Production Planning Problems with Joint Service-Level Guarantee: A Computational Study

Transcription

1 Producion Planning Problems wih Join Service-Level Guaranee: A Compuaional Sudy Yuchen Jiang, Juan Xu :, Siqian Shen, Cong Shi Indusrial and Operaions Engineering, Universiy of Michigan, Ann Arbor, MI {ycjiang, siqian, shicong}@umich.edu : Indusrial and Enerprise Sysems Engineering, Universiy of Illinois a Urbana-Champaign, Champaign, IL juanxu2@illinois.edu Absrac We consider a class of single-sage muli-period producion planning problems under demand uncerainy. The main feaure of our paper is o incorporae a join service-level consrain o resric he join probabiliy of having backorders in any period. This is moivaed by manufacuring and reailing applicaions, in which firms need o decide he producion quaniies ex ane, and also have sringen service-level agreemens. The inflexibiliy of dynamically alering he pre-deermined producion schedule may be due o conracual agreemen wih exernal suppliers or oher economic facors such as enormously large fixed coss and long lead ime. We focus on wo sochasic varians of his problem, wih or wihou pricing decisions, boh subjec o a join service-level guaranee. The demand disribuion could be nonsaionary and correlaed across differen periods. Using he sample average approximaion (SAA) approach for solving chance-consrained programs, we re-formulae he wo varians as mixed-ineger linear programs (MILPs). Via compuaions of diverse insances, we demonsrae he effeciveness of he SAA approach, analyze he soluion feasibiliy and objecive bounds, and conduc sensiiviy analysis for he wo MILPs. The approaches can be generalized o a wide variey of producion planning problems, and he resuling MILPs can be efficienly compued by commercial solvers. Key words: producion planning; sochasic programming; mixed ineger linear programming; join service-level consrain; sample average approximaion. Hisory: Received Sepember 2015; revisions received March 2016; acceped May

2 1 Inroducion In his paper, we sudy a class of producion planning problems subjec o a join servicelevel consrain. The problems fall ino he caegory of single-sage muli-period sochasic opimizaion problems wih no recourse decisions (i.e., firms need o plan heir producion and/or pricing decisions ex ane and canno change hem in subsequen periods). This class of problems is primarily moivaed by manufacuring or reailing applicaions, in which firms have sringen service-level requiremens, bu do no have sufficien flexibiliy of alering heir decisions due o he relaed issues, such as conracual agreemen wih ouside suppliers, enormously large fixed coss and long lead ime. We describe several moivaing examples for conducing he research. In he firs example, CEMEX, a mulinaional building maerials company ha ofen signs conracs wih large even organizers. The firm was he key supplier of cemen for muliple large evens in 2014, including noably a conrac wih Fédéraion Inernaionale de Fooball Associaion (FIFA) o supply ons of cemen for he new soccer sadium in Manaus [4]. The firm has o plan producion quaniies and prices for he commied projecs long before saring he projec. Moreover, guaraneeing an adequae service-level is absoluely essenial for successful compleion of hese projecs. The second example arises in he U.S. auomoive indusry. The lead ime for building a new car model is ypically 52 monhs [8] due o lenghy design and esing cycles, and he lead ime for manufacuring an exising car model is ypically 10 o 18 monhs [2]. The fixed cos relaed o alering a producion schedule is also quie high. Due o hese reasons, car manufacurers usually make heir producion plans ahead of he selling season and heir planning decisions canno be easily changed aferwards. Meanwhile, an adequae service-level is imporan o mainain firms revenue and goodwill. To capure he aforemenioned applicaions sylisically, we propose wo single-sage muli-period models (wih or wihou pricing decisions), subjec o a join service-level guaranee. The firs model concerns he producion decisions while he second model concerns boh he producion and pricing decisions. For he firs model, he demands are random, which can be non-saionary and correlaed across differen periods. Our goal is o minimize he expeced oal cos, including (linear) producion coss, invenory holding coss and backorder penaly coss, subjec o a join service-level consrain over he whole periods o resric he probabiliy of having unme demands during he planning horizon. For he second model, we assume a classical addiive demand model in which he demand depends linearly on he price plus a random disurbance erm (see, e.g., Chen and Simchi-Levi [6]). We consider discree pricing and coninuous pricing opions, in which prices are chosen from a given finie se of values or from a bounded price range, respecively. Our goal is o maximize he oal expeced profi, also subjec o a join service-level consrain. We remark again ha boh problems considered in his paper belong o he caegory 2

