A Branch-and-Cut Method for Dynamic Decision Making under Joint Chance Constraints

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1 Submied o Managemen Science manuscrip hp://dx.doi.org/ /mnsc A Branch-and-Cu Mehod for Dynamic Decision Making under Join Chance Consrains Minjiao Zhang, Simge Küçükyavuz and Saumya Goel Deparmen of Inegraed Sysems Engineering, The Ohio Sae Universiy 1971 Neil Avenue, 210 Baker Sysems, Columbus, OH {zhang.769,kucukyavuz.2, goel.43}@osu.edu In his paper, we consider a finie-horizon sochasic mixed-ineger program involving dynamic decisions under a consrain on he overall performance or reliabiliy of he sysem. We formulae his problem as a muli-sage (dynamic) chance-consrained program, whose deerminisic equivalen is a large-scale mixedineger program. We sudy he srucure of he formulaion, and develop a branch-and-cu mehod for is soluion. We illusrae he efficacy of he proposed model and mehod on a dynamic invenory conrol problem wih sochasic demand in which a specific service level mus be me over he enire planning horizon. We compare our dynamic model wih a saic chance-consrained model, a dynamic risk-averse opimizaion model, a robus opimizaion model, and a pseudo-dynamic approach, and show ha significan cos savings can be achieved a high service levels using our model. Key words : Chance consrains, branch-and-cu, muli-sage, probabilisic lo sizing, service levels 1. Inroducion In his paper, we consider finie-horizon dynamic (muli-sage) decision-making problems under uncerainy such ha a risk, reliabiliy or service level requiremen mus be me over he enire planning horizon. Such a resricion could be due o sipulaions in a cusomer conrac or cerain regulaions. Examples include resricing he loss-of-load probabiliies in power sysems, saisfying conracual service levels in invenory managemen, and limiing risk in cerain financial porfolios. A common approach o limiing violaions of such a requiremen is o add a penaly erm o he objecive funcion. However, such penalies are inangible and inherenly hard o esimae. The resuling soluions may be overly conservaive under high penalies or may resul in low reliabiliy and conrac violaions under low penalies. A more direc approach is o inroduce probabilisic (chance) consrains in he associaed opimizaion model o ensure ha he required reliabiliy or service level is me. Charnes e al. (1958), Charnes and Cooper (1959, 1963) were firs o sudy an opimizaion problem wih individual chance consrains. In a dynamic problem, saisfying a service level in each ime period may resul in a poor service level over he enire horizon. In addiion, ypically here 1

2 2 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc are correlaions beween he random variables in muliple ime periods. As a resul, a join chance consrain may be more appropriae o ensure an overall high service level. Miller and Wagner (1965) sudy probabilisic programming wih join chance consrains and independen random variables. Join chance consrains wih dependen random variables were inroduced in Prékopa (1973). Boh Miller and Wagner (1965) and Prékopa (1973) show ha under cerain assumpions on he disribuion of he righ-hand side vecor in he chance consrain, he deerminisic equivalen can be formulaed as a convex program. Some recen applicaions of opimizaion problems wih chance consrains include probabilisic se covering (Beraldi and Ruszczyński 2002b, Saxena e al. 2010), probabilisic producion and disribuion planning (Lejeune and Ruszczyński 2007), call cener saffing (Gurvich e al. 2010), insuring criical pahs (Shen e al. 2010), opimal vaccine allocaion (Tanner and Naimo 2010), and reliable emergency medical service design (Beraldi e al. 2004). For a more horough reamen of opimizaion under chance consrains we refer he reader o Dencheva (2009), Prékopa (2003), Kall and Wallace (1994), Birge and Louveaux (1997), Prékopa (1995). One challenge wih linear programs wih join chance consrains is ha he feasible region is non-convex. Prékopa (1990) inroduces he concep of p-efficien poins, which define he exreme poins of he non-convex feasible region. There are several mehods in he lieraure ha rely on he enumeraion of he exponenially many p-efficien poins. Sen (1992) uses he p-efficien poins o give a disjuncive programming reformulaion of join chance consrains wih finie discree disribuions. Valid inequaliies are proposed based on he exreme poins of he reverse polar of he disjuncive program. Dencheva e al. (2000) use p-efficien poins o obain various reformulaions of probabilisic programs wih discree random variables, and o derive valid bounds on he opimal objecive funcion value. Ruszczyński (2002) uses he concep of p-efficien poins o derive consisen orders on differen scenarios represening he discree disribuion. The consisen ordering is represened wih precedence consrains, and valid inequaliies for he resuling precedence-consrained knapsack se are proposed. Beraldi and Ruszczyński (2002a) propose a branch-and-bound mehod for probabilisic ineger programs using a parial enumeraion of he p-efficien poins. Cheon e al. (2006) give a global opimizaion algorihm ha successively pariions he non-convex feasible region unil a global opimal soluion is obained. Tayur e al. (1995) give an algebraic geomery algorihm for a scheduling problem wih join chance consrains ha solves a series of chance-consrained ineger programs wih varying reliabiliy levels. Alernaively, a deerminisic equivalen model is obained by adding addiional binary variables. The linear programming relaxaion of his formulaion is weak in general, bu can be srenghened by adding valid inequaliies obained from he so-called mixing se subsrucure (Luedke e al. 2010, Küçükyavuz 2012). Combining decomposiion and cuing plane echniques, Luedke (2011)

