Solving a Production Lot-Sizing and Scheduling Problem from an Enhanced Inventory Management Perspective

Abstract

In this study, we consider a production lot-sizing and scheduling problem found in the fruit juice production industry from an enhanced inventory management perspective. The problem can be classified as a P2SMM (two-stage multi-machine lot-scheduling) problem. We extended the classical P2SMM problem by incorporating an additional inventory management aspect of finished products to reflect a possible real-life case problem, specifically regarding the shelf-life concept and limited warehouse capacity, with a possibility of outsourcing the warehousing demand to a third-party logistics company. We developed the mixed integer linear programming (MILP) model to fully represent the considered problem (due to the NP-hard nature of the problem, only small-scale instances could be solved to optimality), and the hybrid variable neighborhood search with linear programming (VNS/LP) model to solve both small and real-life large-scale problem instances. The goal of the developed models is to minimize total costs that consist of the cost of backordering, the cost of planned minimum and maximum stock level violation, the cost of warehouse capacity overflow, the costs of production setup time and unused available production time. The main idea of the VNS/LP model is to solve the scheduling segment of P2SMM (the production sequence) via a VNS heuristic, and the lot-sizing segment of P2SMM via the linear programming (LP) model. Based on the results from five variants of the problem setup, a potential decision maker can have an overview of the impact of different important input parameters (production time costs, warehouse capacity and costs, inventory related costs and production demand) on the total cost of a production process and improve its efficiency in changing conditions.

1. Introduction

Many production companies have a high level of production automatization where different products are made from raw materials with little or no intervention from human labor. However, the human factor is still important in the production planning process where complex decisions regarding scheduling and lot-sizing on production lines are required, while considering a wide range of constraints. Mainly, these constraints are related to production line capacities, demands for finished goods, the synchronization of raw material input and production lines, and production setup times between two successive products, among others. Clark et al. [1] defined lot-sizing and scheduling as a problem in which a decision maker must efficiently allocate production resources while fulfilling customer requirements and market demand, often by trading-off conflicting objectives. The main objective is to determine the moment of production, as well as the production duration (which determines the produced quantity) on multiple production lines. The complexity and presence of this problem in many industries incited academic researchers to develop various models to solve the integrated production scheduling and lot-sizing problem as a helpful tool for production planning. The literature overview of production scheduling and lot-sizing publications in different industries is presented in

Table 1. The literature overview of production scheduling and lot-sizing publications in different industries.

References	Application Area
Ferreira et al. [3,4,5], Pagliarussi et al. [6], Toledo et al. [7,8], Toscano et al. [9]	Soft drink industry
Baldo et al. [10]	Brewery industry
Claassen et al. [11], Shin et al. [12], Soler et al. [13], Lütke et al. [14], Marinelli et al. [15], Clark et al. [16]	Food processing industry
Araujo et al. [17], Wu et al. [18], Hans and Van de Velde [19]	Metal industry
Silva and Magalhaes [20], Cardona-Valdés et al. [21]	Textile industry
Almada-Lobo et al. [22]	Glass container industry
Klement et al. [23]	Molded plastic industry
Luche et al. [24]	Grain industry
Santos and Almada-Lobo [25] and Figueira at al. [26] and Martínez [27], Livia et al. [28]	Paper manufacturing industry
Stadtler [29]	Pharmaceutical industry
Li at al. [30]	Mask production industry
Wang et al. [31], Jans and Deagraeve [32]	Automobile industry
Xiao et al. [33]	Chipset production industry

. For a more detailed insight into practical expertise in the area of planning and scheduling, we recommend the book from Kopanos and Puigjaner [2] with an overview of outstanding publications.

