In many applications, such as high-tech manufacturing, bio-pharmaceutical 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 two-stage stochastic programming,
where denotes the first-stage action, e.g., the investment decision, denotes the second-stage 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 two-stage 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 mixed-integer 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 two-stage optimization via simulation (OvS).
Compared to the classical two-stage optimization with the known response function, such as two-stage stochastic linear programming problems, there exist additional challenges to solve the two-stage 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 two-stage OvS of interest.
It is computationally demanding to solve the two-stage 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 first-stage solution. Given limited computational resource, we often cannot find the true optimal second-stage 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 two-stage OvS problems.
Notice that the problem of interest is different with the existing black-box 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 one-stage OvS,
Given a feasible decision , the system unknown mean response, denoted by , can be assessed by simulation. Various simulation optimization algorithms are proposed to solve one-stage OvS; see Henderson and Nelson [Henderson_Nelson_2006] for a review. In particular, metamodel-assisted 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 one-stage metamodel assisted approaches, in this paper, we propose a global-local metamodel-assisted two-stage OvS. Specifically, at each visited first-stage action , we construct a local GP metamodel for so that we can simultaneously solve a large set of second-stage optimization problems sharing the same first-stage decision . Then, built on the search results from the second-stage optimization problems, we further develop a global metamodel accounting for various sources of errors. Assisted by the global-local metamodel, we introduce a two-stage optimization via simulation approach that can efficiently employ the limited simulation budget to iteratively search for the optimal first- and second-stage 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 global-local 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 second-stage optimization.
Assisted by the global-local metamodel, we develop a two-stage 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 second-stage decisions.
Our approach can guarantee global convergence as the simulation budget increases. The empirical study also demonstrates that it has good and stable finite-sample performance.
The rest of the paper is organized as follows. Section 2 gives the literature review of relevant studies on two-stage stochastic programming and metamodel-assisted simulation optimization. We formally state the problem of interest in Section LABEL:sec:statement. We develop a global-local metamodel-assisted two-stage 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.
To integrate strategic, tactical and operational decisions, the classical two-stage 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 look-ahead approach with GPS, we consider two examples, including a simple two-stage linear optimization problem and a supply chain management example.
2.1 A Two-Stage Linear Optimization Problem
We first consider a two-stage linear stochastic optimization example from Ekin et al. [Ekin_Polson_Soyer_2014],
where follows the lognormal distribution with mean
and variance. It is easy to see that the first-stage 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 first-stage 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 second-stage relative optimality gap and set the initial number of scenarios .
We study the performance of proposed global-local metamodel assisted two-stage OvS, random sampling SAA approach, and deterministic look-ahead policy with GPS (DLH-GPS) 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) ofobtained by using these approaches. The results are estimated based on 100 macro-replications. Given the same simulation budget, our method and deterministic look-ahead 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.
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.
In addition, we further study the two-stage 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 global-local metamodel assisted two-stage OvS demonstrates the similar performance with the deterministic look-ahead 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 bio-pharmaceutical 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 fast-delivered, 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 first-stage soy ordering decision and the second-stage 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 fast-delivery 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,
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 second-stage production cost at the -th week consists of the ordering cost for the fast-delivery chemical, the subcontract and inventory costs. Let the soy ordering price , the chemical ordering price , the inventory cost and the penalty . For the deterministic look-ahead policy, we solve the optimization problem,
We apply Gaussian process-based 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 closed-form 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.