Bounds in Multistage Linear Stochastic Programming - Semantic Scholar

J Optim Theory Appl DOI 10.1007/s10957-013-0450-1

Bounds in Multistage Linear Stochastic Programming Francesca Maggioni · Elisabetta Allevi · Marida Bertocchi

Received: 2 April 2012 / Accepted: 8 October 2013 © Springer Science+Business Media New York 2013

Abstract Multistage stochastic programs, which involve sequences of decisions over time, are usually hard to solve in realistically sized problems. Providing bounds for optimal solution may help in evaluating whether it is worth the additional computations for the stochastic program vs. simplified approaches. In this paper we generalize measures from the two-stage case, based on different levels of available information, to the multistage stochastic programming problems. A set of theorems providing chains of inequalities among the new quantities are proved. Numerical results on a case study related to a simple transportation problem illustrate the described relationships. Keywords Multistage stochastic programming · Expected value problem · Value of stochastic solution · Skeleton solution 1 Introduction Stochastic programs, especially multistage programs, which involve sequences of decisions over time, are usually hard to solve in realistically sized problems. In the F. Maggioni · M. Bertocchi Department of Management, Economics and Quantitative Methods, Bergamo University, Via dei Caniana 2, 24127 Bergamo, Italy F. Maggioni e-mail: [email protected] M. Bertocchi e-mail: [email protected]

B

E. Allevi ( ) Department of Economics and Management, Brescia University, Contrada S. Chiara 50, Brescia 25122, Italy e-mail: [email protected]

J Optim Theory Appl

simpler two-stage case, several approaches and bounds on the optimal objective value have been adopted in the literature (see for instance [1–13]). The standard measure is given by the Value of the Stochastic Solution, VSS [2], which indicates the expected gain from solving a stochastic model rather than its deterministic counterpart, in which the random parameters are replaced with their expected values. A large VSS means that uncertainty greatly affects the optimal solution, and the deterministic solution is “bad.” Bounds on VSS were introduced in [2] by means of the Sum of Pairs Expected Values Solutions, SPEV, and Expectation of Pairs Expected Value, EPEV, which can be calculated by solving pairs of subproblems which are much less complex than the general recourse problem; these bounds may be valuable in determining whether the additional computational complexity of the recourse problem is warranted. These measures have been the starting point of this research and inspiration of this work. In real world problems, even when the VSS is high, justifying the stochastic program formulation, the sheer size of the resulting model can make it very unpractical to solve. In such cases, we may still be able to calculate the deterministic solution. A qualitative understanding of the deterministic solution becomes then important because it could actually lead to a lot of information: in [14] the structure and upgradeability of the deterministic solution have been analyzed in the two-stage case by means of the Loss Using the Skeleton Solution LUSS and the Loss of Upgrading the Deterministic Solution LUDS in relation to the standard VSS. Compared to the VSS, LUSS and LUDS give broader information on the structure of the problem and could be of practical relevance to take a fast “good” decision instead of using expensive direct techniques. The aim of this paper is to extend to the multistage case the measures of information already valid for the two-stage case introduced in [14, 15] and inspired by [3, 16, 17]. Due to the computational complexity of most multistage problems, we believe in usefulness of such setting: 1. To consider various approximations of the recourse problem, i.e. in case the solution of a large multistage stochastic program is quite demanding, one may try to solve simpler problems for finding lower and upper bounds and proceed to find tighter and tighter bounds, if the difference between the first two is quite large. Some upper or lower bounds may also have some interest after the main big problem has been solved (for instance, to convince skeptical people that the full stochastic approach is much better than a deterministic one); 2. To evaluate, at different levels of information, how the deterministic solution performs in the stochastic framework. This approach applies to algorithmic developments as well as to the practical use of models in the industry and government. An extension to multistage of the classical VSS defined for the two-stage setting has been already introduced in [16] through a chain of values VSSt which take into account the information until stage t of the associated deterministic model. These results are valid if, in the formulation of the multistage problem, only the right-hand side constraints are stochastic.

J Optim Theory Appl

The general idea behind construction of bounds we adopt is that for every optimization problem of minimization type, lower bounds to the optimal solution can be found by relaxation of constraints and upper bound to the optimal solution can be found by inserting feasible solutions. We suggest how to quantify the quality of approximations of the optimal stochastic solution such as the Multistage Expected Value of the Reference Scenario, MEVRS, the Multistage Sum of Pairs Expected Values, MSPEV, and the Multistage Expectation of Pairs Expected Value, MEPEV. They are introduced by means of the new concept of auxiliary scenario and redefinition of pairs subproblem probability. The MSPEV idea is to bound the optimal expected objective function by solving sets of pairs subproblems that are less complex than the original one. If it turns out the calculation of the MSPEV value is easily performed, then possibly one may find an upper bound which is good enough to stop the procedure here and not solve the big problem at all. Moreover, those approaches help to quantify whether the increased complexity of solving the initial stochastic program is justified. Beside the standard Value of Stochastic Solution, VSSt , at stage t, measures of the average solution quality such as the Multistage Loss Using Skeleton Solution, MLUSSt , and the Multistage Loss of Upgrading the Deterministic Solution, MLUDSt , are discussed. These measures are important in practice, since they help to qualitatively understand the behavior of the deterministic solution in relation to the stochastic one, and they give a mathematical formulation to some heuristic approaches which are of interest in solving real world problems of large dimension. As pointed out in [16], the generalization of such measures entails several issues: first of all the choice of the variables to be fixed to their deterministic solution’s values. A simple approach, widely used in practice, is to fix just the first-stage variables, leaving the other ones free to adapt to the particular scenario. This procedure can lead to paradoxical results, where the first-stage deterministic solution performs better than the stochastic one. This is due to relaxation of the nonanticipativity constraints in the later stages (see for instance [18]). An alternative approach is to consider a rolling time horizon procedure (see [3, 17]) in order to update the estimations of the solution at each stage taking into account the arrival of new information. The advantage of the rolling horizon procedure in the multistage case with respect to the solution of the recourse problem (RP) is the decrease of the computational complexity since the number of scenarios is reduced by considering the stochasticity only one stage at a time. On this subject, the Rolling Horizon Value of Stochastic Solution, RHVSS, the Rolling Horizon Loss Using Skeleton Solution, RHLUSS, and the Rolling Horizon Loss of Upgrading the Deterministic Solution, RHLUDS, are presented. The chains of inequalities among the values of the new measures can show if and how much it is worth to face the computational complexity of the stochastic program. An example of such chains of relationships is depicted in Fig. 4 for a practical logistic problem. We finally remark that all these types of measures are often used to describe problem types, even though they are instance-dependent. As in the two-stage case [14], we assume that if a particular performance measure is high (or low) for a given set

J Optim Theory Appl

of instances, then it will also be high (or low) for other instances that have similar characteristics, such as larger instances of the same problem. The paper is organized as follows: the notation and basic definitions are introduced in Sect. 2. Section 3 introduces new bounds for the multistage case of the performance measures. Section 4 reports the performance of the new measures in a multistage stochastic model in a real life transportation problem introduced in [18]. Section 5 concludes the paper.

2 Notation and Basic Definitions We introduce the notation to be used throughout the paper. The following mathematical model represents the nested formulation of a multistage linear stochastic program in which a decision maker has to take a sequence of decisions x 1 , x 2 , . . . , x H to minimize (expected) costs,   RP := min Eξ H −1 z x, ξ H −1 x        = min c1 x 1 + Eξ 1 min c2 x 2 ξ 1 + Eξ 2 · · · + Eξ H −1 min cH x H ξ H −1 x1

x2

xH

s.t. Ax 1 = h1 ,         T 1 ξ 1 x 1 + W 2 ξ 1 x 2 ξ 1 = h2 ξ 1 ,

(1)

.. .

