Bi-level Mixed-Integer Data-Driven Optimization of Integrated Planning and Scheduling Problems

Abstract

Supply chain management is an interconnected problem that requires the coordination of various decisions and elements across long-term (i.e., supply chain structure), medium-term (i.e., production planning), and short-term (i.e., production scheduling) operations. Traditionally, decision-making strategies for such problems follow a sequential approach where longer-term decisions are made first and implemented at lower levels, accordingly. However, there are shared variables across different decision layers of the supply chain that are dictating the feasibility and optimality of the overall supply chain performance. Multi-level programming offers a holistic approach that explicitly accounts for this inherent hierarchy and interconnectivity between supply chain elements, however, requires more rigorous solution strategies as they are strongly NP-hard. In this work, we use the DOMINO framework, a data-driven optimization algorithm initially developed to solve single-leader single-follower bi-level mixed-integer optimization problems, and further develop it to address integrated planning and scheduling formulations with multiple follower lower-level problems, which has not received extensive attention in the open literature. By sampling for the production targets over a pre-specified planning horizon, DOMINO deterministically solves the scheduling problem at each planning period per sample, while accounting for the total cost of planning, inventories, and demand satisfaction. This input-output data is then passed onto a data-driven optimizer to recover a guaranteed feasible, near-optimal solution to the integrated planning and scheduling problem. We show the applicability of the proposed approach for the solution of a two-product planning and scheduling case study.

1. Introduction

Planning and scheduling belong to different levels of supply chain management, yet their coordination is essential for feasible decision making. The former determines the production targets by observing the market considerations (i.e., product demands), whereas the latter decides on the sequencing of tasks and their respective assignment to specific units such that the production targets set by the planning level are met.

Typically, these planning and scheduling problems are solved sequentially: initially, the planning level decides on the production targets of finished goods by observing the demand profiles for the products; later, the production targets are set for the scheduling problem, and the respective optimal schedules are calculated. However, such a sequential approach, where the planning decisions are solely based on the product demand profiles are optimistic estimates and may lead to infeasible schedules (Grossmann, 2005). In other words, the process of converting raw materials to finished goods cannot be realized within the given capacity of the scheduling level. Henceforth, the production target set by the planning level cannot be met, and consequently, the demand is not satisfied. This interconnected network of decision making requires an integrated approach where the interdependencies among planning and scheduling levels need to be addressed simultaneously to achieve globally optimal solutions (Maravelias and Sung, 2009).

Bi-level programming enables the integration of planning and scheduling levels and creates holistic models that explicitly account for the inherent hierarchy between different levels of the supply chain. Unfortunately, many algorithmic difficulties arise when solving bi-level programming problems including, NP-hardness, nonconvexity, and discontinuity, even in the most simplistic formulations (Sinha et al., 2018). Especially the mixed-integer formulations in the scheduling level prohibit the use of the KKT transformation at the lower level, creating a necessity to tackle these problems with an algorithmic approach that can guarantee feasibility to bi-level formulations. An approach based on multi-parametric programming has been proposed to address this (Avraamidou and Pistikopoulos, 2018), but it cannot handle a relatively high number of variables. Recently, the DOMINO algorithm is introduced as a data-driven methodology for solving constrained bi-level mixed-integer nonlinear programming (B-MINLP) problems with a higher number of variables and a feasibility guarantee (Beykal et al., 2020). In this work, our goal is to (1) further advance the DOMINO framework for solving integrated planning and scheduling problems with multiple followers; and, (2) utilize this framework to provide guaranteed feasible solutions to integrated supply chain management problems.

2. The DOMINO Algorithm and its Extension to Multi-Follower Systems

DOMINO is a data-driven optimization algorithm that is tailored to solve single-leader single-follower constrained B-MINLP problems by approximating them as single-level grey-box optimization problems through series of sampling, optimization, and surrogate modeling steps (Beykal et al., 2020). It postulates candidate sampling points for the upper-level decision variables and solves the lower-level problem deterministically to global optimality at each sampling point. The collected input-output data is then passed onto a data-driven optimization step where the solver retrieves the optimal solution of the bi-level program. DOMINO ensures the feasibility of a candidate solution by evaluating the constraint violations of grey-box constraints and the optimality of the lower-level problem. If these two conditions are satisfied, the candidate solution is deemed a guaranteed feasible solution of the bi-level program. Previously, DOMINO is shown to locate near-optimal solutions to varying types of single-follower bi-level programming problems in series of benchmark problems (Beykal et al., 2020) and a case study (Avraamidou et al., 2018).