3 of single-sage sochasic opimizaion problems wih no recourse decisions, which should be disinguished from dynamic invenory conrol problems considered in he lieraure (see, e.g., Zipkin [26]). The main feaure of our models is o incorporae a join service-level consrain, which is pracically relevan bu compuaionally inracable. Our approach employs he Sample Average Approximaion (SAA) mehod (see Luedke and Ahmed [16]) o reformulae our chance consrained problems as mixed-ineger linear programming (MILP) models and o compue upper and lower bounds of he opimal objecive values as well as feasible soluions wih cerain confidence levels. 1.1 Relevan Lieraure The radiional sudy of producion planning problem has been focused on deerminisic models wih known demand. Zangwill [25] developed a deerminisic lo-sizing model ha allowed for backlogged demand and proposed a nework approach. They furher proposed dynamic programming algorihms o compue opimal planning policies based on nework formulaions. Poche and Wolsey [23] sudied several srong MILP reformulaions of he uncapaciaed lo-sizing problem wih backlogging. They also described a family of srong valid inequaliies ha can be effecively used in a cu generaion algorihm. Florian e al. [9] sudied capaciaed lo-sizing problem and showed ha he deerminisic problem is NPhard. Recenly, Absi e al. [1] sudied he single iem uncapaciaed lo-sizing problem wih producion ime windows, los sales, early producions and backlogs. They presened MILP formulaions of hese models and developed dynamic programming algorihms o solve hem. González-Ramírez e al. [10] proposed a heurisic algorihm o solve a muli-produc, muli-period capaciaed lo-sizing problem wih pricing, where he deerminisic demand was assumed o be linear in price. For he producion planning problem wih sochasic demand, Mula e al. [21] reviewed some exising lieraure of producion planning under demand uncerainy. Gupa and Maranas [11] proposed a sochasic programming based approach o incorporae demand uncerainy in miderm producion planning. Kazaz [14] sudied producion planning wih random yield and demand using a wo-sage sochasic programming approach. Clark and Scarf [7] considered a periodic-review muli-period producion planning problem wih uncerain demand and hey showed he srucure of opimal policies via dynamic programming approach. In his paper, we focus on sochasic varians of producion planning problems, subjec o a join service-level consrain. There has been limied lieraure on his opic, among which Bookbinder and Tan [3] sudied a muli-period lo-sizing problem ha imposed individual service-level consrain in each period and heir demand disribuions were known. In conras, our model considers a join service-level consrain ha poses more compua- 3

4 ional challenges, and empirical demand samples are given insead of an explici demand disribuion funcion. We reformulae our problem as an MILP model using he SAA approach, which is based on Mone Carlo simulaion of random samples, o approximae he expeced value funcion by he corresponding sample average. Kleyweg e al. [15] sudied he convergence raes, sopping rules and compuaional complexiy of he SAA mehod. They also presened a numerical example for solving he sochasic knapsack problem using he SAA mehod. Verweij e al. [24] formulaed sochasic rouing problems using he SAA approach. They applied decomposiion and branch-and-cu echniques o numerically solve he approximaing problems. Pagnoncelli e al. [22] applied he SAA mehod o solve wo chance consrained problems, namely, linear porfolio selecion problem and blending problem wih a join chance consrain. Recenly, Mancilla and Sorer [17] formulaed a sochasic scheduling problem using he SAA approach and proposed a heurisic mehod based on Benders decomposiion. Our main mehodology builds upon he heory developed in Luedke and Ahmed [16]. They firs proposed o use he SAA approach o find feasible soluions and lower bounds on he opimal objecive value of a general chance-consrained program. Keeping he same required risk level, hey showed ha he corresponding SAA counerpar yields a lower bound of he opimal objecive value. To find a feasible soluion, hey showed ha i suffices o solve a sample-based approximaion wih a smaller risk level. They also mahemaically derived he required sample sizes in heory for having a lower bound or a feasible soluion wih high confidence. Our paper conribues o he lieraure by firs employing he SAA approach o solve a class of producion planning problems subjec o a join service-level consrain, which is ypically compuaionally inensive. 1.2 Conribuions of his paper The main conribuions of his paper are summarized as follows. 1. From he modeling perspecive, we propose wo new producion planning models (wih and wihou pricing decisions) subjec o a join service-level consrain. In classical sochasic producion planning problems, unsaisfied demands are ypically penalized by a linear backorder cos only. However, i is usually imporan for firms o mainain heir reliabiliy or credibiliy by persisenly saisfying all he marke demands in each period wih high probabiliy. Some firms use α-service-level (defined as he probabiliy ha he demand is fully saisfied) o measure heir qualiy of service (QoS). This meric is ye negleced in mos classical producion planning models in he lieraure. This moivaes us o incorporae a join α-service-level consrain ha ensures he marke demands being saisfied in each period wih a sufficienly high probabiliy, so ha he relaed firms can remain compeiive and profiable. 4

5 Moreover, in mos sochasic producion planning models in he exising lieraure, he demand disribuions in each period are given explicily, while in real-life applicaions, i is usually difficul o deduce he rue underlying demand disribuion. The SAA reformulaion can be done wihou knowing he exac demand disribuion; however, a large amoun of empirical daa (more han 5000 samples of demands) is needed o solve he SAA reformulaion under he nominal risk level and such amoun of daa may no be available in realiy. We show ha by using smaller risk parameer in he SAA reformulaion, we are able o obain good feasible soluions (wihin 5% of opimaliy) using much less empirical daa (around 300 samples of demands), which makes he daa-collecion work less demanding. Also, our proposed models allow for nonsaionary and generally correlaed demands. 2. From he compuaional perspecive, we employ he SAA mehod for chance-consrained programming and reformulae he wo producion planning models as MILPs using finie samples. However, due o he large amoun of samples needed in he SAA reformulaion, solving he resuling MILPs exacly is compuaionally inensive. Tuning he risk parameer in he SAA reformulaion smaller han he required service level (i.e., more conservaive), we achieve feasible soluions by solving he resuling MILPs via much fewer samples. The feasible soluions provide an upper (lower) bound on he cos-minimizaion (profi-maximizaion) problem. On he oher hand, we also compue a lower (upper) bound on he cos-minimizaion (profi-maximizaion) problem by seing he risk parameer o be a leas equal o he service level and solving muliple SAA counerpars wih fewer number of samples. 3. A comprehensive numerical sudy has been conduced using hree popular demand models wih differen paerns of demand correlaions among periods (i.e., idenical and independen demand disribuions, Markov modulaed demand process (MMDP) and auoregressive demands (AR models)), which are exensively used in heory and pracice. For each problem insance wih differen demand model and differen required service levels, boh upper and lower bounds are compued and validaed. We hen compare our bounds wih he opimal soluions (for reasonable problem sizes). I can be observed ha he more samples we use, he less gap i has from our bounds o he opimal soluions. Also, he acual sample size needed o achieve a a given confidence level for boh upper and lower bound soluions is a magniude smaller han he heoreic bounds proposed in Luedke and Ahmed [16]. We conduc sensiiviy analysis for producion planning wih pricing and our numerical resuls show ha when he demand is less sensiive o he price, he firm ends o increase he price while keeping he demand a he same level, in order o obain a beer profi. 5