          T H −1 ξ H −1 x H −1 ξ H −1 + W H ξ H −1 x H ξ H −1 = hH ξ H −1 ,   x 1 ≥ 0; x t ξ t−1 ≥ 0, t = 2, . . . , H, where the parameter vectors c1 ∈ Rn1 , h1 ∈ Rm1 and the matrix A ∈ Rm1 ×n1 are known. Eξ t denotes the expectation with respect to a random vector ξ t , defined on a probability space (Ξ t , At , p) with support Ξ t ∈ Rnt and given probability distribution p on the σ -algebra At (with At ⊆ At+1 ). We use the same notation ξ t to denote a random vector and its particular realization. Which of these two meanings will be used in a particular situation will be clear from the context. The two-stage case is obtained for H = 2. Additionally, let us denote the uncertain parameters vectors and matrices: • ht ∈ Rmt , ct ∈ nt , T t−1 ∈ Rmt−1 ×nt−1 , W t ∈ Rmt ×nt , t = 2, . . . , H ; • ξ t := (ξ 1 , . . . , ξ t ), t = 1, . . . , H − 1. The variables are as follows: • x := (x 1 , x 2 , . . . , x H ) with x t ∈ nt , t = 1, . . . , H and x tj the j th component of xt . In general, ct = ct (ξ t−1 ) for t = 2, . . . , H . The decision x t at stage t = 1, . . . , H is At−1 -measurable and it depends on the history up to time t, more precisely the

J Optim Theory Appl

decision process is nonanticipative and it has the form:         decision x 1 → observation ξ 1 → decision x 2 → observation ξ 2       → · · · → decision x t−1 → observation ξ t−1 → decision x t . The solution obtained by solving problem (1) is denoted with x∗ and called here and now solution. The problem expressed in (1) can be formulated through the corresponding dynamic programming equations (see [13]). This formulation takes into account that at the last stage the values of all the problem data ξ H −1 are already known and the values of the earlier decisions x 1 , x 2 , . . . , x H −1 have been already chosen. The problem therefore becomes   min cH x H ξ H −1 xH

s.t.

          T H −1 ξ H −1 x H −1 ξ H −1 + W H ξ H −1 x H ξ H −1 = hH ξ H −1 , (2)   x H ξ H −1 ≥ 0.

The solution value of problem (2), denoted QH (x H −1 , ξ H −1 ), depends on the decision x H −1 at the previous stage and on the realization of the data process ξ H −1 . Problem (1) is then solved recursively computing the cost-to-go functions Qt (x t−1 , ξ t−1 ), going backward in the stages. At stage t = 2, . . . , H − 1 the problem is formulated as follows:      min ct x t ξ t−1 + Eξ t Qt+1 x t , ξ t xt

          s.t. T t−1 ξ t−1 x t−1 ξ t−1 + W t ξ t−1 x t ξ t−1 = ht ξ t−1 ,   x t ξ t−1 ≥ 0 t = 2, . . . , H − 1,

(3)

where Eξ t represents the conditional expectation and its solution value is denoted with Qt (x t−1 , ξ t−1 ). On top of all these problems we have to find the first decision variable x 1 , as the solution of the model    min c1 x 1 + Eξ 1 Q2 x 1 , ξ 1 x1

s.t. Ax 1 = h1 , x ≥ 0. 1

Let us introduce, for later use, the feasible region of problem (4): ⎧ ⎫ ⎪ ⎪ Ax 1 = h1 ⎨ ⎬ 1 2 1 1 K 1 := x Eξ 1 [Q (x , ξ )] < +∞ , ⎪ ⎪ ⎩ ⎭ x1 ≥ 0

(4)

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as well as the feasible region at stage t = 2, . . . , H − 1 of problem (3): ⎧ ⎫ T t−1 (ξ t−1 )x t−1 (ξ t−1 ) + W t (ξ t−1 )x t (ξ t−1 ) = ht (ξ t−1 ) ⎬ ⎨ K t := x t (ξ t−1 ) Eξ t [Qt+1 (x t , ξ t )] < +∞ . ⎩ ⎭ x t (ξ H −1 ) ≥ 0 Problem (1) is now formulated using several different approaches and compared in terms of optimal objective function values. The multistage wait-and-see problem, where the realizations of all the random parameters are known at the first stage, takes the form   WS := Eξ H −1 min z x, ξ H −1     = Eξ H −1 min c1 x 1 ξ H −1 + · · · + cH x H ξ H −1 x 1 (ξ H −1 ),...,x H (ξ H −1 )

(5) Ax 1 = h1 ,           1 1 1 H −1 2 1 2 H −1 2 1 +W ξ x ξ =h ξ , T ξ x ξ .. .           T H −1 ξ H −1 x H −1 ξ H −1 + W H ξ H −1 x H ξ H −1 = hH ξ H −1 ,   x 1 ≥ 0; x t ξ H −1 ≥ 0, t = 2, . . . , H.

s.t.

Note that the decision process is anticipative, since the decisions x 1 , x 2 , . . . , x H depend on the realization of ξ H −1 . This relaxation is obtained by removing the nonanticipativity constraints. For the two-stage case (H = 2), the difference EVPI = RP − WS

(6)

denotes the Expected Value of Perfect Information (see [19]) and compares wait-andsee WS and here-and-now RP approaches. A small EVPI indicates a small reduced cost for reaching the perfect information. The definition can be easily extended to the multistage case. The Expected Value problem EV is obtained by replacing all random parameters by their expected values and solving the deterministic program, with ξ := (ξ¯ 1 , ξ¯ 2 , . . . , ξ¯ H −1 ) = (Eξ 1 , Eξ 2 , . . . , Eξ H −1 ): EV := min z(x, ξ ) x

=

min c1 x 1 + · · · + cH x H

x 1 ,...,x H

s.t. Ax 1 = h1 ,       T 1 ξ¯ 1 x 1 + W 2 ξ¯ 1 x 2 = h2 ξ¯ 1 , .. .       T H −1 ξ¯ H −1 x H −1 + W H ξ¯ H −1 x H = hH ξ¯ H −1 , x t ≥ 0, t = 1, . . . , H.

(7)

J Optim Theory Appl

Theorem 2.1 For the two-stage case (H = 2) the following inequalities hold true (see [8]): WS ≤ RP ≤ EEV,

(8)

where EEV denotes the solution value of the RP model, having the first-stage decision variables fixed at the optimal values obtained by using the expected value of coefficients. Theorem 2.2 For multistage linear stochastic programs with deterministic objective coefficients and constraint matrices, random parameters on the right-hand side h2 (ξ 1 ), . . . , hH (ξ H −1 ), the following inequality is satisfied: EV ≤ WS.

(9)

The proofs of Theorems 2.1 and 2.2 can be easily extended from the two-stage case, H = 2 (see [15]), to the multistage case H > 2. We introduce the concept of Expected result at stage t by using the Expected Value solution EEV t (t = 1, . . . , H − 1), given by the solution value of the RP model where the decision variables until stage t, xt = (x 1 , x 2 , . . . , x t ), t = 1, . . . , H − 1, are fixed at the optimal values obtained by the average scenario ξ t = (ξ¯ 1 , ξ¯ 2 , . . . , ξ¯ t ), t = 1, . . . , H − 1. See in [16] an alternative definition. It is worth to point out that the problems EEV t , t = 1, . . . , H − 1, could be infeasible since too many variables are fixed to their deterministic solution values. The Value of the Stochastic Solution at stage t, VSSt , is then defined as follows: VSSt := EEV t − RP,

t = 1, . . . , H − 1.

(10)

Theorem 2.3 For multistage stochastic linear programs with deterministic objective coefficients and constraint matrices, random parameters on the right-hand side h2 (ξ 1 ), . . . , hH (ξ H −1 ), the following inequalities are satisfied [16]: VSSt ≤ EEV t − EV,

t = 1, . . . , H − 1.