In the case of integrated planning and scheduling problems, the lower-level problem is composed of multiple scheduling problems (i.e., multiple followers) that need to be solved sequentially over the entire planning horizon. For that reason, the DOMINO algorithm cannot be directly implemented, and a subroutine needs to be devised to collect the input-output data. The production targets for the planning horizon are the upper-level decision variables, which are used to create the input data for the sampling stage. To account for the multi-follower nature of the lower-level problem, these input production targets will be fixed at the scheduling level and the schedules for each planning period will be solved sequentially while accounting for the inventories and production cost per period. Finally, the total cost of planning will be calculated from the total inventory and production cost across the planning horizon and this value will serve as the output information that is required to be minimized in the data-driven optimization stage.

3. Case Study for Bi-level Production Planning and Scheduling

The computational case study for the bi-level production planning and scheduling problem is adapted from Li and Ierapetritou (2009) with an implementation of the continuous-time formulation of Example 2 (Ierapetritou and Floudas, 1998) at the lower-level scheduling problem. An overview of this bi-level program is provided in Eq. (1).

(1)

3.1. The Upper-Level Problem: Planning Model

The planning level objective minimizes the total inventory and production cost over the entire planning horizon. The decision variables for the bi-level program are the production targets $\left({\mathit{\text{Prd}}}_s^t\right)$ of the desired products (i.e., states s∈S_p) per planning period t. The inventory accounting is performed at every planning period using Eq. (2), where ${\mathit{\text{Inv}}}_s^t$ is the inventory level of the product state s at the end of planning period t, ${\mathit{\text{Prd}}}_s^t$ is the production level of the product state s in the planning period t, and ${\mathit{\text{Dmd}}}_s^t$ is the demand level for the product state s in the planning planning period t.

$${\mathit{\text{Inv}}}_s^t={\mathit{\text{Inv}}}_s^{t-1}+{\mathit{\text{Prd}}}_s^t-{\mathit{\text{Dmd}}}_s^t\forall s\in S_p,t$$ (2)

If the inventory becomes negative at any period, the minimum inventory of each product is calculated, and all respective inventories are penalized by adding this minimum level of inventory. Also, the total amount of products given by the production target, and the inventory level at the beginning of each planning period should be greater than or equal to the demand to satisfy the need. The planning horizon is set to be 7 days, and the demand for the two products is assumed to be known for the entire week. Besides, the planning problem is assumed to be cyclic where the last day production target should meet the last day demand and produce the 1^st day inventory of the next planning cycle. The upper-level problem contains 14 production target constraints, 14 demand constraints, and 28 positive variables (14 inventory and 14 decision variables). The upper bound on the decision variables is set to be 80-unit materials per product per planning period.

3.2. The Lower-Level Problem: Scheduling Model

In the selected case study, the scheduling problem (Figure 1) considers the production of two products through three reaction steps, a heating step, and a separation step. Three feed and four intermediate components are considered in the production schedule. The scheduling formulation considers material balances, allocation, capacity, storage, duration, time horizon, and sequence constraints (Ierapetritou and Floudas, 1998). To guarantee feasibility to the integrated multi-follower problem, a constraint on the final states of the products is added to ensure the production targets set by the upper-level problem are met at the scheduling level. The objective function of the scheduling problem minimizes the fixed cost of operating tasks in units, as well as the variable cost coming from handling raw materials, intermediates, and final products within the process. The lower-level problem contains 285 continuous variables, 160 binary variables, and 712 constraints. The detailed model equations and parameters can be provided upon request.

Figure 1.

State-task network of the scheduling case study. Adapted from Ierapetritou and Floudas (1998).

3.3. Other Considerations for the Problem Formulation

The integrated bi-level mixed-integer program is solved for two cases: (1) assuming linear inventory and production cost, leading to an LP-MILP bi-level problem; and, (2) assuming cubic inventory cost at the planning level and quadratic production cost at the scheduling level, creating an NLP-MIQP problem. Compared to other bi-level optimization algorithms, DOMINO allows us to solve highly nonlinear systems using its data-driven optimization strategy and retrieve guaranteed feasible near-optimal solutions (Beykal et al., 2020). Specifically, this capability of DOMINO is advantageous because the nonlinear objective functions allow for a more realistic estimate of the variable cost in the integrated planning and scheduling formulation, as linear cost estimates are approximations of the nonlinear behavior. Also, the inclusion of the nonlinear terms in the bi-level formulation yields a very challenging optimization problem which the exact methods like multi-parametric programming cannot address. Data-driven evolutionary algorithms can handle nonlinearities but they fail to provide guaranteed feasible solutions.