6 1.3 Organizaion of he paper The remainder of his paper is organized as follows. In Secion 2, we inroduce he noaion and formulae he join service-level consrained sochasic producion planning problem. Secion 3 formulaes he producion planning problem wih pricing opions. The compuaional resuls and insighs for boh problems wih/wihou pricing opions are given in Secion 4. Finally, Secion 5 concludes he paper and gives fuure research direcions. Throughou he paper, for noaional convenience, we use a capial leer and is lowercase form o disinguish beween a random variable and is realizaion. We use fi o indicae is defined as, and 1pAq is he indicaor funcion aking value 1 if saemen A is rue and 0 oherwise. For any x P R, we denoe x` maxx, 0u. We also use rxs o denoe he smalles ineger ha is no less han x and use xu o denoe he larges ineger no greaer han x. 2 Producion Planning wih a Join Service-Level Consrain 2.1 Mahemaical formulaion Consider a finie horizon of T periods. The classic producion planning problem decides he producion quaniies for each period simulaneously a he beginning of he planning horizon (denoed as q 1, q 2,..., q T ). During each period p 1,..., T q, demands are realized and hree ypes of cos are incurred: producion cos (wih a per-uni cos c ą 0), holding cos for on-hand invenories from period o ` 1 (wih a per-uni cos h ą 0), and penaly cos for backlogged demand (wih a per-uni cos p ą 0). The objecive is o minimize he oal cos over he T periods. Le D 1,..., D T denoe he random demands over he T periods and hey may be independenly disribued or correlaed. Le X and B be random variables ha denoe on-hand invenories and backorders a he end of period 1, 2,..., T, respecively. Clearly, boh X and B mus be nonnegaive. The iniial invenory and backorder levels are denoed by x 0 and b 0, respecively. We formulae he producion planning problem under a join service-level consrain as Tÿ (PP) min pc q ` h ErX s ` p ErB sq (1) 1 s.. X 1 ` q ` B D ` X ` B 1,..., T, (2) PpX B ě 1,..., T q ě 1 θ, (3) q ě 1,..., T. (4) The objecive (1) minimizes he oal ordering cos, expeced invenory cos and expeced backlogging cos. In each period, he incoming iems are X 1, q and B while he ougoing 6

7 iems are X, B 1 and D. To balance hem, we formulae (2) as he flow-balance consrains. Consrain (3) requires ha he probabiliy of saisfying he demands in all T periods is a leas 1 θ, which defines he service level. 2.2 Reformulaion using he SAA approach Consider N samples of demands over T periods denoed by d piq pd piq 1,..., d piq T q (i 1, 2,..., N) where each sample i is equally likely o occur wih probabiliy 1{N. The on-hand invenories and backorders vary according o demand samples, denoed by x piq px piq 0,..., x piq T q and bpiq pb piq 0,..., b piq T q, respecively. The iniial on-hand invenory and backorder are pre-deermined regardless of he realizaion of random demands, i.e., x piq 0 x 0 and b piq 0 b 0 for all i 1, 2,..., N. The ordering quaniies q 1,... q are decided before knowing he demand realizaions, and hus do no depend on he specific samples. In each sample i, he balance consrain (2) is presened as We compue he oal expeced cos as: x piq 1 x piq b piq 1 ` b piq ` q d 1,..., T. (5) Tÿ 1 c q ` 1 N Tÿ Nÿ 1 i 1 h x piq The join service-level consrain (3) is equivalen o Nÿ i 1! 1 x piq ` p b piq. ) ě b 1, 2,..., T ě rp1 θqns. (6) Define binary variables y piq such ha y piq 1 if and only if he i-h scenario is violaed. We hen replace he join service-level consrain (6) by: $ x piq b piq ě M piq y 1,..., T, (7) & Nÿ y piq ď θnu, (8) i 1 % y P 0, 1u N. (9) Noe ha for each period and sample i, he big-m coefficien M piq is a valid upper bound for x piq b piq ` b piq, because x piq pd piq ÿ ď pds piq s 1 q q ` pb piq 1 x piq 1q pb 0 x 0 q ` 7 q s q ` pb 0 x 0 q ÿ s 1 d piq s. x 0 ` b 0 ` ř s 1 dpiq s