(11)

Proof From (10), Theorems 2.1 and 2.2 we have: VSSt = EEV t − RP ≤ EEV t − WS ≤ EEV t − EV, and the above inequality holds.

t = 1, . . . , H − 1,

(12) 

In order to proceed with numerical computations, it is useful to have a discretization of the underlying random process. So, if we assume that ξ H −1 := (ξ 1 , . . . , ξ H −1 ) is a random parameter evolving as a discrete-time stochastic process with finite support, then the information structure can be described in the form of a scenario tree T where at each stage t there is a discrete number of atoms (nodes) jt where a specific realization of the uncertain parameter takes place. There are H levels (stages) in the tree, that correspond to specific time periods. The final nodes jH are called leaf nodes. Each node, except the root,

J Optim Theory Appl

is connected to a unique node at stage t − 1 called the ancestor node and to nodes at stage t + 1 called the successor nodes. Each non-leaf node j is the root of a subtree T (j ). For each node j at stage t, we denote its ancestor with a(j ) and with πa(j ),j the conditional probability of the random process in node j given its history up to the ancestor node a(j ). A scenario is a path through nodes from the root node to a leaf node. We indicate with πs the probability of scenario s passing through nodes j1 , j2 , . . . , jH (where jt , t = 1, . . . , H , is the generic node at stage t) defined by πs := πj1 ,j2 · πj2 ,j3 · · · πjH −1 ,jH . We also indicate with pj the probability of node j (at stage t): if node j at stage t is reachable through node j1 at stage 1, node j2 at stage 2, . . .  , node jt−1 at stage t − 1, it is given by pj := πj1 ,j2 · πj2 ,j3 · · · πjt−1 ,jt . Moret pj = 1 where nt is the number of nodes at stage t. Let ξ1 , . . . , ξnH be the over, nj =1

possible realizations (or scenarios) of ξ H −1 , Ξ the support of possible scenarios and ξst the history of the s-realization, s = 1, . . . , nH , up to stage t, t = 1, . . . , H − 1. Using this scenario notation the multistage wait-and-see problem (5) can be expressed as: WS := Eξs

min

x 1 (ξs ),...,x H (ξs )

c1 x 1 (ξs ) + · · · + cH x H (ξs )

s.t. Ax 1 (ξs ) = h1 ,       T 1 ξs1 x 1 (ξs ) + W 2 ξs1 x 2 (ξs ) = h2 ξs1 ,

(13)

.. .

      T H −1 ξsH −1 x H −1 (ξs ) + W H ξsH −1 x H (ξs ) = hH ξsH −1 , x 1 (ξs ) ≥ 0;

x t (ξs ) ≥ 0,

t = 2, . . . , H,

s = 1, . . . , nH .

A relaxed reduced formulation of problem (1), useful for computational purposes, is given by the two-stage relaxation where the nonanticipativity constraints in the second and other stages are relaxed as follows: we define a new scenario tree where all random parameters until stage t = 2, . . . , H − 1 are estimated by their expected values, and solve the related model. We denote this new scenario tree as   t− ξ¯ := ξ 1 , ξ¯ 2 , . . . , ξ¯ t , t = 2, . . . , H − 1. Additionally we define another scenario tree where all random parameters from stage t = 2, . . . , H − 1 to the last stage H − 1 are estimated by their expected values. We denote for later use this second scenario tree as   t+ ξ¯ := ξ 1 , ξ 2 , . . . , ξ¯ t , . . . , ξ¯ H −1 , t = 2, . . . , H − 1. The two-stage relaxation problem (TP), given by the two-stage model with H time t− periods evaluated on the scenario tree ξ¯ just defined, is as follows:     2−   H −1−  + · · · + cH x H ξ¯ TP := min c1 x 1 + E ¯ H −1− min c2 x 2 ξ 1 + c3 x 3 ξ¯ x1

ξ

s.t. Ax 1 = h1 ,

x 2 ,...,x H

(14)

J Optim Theory Appl

        T 1 ξ 1 x 1 + W 2 ξ 1 x 2 ξ 1 = h2 ξ 1 , .. .

 H −1−  H −1  H −1−   H −1−  H  H −1−   H −1−  T H −1 ξ¯ x ξ¯ + W H ξ¯ x ξ¯ = hH ξ¯ ,   t−1− ≥ 0, t = 2, . . . , H. x 1 ≥ 0; x t ξ¯ 3 Bounds in Multistage Linear Programs In this section we propose bounds for multistage stochastic linear programs. They come from three different measure types: • Measures of information, where the same problem is solved and compared with and without a piece of available information on the future; • Measures of the quality of the deterministic solution; • Rolling horizon measures which update the estimation and add more information at each stage. 3.1 Measures of Information in Multistage Problems First, we intend to generalize some measures introduced in [20] for the deterministic solution of the modified wait-and-see approach and those surveyed in [15] for the stochastic two-stage (T = 2) case. We consider a simplified version of the stochastic program, where only the right-hand side is stochastic (h = h(ξ )). Instead of using a scenario given by the expected values of the random parameters, one may choose a specific realization ξr (the scenario r = 1, . . . , S) of the random variable ξ H −1 , called the reference scenario, and solve problem (1). Let the PAIRS subproblem of scenarios ξr and ξk (k = 1, . . . , S) be defined as follows:   min zP (x, ξr , ξk ) := min c1 x 1 + c2 πr x 2 (ξr ) + (1 − πr )c2 x 2 (ξk ) x 1 ,x 2

s.t.

Ax 1 = h1 , Tr x 1 + Wr x 2 (ξr ) = ξr ,

(15)

Tk x + Wk x (ξk ) = ξk , 1

x 1 ≥ 0;

2

x 2 (ξr ) ≥ 0;

x 2 (ξk ) ≥ 0.

The Sum of Pairs Expected Values, denoted by SPEV [2, 15], is then defined as follows: 1 SPEV := 1 − πr

S 

πk min zP (x, ξr , ξk ).

(16)

k=1,k =r

Since the pairs subproblems (15) are much less complex than the general recourse problem (1), it is worthwhile to generalize the definition of SPEV to multistage

J Optim Theory Appl

Fig. 1 (a) Pair subtree (ξa , ξk ) = (ξ1 , ξ2 ) of the (b) four-stage scenario tree including 75 equiprobable scenarios

stochastic programs in order to evaluate the convenience of the additional computations required by the stochastic programming models vs. a simplified approach. For that purpose, we fix an auxiliary scenario a with the following characteristics: 1. ξa = ξr , i.e. the values of the random parameters of the auxiliary scenario are the same as the values of the random parameters of the reference scenario; 2. Let be Hˆ the first stage where scenario r and the generic scenario k, (k = 1, . . . , S) start to have different values, i.e. the branching stage in the scenario tree. Define:  πjt ,jt+1 :=

πr 1

if t = Hˆ , if t = Hˆ ;

3. πa = πr , since πa = 1 · 1 · 1 · · · πr · 1 · · · 1. Note that the auxiliary scenario a has the same parameters values as scenario r but different probabilities. This strategy allows to easily extend the SPEV measure to the multistage case as shown below. Figure 1(a) depicts an example of probabilities computation on the pair subproblem made by scenarios a = 1 and k = 2 of the fourstage scenario tree depicted in Fig. 1(b). Then, let us define the pair subproblem as follows: min zP (x, ξa , ξk )  := min c1 x 1 +

ˆ −1 H 

x

s.t.

 H    ct xat (ξa ) + πa ct xat (ξa ) + (1 − πa )ct xkt (ξk )

t=2

t=Hˆ

Ax 1 = h1 , Tat−1 xat−1 + Wat xat = hta ,

(17)

Tkt−1 xkt−1 + Wkt xkt = htk , x 1 ≥ 0;

xat ≥ 0,

xkt ≥ 0,

t = 2, . . . , H.

Eventually, the reference scenario need not to correspond to any of the given scenarios.

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Let us define the Multistage Sum of Pairs Expected Values denoted by MSPEV as follows: nH 1  πk min zP (x, ξa , ξk ). 1 − πa

MSPEV :=

(18)

k=1 k =r

Proposition 3.1 If the scenario ξr is not in Ξ , then MSPEV = WS. Proof If scenario ξr is not in Ξ , then πr = 0, consequently, πa = 0 and the pair subproblem zP (x, ξa , ξk ) reduces to z(x, ξk ). Hence, MSPEV =

nH 

 πk min

c1 xk1

k=1 k =r

=

nH 

H    πa ct xat (ξa ) + (1 − πa )ct xkt (ξk ) +



t=2

πk min z(x, ξk ) = WS.