The time discretization concept in lot-sizing and scheduling problems with a two-level structure (macro-periods with multiple micro-periods) was first introduced by Pressmar [34], and later exploited by many authors, including Fleischmann [35,36] and Haase [37]. The idea of the time discretization concept is to divide the total planning horizon into multiple non-overlapping macro-periods (e.g., a macro-period can be one day or one week) that can have multiple non-overlapping micro-periods. These micro-periods can be allocated to production lots (the length and the total number of used micro-periods can be a part of the decision variables). For example, Ferreira et al. [4] considered a production planning horizon of 3 weeks (5 working days per week), where they developed a model with 3 macro-periods (one-week periods) and 25 micro-periods per one macro-period. The authors estimated that in the observed case of drink production, one day can have up to five product changeovers, i.e., five micro-periods.

The production lot-sizing and scheduling problem is very difficult to be solved to optimality and is, in general, classified as an NP-hard optimization problem [38]. A heuristic approach is required to solve real-life large-scale instances, because they cannot be solved to optimality in a reasonable computational time, as is the case in many research papers [32,39,40]. In the most recent review published by Copil et al. [39], the authors provided a classification of models used to solve simultaneous lot-sizing and scheduling problems, including a wide range of well-known metaheuristic approaches: Tabu Search (TS), Simulated Annealing (SA), Genetic Algorithms (GA), Variable Neighborhood Search (VNS) and hybrid models.

A hybrid approach comprising mathematical and heuristic models is used in several papers to solve the production scheduling and lot-sizing problem. The aim of such an approach is to solve the scheduling segment (the production sequence) using a heuristic model, and the lot-sizing segment using some mathematical (LP—linear programming) model. In this case, the LP model consists of only continuous decision variables and is able to solve real-life large-scale instances promptly. This concept was used by Kuik et al. [41], where the authors developed two hybrid models to solve production scheduling and lot-sizing: LP and SA, and LP and TS. In addition, Meyr [42] developed a hybrid model that uses mixed integer programming (MIP) together with threshold accepting (TA) and SA heuristics. Two papers based on hybrid models to solve production scheduling were published by Toledo et al. [7,8], in which the authors developed an LP model for the lot-sizing segment with only continuous variables, and the GA model for the sequencing segment.

The VNS metaheuristic was recently used in several papers to solve production lot-sizing and scheduling with promising results. Almada-Lobo et al. [22] presented a basic and reduced VNS approach to solve real-life instances of the production planning and scheduling problem in the glass container industry. Almada-Lobo et al. [43] developed and tested VNS and TS approaches to solve a general multi-item capacitated lot-sizing and scheduling problem with sequence-dependent setup times and costs. Based on the results from solving medium- to large-sized problems, they concluded that the VNS approach outperforms the TS approach in the long run (due to the more diverse nature of VNS). The VNS-based approach to solving uncapacitated multilevel lot-sizing (period and production quantity must be determined for multiple products) was used by Xiao et al. [44,45,46], where the authors developed a basic VNS model, a reduced VNS model and an improved VNS model. The results from Xiao et al. [44] showed that the improved VNS model outperformed other algorithms (hybrid GA, MAX–MIN ant system, parallel GA, ant colony algorithm, reduced VNS and iterated neighborhood search algorithm), for both small- and large-sized problems. Figueira et al. [26] developed a hybrid VNS approach to solve short-term production planning and scheduling problems for the pulp and paper industry, motivated by a real-world case study. The authors combined a general VNS (used for a machine setup pattern) with a specific heuristic (for digester’s speed) and exact solver (for the segment of the problem with continuous variables).