3 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc proposes a branch-and-cu decomposiion algorihm for solving a wo-sage chance-consrained program, where he recourse decisions incur no addiional coss. Anoher challenge wih he opimizaion problems wih join chance consrains is ha in cases of coninuous disribuions, calculaing he join probabiliy of several evens requires he evaluaion of a muli-dimensional inegral, which is hard o compue accuraely (Ahmed and Shapiro 2008). Ben-Tal and Nemirovski (1998), Calafiore and Campi (2005, 2006), Nemirovski and Shapiro (2005, 2006) approximae he non-convex chance consrain wih convex consrains such ha he soluion o his approximaion is feasible wih a high probabiliy. However, such mehods could yield highly conservaive soluions (Ahmed and Shapiro 2008). In his paper, we assume a finie discree disribuion (or a finie sample from he coninuous disribuion), circumvening he difficuly of evaluaing high dimensional inegrals. Mos of he lieraure on join chance consrains consider saic decisions (Birge and Louveaux 1997). In oher words, he decisions are made once a he beginning of he horizon and are no updaed as he uncerainy is revealed. An excepion is Lulli and Sen (2004) who consider a probabilisic bach-sizing problem under a finie discree demand disribuion. In heir model, non-anicipaiviy of decisions are enforced only for he scenarios ha mee he desired service consrain. Andrieu e al. (2010) sudy chance consrains wih dynamic (muli-sage) decisions ha appear in hydro power reservoir managemen. The auhors assume a coninuous probabiliy disribuion and give a finie dimensional approximaion of he infinie-dimensional chance consrain by discreizing he coninuous decision variables. The auhors sae I is no clear if he recen advances in ineger programming mehods for chance consrains would apply o he muli-sage seing. In his paper, we show ha due o he non-anicipaiviy of he decisions, branch-andcu mehods can indeed be developed for he muli-sage seing. We give valid inequaliies for he dynamic model based on is mixing and coninuous mixing subsrucures. We illusrae ha significan cos savings can be achieved hrough dynamic decision making using an applicaion in invenory managemen wih random demands and coss, and a prescribed service level requiremen for meeing he demand on ime over he planning horizon. Our compuaional experimens illusrae he effeciveness of he branch-and-cu algorihm using he coninuous mixing inequaliies for his invenory managemen problem. Finally, we compare our dynamic chance-consrained opimizaion model for he invenory managemen problem wih a dynamic risk-averse model, a robus opimizaion model, and a pseudo-dynamic approach. 2. Saic Model Even hough our focus is on dynamic decisions, we firs review he more commonly sudied saic models for compleeness.

4 4 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc Consider a dynamic (muli-sage) problem wih n sages. Le ξ and µ denoe N- and B-variae random variables represening he righ hand sides and coss, respecively, wih a known finie discree disribuion funcion. Le A A A = 21 A , x = A n1 A n2 A n3 A nn x 1 x 2. x b, x = x 1 x 2. x n, ξ = ξ 1 ξ 2. ξ η, ξ = where A i is an η b i marix represening he consrain coefficiens of decision vecor x i R b i + in sage, i wih N = n =1 η and B = n =1 b. Noe ha he enries of he coefficien marix A i wih < i are all zero, because he consrains in sage do no depend on decisions afer sage. Le X(ξ) R β 1 + Z β 2 + be a mixed-ineger se dependen of ξ and defined by addiional consrains on x, where β 1 + β 2 = B. Le τ be he required reliabiliy level, 0 τ 1. The saic chanceconsrained program wih join chance consrains is ξ 1 ξ 2. ξ n, min { E (ξ,µ) µ T x : P(Ax ξ) τ, x X(ξ) }. (1) This is referred o as a saic model, because he decisions in sage do no depend on he realizaions in sages i = 1,..., 1. Leing y = Ax, he join chance consrain can be rewrien as P(y ξ) τ. The formulaion wih individual chance consrains: P(y i ξ i ) τ for all = 1,..., n, i = 1,..., η, resuls in an overall reliabiliy of he sysem o be much lower han ha wih join chance consrains. In addiion, in he case of individual chance consrains, we can linearize he consrains using quaniles (c.f., Kall and Wallace (1994)). Therefore, hroughou his paper, we will be ineresed in opimizaion problems wih join chance consrains. Assume ha he random vecor Γ = (ξ, µ) has finiely many realizaions (scenarios) given by (D 1, c 1 ), (D 2, c 2 ),..., (D m, c m ) wih probabiliies π 1, π 2,..., π m, where D i = (D i 1, D i 2,..., D i n), c i = (c i 1, c i 2,..., c i n) and c i = {c i 1,..., c i b }, for i = 1,..., m, = 1,..., n. Le X i represen he feasible region corresponding o D i. Throughou, we le [i, j] := { Z : i j}. We assume y 0 wihou loss of generaliy (c.f., Küçükyavuz (2012)). A deerminisic equivalen of he saic chance-consrained program is min m i=1 n =1 πi c i x s.. y = Ax, (2) y D(1 i z i ) [1, n], i [1, m], (3) m i=1 πi z i 1 τ, (4) x X i, y R N +, (5) z {0, 1} m, (6)

5 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc where z i = 0 implies ha under scenario i we have no violaed chance consrain (i.e., y = Ax D i ) a he soluion (y, x). This formulaion is due o Luedke e al. (2010). The se defined by he inequaliies (3) for a fixed is known as he mixing se. Aamürk e al. (2000), Günlük and Poche (2001) give valid inequaliies called sar or mixing MIR inequaliies for his se and show ha hese inequaliies give he convex hull of soluions. In addiion, Luedke e al. (2010) and Küçükyavuz (2012) give srenghened mixing inequaliies for he mixing se wih an addiional knapsack consrain, Q = {(y, z) R + {0, 1} m : m i=1 πi z i 1 τ, y D i (1 z i ), i [1, m]} as i arises in saic chance-consrained LPs. 3. Dynamic model Mos models in he lieraure, given by he chance-consrained program in Secion 2, are saic in naure. The decisions are made here and now, and do no change as he uncerain parameers are revealed over ime. However, a more flexible and efficien planning model would be o allow he decisions in period o ake he observed daa in periods 1,..., 1 ino accoun. This gives rise o a muli-sage model under join chance consrains in which he decisions a every sage are adapive o he realizaions of he uncerain daa. In his model, he order of evens is as follows: a he beginning of he planning horizon firs-sage decisions, given by he vecor x 1, are made. Nex, he random evens occurring in he firs-sage, Γ 1, are observed. Based on he observed oucomes, he decisions a he second sage, x 2 (Γ 1 ) are made, and so on. More generally, le Γ 1 := (ξ 1, µ 1, ξ 2, µ 2,..., ξ 1, µ 1 ). Le x (Γ 1 ) be he decision vecor a sage [2, n], whose value is deermined afer he random variables Γ 1 are observed. Then insead of he saic chance consrains in (1), we have a dynamic chance consrain: x 1 x 2 (Γ 1 ) P A. x n (Γ n 1 ) ξ τ. (7) Le x i = (x i 1,..., x i n), where x i = (x i 1,..., x i b ) is he vecor of decisions made in sage under scenario i, [1, n], i [1, m]. Also le y i = Ax i for i [1, m], y i = (y i 1,..., y i n), y i = (y i 1,..., y i η ), [1, n]. Noe ha in he dynamic model, he non-anicipaiviy of he decisions mus be enforced, i.e., he scenarios which have he same se of pas oucomes unil ime, should have he same acion in period. Le S l = {k [1, m] : D l j = D k j, c l j = c k j, j [1, 1]}, be he se of scenarios ha share he same hisory wih scenario l unil ime. The deerminisic equivalen of he dynamic model wih chance consrain (7) is given by min m i=1 n =1 πi c i x i (8) y i = Ax i i [1, m], (9)