(19) 

k=1 k =r

For illustrative purposes, throughout the rest of the paper we will assume without loss of generality that ξr ∈ Ξ . Proposition 3.2 WS ≤ MSPEV. ) (2,H ) , xˆ k ) be an optimal solution to the pair subproblem Proof Let xˆ a,k := (xˆk1 , xˆ (2,H a (2,H ) := (xs2 , xs3 , . . . , xsH ), s = 1, . . . , nH , so by defiof scenarios ξa and ξk , where xs nition (18) we have:

nH MSPEV =

k=1 πk min z k =r

P (x, ξ

a , ξk )

1 − πa

 ˆ −1 nH H  1  1 1 = πk c xˆk + ct xˆat (ξa ) 1 − πa k=1 k =r

t=2

H    + πa ct xˆat (ξa ) + (1 − πa )ct xˆkt (ξk )



t=Hˆ

   ˆ −1 nH H H   1  = πk πa c1 xˆk1 + ct xˆat (ξa ) + ct xˆat (ξa ) 1 − πa k=1 k =r

t=2

t=Hˆ

J Optim Theory Appl

+

nH 

 πk c1xˆk1

+

k=1 k =r

ˆ −1 H 

ct xˆat (ξa ) +



H 

ct xˆkt (ξk )

t=Hˆ

t=2

  ˆ −1 nH H H    1  ∗ 1 1 t t t t ≥ πk πa za + πk c xˆk + c xˆa (ξa ) + c xˆk (ξk ) 1 − πa k=1 k =r

k=1 k =r

t=2

t=Hˆ

nH 1  = πk πa za∗ 1 − πa k=1 k =r

+

nH 

 πk c1 xˆk1

+

k=1 k =r

+

H 

ct xˆat (ξa ) +

t=2

ct xˆkt (ξk ) −

t=Hˆ

≥ WS +

ˆ −1 H 

ˆ −1 H 

ˆ −1 H 

ct xˆkt (ξk )

t=2

 ct xˆkt (ξk )

t=2

nH 

 Hˆ −1    t t t πk c xˆa (ξa ) − xˆk (ξk ) .

k=1 k =r

(20)

t=2

Note that the last sum in (20) is nH 

 Hˆ −1    t t t πk c xˆa (ξa ) − xˆk (ξk ) = 0,

k=1 k =r

(21)

t=2

since 1. If scenarios k and a branch at stage Hˆ = 2, then (21) does not appear; 2. If 2 < Hˆ ≤ H , (21) reduces to zero since k and a are defined on the same nodes until stage Hˆ − 1; consequently, the optimal solutions verify xˆat (ξa ) = xˆkt (ξk ). Then the thesis MSPEV ≥ WS is proved. Proposition 3.3 RP ≥ MSPEV. ∗

(2,H )∗

Proof Let (x 1 , xk ), be an optimal solution to the recourse problem RP. ∗ (2,H )∗ (2,H )∗ 1 , xk ) is feasible for the pairs subproblem of ξa , ξk , and: Then (x , xa c xˆ + 1 1

ˆ −1 H  t=2

ct xˆat (ξa ) +

H    πa ct xˆat (ξa ) + (1 − πa )ct xˆkt (ξk ) t=Hˆ



J Optim Theory Appl 1 1∗

≤c x

+

ˆ −1 H 

H    ∗ ∗ πa ct xat (ξa ) + (1 − πa )ct xkt (ξk ) .

∗

ct xat (ξa ) +

t=Hˆ

t=2

Multiplication by πk and summation provide nH 

 πk c1 xˆk1 +

 H    πa ct xˆat (ξa ) + (1 − πa )ct xˆkt (ξk ) ct xˆat (ξa ) +

ˆ −1 H 

k=1 k =r

t=Hˆ

t=2

= (1 − πa )MSPEV   ˆ −1 nH H H      1 1∗ t t∗ t t∗ t t∗ ≤ πa c xa (ξa ) + (1 − πa )c xk (ξk ) πk c x + c xa (ξa ) + k=1 k =r



1 1∗

= (1 − πa ) c x

+

nH 

t=Hˆ

t=2

 πk

+

ˆ −1 H 

 ∗ ct xat (ξa )

t=2 H    ∗ ∗ πa ct xat (ξa ) + (1 − πa )ct xkt (ξk )

t=Hˆ

k=1 k =r



∗

= (1 − πa ) c1 x 1 +

ˆ −1 nH H  

+

∗ πa ct xat (ξa ) +

nH 

t=Hˆ

k=1 k =r

∗

πk ct xat (ξa )

t=2 k=1 k =r H 



πk



H 

 ∗ ct xkt (ξk )

t=Hˆ

= (1 − πa )RP since nH  k=1 k =r

πk

ˆ −1 H 

∗ ct xat (ξa ) =

t=2

Then, the thesis RP ≥ MSPEV is proved.

nH  k=1 k =r

πk

ˆ −1 H 

∗

ct xkt (ξk ).

(22)

t=2



One of the advantages of working with the pair formulation is that it allows to bound the optimal expected objective function by solving sets of pairs subproblems that are less complex than the original one. Despite the WS-like approach, the pair subproblem formulation includes also the stochasticity. Let us proceed to the extension of the classical two-stage upper bound EVRS on RP (see [15]) to the multistage case.

J Optim Theory Appl

We define as Multistage Expected Value of the Reference Scenario,  −1 H H −1  , ,x ,ξ MEVRS := Eξ H −1 min z xˇ H r xH

(23)

−1 := (x¯ 1 , x¯ 2 , . . . , x¯ H −1 ) is the optimal solution until stage H − 1 of the dewhere xˇ H r r r r terministic problem minx z(x, ξr ) under scenario r. The Multistage Value of Stochastic Solution MVSS is defined as follows:

MVSS := MEVRS − RP.

(24)

Note that MVSS ≥ 0 since there are only two alternative situations: −1 is a feasible solution to the recourse problem; 1. xˇ H r H 2. xˇ r −1 is infeasible and in this case MEVRS = +∞.

The definition of MEVRS (23) can be generalized in a sequence of Multistage Expected Value of the Reference Scenario, MEVRS1 , MEVRS2 , . . . , MEVRSt , where MEVRSt can be expressed as follows:   (25) MEVRSt := Eξ H −1 min z xˇ tr , x(t+1,H ) , ξ H −1 , t = 1, . . . , H − 1, x(t+1,H )

where xˇ tr is the optimal solution until stage t of the deterministic problem minx z(x, ξr ) under scenario r and x(t+1,H ) := (x (t+1) , x (t+2) , . . . , x H ) is At -measurable. According to this definition, MEVRS = MEVRSH −1 and MVSSt := MEVRSt − RP,

t = 1, . . . H − 1.

(26)

t = 1, . . . , H − 2.

(27)

Proposition 3.4 MEVRSt+1 ≥ MEVRSt ,

Proof Any feasible solution of the MEVRSt+1 problem is also a solution of MEVRSt (25), since the feasible region of the former is obviously more restricted than the solution of the latter; therefore the relation (27) holds true. If MEVRSt = +∞, the inequality is automatically satisfied.  (2,H ) (2,H ) As before, let xˆ a,k = (xˆk1 , xˆ a , xˆ k ) be the optimal solutions to the pair subproblems of ξa and ξk , k = 1, . . . , nH , k = r. The Multistage Expectation of Pairs Expected Value is defined as follows:    Eξ H −1 min z xˆk1 , x(2,H ) , ξ H −1 . MEPEV := min (28) k=1,...,nH ∪{r}

x(2,H )

MEPEV represents the minimum optimal value among those obtained by solving the original stochastic program (1), using the optimal first-stage solution of each pair subproblem. Proposition 3.5 RP ≤ MEPEV ≤ MEVRS1 .

(29)

J Optim Theory Appl

Proof Let us denote by K := {x|x t ∈ K t , t = 1, . . . , H − 1} the feasibility set of RP, by K ∩{xˆk1 , k = 1, . . . , nH ∪{r}} the feasibility set of MEPEV and by K ∩ xˇr1 = xˆr1 the one of MEVRS1 . These feasibility sets are obviously smaller and smaller, the thesis is therefore proved.  As a consequence of Proposition (3.4) it follows that RP ≤ MEPEV ≤ MEVRS1 ≤ · · · ≤ MEVRSH −1 .

