall and Wallace (1994), Birge and Louveaux (1997) and Escudero et al.(2007), among others, for more details about these concepts.

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To determine the EVPI, we must solve s deterministic wait-and-see problems likeCLC with an objective value given by W

s , such that φ Ij t = φ I

j ts, φ I Ipt = φ I I

pts anddit = dits, for each s = 1, . . . , S. As these problems are feasible, the expectation WSis well-defined, and 0 ≤ EVPI < +∞ holds. The computation of VSS requires thesolution of two problems: EV and EEV. Commonly, EV is a wait-and-see problemof the expected value, i.e., by considering φ I

j t = ∑s πsφ

Ij ts, φ I I

pt = ∑s πsφ

I Ipts

and dit = ∑s πsdits, but it is possible to define the EV problem according to the

worst-case or the most probable scenario, for example. Regardless of the referencescenario, the EV problem is always feasible in our case, but the feasibility of theEEV problem is not always ensured, except under particular conditions, as shown inProposition 1.

Proposition 1 Consider the EEV problem with the first-stage variables fixed accord-ing to EV: Xit = X

i t , Y jt = Y j t and Z jt = Z

j t , for all i, j, t. Then,

C Et ≥ max

⎧⎨

⎩∑

j∈J

(ν I

j Y j t + φ I

j ts Zj t − Ct

),

∑

p∈P

∑

j∈J

(ν I I

p apj Yj t + φ I I

ptsδpj Zj t − Ct

)⎫⎬

⎭ ,∀t, s (19)

is a sufficiency condition to hold feasibility in the EEV problem for any referencescenario.

Proof As I +i ts and I −

i ts are both unbounded variables, then the material balance Eq.(2) is feasible for any fixed X

i t . Also, by assuming that the EV problem is alwaysfeasible, the coupled constraints (3) are feasible for any fixed X

i t and Y j t . Therefore,

the feasibility of the EEV problem depends only on the capacity constraints (14) and(15), whose feasibility set in always non-empty if the extra capacity C E

t is greater thanthe maximum lack of capacity, as shown in (19). ��

5 Risk-averse two-stage stochastic approach for production planningin furniture plants

Model RN is risk-neutral in the sense that it is concerned with the optimization of anexpectation only. This model is also exposed to the variation of the recourse costs, andit does not capture the decision maker’s preferences towards risk. In the following,we propose four alternate formulations to mitigate those drawbacks of the risk-neutralapproach. The first model is based on the minimax regret criteria and optimizes aworst-case scenario perspective without needing the probability of the scenarios. Thesecond formulation uses the conditional value-at-risk as the risk measure to avoidsolutions influenced by a bad scenario with a low probability. The third strategy is amean-risk model based on the upper partial mean that aggregates a risk term in theobjective function. The last approach is a restricted recourse approach, in which the

123


risk preferences are directly considered in the constraints. The motivation for choosingthese approaches is described in Sect. 1.

5.1 Minimax with regret

Minimax models focus on determining the best worst-case deviation from optimality,among all feasible decisions over all scenarios (Kouvelis and Yu 1997). In order toformulate the minimax regret model (minimax), it is necessary to determine the wait-and-see solutions W

s , for all s = 1, . . . , S. Afterwards, we minimize the maximumdifference between the cost and the wait-and-see cost of each scenario (regret) in theobjective function. The minimax regret problem can be posed as follows:

(Minimax) min �

s.t.: � ≥∑

i∈I

∑

t∈T

(cit Xit + h+

i t I +i ts + h−

i t I −i ts

)

+∑

j∈J

∑

t∈Tw j t Y jt +

∑

t∈Tot Ots − W

s , s = 1, . . . , S

� ≥ 0

Deterministic constraints (3), (6), (8) and (10)

Stochastic constraints (13)–(17).

(20)

The objective function combined with the first two constraints ensures that themaximum regret is minimized. The minimax model is primarily indicated to representconservative risk-averse decisions, since only unfavorable deviations are considereddue to � ≥ 0. This model is also appropriate if the probability of the scenarios isnot available. Otherwise, define �s as the regret of scenario s, minimizing

∑s πs�s

with �s ≥ 0. Then, the contribution of each scenario is considered and the solutioncan be less pessimistic. In any case, this type of approach has some disadvantages,in general; see Alonso-Ayuso et al. (2011). It is possible to adopt a reliability levelto avoid pessimistic solutions, as proposed in Daskin et al. (1997) and Chen et al.(2006).

5.2 Conditional value-at-risk

The previous model does not take into account the probability of occurrence of aspecific scenario and, for this reason, the solution can be influenced by a bad scenariowith a low probability. To overcome this issue, we propose using a risk measure widelyused in risk management, which is the conditional value-at-risk (CVaR) introduced byRockafellar and Uryasev (2000, 2002), a measure very close to the mean excess loss,expected shortfall and average value-at-risk measures. Basically, the CVaR minimizesthe Value-at-Risk (VaR), say η, jointly with the weighted expected cost exceeding itat the confidence level α.