6 6 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc y l D l (1 z l ) [1, n], l [1, m], (10) y l = y k [1, n], l [1, m], k S l \ {l}, (11) x l = x k [1, n], l [1, m], k S l \ {l}, (12) m i=1 πi z i 1 τ, (13) y i R N +, z i {0, 1} i [1, m], (14) x i X i i [1, m]. (15) Consrains (11) (12) are he non-anicipaiviy consrains. Noe ha consrains (9) and (12) imply inequaliy (11). However, in he nex secion, we will use inequaliy (11) o obain srong valid inequaliies. Observe ha a more compac represenaion wih variables corresponding o he nodes of a scenario ree could be used in pracice. All of our resuls apply o he more compac model, and we es our resuls on he compac formulaion in Secion 6. In wha follows, we use he non-compac model for noaional convenience. 4. A Branch-and-Cu Mehod for he Dynamic Model In his secion, we give valid inequaliies for he dynamic join chance-consrained model based on wo subsrucures: mixing se and coninuous mixing se. Throughou his secion, for ease of exposiion, we assume ha here is a single consrain in each sage, i.e, η = 1, [1, n] and N = n. Therefore, y i R n +, i [1, m]. Our resuls apply o he general case wih η > 1 by considering one consrain a a ime in each sage Mixing Inequaliies For any fixed sage [1, n] and scenario l [1, m] wih S l, suppose, wihou loss of generaliy, ha S l = {1,..., m l } and D i for all i S l D ml ν l +1 are sored in nonincreasing order as D 1 D 2. As observed by Luedke e al. (2010), if here exiss ν l m l such ha ν l i=1 πi 1 τ and i=1 πi > 1 τ, hen we mus have y l D νl +1. In addiion, le 1, 2,..., m l be a nondecreasing order of scenario probabiliies, i.e, π 1 π 2 π ml. Also le p l be such ha p l i=1 π i 1 τ and p l +1 i=1 π i > 1 τ. Then he exended (knapsack) cover inequaliy is valid (Küçükyavuz 2012). m l i=1 z i p l Proposiion 1. For any fixed and scenario l wih S l, and D 1 D 2 D ml i S l, and for q l Z + such ha q l ν l, le T = {i 1, i 2,..., i a } {1,..., q l } wih i 1 < i 2 < < i a, for

7 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc L S l \ {1,..., q l + 1} and a permuaion of he elemens in L, Π L = {l 1, l 2,..., l p l q l } such ha l j q l j. For ν l < p l, he ( T, Π L ) inequaliies y l + a j=1 (D i j D i j+1 p l ql )z i j + j=1 α j(1 z l j ) D i 1, (16) are valid for he se given by (10) (14), where D i a+1 = D ql +1 if q l + 1 ν l, D i a+1 = 0 if q l + 1 > ν l, and α 1 = D ql +1 D min{νl +1,ql +2}, and for j = 2,..., p l q l α j = max{α j 1, D ql +1 D min{νl +1,ql +1+j} i:i<j and l i q l +1+j α i}. Proof of Proposiion 1 Noe ha for some [1, n] and l [1, m] such ha S l, from (10), (11) and a relaxaion of (13), we obain a se Q l = {(y, l z) R + {0, 1} Sl : m l i=1 πi z i 1 τ, y l D(1 i z i ), i S l } for which ( T, Π L ) inequaliies of Küçükyavuz (2012) given by (16) are valid. In oher words, non-anicipaiviy of decisions allows us o use he valid inequaliies proposed for he saic problem in a dynamic seing. Corollary 1. Given ha y l D νl +1, a special case of inequaliies (16) wih q l = ν l is he so-called mixing inequaliies, y l + a j=1 (D i j D i j+1 )z i j D i 1, (17) where 1 i 1 < i 2 < < i a ν l and D i a+1 = D νl +1, and D i a+1 = 0 if ν l = m l. Noe ha i 1 = 1 is needed for inequaliies (17) o be srong. We use inequaliies (17) in our compuaional experimens in Secion 6, because of heir polynomial ime separaion (c.f., Günlük and Poche (2001)) Coninuous Mixing Inequaliies For a sage [1, n] and scenario l [1, m], if S l, hen we call he pair (, l) a non-anicipaive node of he scenario ree represening he random daa. A a non-anicipaive node (, l), for a fixed sage T [, n], inequaliy (10) can be rewrien as T y l T = A T j x l j + A T j x l j D l T (1 z l ). (18) j=1 j=+1 In his secion, we assume ha T j=+1 A T jx l j 0. This holds, for example, when A T j 0. Le s = j=1 A T jx l j. Noe ha x l j = x k j for k S l, j [1, ]. Therefore, under his assumpion, a relaxaion of he feasible se defined by consrains (9), (10), (11) and (14) wih respec o non-anicipaive node (, l), sage T [, n] and R l S l, is Q l T = {( s, {y i T } i R l, {z i } i R l ) R R Rl + {0, 1} Rl : y i T + D i T z i D i T, y i T s, i R l }. (19) Le D l T := max{d i T : i R l }.