8 When y piq 0, he consrain x piq b piq ě 0 is enforced for each 1, 2,..., T. When y piq 1, he join service-level consrain in he i-h sample can be violaed and he oal number of violaed samples is no more han θnu, ensured by he consrain ř N i 1 ypiq ď θnu. Therefore, we approximae a muli-period plan of opimal ordering quaniies by using he following MILP model. (SAA-PP) min # T ÿ 1 c q ` 1 N s.. (5), (7) (9), Tÿ Nÿ 1 i 1 h x piq + ` p b piq x piq 0 x 0, b piq 0 b 1,..., N, (10) x piq, b piq, q ě 1,..., T, i 1,..., N. (11) We presen a necessary condiion for any opimal producion plan in he following proposiion. Proposiion 1. For any opimal soluion q, x piq piq (, b o he (SAA-PP), we have eiher i, piq 0 or b 0 holds for all i 1, 2,..., N and 1, 2,..., T. x piq Proof. We show he resuls by conradicion. Suppose ha here exiss an opimal soluion wih x piq ą 0 and b piq ą 0 for some P 1, 2,..., T u and i P 1, 2,..., Nu. We can replace x piq b piq and b piq by x piq x piq minx piq, b piq u and b piq minx piq, b piq u while keeping he oher decisions he same. Since he invenory level remains unchanged, i.e., x piq piq b x piq b piq, all he consrains are saisfied under he new soluion. Meanwhile, his new feasible soluion decreases he oal cos by 1 ph N ` p q minpx piq, b piq q ą 0. This conradics wih he fac ha he original soluion is opimal and hus complees he proof. The above proposiion is also rue when no service-level requiremen is presen [see, e.g., 26]. I assers ha even in he case of having a join service-level consrain, here is no incenive o have backorders while holding posiive invenories. This resul is also rue when here is no service-level requiremen. 3 Model Varian wih Pricing Opions 3.1 Noaion and mahemaical formulaion We consider pricing decisions in he above producion planning problem. In his varian, besides he ordering quaniy q for each period 1, 2,..., T, he manager also decides he price r for each period 1, 2,..., T a he beginning of he whole ime horizon. The price r se by he manager affecs he underlying demand disribuion Dpq and hus he realizaion d. The goal is o maximize he oal expeced profi over he T periods. 8

9 We inerpre he random demand by a deerminisic linear funcion in price r plus a noise erm, i.e., D pr q a r ` β ` ɛ, where ɛ is a random variable wih Er ɛ s 0 for 1,..., T, and boh a, β ą 0. This demand model is well known as he addiive demand model in he lieraure (see, e.g., Mills [19]). I allows for correlaed demands over periods, which indicaes ha ɛ u T 1 are no necessarily independen random variables. We furher assume ha once he demand is realized in period, i esablishes a conrac beween he buyer and he reailer wih a uni price of r. In oher words, given ha he realized demand D d, i immediaely incurs a revenue of r d, no maer when he demand is saisfied. The price r pr 1, r 2,..., r T q T is chosen from a given se P Ď R T. We can hen formulae he producion planning problem wih pricing opions as (PO) max Tÿ 1 s.. (2) (4), ˆ r ErD s c q h ErX s p ErB s D a r ` β ` 1,..., T, (12) r P P. In he formulaion above, he consrains (2) (4) are carried from he model wihou pricing opions. The consrain (12) shows he relaionship beween price and demand in each period. 3.2 Sample average approximaion reformulaion We reformulae he (PO) model using N sample daa, namely d piq pd piq 1, d piq 2,..., d piq T qt, i 1, 2,..., N. For he i-h sample, we use ɛ piq o denoe he realizaions of ɛ and oher noaions remain he same. Then he oal expeced profi is given by Tÿ 1 c q ` 1 N Tÿ Nÿ 1 i 1 r d piq h x piq ı p b piq. Hence, using he linear relaionship beween demand and price, i.e., d piq we reformulae he (PO) model as: a r ` β ` ɛ piq, (SAA-PO) max s.. Tÿ 1 ` c q a r 2 ` β r ` 1 N x piq 1 x piq x piq b piq ě s 1 Tÿ Nÿ 1 i 1 `ɛpiq r h x piq p b piq b piq 1 ` b piq ` q a r ` β ` ɛ 1,..., T, i 1,..., N, ˆ ÿ ` x 0 ` b 0 ` as r s ` β s ` ɛ piq s y piq, 9

10 (8) (11), r P 1,..., T, i 1,..., N, (13) Here, he probabilisic consrain (3) in he (PO) model is equivalen o consrains (13), (8) and (9), for which we define new binary variables y piq (i 1,..., N). However, he above model sill involve nonlinear erms r 2 in he objecive funcion and bilinear erms r s y piq in consrain (13). We reformulae i as a linear model for wo specific price ses wih eiher discree or coninuous price choices, where price decisions p, 1,..., T, are independenly deermined for each period. For he former, he price is drawn from a se of finie possible prices, denoed by se R γ 1,..., γ m u. In his case, he price se can be wrien as P Ś T 1 R. For he laer, we consider he possible price r chosen from a price inerval rl, U s given for each period and he price se is specified as P Ś T 1 rl, U s. We will give MILP models for each price se in he following subsecions. Noe ha our model is also capable of describing he relaionship among he prices in each period. For example, if he prices are no allowed o increase from periods o periods, we can simply add he consrain r 1 ě r 2 ě ě r T o our model while he complexiy of he resuling model remains he same Discree price se Consider a finie se R γ 1,..., γ m u from which produc price is chosen in each period. Define a binary decision variable u j o indicae wheher using he j-h price opion, i.e., γ j (j 1,..., m ). In each period, u j 1 if r γ j and u j 0 oherwise. To ensure only one price is used from he se pγ 1, γ 2,..., γ m q in each period, we require ř m j 1 u j 1 for each. Then, he quadraic erm r 2 in objecive funcion can be expressed by a linear erm: ÿm r 2 pγjq 2 u j. (14) Similarly, for any 1,..., T and i 1,..., N, he nonlinear erm r y piq service-level consrain can be rewrien as r y piq j 1 ÿm j 1 in he join γ ju j y piq. (15) To furher linearize he erm u j y piq in (15), we inroduce anoher decision variable v piq j o replace u j y piq and add he following McCormick inequaliies o force v piq j u j y piq (see 10