Contrarily, the VaR measure minimizes the maximum cost over the scenarios butaccepting (1 − α)% scenarios whose cost is above VaR. Notice that this measure

123


requires more computational effort, since one 0–1 variable is added for each scenario.However, the main disadvantage is that it does not consider the cost of the (1−α)% non-wanted scenarios. Readers interested in other studies discussing VaR, CVaR and therelationships between them are referred to Artzner et al. (1999), Schultz and Tiedemann(2006) and Quaranta and Zaffaroni (2008), amongst others. An optimization probleminvolving CVaR at the confidence level α (CVaRα) can be posed as follows:

minx

{CVaRα[ f (x, ξ)]} , where CVaRα(ξ) = infη∈R

{η + 1

1 − αE[ξ − η]+

},

(21)

considering [ξ − η]+ = max(0, ξ − η) and α ∈ (0, 1). By assuming that ξ is discretewith finitely many scenarios s = 1, . . . , S and corresponding probabilities π1, . . . , πS,

then the CVaRα model becomes:

(CVaRα) min η + 1

1 − α

S∑

s=1

πsvs

s.t.: vs ≥∑

i∈I

∑

t∈T

(cit Xit + h+

i t I +i ts + h−

i t I −i ts

)

+∑

j∈J

∑

t∈Tw j t Y jt +

∑

t∈Tot Ots − η, s = 1, . . . , S

vs ≥ 0, s = 1, . . . , S

η ∈ R

Deterministic constraints (3), (6), (8) and (10)Stochastic constraints(13)–(17).

(22)

where vs is a continuous variable that gives the excess cost over the VaR variable forscenario s. Notice that η can be interpreted as the first-stage variable and vs as thesecond-stage ones; see Schultz and Tiedemann (2006). The confidence level α servesto reflect risk preferences: larger values of α indicate more risk aversion, and CVaRcontrols higher expected cost deviation from VaR deviations. For instance, if α = 0.7,

then CVaR constraint can control the largest 30 % of relative deviations. For smallvalues, e.g., α = 0.001, CVaR is very similar to the expected value (in this case, itminimizes the 99.9 % worst scenarios).

5.3 Mean-risk with upper partial mean

Mean-risk models became popular mainly due to Mulvey et al. (1995). The mainidea of these models is to combine expectation and dispersion of the expected costfunction:

minx

{E[ f (x, ξ)] + λ · D[ f (x, ξ)]} , (23)

where the random objective f (x, ξ) is characterized by the expectation E[·] and bythe dispersion statistic D[·], and λ is a a non-negative weight to tradeoff expected cost

123


and risk. By increasing λ, we can generate solutions with a higher expected cost, butwith a lower recourse variability. For instance, Ogryczak and Ruszczynski (1999) andAhmed (2006) discuss several strategies for this type of risk minimization.

Here, we measure risk by quantitative deviations of the random objective from theexpected recourse cost by using the upper partial mean (UPM) proposed in Ahmed andSahinidis (1998). This measure penalizes only the positive deviation of each scenario’scost from the recourse cost, and the corresponding model is still tractable, unlike amodel with a variance-based measure. The upper partial expected excess cost over thescenarios can be expressed as

∑Ss=1 πs�s, where

�s = max

{0,

∑

i∈I

∑

t∈T(h+

i t I +i ts + h−

i t I −i ts) +

∑

t∈Tot Ots

−S∑

s′=1

πs′

[∑

i∈I

∑

t∈T(h+

i t I +i ts′ + h−

i t I −i ts′) +

∑

t∈Tot Ots′

]},

s = 1, . . . , S. (24)

The corresponding mean-risk (MR) model with upper partial mean (UPM) is:

(MRλ) min∑

i∈I

∑

t∈Tcit Xit +

∑

j∈I

∑

t∈Tw j t Y jt

+S∑

s=1

πs

[∑

i∈I

∑

t∈T

(h+

i t I +i ts + h−

i t I −i ts

) +∑

t∈Tot Ots

]+ λ

S∑

s=1

πs�s

s.t.: �s ≥∑

i∈I

∑

t∈T(h+

i t I +i ts + h−

i t I −i ts) +

∑

t∈Tot Ots

−S∑

s′=1

πs′

[∑

i∈I

∑

t∈T

(h+


i t I −i ts′

) +∑

t∈Tot Ots′

]

�s ≥ 0, s = 1, . . . , SDeterministic constraints, (6), (8) and (10)Stochastic constraints(13)–(17).

(25)

The constraint type (25) computes the positive deviation of each scenario’s costfrom the expected cost, for all s = 1, . . . , S. This constraint plus �s ≥ 0 withthe minimization of the last term in the objective function satisfies the definition ofthe upper partial mean introduced in Ahmed and Sahinidis (1998). For notationalconvenience, we denote the expected deviation

∑s πs�s as UPM.

The risk factor λ provides the relative importance between the expected variability,∑s πs�s, and the expectation, i.e., the remaining expression in the objective function

of (25). Notice that∑

s πs�s is itself an expected cost, consequently the risk termλ

∑s πs�s is given in monetary units as well. The case for λ = 0 represents the

risk-neutral model RN, where the objective is to minimize the expected cost only,

123


irrespective of the recourse variability. This case is preferable by a risk-neutral decisionmaker. But if the decision-maker is risk-averse, he/she would prefer larger values ofλ to obtain solutions with lower recourse variability. By assigning different values forλ, we can generate tradeoff curves between cost and risk minimization.