To assess the consistency and accuracy of the DOMINO algorithm in finding the best solution for the integrated planning and scheduling problem, the algorithm is randomly executed 10 times with the NOMAD algorithm (Le Digabel, 2011) chosen as the data-driven optimizer. The results retrieved from DOMINO are presented in the next section.

4. Results

4.1. LP-MILP Solution

The best solution of the 10 random runs for the LP-MILP integrated planning and scheduling formulation is summarized in Figure 2. The results show that the demand for both products is satisfied over the 7-day planning period where the corresponding production targets are met with globally optimal schedules. We also observe that at the start of the planning period (Day 1), the system produces more than the minimum required level to sustain the inventory levels for Product 1 whereas a lower production target is set for Product 2 and the remaining demand is supplemented from the starting inventory. On Day 2, both production targets are relatively higher than the demand. This higher level of production enables Product 1 to sustain its starting inventory levels whereas Product 2 makes up for the lost inventory on Day 1. Later, when peak demand is expected on Day 4 for both products, the production target is supplemented with this accumulated inventory to satisfy the demand. On Day 6 as the demand is lower, the production can meet this demand without requiring any extra supply from the inventory. On the final day of the planning period, the production levels are increased to satisfy the 7^th day demand as well as to produce the starting inventory of the 1^st day of the next planning cycle.

Figure 2.

Demand, production, and starting inventory profiles for (A) Product 1; and (B) Product 2.

The results of the other 9 random cases also showed very consistent production and inventory levels in Days 3–7 for Product 1 and in Days 4–5, 7 for Product 2. Some variability is observed in the early days of the week where in one instance a significantly higher production is observed for Day 1 for both products which led to no production in Day 2, essentially pushing the system to spend the accumulated inventory more quickly than the best solution. In the other two instances, Day 1 production is observed to be zero where the system relied on having a higher start inventory by producing more at the end of the planning period when the demand is lower. Nonetheless, the final cost objective value for all runs was within ± 0.0386 standard error and all final solutions were guaranteed feasible with globally optimal schedules.

4.2. NLP-MIQP Solution

The best solution of the 10 random runs for the NLP-MIQP formulation is summarized in Figure 3. The nonlinear integrated planning and scheduling results show very similar production and starting inventory profiles to the LP-MILP case study where the demand is satisfied for the entire planning horizon of 7 days with a globally optimal lower-level solution. Only on Day 7, a slightly higher starting inventory level is observed for Product 1 because the production target of the prior day is higher in the NLP-MIQP solution. Furthermore, the DOMINO solution is very consistent across all runs for the NLP-MIQP formulation. For Product 1, we observe that the same starting inventory and production levels are determined for Days 1–5 whereas a slight deviation is observed for Days 6 and 7. For Product 2, some variability is observed at the start and end of the planning period, but the production and inventory levels are consistent for Days 3 and 4. The consistency of the solutions is also reflected in the final objective values where for all runs the objective value was within ± 0.1465 standard error.

Figure 3.

Demand, production, and starting inventory profiles for (A) Product 1; and (B) Product 2 in the NLP-MIQP solution. The error bars indicate the minimum and maximum deviation observed across all runs with respect to the best-found solution.

5. Conclusions and Future Work

In this work, we present a data-driven approach to solve bi-level multi-follower mixed-integer formulations of integrated planning and scheduling problems. By extending the DOMINO algorithm to solve multi-follower bi-level optimization problems and utilizing the data-driven and deterministic optimization capabilities of this framework, we solve the integrated problems to guaranteed feasibility. For all the tested cases, DOMINO identified solutions that meet the product demand and have globally optimal schedules at the lower level, which ensures meeting the production targets, DOMINO also found consistent feasible solutions for both the linear and nonlinear formulations. In the future, the results of the linear formulation will be compared to the deterministic algorithm developed by Avraamidou and Pistikopoulos, 2019. This research was funded by the U.S. National Institutes of Health (NIH) grant P42 ES027704.