8 8 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc Proposiion 2. A a non-anicipaive node (, l) wih R l S l, for a fixed sage [1, n] and T [, n], le σ = s D T l ; r i = yi T s ; z i = z i for i R l D T l wih DT i < DT l, and z i = z i 1 for i R l wih for i R l wih DT i < DT l, and f i = 0 for i R l wih DT i = DT l. Also le DT i = DT l ; and f i = Di T D T l he digraph G = (V, E), where V = R l and E = {(i, j) : i, j R, l i j, f i f j } {(i, i) : i R}. l The arc lengh φ jk (σ, r, z) associaed wih (j, k) E wih respec o he poin ( s, {y i T } i R l, {z i } i R l ) Q l T is Then he inequaliy σ + r j + (f j f k + 1) z j f k for (j, k) E if f j < f k, φ jk (σ, r, z) = r j + (f j f k ) z j for (j, k) E if f j > f k, σ + r j + z j f j for (j, k) E if j = k. (j,k) C where C E is an elemenary cycle in G, is valid for Q l T. φ jk (σ, r, z) 0, (20) Proof of Proposiion 2 se of feasible soluions: Weakening he coefficiens of z i, i R l in (19), we obain an equivalen Q l T = {( s, {y i T } i R l, {z i } i R l ) R R Rl + {0, 1} Rl : y i T + D l T z i D i T, y i T s, i R l }, or equivalenly, Q l T = {( s, {y i T } i R l, {z i } i R l ) R R Rl + {0, 1} Rl : Noe ha Q l T s D l T + yi T s D l T + z i Di T, y i DT l T s, i R}. l (21) given by (21) is he firs ype of coninuous mixing se (Van Vyve 2005), which is defined as P CMIX = {(σ, r, z) R R Rl + Z Rl : σ + r i + z i f i, i R l }, where 0 f j < 1, j R l. Van Vyve (2005) proposes a linear descripion of conv(p CMIX ). The linear descripion includes bound consrains r 0, and cycle inequaliies given by (20). A a non-anicipaive node (, l) for a fixed sage [1, n], T [, n] and l [1, m], if a lower bound for s = j=1 A T jx l j is known, hen coninuous mixing inequaliies can be srenghened. In oher words, suppose ha as in Secion 4.1, we have s = j=1 A T jx l j D l in all feasible soluions, for some D l 0. Then for R l {k S l : D k T D l }, le Q l T = {( s, {y i T } i R l, {z i } i R l ) R R Rl + {0, 1} Rl : y i T + D i T z i D i T, y i T s D l, i R l }. (22) Le D l T := max{d i T : i R l } D l. Then (22) is equivalen o Q l T = {( s, {y i T } i R l, {z i } i R l ) R R Rl + {0, 1} Rl : ( s D l ) D l T + yi T s D l T + z i (Di T D ) l, y D i T l T s D, l i R}. l

9 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc Proposiion 3. A a non-anicipaive node (, l) wih R l {k S l : D k T D l }, for a fixed sage [1, n] and T [, n], le σ = ( s D l ) D l T ; r i = yi T s ; z D i = z i for i R l T l wih DT i D l < D T l, for i R l wih DT i D l < D T l, and z i = z i 1 for i R l wih DT i D l = D T l ; and f i = (Di T D l ) D T l and f i = 0 for i R l wih DT i D l = D T l ; and f 0 = 0. Also le he digraph G + = (V +, E + ), where V + = {0} R l and E + = {(i, j) : i, j V +, i j, f i f j } {(i, i) : i R}. l The arc lengh φ + jk (σ, r, z) associaed wih each arc wih respec o he poin ( s, {y i T } i R l, {z i } i R l ) Q l T φ + jk σ + r j + (f j f k + 1) z j f k for (j, k) E + if f j < f k, j 0, r (σ, r, z) = j + (f j f k ) z j for (j, k) E + if f j > f k, σ f k for (0, k) E + σ + r j + z j f j for (j, k) E + if j = k where z 0 = r 0 = f 0 = 0. Then he cycle inequaliy (j,k) C where C E + is an elemenary cycle in G +, is valid for Q l T. φ + jk (σ, r, z) 0, (23) Proof of Proposiion 3 Noe ha in his case, because σ 0 in all feasible soluions o Q l T, we have he second ype of coninuous mixing se (Van Vyve 2005), which is defined as P CMIX + = {(σ, r, z) R + R Rl + Z Rl : σ + r i + z i f i, i R l }, where 0 f j < 1, j R l. Van Vyve (2005) proposes a linear descripion of conv(p CMIX + ), which includes bound consrains σ, r 0 and cycle inequaliies defined on he digraph G + given by inequaliies (23). We refer o cycle inequaliies (20) and (23) as coninuous mixing inequaliies (cus). Noe ha coninuous mixing inequaliies (20) and (23) subsume mixing inequaliies (17) (Van Vyve 2005). For = T, we have y i T = s, and he coninuous mixing inequaliies reduce o he mixing inequaliies. The separaion problem of inequaliies (20) and (23) for a given se R l is is o find a negaive cos cycle in graphs G and G +, respecively (Van Vyve 2005). Therefore, i can be solved in polynomial ime using he Bellman-Ford algorihm (Ahuja e al. 1993). In graph G +, for a given se S l, o selec he bes subse R l {k S l : DT k D } l = {k 1,..., k v }, where D k i T Dk i+1 T i [1, v 1], also akes polynomial ime, because we only need o consider v possibiliies of R l, which are {k 1 }, {k 1, k 2 },..., {k 1,..., k v }. In oher words, if here exiss a violaed coninuous mixing inequaliy corresponding o a se R l {k S l : D k T D l }, hen i is also a violaed coninuous mixing inequaliy corresponding o se {k 1,..., k h(r l ) }, where h(rl ) = max{i : k i R l }, because he graph corresponding o he se R l is a subgraph of he graph corresponding o se {k 1,..., k h(r l ) }. The algorihm o find he bes subse R l of a given S l is similar for he graph G. Nex, we sudy a dynamic probabilisic lo-sizing (DPLS) problem o illusrae addiional modeling consideraions for dynamic problems under join chance consrains. In Secion 6, we presen our for