5.4 Restricted recourse

It is still possible to limit �s to a prescribed level �max to incorporate risk-averse conditions over the second-stage variables. This approach was proposed byVladimirou and Zenios (1997) and its main appeal is to ensure the stability of therecourse decisions in the presence of multiple scenarios, without incorporating explicitrisk preferences; see also Beraldi and Triki (1998) and Ahmed and Sahinidis (1998).In this paper, we consider an upper bound on the deviation of the risk-neutral problem,say �0, and define another parameter γ ∈ [0, 1] to progressively reduce �0, such that�max = γ�0. The restricted recourse model RR becomes:

(RRγ ) min∑

i∈I

∑

t∈Tcit Xit +

∑

j∈I

∑

t∈Tw j t Y jt

+S∑

s=1

πs

[∑

i∈I

∑

t∈T

(h+

i t I +i ts + h−

i t I −i ts

) +∑

t∈Tot Ots

]

s.t.: �s ≥∑

i∈I

∑

t∈T(h+

i t I +i ts + h−

i t I −i ts) +

∑

t∈Tot Ots

−S∑

s′=1

πs′

[∑

i∈I

∑

t∈T

(h+


i t I −i ts′

) +∑

t∈Tot Ots′

]

S∑

s=1

πs�s ≤ γ�0,

�s ≥ 0, s = 1, . . . , SDeterministic constraints (3), (6), (8) and (10)Stochastic constraints(13)–(17).

(26)

The constraint∑

s πs�s ≤ γ�0 bounds the dispersion of the recourse decisions.By reducing the bound �0, a sequence of solutions less sensitive to realizations of thedata in a scenario set is progressively generated. Although the models MR and RRpresent different structures (objective function and constraints), they can be equivalent,as shown in Ahmed and Sahinidis (1998).

Table 1 shows the model dimensions in the usual way for the risk-neutral strategyand the risk-averse strategies for the two scenario tree cases that have been consideredin this study. The headings are as follows: m, number of constraints; n01, number of0–1 variables; nc, number of continuous variables; nel, number of non-zero elementsin the constraint matrix; and Den, matrix density (in %). Notice that the number ofconstraints and continuous variables can become prohibitive if the number of scenariosis large.

123


Table 1 Model dimensions with respect to the number of constraints (m), number of continuous variables(nc), number of 0–1 variables (n01), number of non-zero elements in the constraint matrix (nel) and matrixdensity (Den in %)

Model # Scenarios m nc n01 nel Den (%)

RN 27 2,364 2,212 648 119,554 1.768

Minimax 27 2,391 2,213 648 137,054 2.004

CVaR 27 2,364 2,213 648 96,257 1.423

MR 27 2,364 2,212 648 119,581 1.769

RR 27 2,365 2,213 648 119,583 1.767

RN 125 7,166 7,798 648 413,848 0.6838

Minimax 125 7,291 7,799 648 502,790 0.8164

CVaR 125 7,166 7,799 648 308,329 0.5094

MR 125 7,166 7,798 648 413,875 0.6838

RR 125 7,167 7,799 648 413,877 0.6836

6 Computational experience

In this section, we present a series of computational experiments using real-life datafrom a typical small-scale furniture plant in the northwest of São Paulo State (Brazil).The primary goal of these experiments is to study how the strategies perceive riskaversion, by comparing risk-neutral and risk-averse solutions with respect to averagecost and risk reduction mainly. The secondary goal is to analyze service levels andthe computational performance of the strategies. The models were coded in GAMS(Brooke et al. 1998) and solved using the optimization system ILOG-CPLEX 11.0under default setting (ILOG 2008). All numerical experiments were conducted on aPC Core i7 2.8 GHz, 8.0 GB RAM and Windows 7 operating system.

6.1 Problem setting

Data on products, pieces, cutting patterns, machines and activities were provided bythe manager responsible for the production planning of the plant. We consider threefinal products: 5-door wardrobes (5dw), dressers (drs) and bedside tables (bdt). Theraw material used in the production process combines two sizes of MDF boards:2.75 × 1.83 m and 2.75 × 1.85 m. The first size has five different thicknesses: 3,9, 12, 20 and 25 mm, whereas the second has only one thickness of 15 mm. Finalproducts consist of 49 distinct rectangular pieces by cutting MDF boards accord-ing to 81 feasible cutting patterns, which are often used by the furniture company.On average, the waste of material associated with non-homogeneous and homoge-neous cutting patterns is up to 7 and 17 %, respectively. Cutting times are estimatedbased on the thickness of each cutting pattern. We consider the processing time ofthe automatic drilling machine at 2.5 sec per piece. Deterministic setup times areestimated in 600 and 900 sec for cutting and drilling machines, respectively. Capac-ities in both phases are estimated in 158,400 sec for each period, i.e., according to

123


the machine availabilities for five work-shifts per week. The extra capacity of eachperiod is 79,200 sec, as the company has the flexibility to contract additional work-shifts per week. We consider a planning time horizon of 8 periods and each periodof the planning horizon refers to one production week. Costs are non-stationary withan interest rate equal to 0.01 % per month. Further details are available in Alem(2011).