11 McCormick [18]): $ & % v piq j ď u j v piq j ď y piq v piq j ě u j ` y piq 1 v piq j ě 0. Here boh u j and y piq are binary variables. If u j 1 and y piq 1, hen v piq j 1; oherwise, he consrains enforce v piq j 0. Therefore, for discree pricing opions, using equaliies (14) and (15), we can formulae he following (PP-D) model: (16) (PP-D) max s.. Tÿ ˆ ÿm ÿm c q a pγjq 2 u j ` β γju j 1 j 1 x piq 1 x piq b piq 1 ` b piq x piq b piq (8) (11), ě px 0 b 0 qy piq ` ` q a j 1 m ÿ j 1 ÿ ÿm s ˆa s s 1 j 1 ` 1 N Tÿ Nÿ 1 i 1 ˆ ɛ piq ÿm j 1 γ ju j h x piq p b piq γ ju j ` β ` ɛ 1,..., T, i 1,..., N, γ s j v piq js β sy piq ɛ piq s y piq, 1,..., T, i 1,..., N, j 1,..., m, ÿm u j 1,..., T, j 1,..., T, i 1,..., N, u j P 0, 1u, v piq j ě 1,..., T, i 1,..., N, j 1,..., m Coninuous price se For coninuous pricing opions, he price in period is chosen from he se P rl, R s. We firs noe ha he price r mus be non-negaive, hence L ě 0. Also, r should be bounded above by β {a ; oherwise he expeced demand a r `β ă 0, which is unlikely o happen in realiy. Hence, he reailer will never se a price higher han β {a, and herefore, U ď β {a. For he nonlinear erm r y piq presened in he join service-level consrain, we linearize i by inroducing a new decision variable w piq. The following ses of linear inequaliies enforce r y piq when y piq is a binary: w piq $ & % w piq w piq w piq ď r ď U y piq ě r ` U py piq 1q w piq ě (17)

12 Finally we can formulae he following (PP-C) model wih quadraic objecive for producion planning problem wih coninuous pricing opions as follows (PP-C) max s.. Tÿ 1 ` c q a r 2 ` β r ` 1 N x piq 1 x piq b piq 1 ` b piq x piq b piq (8) (11), s 1 Tÿ Nÿ 1 i 1 `ɛpiq r h x piq p b piq ` q a r ` β ` ɛ 1,..., T, i 1,..., N, ÿ ě px 0 b 0 qy piq ` `as ws piq β s y piq ɛ piq s y 1,..., T, i 1,..., N, 1,..., T, i 1,..., N, y piq P 0, 1,..., N, L ď r ď 1,..., T. 4 Compuaional Resuls 4.1 Soluion mehods In general, an MILP reformulaion of a chance-consrained program is compuaionally inracable since i usually requires a large number of Mone Carlo samples o aain soluion accuracy. Luedke and Ahmed [16] suggesed using he SAA approach for solving general chance-consrained programs, and derived heocraical sample-size bounds for obaining soluions ha saisfy he chance consrains wih cerain confidence for specific risk levels. Specifically, consider a generic chance-consrained program: pp θ q : z θ minfpxq : x P X θ u, where X θ! x P X : P Gpx, ξq ě 0u ě 1 θ ). Here X Ď R n represens a deerminisic feasible region (i.e., given by he consrains (5), (10), (11)), f : R n Ñ R represens he objecive o be minimized, ξ is a random vecor wih suppor Ξ Ď R d, G : R n ˆ R d Ñ R m is a given consrain mapping and θ is a risk parameer of service level. We assume ha z θ exiss and is finie. The SAA counerpar of he chance-consrained problem pp θ q wih risk parameer α is defined as pp N α q : ẑα N minfpxq : x P X α u,! ř 1 ) where X α x P X : 1 N N i 1 Gpx, ξ i q ě 0 ě 1 α. Feasible soluions: To obain feasible soluions of pp θ q, one can choose a smaller risk parameer α ă θ and solve he SAA counerpar pp N α q. As shown in Luedke and Ahmed 12

13 [16], if he sample size N ě 1 2pθ αq 2 log ˆ XzXθ δ, (18) hen solving pp N α q will yield a feasible soluion o pp θ q wih probabiliy a leas 1 δ. This gives a heoreical sample size o guaranee a feasible soluion for P θ using he SAA approach wih a confidence level 1 δ. Noe ha he exac value of XzX θ is ofen no available, and we can esimae he leas value of he righ-hand side in (18) by seing XzX θ 1. For example, When θ 0.02 and α 0, o guaranee a feasible soluion wih probabiliy a leas 90%, we need a leas N ě 1 2ˆ logp1{0.1q «2878 number of samples. Similarly, when θ 0.05 and α 0, o guaranee a feasible soluion wih probabiliy a leas 90%, we need a leas N ě 1 2ˆ logp1{0.1q «460 number of samples. In our laer compuaional sudies, we es differen values of N ha are much smaller han he suggesed heoreical sample sizes, and demonsrae he performance of he resuling soluions in ou-of-sample simulaion ess. Lower bounds: To obain lower bounds on he original opimizaion problem pp θ q, we se α θ and o generae M SAA problems, namely, pp N θ,iq (i 1, 2,..., M). Then we solve each sample-based problem and obain a se of opimal objecive values, denoed by ẑ N θ,i (i 1, 2,..., M). The L-h minimum value among all M opimal objecive value is denoed by ẑθ,rls N. Then, according o Luedke and Ahmed [16], he following resul holds: L 1 ÿ ˆM Ppẑθ,rLs N ď z θ q ě 1 p1{2q M (19) i for large enough N relaive o ɛ (e.g., Nɛ ě 10). Hence, we can say ha ẑθ,rls N is a lower bound of he objecive value wih a confidence level 1 řl 1 `M i 0 p1{2q M. i We es he effeciveness of he SAA approach applied o boh models in his paper wih and wihou he pricing opion. We use CPLEX for solving all MILP models. All he compuaions are performed on a 3.40GHz Inel(R) Xeon(R) CPU. i Sochasic producion planning problem We presen numerical resuls on he sochasic producion planning problem. We numerically solve he appropriae SAA counerpar problems and compue boh he upper bounds and he lower bounds for opimal objecive values. We also compue he required sample size in pracice o show he effeciveness of our approach Parameer seing We use randomly generaed insances based on differen demand disribuions o demonsrae he general feaures of our models and approaches. We es boh i.i.d. demand and correlaed 13