10 10 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc compuaional experience wih a branch-and-cu mehod using mixing inequaliies (17) and coninuous mixing inequaliies (23) for DPLS. Nex, we compare he dynamic join chance-consrained program wih is saic counerpar, a risk-averse opimizaion model, a robus opimizaion model, and a pseudo-dynamic approach for an applicaion in invenory conrol. 5. An Applicaion: Probabilisic Lo Sizing wih Service Levels Saring wih he seminal work of Wagner and Whiin (1958), dynamic invenory conrol models ypically rely on he assumpion ha demand is known for all successive ime periods a priori wih cerainy. This is a resricive assumpion because he fuure demand can be influenced by many facors, mos of which canno be quanified ahead of ime, such as recession, energy prices ec. In addiion, mos deerminisic invenory conrol models assume ha in case backlogging is allowed, i is penalized wih a shorage cos (Zangwill 1966, Poche and Wolsey 1988, Küçükyavuz and Poche 2009) wih he excepion of Gade and Küçükyavuz (2013) who limi he number of periods in which shorages occur. Sochasic lo-sizing models address randomness in demands and coss. Guan and Miller (2008) sudy he sochasic uncapaciaed lo-sizing problem wih zero lead imes and give a backward dynamic programming recursive algorihm which is polynomial in ime wih respec o he size of he scenario ree for cases when backlogging is no allowed or is prohibiively expensive. Huang and Küçükyavuz (2008) sudy sochasic lo-sizing problem wih random lead imes and give an algorihm which is polynomial wih respec o he size of he scenario ree for cases wih no backlogging. Guan (2011) sudy he sochasic capaciaed lo-sizing problems wih zero lead imes and give a dynamic programming algorihm for cases in which backlogging is allowed and is penalized wih backlogging coss. The calculaion of back-ordering cos involves cerain inangible facors which are difficul o quanify, such as he cos of los cusomer goodwill. In classical sochasic invenory conrol lieraure, an alernaive model is o se a required service level corresponding o a maximum sock-ou probabiliy (Nahmias 2005). The service level, τ, indicaes he overall join probabiliy of meeing demand on ime and can be a very crucial elemen of he cusomer conrac as i limis he demand ha is backlogged. Biran and Yanasse (1984) give deerminisic approximaions for capaciaed and saic probabilisic producion problems using hese service levels. Lasserre e al. (1985) use a sochasic opimal conrol approach for probabilisic lo-sizing problems based on required service levels. In heir deerminisic equivalen model, violaions of chance-consrains are penalized in he cos funcion. In our research, we propose exac mehods for dynamic probabilisic lo-sizing models ha mee a required service level under a finie discree disribuion on he demands and coss. Unlike classical invenory conrol models, we assume ha he discree demand disribuion over he planning horizon may be non-saionary and correlaed.

11 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc In more recen lieraure, Beraldi and Ruszczyński (2002a) discuss a saic, probabilisic version of he lo-sizing problem wih a required service level and give a branch-and-bound algorihm for solving i. Demand is assumed o follow a finie and discree disribuion and he producion schedule is deermined a he beginning of he planning horizon. (See also Lejeune and Ruszczyński (2007) for a more general probabilisic producion and disribuion planning problem.) This is a saic model because i assumes ha he producion schedule canno be updaed during he planning horizon based on how he demands and coss unfold over ime. In realiy, many producion schedules have he flexibiliy ha he producion level can be updaed depending on he demands and coss observed in he pas periods. In his secion, we consider a single-produc muli-sage probabilisic lo-sizing problem in a dynamic seing. Based on he demands and coss we observed in he previous ime periods, we deermine an order schedule ha minimizes he oal expeced cos while saisfying boh he flow balance and service level consrains. This wai-and-see approach allows for he order quaniies o be deermined as demand and cos evolve. Bookbinder and Tan (1988) presen an inermediae saic-dynamic approach which uses a heurisic based algorihm o yield approximae resuls. In bach-sizing problems, producion occurs in inegral muliples of a given bach-size. Lulli and Sen (2004) sudy a relaed dynamic probabilisic bach-sizing problem and propose a branch-and-price algorihm for is soluion. We discuss he assumpions of heir model in more deail in Secion 5.2. We evaluae he overall probabiliy of socking ou over he horizon, insead of mainaining he service level for each period individually. In oher words, we consider join chance consrains insead of he much easier case of individual chance consrains which call for quanile-based linear reformulaions. Alhough our focus is on he dynamic probabilisic lo sizing (DPLS) in his secion, for he sake of compleeness we firs review he saic version of he probabilisic lo-sizing (SPLS) problem Saic Probabilisic Lo Sizing Beraldi and Ruszczyński (2002a) sudy sochasic ineger problems under probabilisic consrains and presen SPLS as an example. In heir model, hey approximae he oal expeced cos by eliminaing he holding cos and invenory variables from he objecive funcion. The model wih he invenory coss is solved using a branch-and-cu algorihm in Küçükyavuz (2012). In SPLS, he order quaniies, x, = 1,..., n, are deermined a he beginning of he planning horizon. I is assumed ha hese order quaniies canno be changed during he planning horizon as some of he demands and coss are revealed. The objecive is o minimize he expeced oal cos while complying wih he invenory balance and join probabilisic service level consrains. Le δ, ξ, µ, γ, ν, be he random variables for demand, cumulaive demand, variable and fixed order coss, and he holding cos in period, [1, n], respecively. Thus, ξ = j=1 δ j for [1, n].

12 12 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc Suppose ha he join disribuion of hese random variables is discree and has finie suppor. Then he probabiliy disribuion can be represened by a finie number, m, of scenarios, wih probabiliies π 1,..., π m. The sochasic program for SPLS is: min E Γ ( n (µ =1 x + ν s (ξ ) + γ w )) x 1 ξ 1 x 1 + x 2 ξ 2 s.. P x 1 + x 2 + x 3 ξ 3... τ, x 1 + x 2 + x x n ξ n 0 x M w [1, n], s (ξ ) j=1 x j ξ [1, n 1], (24) s n (ξ n ) = n j=1 x j ξ n, (25) s (ξ ) 0 [1, n], (26) w {0, 1} [1, n], where s (ξ ) is he invenory a he end of period and M is he order capaciy in period. Inequaliies (24) and (26) ensure ha s (ξ ) = ( j=1 x j ξ ) +, for [1, n 1]. Inequaliy (25) is he end of horizon invenory consrain, which ogeher wih (26), ensures ha all demand is delivered by he end of planning horizon. Observe ha he random righ hand sides in he chance consrain, ξ 1,..., ξ n, are highly correlaed for his problem. Le D i be he cumulaive demand unil period, and c i and g i be he variable and fixed coss of ordering, and h i be he variable holding cos in period under scenario i. Finally, le x be he decision variable represening order quaniy in period, s i be he invenory level a he end of period under scenario i, and w be 1 if an order seup is made, and 0 oherwise. The deerminisic equivalen of he SPLS model is: min m i=1 n =1 πi (c i x + h i s i + g i w ) (27) s.. y = i=1 i D(1 i z i ) [1, n], i [1, m], (28) n x j=1 j Dn i i [1, m], (29) s i j=1 x j D i [1, n], i [1, m], (30) 0 x M w [1, n], (31) m i=1 πi z i 1 τ, (32) s i 0 [1, n], i [1, m], (33)