There are various schemes for generating scenarios for specific problems: enumer-ation, scenario tree, Monte Carlo simulation, time series, regression models, amongstothers. The reader is referred to Dupacová et al. (2000) for an overview on scenariogeneration in a multistage context, and to Beraldi et al. (2010) and Kaut and Wallace(2011) for more recent contributions. This paper considered a scenario tree composedby the combination of the tree stochastic parameters: demand, setup time of the cut-ting machine and setup time of the drilling machine. For each stochastic parameter,we tested one case with three possible realizations, low, normal, or high (27-scen.tree), and another case with five generic realizations (125-scen. tree), as shown inFig. 4. The probability of each scenario in the 27-scenario tree case was determinedby the product between the probabilities of each uncertain parameter, by assumingstatistical independence between them. The individual probability of each possiblerealization (low, normal and high) was illustrated in four different cases: (i) moderatecase (M): (0.25, 0.5, 0.25); (ii) equiprobable case (E): (1/3, 1/3, 1/3); (iii) optimisticcase (O): (0.6, 0.3, 0.1) for demands and (0.5, 0.4, 0.1) for setup times; (iv) pes-simistic case (P): (0.1, 0.3, 0.6) for demands and (0.1, 0.4, 0.5) for setup times. Therealizations for demands and setup times in low, normal and high scenarios weregenerated according to a discrete (continuous) uniform distribution in the followingintervals: [0.7θ, 0.95θ ], [0.9θ, 1.05θ ] and [1.05θ, 1.3θ ], respectively, where θ is thecorresponding deterministic parameter. In the 125-scenario tree case, all equiprobablescenarios were generated in the interval [0.7θ, 1.3θ ].

Depending on the number of uncertain parameters and/or realizations, the numberof scenarios may be huge and the resulting program might become intractable. Inthis case, we can use some scenario reduction strategy to define a new scenario treedistribution, whose dimension is smaller than the original one. To remove a subset ofscenarios, usually scenario reduction methods use probability metrics (metric distanceson spaces of probability measures) to measure the distance between a distribution P ′of a reduced tree and a distribution P of the original tree; see, e.g., Dupacová et al.(2003); Heitsch and Römisch (2003, 2007, 2009), among others.

6.2 Results and discussion

We tested the risk-averse approaches by considering the two types of scenariotree described in Sect. 6.1 under different probability values. We reported only the27-scenario tree in the moderate case, and the 125-scenario tree in the equiproba-ble case, but the results for different probabilities are similar. For each type of sce-nario tree, we generated a sample of 100 instances, totaling 3,800 simulation runs.The instances were solved for the risk-neutral and minimax approaches, CVaRα

for α ∈ {0.1, 0.3, 0.5, 0.7, 0.9, 0.95}, MRλ for λ ∈ {1, 3, 5, 7, 10}, and RRγ for

123


Fig. 4 27-scenario tree and 125-scenario tree formed by combining three and five realizations, respectively,for each independent stochastic parameter: demands, setup time of the cutting machine and setup time ofthe drilling machine

γ ∈ {0.9, 0.8, 0.7, 0.5, 0.45, 0}. We limited in 100 sec (200 sec) each simulation runof the 27-scenario tree (125-scenario tree) instance and set the optimality gap relativeto the best lower bound in 0.001. In order to analyze the proposed models, we presentthe incumbent solution value (Zmip); elapsed times (in sec); the average cost (μ) andthe corresponding standard deviation (σ ) over all scenarios; the relative cost savingsby using the risk-averse rather than the risk-neutral (RN) approach to determine the

123


Table 2 Stochastic solution analysis over a scenario-tree sample of 100 instances

# Scenarios RN WS EEV EVPI EVPIRN × 100 % VSS VSS

RN × 100 %

27 254,292 189,054 427,154 65,238 25.65 172,862 67.97

125 230,747 189,106 365,720 41,641 18.04 134,931 58.46

Table 3 Average results for the Minimax model (20) over 100 samples for each scenario tree proposed

# Scenarios Zmip

regretRNregret

Save

=(

ZmipRN −1

)

× 100 %

Averagecost (μ)

Price=

(μ(minimax)

μ(RN)−1

)

× 100 %

Cost std.dev. (σ )

B(%)

Elapsedtimes(sec)

27 110,608 127,036 −12.93 282,081 9.344 38,504 96.66 17.87

125 77,055 168,211 −54.19 254,577 10.33 18,878 98.30 192.8

maximum regret (or the minimum CVaR); the relative price by using the risk-averserather than the risk-neutral strategy; the fill rate service level (or type II service level),evaluated as B = 1 − ∑

s πs I −T s/

∑s πs Ds, where I −

T s = ∑i I −

is is the backloggingin the last period T, and Ds = ∑

i t dits is the cumulative demand. For comparisonpurposes, we evaluated the average cost as the objective function of the risk-neutralapproach under equiprobable values for the scenarios.

EVPI and VSS. Both EVPI and VSS were determined for the risk-neutral optimiza-tion model RN. All problems (wait-and-see, EV and EEV) were solved in less thanone second on average (standard deviation over the instances less than 0.5 sec). Theresults suggested that it is possible to obtain significant savings by acquiring perfectinformation about the future (Table 2), as the expected value of perfect informationrepresents around 25 and 18 % of the stochastic solution value for the 27-scen. and125-scen. tree samples, respectively. Moreover, by adopting the EV problem based onthe mean scenario, we found that the EEV solution value is much worse than the RNsolution value, which provided a relative VSS of 67.97 and 58.46 %, for the sampleswith 27 and 125 scenarios, respectively. It is worthy of mentioning that for the 125-scen. tree, some EEV instances were proven infeasible (in this case, we assume thatVSS → ∞; see Proposition 1). Thus, VSS was evaluated by taking into account onlythe feasible instances. As a result, the expected value problems are not reasonableapproximations to the RN ones, as they lead to a high expected solution values andpotentially infeasible plans.