14 demand. We consider he oal number of periods T 5, and assume saionary uni ordering cos, uni holding cos, and uni backlogging cos in each period, which are c 5, h 1, and p 10, respecively. The i.i.d demand in each period follows Poisson disribuion wih mean value 20. For he correlaed demand, we consider boh Markov Modulaed Demand Process (MMDP) (see, e.g., Chen and Song [5]) and Auoregressive Model of Order 1 (AR(1)) (see, e.g., Mills [20]). The demands generaed from MMDP have hree saes corresponding o he sae of economy: poor (1), fair (2), and good (3). In each period, given ha he curren sae is i P 1, 2, 3u, we es cases where he demand disribuions are Poisson wih mean value 10i. We also assume ha he sae of he economy follows a Markov chain wih he iniial sae 1 (i.e., poor) and he ransiion probabiliy marix P For he AR(1) demand case, he demands in period saisfy d d 1 ` η, where he noise erm η is normally disribued wih mean 0 and sandard deviaion 1. We se he iniial demand d In all our compuaions, we es he problems wih required service-level θ 0.02 and Feasible soluions We aim a demonsraing he effeciveness of he SAA approach for finding feasible soluions. To compue for feasible soluions, we se he risk level α 0. This gives us a more conservaive SAA counerpar problem. However, a relaively small sample size N is needed o compue feasible soluions. Also, we numerically compue he soluion o he SAA counerpar which uses he required service level as he risk parameer (i.e., α θ). We compare he saisics of using hese wo differen values for he risk level α. Our numerical es consiss of wo pars. Firs, we generae N samples and solve he corresponding SAA insances each ime. The above process is repeaed M 10 imes so ha we obain 10 soluions for he same problem. Second, o validae if all hese 10 soluions are feasible, we conduc a poseriori check o compue he risk for each soluion: we generae a simulaion sample wih N 1 10, 000 scenarios, and check he number of scenarios ha are violaed under he larger problem for each given soluion. The soluion risk is hen calculaed as number of violaed scenarios R. N 1 If R ă θ, he service-level requiremens are saisfied; oherwise, he soluion is no feasible. For soluion risk, we repor he average (Avg), minimum (Min), maximum (Max), and 14

15 sample sandard deviaion (σ) over he soluions given by he 10 SAA problems. We also repor he number of feasible soluions as well as he average, minimum, maximum, and sample sandard deviaion of he cos over hese feasible soluions. Tables 1 7 summarize our compuaional resuls for finding feasible soluions o he sochasic producion planning problem. When no applicable, we indicae *** in he corresponding enry of each able. Our observaions are summarized as follows: 1. From each able, we observe ha as he sample size N grows, he average soluion risk decreases and he number of feasible soluions increases. This is because as more samples are used, more consrains are being enforced ino he model, which leads o a smaller feasible region. Hence, as he sample size grows, we can obain more conservaive soluions by solving he SAA problems which have lower soluion risks and a higher likelihood o be feasible a he nominal risk level θ. 2. We observe ha using α 0 requires much fewer samples o achieve a feasible soluion han using α θ. For example, in Table 1, we only need 300 samples o ge a feasible soluion wih confidence level 90% by using α 0. However, solving he SAA reformulaion a he nominal risk level α 0.02 requires a leas 3000 samples o have a confidence level of 80%. Noe ha he problem size grows as he number of samples increases, we conclude ha solving he SAA problem by seing risk level α 0 is more efficien han solving he original SAA problem a he nominal risk level in erms of obaining feasible soluions. 3. We also noice ha he required sample sizes in our ess are also smaller han he heoreical bound given in Luedke and Ahmed [16]. For insance, o achieve 90% confidence level, he heoreical required sample sizes calculaed by (18) in Secion 4.1 are N and N for θ 0.02 and θ 0.05, respecively; however, from Tables 3 and 4, we can see ha he acual sample sizes in pracice are N and N , respecively. The smaller sample size no only makes he compuaion more efficien, bu also makes he daa-collecion work less demanding. 4. In erms of he coss for feasible soluions, we observe ha using α 0 yields a higher average cos and a higher variance among all feasible soluions han using α θ. For example, in Table 6 he average cos for feasible soluions is using α 0 and N 50 samples, as compared o he average cos of by using α θ and N Hence, alhough using a smaller risk level α 0 is more efficien o compue a feasible soluion under a given confidence level, i migh yield a more conservaive soluion ha has a higher cos han solving he SAA problems under he nominal risk level because no scenarios can be violaed when risk level α is se o be 0. Therefore, 15

