1 Introduction
In many applications, such as hightech manufacturing, biopharmaceutical supply chains, and smart power grids with renewable energy, we often need to integrate the strategic, tactical and operational decisions. For example, in the semiconductor manufacturing, the managers need to consider the facility planning and the ensuing production scheduling. The planning decision is made “here and now", and the production scheduling is a “wait and see" decision, which depends on the investment decision and also the realization of demand. To guide the dynamic decisions making, in this paper, we consider the twostage stochastic programming,
(1) 
where denotes the firststage action, e.g., the investment decision, denotes the secondstage decision, e.g., production scheduling, denotes the random inputs, e.g., demands, and represent the feasible decision sets. The overall cost includes the investment cost and the expected production cost occurring in the planning horizon.
The existing twostage optimization approaches often assume that the response function is known [Beale_1955, Dantzig_1955, Shapiro_Philpott_2007, shapiro_2009]. For example, it can be a linear or mixedinteger function. However, for many complex real systems, the response function could be unknown. For example, in the semiconductor manufacturing, the production processes can involve thousands of steps and the production cost function is unknown [Liu_etal_2011]. We resort to simulation for the unknown response under different scenarios and decisions . Thus, in this paper, we consider the twostage optimization via simulation (OvS).
Compared to the classical twostage optimization with the known response function, such as twostage stochastic linear programming problems, there exist additional challenges to solve the twostage OvS listed as follows.

When the response function is known, various algorithms exploiting the structural information haven been developed in the optimization community to search for the optimal solution, including Benders decomposition algorithm and stochastic decomposition [Ekin_Polson_Soyer_2014, Frauendorfer_1988, Sen_Higle_1991, Higle_Sen_1994]. However, they cannot be employed and extended to the twostage OvS of interest.

It is computationally demanding to solve the twostage OvS. For complex stochastic systems, each simulation run could be computationally expensive. In addition, there could exist high prediction uncertainty and the number of scenarios used by the Sample Average Approximation (SAA) to approximate the expected future cost needs to be large [Ahmed_Shapiro_2002, shapiro_2009]. Hence, there exists tremendous computational burden.

It is challenging to search for the optimal firststage solution. Given limited computational resource, we often cannot find the true optimal secondstage decisions, which leads to the optimality gap. Thus, besides the finite sampling error introduced by SAA, this optimality gap can further lead to a biased estimate of the expected future cost.
Hence, in this paper, we introduce a new simulation optimization approach that allows us to efficiently solve the complex twostage OvS problems.
Notice that the problem of interest is different with the existing blackbox OvS problems studied in the simulation literature [Spall_1992, Jones_1998, Hong_Nelson_2006, Huang_etal_2006G, Sun_etal_2014]. Existing studies tend to focus on the stochastic onestage OvS,
(2) 
Given a feasible decision , the system unknown mean response, denoted by , can be assessed by simulation. Various simulation optimization algorithms are proposed to solve onestage OvS; see Henderson and Nelson [Henderson_Nelson_2006] for a review. In particular, metamodelassisted optimization approaches can efficiently employ the simulation resource for the search of optimal solution [Henderson_Nelson_2006, Fu_2014]. When there is no strong prior information on the mean response surface, the Gaussian process (GP) can be used to characterize the remaining metamodel estimation uncertainty. To balance exploration and exploitation, Sun et al. [Sun_etal_2014] proposed a GP based search (GPS) algorithm for discrete optimization problems, and it can efficiently use the simulation resource and guarantee global convergence.
Inspired by those onestage metamodel assisted approaches, in this paper, we propose a globallocal metamodelassisted twostage OvS. Specifically, at each visited firststage action , we construct a local GP metamodel for so that we can simultaneously solve a large set of secondstage optimization problems sharing the same firststage decision . Then, built on the search results from the secondstage optimization problems, we further develop a global metamodel accounting for various sources of errors. Assisted by the globallocal metamodel, we introduce a twostage optimization via simulation approach that can efficiently employ the limited simulation budget to iteratively search for the optimal first and secondstage decisions. Here, suppose each simulation run could be computationally expensive, say taking about a few days.
Therefore, the main contributions of this paper are as follows.

We propose a globallocal metamodel accounting for the finite sampling error introduced by using a finite number of scenarios to approximate the expected future cost and also the bias introduced by the optimality gap from the secondstage optimization.

Assisted by the globallocal metamodel, we develop a twostage optimization via simulation approach that can simultaneously control the impact from various sources of error and efficiently employ the simulation budget to search for the optimal first and secondstage decisions.