Minimax model. As the minimax approach ignores the different probabilities of the sce-narios, only the most unfavorable deviation is considered for computing optimal val-ues. This often leads to average costs higher than those from the risk-neutral approach.In this case, the price of this conservativeness varies in 9.344–10.33 %, as shown in the“Price” column of Table 3. However, this price is compensated by its regret, which is12 % lower than the one provided by the risk-neutral model, considering the instanceswith 27 scenarios. The cost savings attained for the 125-scen. tree sample is much

123


(a) (b)

(c) (d)

(e)

Fig. 5 Histograms of the recourse cost over a sample of 2,700 scenarios from the 27-scenario tree samplein (a) risk-neutral, (b) minimax, (c) CVaR0.95, (d) MR10, (e) RR0.45, and (f) RR0 strategies

higher (54 %), suggesting that the regret of the risk-neutral approach can be evenworse when more scenarios are considered. Moreover, histograms “a” of Figs. 5 and6 illustrate that the ranged spanned by the recourse cost in the risk-neutral approachis wider, indicating that the probability of more pessimistic (non-wanted) scenarios inRN is higher than in minimax.

CVaR model. As expected, increasing α means higher CVaR values, since larger real-izations of the total cost are considered and costly scenarios have more importancethan cheaper ones. To compare CVaR and RN, we also computed the CVaR risk mea-sure for the RN solution (column four of Table 4), and evaluated the relative costsavings, according to the expression in column five of Table 4. Noticed that the priceof the risk aversion is rather negligible compared to the cost savings obtained in theCVaR risk-averse strategy (columns seven and five of Table 4, respectively). As theconditional value-at-risk represents the possible average potential losses over a certainvalue, we can conclude that the proposed CVaR model is more efficient in minimizingthe losses than the risk-neutral approach for all confidence levels tested in the 27-scen.tree sample. In the 125-scen. tree sample, the risk-averse CVaR is a good alternativeonly from α = 0.7 on, indicating that with a relatively large number of scenarios andfor small values of α, CVaR and RN can be similar. This happens because CVaR isnot a real risk measure for α < 0.5 (Artzner et al. 1999). The solution for α = 0.9 inthe 27-scen. tree (125-scen. tree) sample is particularly attractive, as the cost savingsattain 6.0 % (6.4 %) and the cost standard deviation is reduced by 21.99 % (62.82 %),leading to a less than 1 % (3.8 %) of increasing in the average cost. Histograms “c”of Figs. 5 and 6 also show that the interval spanned by the cost in the CVaR approachis much smaller than in the risk-neutral, which indicates that it is possible to reduce

123


(a) (b)

(c) (d)

(e)

Fig. 6 Histograms of the recourse cost over a sample of 12,500 scenarios from the 125-scenario-tree samplein (a) risk-neutral, (b) minimax, (c) CVaR0.95, (d) MR10, (e) RR0.45, and (f) RR0 strategies

the probability of higher costs on non-wanted scenarios by enforcing risk aversion.Finally, the bi-modal solutions presented in histogram “c” of Fig. 5 suggest that CVaRis reasonably good; however, there are many scenarios with a great recourse cost.These solutions could be improved possibly combining CVaR with the expected valuein a mean-risk model.

Mean-risk model. As the risk parameter (λ) increases, large risk reductions are obtainedat the expenses of minor increases in the average costs (Table 5). As expected, the solu-tion for λ = 10 attains the lowest risk and the highest expected cost: the cost standarddeviation is reduced from 85.08 to 91.02 % and UPM, from 92.63 to 97.68 %, lead-ing to a (low) price of the risk aversion that varies between 5.587 and 6.645 % ofthe risk-neutral solution value. More notable is that the obtained risk-averse solutionseven for λ = 1 diminish considerably the risk and the average costs are only mar-ginally affected in both scenario samples. Indeed, whereas the average cost changesfrom −0.1627 to 0.1824 %, the cost standard deviation is decreased by 8.452 %, andUPM is 12.25 % lower in the 27-scen. tree sample. With 125 scenarios, both riskmeasures are approx. 20 % lower. The efficient frontier of the average cost versus therisk measure (cost standard deviation and UPM) for the 27-scenario tree sample isdepicted in Fig. 7. The distribution of the recourse cost shows that MR mitigates thevariability of the stochastic parameters (histograms “d” in Figs. 5 and 6), by concen-trating the risk-averse solutions in narrow intervals [54, 337; 59, 313] (27-scenario)and [39, 201; 39, 495] (125-scen.) with 95 % of probability. Also, the MR approachhas a lower probability of exceeding a more pessimistic cost (e.g. $ 70,000) than theRN strategy.