16 using a smaller risk level α 0 produces a feasible soluion and an upper bound on he objecive values efficienly. 5. Anoher observaion from he ables is ha he average feasible soluions cos and heir sandard deviaions are more erraic when we use a risk level α 0, as compared o α θ. The reason is ha when using a smaller risk level α 0, he feasible soluions cos are more sensiive wih respec o he sample size. This also explains why a smaller sample size is required when we have a smaller risk level. Table 1: Soluion resuls of i.i.d. demand for sochasic producion planning problem wihou pricing for θ 0.02 Soluion Risk Feasible Soluions Cos α N Avg Min Max σ # Avg Min Max σ *** *** Table 2: Soluion resuls of i.i.d. demand for sochasic producion planning problem wihou pricing for θ 0.05 Soluion Risk Feasible Soluions Cos α N Avg Min Max σ # Avg Min Max σ ***

17 Table 3: Soluion resuls of MMDP for sochasic producion planning problem wihou pricing decisions for θ 0.02 Soluion Risk Feasible Soluions Cos α N Avg Min Max σ # Avg Min Max σ *** Table 4: Soluion resuls of MMDP for sochasic producion planning problem wihou pricing decisions for θ 0.05 Soluion Risk Feasible Soluions Cos α N Avg Min Max σ # Avg Min Max σ ***

18 Table 5: Soluion resuls of AR(1) demand for sochasic producion planning problem wihou pricing decisions for θ 0.02 Soluion Risk Feasible Soluions Cos α N Avg Min Max σ # Avg Min Max σ *** *** Table 6: Soluion resuls of AR(1) demand for sochasic producion planning problem wihou pricing decisions for θ 0.05 Soluion Risk Feasible Soluions Cos α N Avg Min Max σ # Avg Min Max σ

19 Table 7: Soluion resuls of AR(1) demand for sochasic producion planning problem wihou pricing decisions for θ 0.08 Soluion Risk Feasible Soluions Cos α N Avg Min Max σ # Avg Min Max σ Lower bounds To obain he lower bounds for he sochasic producion planning problem, we follow he same seings in Luedke and Ahmed [16] by choosing α θ and M 10. We hen ake he L-h minimum opimal objecive value among all M 10 runs. According o (19), he confidence levels of using L 1,..., 4 are 0.999, 0.989, and 0.828, respecively. In addiion o he lower bounds compued a each confidence level, we also repor opimaliy gaps, defined as he percenage ha he lower bound is below he cos of bes feasible soluion (i.e., he minimum cos among all feasible soluions, given by Tables 1 6). Tables 8 14 repor he es resuls. Combining he es resuls on he lower bounds and he resuls of feasible soluions in Secion 4.2.2, we can obain he range of he opimal cos. For example, in he i.i.d. demand case wih service level θ 0.02, solving M 10 SAA insances wih sample size N 250 yields a feasible soluion of cos shown in Table 1 while geing a lower bound wih a confidence level shown in Table 8. This means ha we have a leas 99.9% confidence o say ha he opimal soluion is a mos p q{ ˆ 100% «6.96% less cosly han he bes feasible soluion we ge. Similarly, we can analyze he problem wih oher demand cases and differen service level parameers using corresponding ables. From hese resuls, we observe ha as sample size N becomes larger, he lower bound becomes larger and he gap becomes smaller a each confidence level. When he gap reaches zero, we come o a conclusion ha he bes feasible soluion is he opimal soluion wih he 19

20 corresponding confidence level. For example, as we noice from Table 13, when he sample size N 2000, we have confidence a leas 82.8% ha he feasible soluion of cos is opimal; when he sample size raises o N 3000, our confidence increases from 82.8% o 99.9%. Thus, for a cerain confidence level, we can make a beer esimaion of he opimal soluion of an SAA problem as we increase he sample size N. Table 8: Lower bounds of i.i.d. demand for sochasic producion planning problem wihou pricing for α θ 0.02 LB wih confidence a leas Gap wih confidence a leas N % 5.88% 5.84% 5.68% % 2.49% 2.41% 2.27% % 2.67% 2.62% 1.94% % 1.68% 1.56% 1.17% % 0.65% 0.00% 0.29% Table 9: Lower bounds of i.i.d. demand for sochasic producion planning problem wihou pricing for α θ 0.05 LB wih confidence a leas Gap wih confidence a leas N % 2.36% 1.96% 1.63% % 1.45% 1.44% 0.87% % 1.64% 1.43% 0.97% % 0.36% 0.00% 0.27% 4.3 Producion planning wih pricing opions In his secion, we repor he compuaional resuls of muli-period join service-level consrained producion planning wih pricing opions. We also conduc sensiiviy analysis for his model Parameer seing Consider he coninuous pricing in he es insances. The seing of cos parameers is he same as hose in Secion Moreover, we se a 5 and β 200 in he funcion d pr q a r ` β ` ɛ for all 1,..., T. The noise erm ɛ follows normal disribuion 20

21 Table 10: Lower bounds of MMDP for sochasic producion planning problem wihou pricing for α θ 0.02 LB wih confidence a leas Gap wih confidence a leas N % 4.88% 4.35% 3.77% % 3.14% 2.60% 1.85% % 1.04% 0.98% 0.80% % 0.82% 0.62% 0.59% Table 11: Lower bounds of MMDP for sochasic producion planning problem wihou pricing for α θ 0.05 LB wih confidence a leas Gap wih confidence a leas N % 3.47% 3.42% 2.93% % 1.57% 1.54% 1.38% % 1.64% 1.12% 1.09% Table 12: Lower bounds of AR(1) demand for sochasic producion planning problem wihou pricing for α θ 0.02 LB wih confidence a leas Gap wih confidence a leas N % 2.64% 2.42% 2.20% % 0.99% 0.90% 0.80% % 0.12% 0.00% 0.05% % 0.15% 0.08% 0.00% % 0.10% 0.18% 0.18% 21