13 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc z i {0, 1} i [1, m], (34) w {0, 1} [1, n], (35) where z i is 1 if here is unme demand in scenario i and is 0 oherwise, following he convenion of Luedke e al. (2010). This formulaion has he mixing se (28) and a knapsack consrain (32) as is subsrucure for which srong valid inequaliies are described in Secion Dynamic Probabilisic Lo Sizing In DPLS, he order decision for a ime period = 1,..., n is dependen on he demands and coss revealed in periods 1 o 1. This gives us he flexibiliy o updae he order schedule over ime, based on wha we have already observed. Le Γ 1 := (ξ 1, µ 1, γ 1, ν 1,..., ξ 1, µ 1, γ 1, ν 1 ), be he random vecor represening he demands and coss unil ime [2, n]. Le x (Γ 1 ) be he decision variable a sage [2, n], whose value is deermined afer he random variables, Γ 1, are observed, and x 1 be he iniial order quaniy. Then he chance consrain is updaed as: x 1 ξ 1 x 1 + x 2 (Γ 1 ) ξ 2 P x 1 + x 2 (Γ 1 ) + x 3 (Γ 2 ) ξ 3 τ. (36)... x 1 + x 2 (Γ 1 ) + x 3 (Γ 2 ) + + x n (Γ n 1 ) ξ n Le x i represen he quaniy ordered in period in scenario i o allow order decisions o be dependen on he demand and cos realizaions unil ime. Also le w i = 1 if an order seup is made, and w i = 0 oherwise. Le S l = {k [1, m] : D l j = D k j, h l j = h k j, g l j = g k j, c l j = c k j, j [1, 1]}, be he se of scenarios ha share he same demand and cos hisory wih scenario l unil period. The deerminisic equivalen of he DPLS model is min m i=1 n =1 πi (c i x i + g i w i + h i s i ) s.. (32) (34), y i = j=1 xi j D(1 i z i ) [1, n], i [1, m], (37) n j=1 xi j Dn i i [1, m], (38) s i j=1 xi j D i [1, n], i [1, m], (39) 0 x i M w i [1, n], i [1, m], (40) x l = x k [1, n], l [1, m], k S l \ {l}, (41) w i {0, 1} [1, n], i [1, m]. (42) Consrains (37) (40) are he dynamic versions of consrains (28) (31). Consrains (41) enforce non-anicipaiviy. Consrains (37) for a fixed T [1, n] and (41) for some T wih S l have

14 14 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc he coninuous mixing se as is subsrucure for which srong valid inequaliies are described in Secion 4.2. Noe ha, for his model, for 1 T n, we have s = A j=1 T jx l j = y l D l = D νl +1, where ν l is as defined in Secion 2. Also, T j=+1 A T jx l j = T j=+1 xl j 0. Therefore, he sronger coninuous mixing inequaliies (23) are valid. Example 1. In order o highligh he difference beween saic and dynamic models, le us consider a small es case wih 5 ime periods and 5 scenarios as given in Figure 1. The wo numbers in he nodes of he scenario ree indicae he demand and uni order cos for each scenario in ha ime period. For example, he demand and uni order cos a period 2 in scenario 1 are 73 and 48, respecively. Each oucome of he fuure is represened by a scenario pah. Noe ha S j 1 = {1, 2,..., 5}, for j = 1,..., 5 (because a he firs period we did no see any demand), S 1 2 = S 2 2 = {1, 2}, S 3 2 = S 4 2 = {3, 4} and S 3 3 = S 4 3 = {3, 4}. Le he probabiliy of occurrence of each scenario be π = (0.25, 0.15, 0.10, 0.10, 0.40). Le he holding cos be 10 percen of he uni order cos and he required service level τ be 85 percen. Finally, suppose ha here are no fixed order coss. We repor he opimal order quaniies for he SPLS and DPLS models in Table 1. The opimal cos given by SPLS model is In addiion, because he order quaniies are decided ahead of ime and are independen of he scenario pah realized, he observed service level is 100%, even hough backorders are accepable. On he oher hand, he opimal cos given by he DPLS model is and he observed service level is 85%. As a resul, significan cos savings are achieved when order quaniies are deermined based on he demand and cos hisory. Lulli and Sen (2004) consider he dynamic probabilisic bach-sizing problem (DPBS) and propose a branch-and-price algorihm for solving i. In he Danzig-Wolfe reformulaion of DPBS, he auhors inroduce one variable for each feasible order schedule for every scenario, and anoher variable which is equal o 1 if a scenario is no violaed, and 0 oherwise. Then he maser problem enforces non-anicipaiviy and calculaes cos only for he scenarios which are no violaed. In conras, in our model, we assume ha he non-anicipaiviy holds even for scenarios which have a sock-ou, which resuls in soluions ha are implemenable. We observe from Figure 1 ha scenarios 3 and 4 have same demand and cos hisory unil period 3. Therefore same decisions mus be made for hese scenarios unil ime period 3. If we do no enforce he non-anicipaiviy consrains (41) for he scenarios ha are violaed, we ge he order quaniies shown in Table 2. Noe ha his order schedule is no implemenable. The order quaniy a he firs ime period has o be he same across all scenarios because so far we have seen no demand and order coss. Therefore, a his poin (he firs ime period) our decision should be independen of which scenario pah we will follow in fuure.

15 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc In addiion, we assume ha all demand has o be me a he end of he ime horizon in every scenario by inroducing he end of horizon consrains (38). In Table 3 we show he order quaniies in DPLS model wih no end of horizon consrains. As shown in he able, he scenarios which have sock-ous also have unfulfilled demand a he end of planning horizon. For example, oal demand in scenario 2 is 334 unis whereas oal order quaniy in scenario 2 is only 142 unis leading o a shorage a he end. Once a sock-ou occurs a period 3, no furher orders ake place and all fuure demand is los. Therefore, we need o add hese consrains o ensure ha all demand is saisfied a he end of conrac horizon for all scenarios. 73,48 68,85 30,63 80,95 69,170 80,108 18,87 67,79 100, ,129 15,60 87,77 55,48 86,26 80,161 72,154 16,38 35,192 59,67 79,155 36,48 47,23 Figure 1 Scenario ree represenaion Table 1 Quaniies produced in DPLS and SPLS models DPLS Time Scenario SPLS Nex we give an example for a valid mixing inequaliy. Le = 2, l = 1, S 1 2 = S 2 2 = {1, 2}, T = {2}, D νl +1 = , hen a valid mixing inequaliy (17) is x x [( ) ( )]z To see he validiy of his inequaliy, noe ha if z 2 = 0, hen he demand is no backlogged in scenario 2. Therefore, we mus have x x On he oher hand, if z 2 = 1, hen he demand