123


Tabl

e4

Ave

rage

resu

ltsfo

rth

eC

VaR

αm

odel

(22)

over

100

sam

ples

for

each

scen

ario

tree

prop

osed

#Sc

enar

ios

αZ

mip

RN

Save

Ave

rage

Pric

e=

Cos

tstd

.R

educ

tion

BE

laps

ed

CV

aRα

CV

aR=

(Z

mip

RN

−1)

cost

(μ)

( μ(C

VaR

)

μ(R

N)

−1)

dev.

(σ)

=( 1

−σ(C

VaR

)σ(R

N)

)(%

)tim

es

×100

%×1

00%

×100

%(s

ec)

270.

125

5,84

626

0,60

6−1

.861

258,

322

0.13

4515

,855

4.97

499

.35

10.8

1

270.

325

9,06

026

6,05

6−2

.701

258,

024

0.01

9014

,929

10.5

299

.42

9.29

8

270.

526

3,64

427

1,99

3−3

.167

257,

838

−0.0

532

13,3

5719

.94

99.4

99.

824

270.

727

0,84

627

6,54

9−2

.105

259,

394

0.55

0012

,650

24.1

899

.21

9.26

6

270.

927

3,88

329

0,55

7−6

.088

259,

705

0.67

0613

,016

21.9

999

.34

9.90

1

270.

9527

4,57

330

0,67

0−9

.505

260,

032

0.79

7113

,251

20.5

899

.33

10.6

9

125

0.1

232,

460

231,

250

0.52

3323

0,80

30.

0244

20,2

652.

570

99.3

913

8.6

125

0.3

235,

753

234,

621

0.48

2823

1,26

40.

2242

16,4

9320

.71

99.4

813

5.1

125

0.5

239,

463

238,

977

0.20

3323

2,45

50.

7404

12,6

5539

.16

99.5

412

7.0

125

0.7

244,

064

246,

108

0.83

0523

4,38

01.

574

11,0

6546

.80

99.6

012

4.1

125

0.9

250,

091

267,

263

6.42

523

9,40

33.

752

7,73

362

.82

99.6

511

6.2

125

0.95

252,

513

300,

332

−15.

9224

1,00

14.

444

8,17

660

.69

99.6

812

1.1

123


Tabl

e5

Ave

rage

resu

ltsfo

rth

eM

Rλ

mod

el(2

5)ov

er10

0sa

mpl

esfo

rea

chsc

enar

iotr

eepr

opos

ed

#Sc

enar

ios

λZ

mip

Ave

rage

Pric

eC

osts

td.

Red

uctio

nU

PMU

PMB

Ela

psed

cost

(μ)

=( μ

(MR

)

μ(R

N)

−1)

dev.

(σ)

=( 1

−σ(M

R)

σ(R

N)

)re

duct

ion

(%)

times

(sec

)

×100

%×1

00%

(%)

270

254,

292

257,

975

–16

,685

–6,

206

–99

.39

8.58

8

271

259,

971

257,

556

–0.1

627

15,2

758.

452

5,44

612

.25

99.4

98.

973

273

270,

181

258,

164

0.07

2912

,930

22.5

14,

730

23.7

999

.46

9.00

5

275

272,

420

269,

248

4.37

04,

509

72.9

759

8.5

90.3

698

.88

9.58

9

277

273,

274

271,

452

5.22

42,

338

85.9

925

0.6

95.9

698

.89

9.76

7

2710

273,

871

272,

387

5.58

71,

498

91.0

214

3.9

97.6

898

.82

10.1

0

125

023

0,74

723

0,74

7–

20,8

00–

5,64

4–

99.3

511

7.3

125

123

5,64

723

1,16

80.

1824

16,2

0122

.11

4,47

920

.63

99.5

015

0.9

125

324

3,28

623

4,55

61.

651

11,0

6446

.81

2,91

048

.44

99.5

916

6.7

125

524

6,84

824

1,05

44.

467

5,62

372

.97

1,15

979

.47

99.5

718

4.9

125

724

8,71

824

3,82

15.

666

4,19

779

.82

700

87.6

099

.56

175.

0

125

1025

0,24

124

6,08

16.

645

3,10

485

.08

416

92.6

399

.55

182.

7

123


25.6 25.8 26.0 26.2 26.4 26.6 26.8 27.0 27.2 27.4

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

(x104)

λ = 10

λ = 7

λ = 5

λ = 3

λ = 1

λ = 0 Standard deviation Upper partial mean

Average costs ($)

Cos

t sta

ndar

d de

viat

ion

and/

or U

PM

($)

Fig. 7 Tradeoff between average costs and risk for the 27-scen. tree sample obtained by MR model

Restricted recourse model. It is also possible to obtain large risk reductions in RRmodel, mainly for γ ≤ 0.5, when the standard deviation decreases from 32.67 to48.54 % and the UPM reduces by 50 %, causing an increase of less than 2 % in theaverage costs (Table 6). For less conservative managers, lower values for γ may begood alternatives to obtain relevant risk reductions by paying a negligible price though,e.g., γ = 0.8 provides around 20 % of reduction in both risk measures at the expensesof less than 0.5 % in the average costs. It is remarkable that in the RR strategy, anyrisk can be totally eliminated for γ = 0. In this case, both the standard deviation andthe UPM are reduced by 100 %, leading to a price that varies from 6.595 to 10.36 %.Moreover, the distribution of the expected cost of each instance is a singleton bar, asthe worst and best scenarios have identical solution values that coincide with theiraverage cost. Consequently, the interval spanned by the recourse cost is the narrowestone, as depicted in histograms “e” of Figs. 5 and 6. The efficient frontier of the averagecost versus the risk measure (cost standard deviation and UPM) for the 27-scenariotree sample is depicted in Fig. 8.