22 Table 13: Lower bounds of AR(1) demand for sochasic producion planning problem wihou pricing for α θ 0.05 LB wih confidence a leas Gap wih confidence a leas N % 1.23% 0.66% 0.64% % 0.66% 0.57% 0.33% % 0.39% 0.23% 0.00% % 0.06% 0.12% 0.14% % 0.06% 0.15% 0.18% Table 14: Lower bounds of AR(1) demand for sochasic producion planning problem wihou pricing for α θ 0.08 LB wih confidence a leas Gap wih confidence a leas N % 0.95% 0.54% 0.51% % 0.33% 0.13% 0.00% % 0.81% 0.00% -0.11% % 0.00% -0.01% -0.05% % 0.00% -0.01% -0.08% 22

23 wih mean 0 and sandard deviaion 22 for all 1,..., T. We also assume ha he pricing range in each period is beween W L θ and W U 40. We fix he required service level Feasible soluions Table 15 repors saisics of he soluions of i.i.d. demand for producion planning wih pricing opions. Table 16 repors saisics of he soluions of AR(1) demand for producion planning wih pricing opions. The insighs of our numerical resuls are summarized as follows: 1. As he sample size N grows, he average soluion risk decreases and he number of feasible soluions increases since more consrains are being enforced ino he model, which leads o a smaller feasible region. Hence, as he sample size increases, solving he SAA counerpar under any fixed risk level yields a lower soluion risk and a higher likelihood o be feasible a he nominal risk level θ. 2. From Table 15 and Table 16, we observe ha using α 0 requires much fewer samples o achieve a feasible soluion han using α θ. For example, in Table 15, we generae 300 samples o obain a feasible soluion wih confidence level 100% by using α 0. However, even using 2500 samples in he SAA reformulaion, wih risk level α 0.02, can only find a feasible soluion a a confidence level of 60%. In our numerical es, solving an SAA reformulaion ha involves more han 2500 samples is compuaionally inracable (more han hree CPU minues for each insance). Therefore, an efficien way o compue a feasible soluion is o solve he SAA reformulaion wih a more conservaive risk level α 0. The smaller sample size no only makes he compuaion more efficien, bu also makes he daa-collecion work less demanding. 3. In erms of he profi for feasible soluions, we observe ha using α 0 yields a lower average profi and a higher variance among all feasible soluions han using α For example, in Table 16 he average profi for feasible soluion is using α 0 and N 200 samples, as compared o he average profi of by seing α 0.02 and N Hence, alhough using a smaller risk level α 0 is more efficien o compue a feasible soluion under a given confidence level, i will yield a more conservaive soluion ha has a lower profi han solving he SAA problems under he nominal risk level. Therefore, using a smaller risk level α 0 helps us o find a feasible soluion and a lower bound on he oal profi efficienly. 23

24 Table 15: Soluion resuls of i.i.d. demand for producion planning wih pricing for θ 0.02 Soluion Risk Profi for Feasible Soluions α N Avg Min Max σ # Avg Min Max σ *** *** Table 16: Soluion resuls of AR(1) demand for producion planning wih pricing for θ 0.02 Soluion Risk Profi for Feasible Soluions α N Avg Min Max σ # Avg Min Max σ ***

25 4.3.3 Upper bounds We check he upper bounds for producion planning wih pricing opions when α θ The gaps are defined as he percen by which he upper bound is above he bes feasible soluion (i.e., he maximum profi among all feasible soluions in his case). We use L 1,..., 4 o generae bounds by he opimal objecive values of he M 10 SAA problems and he corresponding confidence levels given by (19) are 0.999, 0.989, 0.945, 0.828, respecively. Table 17 shows he upper bounds for i.i.d. demand for producion planning wih pricing opions when α θ Table 18 shows he upper bounds for AR(1) demand for producion planning wih pricing opions when α θ Combining he es resuls on he upper bounds and he resuls of feasible soluions in Secion 4.3.2, we can obain he range of he opimal profi. For example, in he i.i.d. demand case, solving M 10 SAA insances wih sample size N 1000 yields a bes feasible soluion wih profi , as shown in Table 15; we also ge an upper bound of wih confidence 98.9% (shown in Table 17). This means ha we have a leas 98.9% confidence o say ha he opimal profi is a mos p q{ «0.44% greaer han he bes feasible soluion We can make a similar analysis wih oher demand cases using corresponding ables. From hese resuls, we observe ha as sample size N becomes larger, he upper bound becomes smaller and he gap becomes smaller a each confidence level. When he gap reaches zero, we can conclude ha he bes feasible soluion is he opimal soluion wih he corresponding confidence level. For example, as we observe from Table 18, when he sample size N 1000, we have confidence a leas 94.5% o say ha he opimal profi is a mos 0.1% greaer han ; when he sample size increases o N 2000, we have confidence a leas 94.5% o say ha he feasible soluion wih profi is opimal. Thus, for a cerain confidence level, we can make a beer esimaion of he opimal soluion of an SAA problem as we increase he sample size N. Table 17: Upper bounds of i.i.d. demand for producion planning wih pricing for α θ 0.02 UB wih confidence a leas Gap wih confidence a leas N % 1.02% 0.60% 0.56% % 0.44% 0.00% 0.34% % 1.18% 1.11% 0.79% % 0.44% 0.18% 0.00% % 0.16% 0.02% 0.00% 25