Our approach can guarantee global convergence as the simulation budget increases. The empirical study also demonstrates that it has good and stable finitesample performance.
The rest of the paper is organized as follows. Section 2 gives the literature review of relevant studies on twostage stochastic programming and metamodelassisted simulation optimization. We formally state the problem of interest in Section LABEL:sec:statement. We develop a globallocal metamodelassisted twostage OvS approach for complex stochastic systems in Section LABEL:sec:_metamodel. We study the finite sample performance of our approach in Section LABEL:sec:example, and conclude the paper in Section LABEL:sec:conclusion.
2 Background
To integrate strategic, tactical and operational decisions, the classical twostage stochastic optimization was introduced [Beale_1955, Dantzig_1955].
To study the finite sample performance of proposed framework and compare it with the random sampling SAA and the deterministic lookahead approach with GPS, we consider two examples, including a simple twostage linear optimization problem and a supply chain management example.
2.1 A TwoStage Linear Optimization Problem
We first consider a twostage linear stochastic optimization example from Ekin et al. [Ekin_Polson_Soyer_2014],
(3) 
where follows the lognormal distribution with mean
and variance
. It is easy to see that the firststage optimal solution is . Notice that the objective function in (3) is monotonic in , which means that the closer a solution is to the true optimal , the better quality it has. Thus, we can examine the algorithm’s performance by directly checking the value of optimal firststage decision . To study the performance of our approach, we pretend that the objective function is unknown, and it is estimated by simulation. We discretize the solution spaces of and with an increment . In our optimization procedure, we set the initial threshold for the secondstage relative optimality gap and set the initial number of scenarios .We study the performance of proposed globallocal metamodel assisted twostage OvS, random sampling SAA approach, and deterministic lookahead policy with GPS (DLHGPS) under different simulation budget . For the random sampling SAA approach, we consider two representative settings for and : and . For the GPS, we set when the budget and set when . Following the setting of Section (5.1) in [Sun_etal_2014], we set the spatial variance of the GP to be . Table 1
records mean and standard deviation (SD) of
obtained by using these approaches. The results are estimated based on 100 macroreplications. Given the same simulation budget, our method and deterministic lookahead with GPS provide much higher quality solutions in terms of means and standard deviations of . They deliver very close to with all three budget levels. By contrast, the optimal decision obtained by the random sampling SAA approach has low quality and high estimation uncertainty, and it shows only a small improvement as increases.Our approach  DLHGPS 



mean  SD  mean  SD  mean  SD  mean  SD  
2.91  0.07  2.91  0.07  2.69  0.28  2.67  0.27  
2.92  0.05  2.92  0.06  2.68  0.29  2.67  0.31  
2.94  0.04  2.93  0.05  2.73  0.22  2.73  0.22 
We also examine the estimation accuracy of , the estimator of corresponding objective of the obtained optimal solution . Since for each with , we can calculate the relative estimation error . Table 2 documents mean and SD of in obtained by using these three approaches. Our method can provide reliable estimates of . The random sampling SAA method shows its deficiency in the objective value estimation accuracy and it has only a small improvement as increases.
Our approach  DLHGPS 



mean  SD  mean  SD  mean  SD  mean  SD  
5.4  3.0  5.5  3.2  24.2  5.0  23.1  5.6  
5.0  2.9  5.2  2.8  24.1  6.0  21.7  6.2  
4.0  3.0  4.0  3.0  22.1  4.0  22.0  6.0 
In addition, we further study the twostage linear optimization problem in (3) and note that is monotonically increasing in for . Since follows the lognormal distribution, it means that the optimal value doesn’t depend on . That explains why the proposed globallocal metamodel assisted twostage OvS demonstrates the similar performance with the deterministic lookahead approach with GPS.
2.2 A Supply Chain Management Example
In this section, we use a supply chain management example to study the performance of our approach. It is inspired by our research collaboration with a biopharmaceutical manufacturing company. The company produces various commercial and clinical products, which requires some common vital raw materials, including soy and other chemical raw materials. For simplification, we only consider the soy raw material and one type of raw chemical material used for producing the key clinical product. The company orders soy and chemical raw material from outside vendors. While the chemical raw material can be fastdelivered, due to the regulations and long testing cycles, soy has long lead time. Since the clinical demand has high prediction uncertainty [Kaminsky_Wang_2015], the company faces high fluctuations in the total cost. Thus, the company is interested in finding the firststage soy ordering decision and the secondstage decisions , including production scheduling and inventory control for the raw chemical material (specified by review policy), to minimize the expected overall cost.
Considering the long lead time of soy delivery, the company first forecasts the clinical demand and places the soy order in advance. Suppose that is within the range with an increment . Then, after the clinical demand is the realized, the company needs to make two types of decisions: the inventory control for the chemical material and the daily production decision . Due to the fastdelivery nature, the company excises the daily review policy for the chemical raw material satisfying , and . Here, we consider a variety of choices: , , ,, ,,,, , and . For simplification, suppose that the chemical raw material orders have zero lead time.
The production planning horizon has four weeks, and each week has five work days. Let , where denotes the aggregated clinical demand occurring in the th week with . If the production can not fully meet the demand , the unmet demand will be subcontracted at a much higher price per unit. If the company produces more than needed, the additional products will be stored with the holding cost as per unit. The goal is to minimize the expected total cost,
S.t.  
where and are the unit ordering costs for soy and chemical raw material, denotes the inventory left at week , denotes the unmet demand for week , and denote the raw chemical material ordering decision and the inventory in the th day of th week. Let starting chemical raw material inventory . Thus, the secondstage production cost at the th week consists of the ordering cost for the fastdelivery chemical, the subcontract and inventory costs. Let the soy ordering price , the chemical ordering price , the inventory cost and the penalty . For the deterministic lookahead policy, we solve the optimization problem,
We apply Gaussian processbased search approach (GPS) with same setting as 2.1 except the standard deviation of the Gaussian process following Section (5.2) in [Sun_etal_2014].
Given any , since there is no closedform of , we use SAA with scenarios to correctly estimate the mean response, which will be used to assess the performance of optimal solutions obtained by the different candidate approaches.
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