General Comments and managerial insights. All risk-averse policies provided highservice levels (approx. 98–99 %), but the best performance was observed in CVaR,mainly in the 125-scen. tree sample, whereB% is successively improved asα increases.On the other hand, minimax produced the worst service level for the 27-scen. treesample (96.66 %). In MR and RR models, service levels are slightly deteriorated forlarger λ and γ. In most cases, risk-averse solutions were achieved by augmentingproduction rate (thus, increasing inventory levels and trim-loss) by using overtimelabor and avoiding to incur in backlogged demand. One exception is when γ = 0 in

123


Tabl

e6

Ave

rage

resu

ltsfo

rth

eR

Rγ

mod

el(2

6)ov

er10

0sa

mpl

esfo

rea

chsc

enar

iotr

eepr

opos

ed

#Sc

enar

ios

γZ

mip

Ave

rage

Pric

eC

osts

td.

Red

uctio

nU

PMU

PMB

Ela

psed

cost

(μ)

=( μ

(MR

)

μ(R

N)

−1)

dev.

(σ)

=( 1

−σ(M

R)

σ(R

N)

)re

duct

ion

(%)

times

×100

%×1

00%

(%)

(sec

)

270.

925

4,62

025

7,70

5−1

.299

15,0

679.

185

5,58

510

.00

99.4

510

.47

270.

825

5,69

925

8,04

60.

0290

13,6

5016

.84

4,96

520

.00

99.4

59.

938

270.

725

7,25

725

8,89

10.

3571

12,4

8422

.91

4,34

430

.00

99.4

19.

745

270.

526

0,93

526

1,38

41.

323

10,8

7232

.67

3,10

350

.00

99.2

310

.40

270.

4526

1,88

026

2,11

41.

606

10,4

0735

.61

2,79

355

.00

99.1

69.

966

270

274,

785

274,

989

6.59

50.

0000

100.

00.

0000

100.

098

.72

9.64

5

125

0.9

230,

892

230,

892

0.06

2918

,635

10.4

15,

080

10.0

099

.42

174.

5

125

0.8

231,

270

231,

270

0.22

6716

,211

22.0

64,

515

20.0

099

.49

175.

7

125

0.7

232,

060

232,

060

0.56

8913

,571

34.7

63,

951

30.0

099

.56

179.

2

125

0.5

234,

973

234,

973

1.83

110

,704

48.5

42,

822

50.0

099

.60

183.

5

125

0.45

235,

811

235,

811

2.19

49,

961

52.1

12,

540

55.0

099

.59

183.

9

125

025

4,65

925

4,65

910

.36

0.00

0010

0.0

0.00

0010

0.0

99.4

019

6.2

123


25.6 25.8 26.0 26.2 26.4 26.6 26.8 27.0 27.2 27.4 27.6

0

2000

4000

6000

8000

10000

12000

14000

16000

0

2000

4000

6000

8000

10000

12000

14000

16000

(x104)

γ = 0

γ = 0.45γ = 0.5

γ = 0.7γ = 0.8

γ = 0.9 Standard deviation Upper partial mean

Cos

t sta

ndar

d de

viat

ion

and/

or U

PM

($)

Average cost ($)

Fig. 8 Tradeoff between average costs and risk for the 27-scen. tree sample obtained by RR model

RR strategy, when the backlogging considerably increased during the planning timehorizon. CVaR also yielded the most stable first- and second-stage solutions as therisk parameter increased. From a factory standpoint, this feature is important becausean operational plan can be easily re-implemented for different risk attitudes.

Table 7 reports the value of the gaps equal to (Zmip/ZLP−1)100 %; the elapsed timesfor obtaining the incumbent solution; the number of explored nodes in the branch-and-bound tree; and the total number of cuts added during the branch-and-cut algorithm(clique, flow, cover, Gomory, etc.). Observations on Tables 1, 3–6 (last columns) andTable 7 are summarized as follows: It seems that the computational effort increases asrisk aversion is enforced in MR model. This behavior is also evident in RR model with125 scenarios. As expected, all strategies experienced an increase in elapsed times asthe number of scenarios increased. However, minimax required up to five times moreeffort than the other approaches, which resulted in higher gaps. On the other hand,CVaR presented the most stable performance for relatively large s and better gaps.Not surprisingly, the algorithm explored more nodes while solving minimax thansolving the remaining problems. As minimax presents the least sparse matrix andCVaR the most sparse one (Table 1), it is reasonable that the former presented theworst computational effort (Bixby 2002). In our test cases, there is no clear trendamong the number of scenarios and the number of cuts added during the branch-and-cut algorithm. Another relevant point is that RN, MR and RR approaches experiencedsimilar computational performance, suggesting that effective methods for solving the

123


Table 7 Computational performance of the risk-neutral and risk-averse strategies concerning gaps, elapsedtimes, number of nodes and cuts for each number of scenarios tested