16 16 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc Table 2 Quaniies produced in DPLS model wih relaxed (41) Scenario Time Table 3 Quaniies produced in DPLS model wih relaxed (38) Scenario Time can be backlogged in scenario 2, however, because π 1 = 0.25 and τ = 0.85, demand canno be backlogged in scenario 1. Hence, we mus have x x Now we give an example for a valid coninuous mixing inequaliy. Le = 1, l = 1, T = 3, S 1 1 = {1, 2, 3, 4, 5}. Noe ha D 1 1 = 69. Consider he consrains associaed wih se R l = {2, 3} S 1 1: or x x x z 2 167, x x x z Noe ha D 1 1,2 = max{167, 114} D 1 1 = 98. Then where x 2 1 = x 3 1 due o non-anicipaiviy. x x x z = 98, x x x z = 45, x (x2 2 + x 2 3) + z 2 1 0, x (x3 2 + x 3 3) + z , Consider he graph G + = (V +, E + ), where V + = {0, 2, 3} and E + = {(0, 3), (3, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 3), (3, 2)} and he non-elemenary cycle {(2, 3), (3, 2)} E +. The lengh of arc (2, 3) is x (x2 2 +x2 3 ) + ( )(z2 1) 45, because f 2 = 0 < 45 = f The lengh of arc (3, 2) is (x3 2 +x3 3 ) + ( )z3, because f 3 = 45 = f 2 > 0. The corresponding 98 coninuous mixing inequaliy (23) is x (x2 2 + x 2 3) 98 + ( )(z2 1) (x3 2 + x 3 3) z3 0,

17 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc or x x x z 2 + x x z (43) To see he validiy of inequaliy (43), noe ha we canno have z 2 = z 3 = 1, because he oal probabiliy of scenarios 2 and 3 is τ. If z 2 = 0, hen we mus have x x x , hus, inequaliy (43) is saisfied. Finally, if z 2 = 1 and z 3 = 0, hen x 2 1 +x 3 2 +x 3 3 = x 3 1 +x 3 2 +x , where he firs equaliy follows from non-anicipaiviy. Observe ha for he same choice of, T, l and R l, inequaliy (20) is given by which is clearly weaker han (43). x x x z 2 + x x z 3 167, 5.3. Comparison wih a Dynamic Risk-Averse Opimizaion Model Risk-averse opimizaion has been receiving increasing aenion in recen years (Ruszczyński and Shapiro 2006a,b, Shapiro 2009). In dynamic risk-averse opimizaion, decisions are made in muliple sages such ha a nesed dynamic risk measure ofen involving he coss is minimized (c.f. Ruszczyński and Shapiro (2009)). In conras, he dynamic join chance-consrained model (8) (15), minimizes a risk-neural objecive (he expeced oal cos) while conrolling he risk by a consrain ha ensures ha every feasible soluion mees he required service level. In anoher line of research, Dencheva and Ruszczyński (2008) propose a sochasic dynamic opimizaion problem in which he risk aversion is expressed by a sochasic ordering consrain. We now compare a dynamic risk-averse opimizaion model for muli-sage invenory problems (Ahmed e al. 2007) wih he join chance-consrained DPLS model. As in previous secions, δ, ξ, µ, γ and ν denoe he random demand, cumulaive demand, variable cos, fixed charge and holding cos in period. Le d denoe he realizaion of δ in period. We assume ha δ has a discree disribuion and is independen of (δ 1,..., δ 1 ). In addiion, le ζ denoe he random backlogging cos in period, whose realizaion is denoed as b. Ahmed e al. (2007) propose a muli-sage risk-averse invenory model, where he decision makers decide he order quaniy a he beginning of each period afer observing he invenory level a he end of he previous period. Based on he model in Ahmed e al. (2007), bu using he same decision variables x and w, [1, n] as in our SPLS model, consider he dynamic risk-averse opimizaion model min 0 x Mw, [1,n] n=1 x ξ n y = i=1 x i, [1,n] w {0,1}, [1,n] ρ 1 [ µ 1 x 1 + γ 1 w 1 + ν 1 (y 1 ξ 1 ) + + ζ 1 (ξ 1 y 1 ) + + ρ 2 Γ1 [µ 2 x 2 + γ 2 w 2 + ν 2 (y 2 ξ 2 ) + + ζ 2 (ξ 2 y 2 ) + +

18 18 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc ρ Γ 1 [µ x + γ w + ν (y ξ ) + + ζ (ξ y ) + + [ ] ] ]] + ρ n Γn 1 µ n x n + γ n w n + ν n (y n ξ n ) + + ζ n (ξ n y n ) +, (44) where ρ Γ 1 (Z) is a risk measure. In he following example, we use he mean-absolue deviaion (MAD) as he risk measure, which is defined as MAD λ (Z) := E[Z] + λe Z E[Z], λ [0, 1/2]. Example 1. (coninued) Le us solve he five-period es case wih he risk-averse model (44). To illusrae he effec of he backlogging cos, we consider alernaive backlogging coss given by 0, 0.5µ, 1.50µ, or 2µ. The comparison beween he soluions, service levels and he expeced oal order and holding coss of he DPLS, SPLS and he risk-averse models is shown in Table 4. In our expeced cos comparison, we recalculae he expeced oal cos of he risk-averse model as m n i=1 =1 πi (c i x + gw i + h i s ) o exclude he arificial backlogging coss. When he backlogging coss are aken as zero, we see ha opimizing he muli-sage risk measure on he order and holding coss canno guaranee a feasible soluion ha mees he required service level. Increasing he backlogging cos resrics backlogging quaniy o some exen. For example, he service level increases from 0 o 25% when he backlogging cos increases from 0 o 0.5µ. However, even when we le he backlogging coss equal 1.5 imes he uni order coss (1.5µ ), we are no able o find a feasible soluion o our original problem because he service level is only a 60%. We canno find a feasible soluion unil we increase he backlogging coss o wice he uni order cos (2µ ), bu he expeced order and holding cos is 23.49% higher han ha of DPLS. Clearly, he risk-averse model opimizes a differen objecive han he DPLS model. However, we provide he expeced cos comparison o show ha even when a high enough backlogging penaly is used in he riskaverse model ha yields a feasible soluion o DPLS, he soluion qualiy wih respec o he original objecive can be far from opimal. Comparing he soluions o he DPLS, SPLS and he risk-averse models, we can conclude ha he decisions made by he DPLS model always saisfy he required service level a significanly lower expeced order and holding coss. I is worh noing ha in our model, he risk measure of meeing a service level over he enire horizon is a saic risk measure, unlike hose in dynamic risk averse opimizaion. Shapiro (2009) discusses a desirable ime-invariance propery for cerain dynamic risk measures. Nex we consider a pseudo-dynamic approach, which solves a series of saic join chanceconsrained programs in an effor o obain feasible soluions ha are adapive o he pas observaions wih less compuaional effor han he fully dynamic model Comparison wih a Pseudo-Dynamic Approach Rolling horizon mehod (Baker 1977) is one of he widely used mehodologies o solve a mulisage problem. In solving a problem wih n sages, a he beginning of he firs period, we solve a subproblem involving periods 1 o 1+n 1, 1 + n 1 n, and fix he soluion o he firs period. Then