(30)

Putting together the previous relations results in the following theorem: Theorem 3.1 0 ≤ MEVRSt − MEPEV ≤ MVSSt

(31)

≤ MEVRS − MSPEV ≤ MEVRS − WS, t

t

t = 1, . . . , H − 1.

In conclusion, the new measures MSPEV and MEPEV as described in Propositions 3.3 and 3.5, give respectively lower and upper bounds on the value of RP using the PAIR subproblem approach. 3.2 Measures of the Quality of Deterministic Solution in Multistage Problems Measures of the structure and upgradeability of the deterministic solution for the two-stage case, such as the Loss Using the Skeleton Solution, LUSS, and the Loss of Upgrading the Deterministic Solution, LUDS, has been introduced in [14], in relation to the standard VSS for the two-stage case. We point out that the scope of this section is to give a mathematical formulation to some heuristic approaches which are of interest in solving practical real problems of large dimension. The aim of the measures is to find out, even when VSS is large, if the deterministic solution carries useful information for the stochastic case. The contribution of this section consists in extending the definition given in [14] to the multistage-case by introducing the Multistage Loss Using the Skeleton Solution until stage t, say MLUSSt , and the Multistage Loss of Upgrading the Deterministic Solution until stage t, say MLUDSt , in relation to VSSt , t = 1, . . . , H , defined by (10). The definition of MLUSSt for t = 1, . . . , H − 1 is based on the following procedure: 1. Fixing at zero (or at the lower bound) all the variables which are at zero (or at the lower bound) in the expected value model until stage t; 2. Solving the related stochastic program. Let J t , t = 1, . . . , H − 1, be the set of indices for which the components of the expected value solution x¯ t , up to stage t, are at zero or at their lower bound. Then let

J Optim Theory Appl

x˜ be the solution of the model   min Eξ H −1 z x, ξ H −1 x

 v s.t. x vj = x¯ vj ξ¯ ,

j ∈ J t , v = 1, . . . , t.

(32)

We then compute the Multistage Expected Skeleton Solution Value at stage t as   MESSV t := Eξ H −1 min z x˜ t , x(t+1,H −1) , ξ H −1 , t = 1, . . . , H − 1, (33) x(t+1,H −1)

with x(t+1,H −1) At -measurable. Then the Multistage Loss Using Skeleton Solution until stage t (MLUSSt ) can be expressed as MLUSSt := MESSV t − RP,

t = 1, . . . , H − 1.

(34)

Note that MLUSSt ≥ 0, t = 1, . . . , H − 1, since x˜ t is either feasible solution to the recourse problem RP or infeasible such that MLUSSt = +∞. Having an MLUSSt close to zero suggests that the variables chosen by the expected value model until stage t are the correct ones. Consider the following proposition: Proposition 3.6 MLUSSt+1 ≥ MLUSSt ,

t = 1, . . . , H − 2.

(35)

Proof Any feasible solution of MESSV t+1 problem is also a solution of MESSV t , since the former is more restricted than the latter, and then the relation (35) holds true. If MLUSSt = +∞, the inequality is automatically satisfied.  Then, it results in RP ≤ MESSV t ≤ EEV t ,

(36)

VSSt ≥ MLUSSt ≥ 0.

(37)

and, consequently,

For multistage stochastic linear programs with deterministic constraint matrices and deterministic objective coefficients, the following inequalities are satisfied (see [16]): EEV t − EV ≥ VSSt .

(38)

It can be noted that MLUSSt = 0 (i.e. MESSV t = RP) corresponds to the perfect skeleton solution until stage t in which the condition  v x vj = x¯ vj ξ¯ , j ∈ J t , v = 1, . . . , t is satisfied by the stochastic solution even without being enforced by the set of constraints.

J Optim Theory Appl t MLUDSt , t = 1, . . . , H − 1, measures if the solution x¯ t (ξ¯ ) of the expected value model until stage t can be considered as a starting point (if not optimal) in the stochast tic setting. This is equivalent to adding to problem (1) the constraint xt ≥ x¯ t (ξ¯ ) and, hence, solving the problem   min Eξ H −1 z x, ξ H −1 x

 t s.t. xt ≥ x¯ t ξ¯ ,

(39)

for obtaining the optimal solution x¯¯ . We then compute the Multistage Expected Input Value until stage t as   (40) MEIV t := Eξ H −1 min z x¯¯ t , x(t+1,H −1) , ξ H −1 , x(t+1,H −1)

with x(t+1,H −1) At -measurable. From these measures, we define the Multistage Loss of Upgrading the Deterministic Solution until stage t (MLUDSt ) as follows: MLUDSt := MEIV t − RP.

(41)

As in the case of MLUSSt , the following proposition holds: Proposition 3.7 MLUDSt+1 ≥ MLUDSt ,

t = 1, . . . , H − 2,

EEV t ≥ MEIV t ≥ RP,

t = 1, . . . , H − 1,

EEV − EV ≥ VSS ≥ MLUDS ≥ 0, t

t

t

t = 1, . . . , H − 1.

Proof See the proof of Proposition 3.6.

(42) (43) (44) 

It can be noted that having an MLUDSt = 0 (i.e. MEIV t = RP) corresponds to t the case where the conditions xt ≥ x¯ t (ξ¯ ) are satisfied by the stochastic solution, obviously without being enforced (under the assumption that the stochastic first-stage decision is unique). 3.3 Rolling Horizon Measures in Multistage Problems Multistage problems such as MEVRSt , MESSV t and MEIV t are often infeasible since they require to fix many variables to their value obtained via the expected value or the reference-scenario models. An alternative approach to consider is the rolling time horizon procedure (RHP) presented in [3, 17] in order to update the estimations of the solution at each stage taking into account the arrival of new information. Moreover, the advantage of RHP in the multistage case with respect to the solution of the RP problem consists in reducing the computational complexity since the number of scenarios is reduced by considering the stochasticity only one stage at a time. We propose the following methodology

J Optim Theory Appl

for the evaluation of the reference-scenario model; see [18, 21] for an extension of the expected value model. 1. Solve the reference-scenario r model and store the first-stage decision variables xˇr1 ; 2. Define a new scenario tree T 2,ev where all random parameters in stages 2+ 2, . . . , H − 1 are replaced by their expected values ξ¯ = (ξ 1 , ξ¯ 2 , . . . , ξ¯ H −1 ) and 1 1 2 . solve the related model with x = xˇr . Store all the second-stage variables xˇr,ev 3. At stage t (t = 2, . . . , H − 1) define a new scenario tree T t+1,ev where all random parameters in stages t + 1, . . . , H − 1 are replaced with their expected value t+1+ ξ¯ = (ξ 1 , ξ 2 , . . . , ξ¯ t+1 , . . . , ξ¯ H −1 ), t = 2, . . . , H − 1, and solve the associated model with     2 t = xˇ tr,ev . , . . . , xˇr,ev xt = x 1 , x 2 , . . . , x t = xˇr1 , xˇr,ev t+1 . Store all the t + 1 stage variables xˇr,ev 4. Finally, solve the model on the initial scenario tree T with all the tth variables −1 (t = 1, 2, . . . , H − 1) fixed to the stored values xH −1 = xˇ H r,ev .

So, the Rolling Horizon Value of the Reference Scenario can be expressed as follows:  −1 H H −1  , (45) RHVRS := Eξ H −1 min z xˇ H r,ev , x , ξ xH

and the Rolling Horizon Value of Stochastic Solution with respect to reference scenario is expressed as RHVSS := RHVRS − RP.