Strategy # Scenarios GAP (%) Elapsed times (sec) No. of nodes No. of cuts

27 0.2242 6.070 495 220

50 0.2303 24.42 541 190

100 0.2729 82.40 607 316

RN 125 0.2427 116.5 664 499

200 0.2162 165.1 487 199

300 0.2335 716.1 714 511

400 0.2414 1,246 812 256

Average – 0.2373 336.6 617.1 313.0

27 0.9624 12.92 630 144

50 0.7228 84.53 1,771 202

100 0.5679 395.2 4,210 326

Minimax 125 0.4692 645.8 4,237 317

200 0.3642 2,633 929 456

300 0.5195 2,865 498 257

400 0.4559 5,476 510 187

Average – 0.5803 1,730 1,826 269.9

27 0.2353 6.350 648 175

50 0.2530 33.67 611 148

100 0.2298 111.7 740 228

CVaR0.95 125 0.1107 167.6 784 305

200 0.2159 368.1 511 172

300 0.2330 663.3 610 210

400 0.2470 958.0 599 206

Average – 0.2178 329.8 643.3 206.3

27 0.2390 6.360 495 261

50 0.2230 22.05 507 211

100 0.2491 49.52 580 368

MR10 125 0.2618 112.8 613 187

200 0.2396 280.1 660 193

300 0.2335 706.1 714 511

400 0.2414 1,233 812 256

Average – 0.2410 344.3 626 283.9

27 0.2231 6.090 522 246

50 0.2799 25.16 543 220

100 0.2541 66.22 507 342

RR0 125 0.2691 123.1 556 186

200 0.2396 283.5 660 193

300 0.2335 715.2 714 511

400 0.2414 1,245 812 256

Average – 0.2487 352 616 279.1

123


risk-neutral problem might perform well under certain types of risk-averse constraintsas well.

Remark. Although we basically omitted the results for the optimistic and pessimisticscenario trees for the sake of similarity, it is worth noting that the structure of theoptimal solutions is not particularly sensitive to variations on the probabilities, but theoptimal values are. An implication of this result was noticed in the EVPI and VSSanalysis. The absolute values for EVPI suggested that more pessimistic situationsgenerate larger EVPI’s, possibly because worse realizations have higher probabilitiesin these cases. Thus, the randomness of more unfavorable scenarios has more impor-tance, which reflects in the EVPI. Also, the expected value solution leads to infeasibleinventory levels, mainly in more pessimistic situations. In general, the optimistic caseprovides slightly worse risk-averse solutions as risk reduction is rather lower, however,at the expense of higher costs. On the other hand, in pessimistic situations the riskreduction is higher as well as the price of their risk-averse solutions. Further tests alsoanalyzed the proposed strategies under a normal distribution for the setup times anda truncated log-normal distribution for the demands; and by increasing the standarddeviation of demands in order to generate higher (lower) values for the random vari-ables. The results indicated that, in general, minimax and MR were the most sensitivestrategies, leading to a little worse performance in risk reduction and service levels,at the expenses of higher relative prices. CVaR exhibited the most stable behavior onaverage, mainly for high-variability scenarios, followed by the RR approach.

7 Concluding remarks

The idea of jointly reducing the variability of the expected costs and hedging againstthe occurrence of undesirable scenarios can be part of the solution for improving theproduction planning in small furniture plants under data uncertainty. The analysis ofthe four risk-averse strategies for minimizing the risk indicated that all of them perceiverisk aversion differently and outperformed the risk-neutral approach in achieving less,though more costly, risky solutions. Based on our numerical experience, it is note-worthy that the minimax approach provided the least encouraged results in reducingthe risk and in terms of computational effort, although this strategy can be usefulin the absence of the probability distribution. Under appropriate confidence levels,CVaR is especially relevant for minimizing the losses of worse scenarios and reduc-ing the cost standard deviation. Also, CVaR is the most stable and computationallyfaster strategy. MR and RR approaches have similar and attractive features, in thesense that both are capable to diminish simultaneously the cost standard deviationand UPM, paying a reasonable price for this. In addition, less conservative managerswould be more willing to adopt MR or RR solutions to ensure major risk reductionsat a negligible price. Although such results are dependent on our particular applica-tion, an interesting line of research would be to extend the framework for studyingthe effectiveness of the risk aversion in different industrial contexts. Further researchmight include: developing minimax and CVaR mean-risk models; explicitly model-ing the service level constraints as studied in Yildirim et al. (2005) and Tempelmeier(2007); exploring first and second-order dominance stochastic constraint strategies

123


(Schultz and Tiedemann 2003; Gollmer et al. 2008, 2011; Alonso-Ayuso et al. 2011);extending the two-stage model to a multistage environment; and developing exact andheuristics methods to solve larger instances, e.g., as in Alonso-Ayuso et al. (2003,2007), Beraldi et al. (2006) and Escudero et al. (2009).

Acknowledgments We are indebted to three anonymous referees whose valuable comments helped toimprove both the content and presentation of this paper. The authors are grateful to the Luapa Company forcollaborating with this research, Prof. Alyne Toscano Martins (Federal University of Triângulo Mineiro)for providing the technical report about Brazilian furniture and Prof. Paulo A. V. Ferreira (State Universityof Campinas) for suggesting an additional mean-risk discussion.

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Risk-averse two-stage stochastic programs in furniture plants

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