19 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc Table 4 Order quaniies for DPLS, SPLS, and risk-averse models in Example 1 DPLS risk-averse Time Scenario SPLS ζ /µ Service level 85% 100% 0 25% 60% 100% Cos increase % % % 7.48% 23.49% from DPLS he inpu o he second period is updaed. A he beginning of he second period, we solve a subproblem on periods 2 o 2+n 2, where 2 + n 2 n and n 2 is no necessarily equal o n 1, and fix he soluion o he second period. We repea his process unil he soluion o he las period is fixed. The rolling horizon mehod significanly reduces he compuaional burden for problems wih a long or infinie horizon. Rolling horizon mehod has been applied o he lo-sizing problem wih ime-varying coss and demands (see, for example, Moron (1977) and Blackburn and Millen (1980)). In his subsecion, we apply a pseudo-dynamic approach ha is inspired by he rolling horizon mehod o he probabilisic sochasic lo-sizing problem. Iniially, we solve he SPLS model and obain he saic decisions x = (x 1, x 2,..., x n ). We choose x 1 as he order quaniy in he firs period. In period [2, n], for each non-anicipaive node in period, we consider a similar saic decision making problem over he planning sub-horizon over periods o n, and we choose x as he order quaniy in period. Noe ha he resuling decisions are dynamic (adapive o he pas observaions), bu we solve a series of saic decision-making models, which are easier o solve. During his process, we updae he required service level for each subproblem. In period [1, n], given he soluion x i j for j [1, 1] and i [1, m], we define a se V := {i : i [1, m], j k=1 xi k D i j for j [1, 1]} as he se of scenarios ha do no have sockou in periods 1 o 1. Moreover, we define he updaed required service level for he non-anicipaive se S l l V, oherwise τ l = 1 ( i V π i τ) +. as τ l, where τ l = 0 if For a given [1, n] and l [1, m], he iniial invenory/sockou quaniy is s l 1 := 1 j=1 x j D 1. Noe ha s l 1 is negaive when here is sockou. A each non-anicipaive node associaed wih S l, given V and s 1, he problem we need o solve, SP l ( s 1, τ l ), is defined as: min n π i (c i jx j + h i js i j + gjw i j ) i S l j=

20 20 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc s. s l 1 + s l 1 + s i j j j x k d i k(1 z i ) j [, n], i S l, k= n x k k= k= n d i k i S l, k= j (x k d i k) + s l 1 j [, n], i S l, k= 0 x j M j w j j [, n], π i z i 1 τ l, i S l s i j 0 j [, n], i S l, z i {0, 1} i S l, w j {0, 1} j [, n], where π i is he probabiliy of scenario i. There are wo opions for π i : (1) we can keep π i = π i when we solve every sub-problem; (2) we can updae he probabiliy of scenario i in S l for he problem SP l ( s 1, τ l ) by he condiional probabiliy π i = π i /( j S l πj ). Now we look ino hese wo cases. Case 1. Table 5 gives he soluion o he pseudo-dynamic model when we do no updae he probabiliies of each scenario a each sage. The service level reached is 75%, which is lower han he required service level. The expeced oal cos is From his case, we can reach he conclusion ha using he pseudo-dynamic approach wihou updaing he probabiliies of each scenario, we may obain infeasible soluions violaing he required service level consrain. Table 5 Quaniies produced by he pseudo-dynamic approach wihou updaing probabiliies DPLS Time Scenario Case 2. Table 6 gives he soluion o he pseudo-dynamic model when we updae he probabiliies of each scenario in each sage using condiional probabiliies. The service level reached is 100%, and he expeced oal cos is , which is higher han ha of he DPLS model, bu lower han ha of he SPLS model. Thus, for his insance, he pseudo-dynamic approach finds a more conservaive decision compared wih he DPLS model and a less conservaive decision compared wih he SPLS model.

21 Aricle submied o Managemen Science; manuscrip no. hp://dx.doi.org/ /mnsc Table 6 Quaniies produced by he pseudo-dynamic approach wih updaed probabiliies DPLS Time Scenario Ineresingly, he soluion given by he pseudo-dynamic approach using condiional probabiliies can give more conservaive soluions han ha of SPLS. For example, if we consider he insance shown in Table 7 wih wo scenarios and hree periods wih he required service level a 80%, he soluion given by he saic model is x = (2, 0, 10) wih he expeced oal cos of 13. The soluion given by he pseudo-dynamic approach is x 1 = (2, 9, 1), x 2 = (2, 0, 1), wih he expeced oal cos of 22. This comparison indicaes ha, alhough he pseudo-dynamic approach requires more compuaional effor han he saic model, i canno guaranee a beer soluion han ha of he saic model. Table 7 Coss and demands for a wo-scenario insance Scenario 1 Scenario 2 Time Time Parameers c d f h π Comparison wih a Robus Opimizaion Model In he dynamic join chance-consrained model (8) (15), we assume ha he random variables have finiely many realizaions, i.e., hey have known discree disribuions. Anoher imporan mehod addressing he uncerainy of daa is robus opimizaion, which assumes ha he random variables belong o an uncerainy se and opimizes agains he wors case. For example, Soyser (1973) proposes a formulaion for a convex mahemaical programming problem in which random variables can ake any value in he given ranges. Ben-Tal and Nemirovski (1998, 1999, 2000), El Ghaoui and Lebre (1998), and El Ghaoui e al. (1997) assume ha he uncerain variables are in ellipsoidal uncerainy ses. Erdoğan and Iyengar (2006) sudy he ambiguous chance-consrained program where he disribuions of he random variables are uncerain. They propose a robus sampled problem o approximae he ambiguous chance-consrained program wih a high probabiliy. To