(46)

In a similar way, the Rolling Horizon Expected Skeleton Solution Value RHESSV can be obtained as follows: 1. Solve the expected value model and store the first-stage decision variables x¯ 1j , j ∈ J 1 , which are at zero or at their lower bound; 2. Define a new scenario tree T 2,ev where all random elements of stages 2, . . . , H −1 are replaced with their expected values and solve the related model with x 1j = x¯ 1j , j ∈ J 1 . Store all the second-stage variables which are at zero or at their lower bound x¯ 2j , j ∈ J 2 . 3. At stage t (t = 2, . . . , H − 1) define a new scenario tree T t+1,ev where all random parameters in stages t + 1, . . . , H − 1 are replaced by their expected value and solve the related associated model where the decision variables of the expected value model are fixed at zero or at their lower bound. Store all the t + 1 stage variables which are at zero or at their lower bound x¯ t+1j , j ∈ J t+1 . 4. Finally, solve the original model (with the initial scenario tree T ) where all the j -components, j ∈ J t , up to stage t (t = 1, 2, . . . , H − 1) are fixed to zero or to their lower bound. So the Rolling Horizon Expected Skeleton Solution Value can be expressed as   RHESSV = Eξ H −1 min z x¯ H −1j , x H , ξ H −1 , j ∈ J t , t = 1, . . . , H − 1, xH

(47)

J Optim Theory Appl

and the Rolling Horizon Loss Using Skeleton Solution is expressed as RHLUSS := RHESSV − RP.

(48)

Starting from the definition of MEIV t , we can analogously define the Rolling Horizon Expected Input Value RHEIV. We can then define the Rolling Horizon Loss of Upgrading the Deterministic Solution as follows: RHLUDS := RHEIV − RP.

(49)

4 Case Study: a Multistage Stochastic Optimization Model for a Single-Sink Transportation Problem We consider a real case of clinker replenishment in Sicily, provided by the primary Italian cement producer. The problem has been already analyzed in detail in [18]. The logistics system is organized as follows: in Catania there is a warehouse to be replenished by clinker produced by four plants located in Palermo (PA), Agrigento (AG), Cosenza (CS) and Vibo Valentia (VV). The monthly demand of the single customer in Catania as well as the production capacities of the four plants are stochastic. All the vehicles must be booked in advance from an external transportation company, before the demand and production capacities are revealed. We assume that the transportation company has an unlimited fleet and that only full load shipments are allowed. When the demand and the production capacity are revealed, there is an option to cancel some of the reservations against a cancellation fee. If the quantity delivered from the four suppliers is not sufficient to satisfy the demand, the residual quantity is purchased from an external company at a higher price b. The problem consists in determining the number of vehicles for each supplier to book at the beginning of the month, in order to minimize the total costs, given by the sum of the transportation costs (including the cancellation fee for vehicles booked but not used) and the costs of the product purchased from the external company. Let assume the following notation. Sets: I = {i : i = 1, . . . , I }, J t = {j : j = 1, . . . , nt },

set of suppliers (AG, CS, PA, VV); set of ordered nodes of the tree at stage t = 1, . . . , H,

where nt is the number of nodes at stage (month) t. Deterministic parameters: ti , unit transportation costs of supplier i ∈ I; b, buying cost from an external source; q, vehicle capacity; g, unloading capacity at the customer; l0 , initial inventory level at the customer;

J Optim Theory Appl

lmax , storage capacity at the customer; α, cancellation fee; J = {0}, root of the tree; 1

a(j ), ancestor of the node j ∈ J t , t = 2, . . . , H in the scenario tree. Stochastic related parameters: pj , probability of node j ∈ J t , t = 1, . . . , H ; vi,j , production capacity of supplier i ∈ I in node j ∈ J t , t = 2, . . . , H ; dj , customer demand at node j ∈ J t , t = 2, . . . , H. Note that b is fixed on the basis of the known production and transportation costs from each supplier. In our case we suppose b > maxi (ti + ci ) where ci is the unit production cost of supplier i ∈ I. Variables for each node (i.e. scenario group) in the scenario tree: xi,j ∈ N, number of vehicles booked from supplier i ∈ I, for j ∈ J t , t = 1, . . . , H − 1; zi,j ∈ N, number of vehicles actually used from supplier i ∈ I, for j ∈ J t , t = 2, . . . , H ; yj ∈ R, volume of product to purchase from an external source for j ∈ J t , t = 2, . . . , H ; lj ∈ R, inventory level of the customer at node j : lj = la(j ) + q

I 

zi,j + yj − dj ,

j ∈ J t , t = 2, . . . , H.

(50)

i=1

The multistage risk neutral based model is formulated as follows: min

nt H −1  

 pj q

t=1 j =1

I 

 ti xi,j +

nt H   t=2 j =1

i=1

 pj byj − (1 − α)q

I 

 ti (xi,a(j ) − zi,j )

i=1

(51) subject to q

I 

xi,j ≤ g,

j ∈ J t , t = 1, . . . , H − 1

(52)

j ∈ J t , t = 2, . . . , H

(53)

i=1

la(j ) + q

I  i=1

zi,j + yj − dj ≥ 0,

J Optim Theory Appl

la(j ) + q

I 

zi,j + yj − dj ≤ lmax ,

j ∈ J t , t = 2, . . . , H

(54)

i=1

zi,j ≤ xi,a(j ) ,

i ∈ I, j ∈ J t , t = 2, . . . , H

(55)

qzi,j ≤ vi,j ,

i ∈ I, j ∈ J , t = 2, . . . , H

(56)

xi,j ∈ N,

i ∈ I, j ∈ J t , t = 1, . . . , H − 1

(57)

j ∈ J , t = 2, . . . , H

(58)

i ∈ I, j ∈ J t , t = 2, . . . , H.

(59)

yj ≥ 0, zi,j ∈ N,

t

t

The first sum in the objective function (51) denotes the expected booking costs of the vehicles, while the second sum represents the expected recourse actions, consisting of buying clinker from external sources (yj ) and canceling unwanted vehicles. Constraint (52) guarantees that the total quantity delivered from the suppliers to the customer is not greater than the customer’s unloading capacity g, inducing thus an upper bound on the total number of vehicles. Constraints (53) and (54) ensure that the storage levels are between zero and lmax . Constraint (55) guarantees that the number of vehicles serving supplier i is at most equal to the number booked in advance and (56) controls that the quantity of clinker delivered from supplier i does not exceed its production capacity vi,j . Finally, (57)–(59) define the decision variables of the problem. Note that the stochasticity appears only in the random right-hand side coefficients dj and vi,j . 4.1 Computation of Measures for “A Single Sink Transportation Problem” This section presents our computational tests about the performance measures presented in Sects. 2 and 3 on the single-sink transportation problem (51)–(59). We analyze the multistage stochastic one by means of a four-stage scenario tree defined by the user, where a stage represents a month and the root the month of January. We use Ampl environment along the callable library CPLEX 11.2.0 to solve the linear MIP problem derived from our case study. All the computations have been done on a 64-bit machine with 12 GB of RAM and a 2.90 GHz processor. We refer to [18] for the data used in the simulation. Our case-study is characterized by mixed-integer variables in all the stages. Summary statistics on the size of the optimization problems that Ampl generates are reported in Table 1 for the four-stage case. 4.1.1 Multistage Case In this section we compute the performance measures presented in Sects. 2 and 3 on the single sink transportation problem in a four-stage formulation. For this purpose, we consider the scenario tree shown in Fig. 3: this is a four-stage tree with 5 branches from the root, 5 from each of the second-stage nodes, and three from each of the third-stage nodes, resulting in S = 5 × 5 × 3 = 75 scenarios and 106 nodes. We use it as a benchmark to evaluate the cost of optimal solutions obtained using the other reduced scenario trees (see Fig. 2). The results are presented in Table 3 and

J Optim Theory Appl Table 1 Number of discrete and continuous variables used in computing the bounds in the four-stage case of the “single-sink transportation problem”

EV (mean scenario = ξ¯ ) RP TP EEV 1 (w.r.t. ξ¯ ) EEV 2 (w.r.t. ξ¯ ) fixing xi,j EEV 3 (w.r.t. ξ¯ ) fixing xi,j MEVRS1 (w.r.t. ξ44 ) MEVRS2 (w.r.t. ξ44 ) fixing xi,j MEVRS3 (w.r.t. ξ44 ) fixing xi,j MESSV 1 MESSV 2 fixing xi,j MESSV 3 fixing xi,j MEIV 1 MEIV 2 lower bound on xi,j MEIV 3 lower bound on xi,j WS (each subproblem) MEPEV (each pair subproblem) MSPEV (each pair subproblem)

Integer variables

Continuous variables

24 544 104 540 520 420 540 520 420 542 537 487 544 544 544 24 44 44

7 211 31 211 211 211 211 211 211 211 211 211 211 211 211 7 13 13

Fig. 2 Reduced scenario trees used respectively for the mean-scenario model (EV), the two-stage relaxation (TP) and the computations of the rolling horizon values reported in Tables 2 and 3 for the single-sink transportation problem

Fig. 3 Four-stage scenario tree considered for the single-sink transportation problem

the relevant performance measures are shown in Table 4. We can observe in Table 3 that a better description of the stochasticity leads to larger bookings in the first stage. Actually, in the four-stage scenario tree, the total number of booked vehicles is 1260, which equals the customer’s total unloading capacity. This is due to the low initial inventory level l0 = 2 000 at the customer (arising from real data).

J Optim Theory Appl Table 2 Optimal solution value for the deterministic (mean scenario) four-stage case of the “single-sink transportation problem” Node

AG

CS

PA

VV

yj

Costs (e)

0

158

0

647

0

–

327 717

1

264

0

416

143

0

345 621

2

315

0

518

0

0

326 844

3

–

–

–

–

0

0

The total costs from the two deterministic models, mean scenario (EV) (see Table 2) and worst scenario ξ44 , the recourse problem (RP) and the two-stage relaxation (TP) (see Table 3), are not directly comparable because they refer to different scenario trees. The optimal solutions are then compared on the scenario tree depicted in Fig. 3 that is used as a benchmark. First of all we evaluate the optimal solutions of the average scenario model by fixing just the number xi,j of vehicles booked from supplier i ∈ I in nodes j ∈ J t until stage t (t = 1, 2, 3) in the stochastic framework by means of EEV t : EEV 1 = 1364343.58 < EEV 2 = 1431896.24 < EEV 3 = 1579825.92. The associated chain VSS1 = 91 267.32 < VSS2 = 185 002.22 < VSS3 = 299 449.62 shows the losses obtained by booking the number of vehicles suggested by the deterministic solution (see Table 2). The low first-stage deterministic booking is compensated at each of the second-stage nodes of the stochastic framework, by a reservation almost equal to the customer’s unloading capacity and by buying extra clinker at a higher price. The evaluation of the deterministic solution on a rolling-horizon basis (see Fig. 2) allows to update the estimations and add more information step by step as measured by RHEEV = 1 540 248.22 < EEV 3 = 1 579 825.92. Note that by fixing all the decision variables xi,j , yj and zi,j from the averagescenario model until the second stage, we get EEV 2 = ∞ and consequently EEV 3 = ∞ (EEV 2 ≤ EEV 3 ) and, thus, revealing the inappropriateness. We will investigate later by means of MLUSSt and MLUDSt the reason of this infeasibility. The same considerations can be applied by evaluating the worst-scenario model ξ44 in the stochastic framework by means of MEVRSt . In particular by fixing just the number xi,j of vehicles booked from supplier i ∈ I in nodes j ∈ J t until stage t, we get MEVRS1 = 1 372 285.48 < MEVRS2 = 1 433 501.32 < MEVRS3 = 1 436 997.78.

J Optim Theory Appl

The evaluation of the worst-case solution on a rolling-horizon basis is measured by RHVRS = 1 535 476.52. MESSV t allows the evaluation of the structure of the deterministic solution until stage t. Following the structure of the deterministic solution, the booking of vehicles from CS is not allowed in any of the three stages, nor from VV at the root and at stage t = 3. We can check it in Table 2 where it shows the number of booked vehicles (equal to the optimal number of used vehicles at node j + 1) for each supplier, the clinker purchased yj at each stage j = 1, 2, 3 and partial optimal costs. So, it results in MESSV 1 = 1 299 327.68 ≤ MESSV 2 = 1 301 017.28 ≤ MESSV 3 = 1 404 215.16, with a consequent chain of measures MLUSS1 = 26 282.62 ≤ MLUSS2 = 27 863.9 ≤ MLUSS3 = 131 140.86. Note that they measure the loss by booking vehicles coming only from the suppliers suggested by the deterministic model. We can conclude that the deterministic solution xi,j is bad, since it books the wrong number of vehicles from the wrong suppliers already from the first stage. The evaluation of the deterministic skeleton solution on a rolling-horizon basis allows to update the estimations in the following way: RHESSV = 1 302 705.36 ≤ MESSV 3 = 1 404 215.16. As before, by fixing at zero the clinker purchased yj and the vehicles actually used zi,j , it results an infeasibility already at the second stage, MESSV 2 = MESSV 3 = ∞ and then revealing the unsuitability of the structure of the full deterministic solution. We then consider the vehicles booked in the average-scenario model as an input in the stochastic setting and we check if the solution can be upgraded. It is worth noting that, in the root node, the booked number of vehicles for all four suppliers is higher in the stochastic solution than in the deterministic one. We have MLUDS1 = 0. We can observe that the condition is no longer satisfied at stage 2 for suppliers PA and VV with MLUDS2 = 2707.4 and at stage 3 for suppliers AG and PA with MLUDS3 = 27552.22. Note that the chain (43) in Proposition 3.7, MLUDS1 = 0 ≤ MLUDS2 = 2707.4 ≤ MLUDS3 = 27552.22, holds true. We can conclude that the deterministic solution can be taken as input in the stochastic model only in the first stage. An alternative approach consists in solving pairs subproblems of the initial stochastic program with respect to the worst scenario ξ44 . The best pair subproblem is given by the couple (ξ44 , ξ2 ) with MEPEV = 1 313 983.3

J Optim Theory Appl

which satisfies the chain of inequalities (30): RP = 1 273 074.3 < MEPEV = 1 313 983.3 < MEVRS1 = 1 372 285.48 < MEVRS2 = 1 433 501.32 < MEVRS3 = 1 436 997.78. This means that the optimal first-stage solution of the pair subproblem (ξ44 , ξ2 ) performs better than the deterministic one (mean or worst scenario), and it can be chosen instead of the direct deterministic solution, in case of large scale problems. When the auxiliary scenario does not belong to the scenario tree, then MSPEV = WS = 1037820 (see Proposition 3.1). If the auxiliary scenario (the worst one) belongs to the scenario tree, then WS = 1037820 < 1041627.34 = MSPEV (see Proposition 3.2). Note that the sum (21) is zero: if k = 1, . . . , 30 or k = 46, . . . , 75, scenarios k and the auxiliary a, branch at stage Hˆ = 2 and (21) is not defined; if k = 31 . . . , 43, scenarios k and a branch at 2 (ξ ) = xˆ 2 (ξ ). The same Hˆ = 3 and at stage 2 are both defined on node 3 with xˆ44 44 k k argument can be applied for scenarios 43 and 46 which branch with scenario 44 at Hˆ = 4 and are defined on the same node 20 at stage 3. If we choose as auxiliary scenario the best one, i.e. ξ1 (it gives the minimum cost over all the scenarios in the tree), then WS = 1037820 < MSPEV = 1039068.91. Finally, Proposition 3.3 is satisfied being RP = 1273074.3 > MSPEV = 1039068.91, and Theorem 3.1 is verified. Figure 4 shows the percentage deviation of the new approaches (denoted by ×) from the optimal RP cost. Most of them are better than those known in the literature (gray circles) [15, 16]. For instance, MSPEV is a better lower bound to RP than the classical WS. All the measures between MEIV 1 to MEPEV are better upper bounds to RP than EEV 1 . We observe that the best interval from the optimal stochastic value RP is represented by the range [TP, MEIV 2 ] (≈ 13.64 %). The worst one is represented by the range [EV, EEV 3 ] (≈ 45.53 %) which measures the bad performance of the deterministic model in the multistage stochastic environment, due to the low number of booked vehicles in all stages, starting from the first. Tighter and tighter intervals, presenting a decreasing deviation from RP, include all our new approximation measures. Figure 5 shows the percentage deviation from the multistage stochastic recourse problem RP for increasing values of complexity of calculation measured in CPU seconds. This graphic allows to combine the performance of the new bounds both in terms of deviation from RP and computing time. The lower bound MSPEV which shows a minor deviation from RP with respect to WS is more expensive in terms of computing time (1.59 vs. 0.63 CPU seconds). The deterministic solution (EV),

J Optim Theory Appl

Fig. 4 Percentage deviation from the multistage stochastic recourse problem RP of different approaches as in Table 3, reported for increasing values. The black circle denotes RP, the squares the values of the new performance measures and the gray circles are the values of measures already known in the literature

Fig. 5 Percentage deviation from the multistage stochastic recourse problem RP for increasing values of complexity of calculation measured in CPU seconds

which is the less expensive in terms of computing time is the worst lower bound approximation of the RP value. The upper bound MEPEV is a good approximation to RP in terms of objective function value but requires the longest computing time.

J Optim Theory Appl

Fig. 6 Zoom of Fig. 5 in the range [EV, RHESSV] ([0, 0.1] CPU seconds). Percentage deviation from the multistage stochastic recourse problem RP for increasing values of complexity of calculation measured in CPU seconds

A zoom in the range ([0, 0.1] CPU seconds) of Fig. 5 is shown in Fig. 6 corresponding to the interval measures [EV, RHESSV]. MESSV 2 represents the best upper bound to RP solution with respect to both computing time and cost performance. The positive values of MLUSSt lead us to understand that the deterministic solution is structurally wrong, since it forces booking vehicles from the wrong suppliers. However, the deterministic solution value should be considered as a lower bound for the first-stage stochastic solution value in multistage stochastic setting. This might be useful information for the user that does not want to solve the general recourse problem. On the contrary, EEV 3 is the worst upper bound to RP solution but it requires a low computing time. The rolling-horizon approach should be also considered as a useful alternative to the standard methods allowing to update the estimation at each time period but it requires a large computing time. Table 3 shows the optimal number of booked vehicles for each supplier and total optimal costs. Table 4 shows the values of the performance measures for the fourstage case. 5 Conclusions The paper develops lower and upper bound for linear multistage programs by using different levels of information, of quality of solution and rolling horizon approaches. Our generalization is both of theoretical and practical importance. The theoretical importance comes from the extensions of bounds in multistage framework. The practical one arises when solving problems of large instances where it is fundamental

J Optim Theory Appl Table 3 Optimal solutions and comparison measures for the four-stage case of the “single-sink transportation problem” AG

CS

PA

VV

Objective value (e)

158

0

647

0

1 000 182

deterministic (worst scenario = ξ44 )

0

299

533

116

1 342 803

RP

389

0

755

116

1 273 074.30

TP

401

0

641

116

1 104 279.60

EEV 1 (w.r.t. ξ¯ ) EEV 2 (w.r.t. ξ¯ ) fixing xi,j

158

0

647

0

1 364 343.58

158

0

647

0

1 431 896.24

EEV 2 fixing all 1st and 2nd stage var. EEV 3 (w.r.t. ξ¯ ) fixing xi,j

158

0

647

0

∞

158

0

647

0

1 579 825.92

EEV 3 fixing all 1st, 2nd and 3rd stage var.

158

0

647

0

∞

MEVRS1 (w.r.t. ξ44 ) MEVRS2 (w.r.t. ξ44 ) fixing xi,j

0

299

533

116

1 372 285.48

0

299

533

116

1 433 501.32

MEVRS2 fixing xi,j and yj MEVRS2 fixing all 1st and 2nd stage var.

0

299

533

116

∞

EV (mean scenario = ξ¯ )

0

299

533

116

∞

MEVRS3 (w.r.t. ξ44 ) fixing xi,j MEVRS3 fixing xi,j and yj

0

299

533

116

1 436 997.78

0

299

533

116

∞

MEVRS3 fixing all 1st, 2nd and 3rd stage var.

0

299

533

116

∞

MESSV 1

389

0

871

0

1 299 327.68

MESSV 2 fixing xi,j

389

0

871

0

MESSV 2 MESSV 3 fixing xi,j

389

0

871

0

MESSV 3

1 301 017.28 ∞

MEIV 1

389

0

755

116

MEIV 2 lower bound on xi,j

401

0

743

116

MEIV 2 MEIV 3 lower bound on xi,j

1 301 017.28 ∞

1 273 074.30 1 278 053.08 ∞

401

0

743

116

1 299 784.80

MEIV 3

∞

WS

1 037 820

MEPEV (pair subproblem (ξ44 , ξ2 ))

303

0

516

116

1 313 983.30

MSPEV (w.r.t. ξ44 )

1 041 627.34

MSPEV (w.r.t. ξ1 ) MSPEV (w.r.t. ξ¯ )

1 039 068.91 1 037 820

RHEEV (w.r.t. ξ¯ fixing xi,j )

158

0

647

0

RHVRS (w.r.t. ξ44 fixing xi,j )

0

299

533

116

1 531 020.74

RHESSV (fixing xi,j )

401

0

859

0

1 302 705.36

RHEIV (lower bound on xi,j )

401

0

743

116

1 290 604.32

1 540 248.22

to have approximations of the optimal solution. The proposed methodologies have been implemented and tested on a real problem. Our main contribution is in solving a series of subproblems more computationally tractable than the initial one, under the assumption that some additional information on the future development of a random

J Optim Theory Appl Table 4 Performance measures for the four-stage case of the “single-sink transportation problem”

Performance measures

Value (e)

VSS1

91 267.32

VSS2 fixing xi,j VSS2

185 002.22

VSS3 fixing xi,j VSS3

299 449.62

MEVRS1 − RP = MVSS1

103 463.38

MEVRS2 − RP = MVSS2 fixing xi,j MEVRS2 − RP = MVSS2

164 122.76

∞ ∞

∞

MEVRS3 − RP = MVSS fixing xi,j

168 187.02

MEVRS3 − RP = MVSS

∞

MLUSS1

26 282.62

MLUSS2 fixing xi,j

27 863.90

MLUSS2

∞

MLUSS3 fixing xi,j

27 863.90

MLUSS3

∞

MLUDS1

0

MLUDS2 lower bound on xi,j

2 707.40

MLUDS2

∞

MLUDS3 lower bound on xi,j

27 552.22

MLUDS3

∞

RHVSS (w.r.t. ξ¯ )

267 173.92

RHVSS (w.r.t. ξ44 )

262 402.22

RHLUSS

29 631.06

RHLUDS

17 530.02

variable is available. This extension has been done by introducing the new concept of auxiliary scenario and by the redefinition of probability of pairs subproblem. The results show that for lower bound a better alternative than choosing the deterministic solution is given by the MSPEV approach which performs worse in time than WS. For upper bounds measures of quality of the expected value solution in terms of structure and upgradeability give good results with also the best pair subproblem solution as measured by MEPEV. The above measures are also defined in a rolling horizon framework by means of the Rolling Horizon Value of Stochastic Solution, RHVVS, as well as the Rolling Horizon Loss Using Skeleton Solution and Rolling Horizon Loss of Upgrading the Deterministic Solution, RHLUDS. In these rolling horizon approaches, the estimations of the solution are updated at each stage taking into account the availability of new information and the stochasticity is considered only at one stage at a time. Chains of inequalities among the new measures are proved and tested on a stochastic multistage single-sink transportation problem. Differences among the values in the chains indicate the distance, at stage t, of the proposed approach to the stochastic one and give insight of what is potentially problematic in the solution obtained by using the approximated method of choice.

J Optim Theory Appl

We conclude the paper with possible future research directions consisting in extending the analysis to the case of mixed-integer multistage linear programs and to the general non-linear case. Acknowledgements The work has been supported under grant by Regione Lombardia: “Metodi di integrazione delle fonti energetiche rinnovabili e monitoraggio satellitare dell’impatto ambientale”, EN-17, ID 17369.10 and by Bergamo and Brescia University grants 2010–2011. We thank the referees for their helpful comments that improved the quality of the paper.

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