The soft drink (including beverages, brewery and fruit juice) production problem includes an integrated production lot-sizing and scheduling problem as the most important segment. In general, the production process starts with tanks for raw materials, from which raw materials are sent to production lines for bottling. The sequence and lot size of each batch must be determined for all products (bottled flavors). Numerous authors have contributed to this field, publishing papers that are presented below. Ferreira et al. [5] considered a small-scale soft drink plant and developed a MIP model with relax and fix heuristics to solve integrated lot-sizing and scheduling (applicable to small-scale problems). To improve obtained solutions, the authors recommended the use of VNS in future research. Ferreira et al. [3] continued the previous research and proposed four single-stage formulations to solve the synchronized two-stage lot-sizing and scheduling problem. Baldo et al. [10] considered lot-sizing and scheduling in the brewery industry and developed a MIP-based heuristic to solve real-world problem instances. Pagliarussi et al. [6] developed a MIP model to solve lot-sizing and scheduling problems of fruit juice beverage production of small-scale dimensions (the CPLEX solver was not capable of obtaining solutions within an acceptable computational time for most test instances). Toledo et al. [8] developed the GA model that was compared to the exact model developed in the CPLEX solver to solve a synchronized and integrated two-level lot-sizing and scheduling problem (SITLSP) in soft drink production. The extension of the aforementioned research was published by Toledo et al. [7], where the authors developed a hybrid approach with two main segments: the LP model for the lot-sizing segment, and the GA heuristic for the sequencing segment. For real-world problem instances, the hybrid approach outperformed other models available in the literature in terms of objective function value and run times. Toscano et al. [9] observed a two-stage fruit-based beverage lot-sizing and scheduling problem. The authors developed two variants of a two-phase heuristic algorithm based on MIP and constructive heuristics. All of the papers from the soft drink production lot-sizing and scheduling research area [3,5,6,7,8,9,10] observed the three segments in the objective function (inventories, backorder, production setup). A comparative analysis of relevant research papers addressing the similar problem observed in this paper, with research gaps, is presented in

Table 2. A comparative analysis of relevant research papers addressing the similar problem observed in this paper, with research gaps.

Reference	Solution Approach	Specific Objective Function Segments
		Inventory	Backorder	Prod. Setup Time	Unused Prod. Time	Min. and Max. Stock Level Violation	Warehouse Capacity Overflow	Problem Scale
Ferreira et al. [5]	MIP, relax and fix heuristics	✓	✓	✓				Reduced to stage II
Ferreira et al. [3]	MIP, Asymmetric Travelling Salesman Problem	✓	✓	✓				Stage I and II
Baldo et al. [10]	MIP, MIP-based heuristic	✓	✓	✓				Stage I and II
Pagliarussi et al. [6]	MIP	✓	✓	✓				Reduced to stage II
Toledo et al. [8]	MIP branch and cut, GA	✓	✓	✓				Stage I and II
Toledo et al. [7]	MIP, GA/mathematical programming approach	✓	✓	✓				Stage I and II
Toscano et al. [9]	MIP/two-phase heuristic	✓	✓	✓				Stage I and II
This paper	MILP, VNS/LP heuristic	✓ *	✓	✓	✓	✓	✓	Stage I and II

* indirectly.

.

The problem observed in this paper can be classified as P2SMM (two-stage multi-machine lot-scheduling), which represents a variant of SITLSP. In P2SMM, each production line (bottling line in our case) has one dedicated tank with raw materials. The P2SMM problem formulation is also an NP-hard problem [7]. We extended the classical production lot-sizing and scheduling problem in the soft drink industry, with a special emphasis on the inventory management of finished products. This extension was accomplished by introducing the shelf-life concept, limited warehouse capacity and the possibility of outsourcing the excessive warehousing demand to a third-party logistics (3PL) company. Additionally, we introduced the cost of unused production time with the goal of increasing production capacity utilization in the case of lower than usual demand. The increase in production capacity utilization is limited by the maximum stock levels or available inhouse warehouse capacity. Overproduction that incurs storage outsourcing is penalized by the associated cost of outsourcing. This represents an original and novel approach to tackle production scheduling with a wider set of managerial objectives that are important from a production, as well as a logistics point of view. The proposed approach is described in the problem description section in more detail. In this paper, we developed two models to solve a P2SMM from an enhanced inventory management perspective: the mixed integer linear programming (MILP) model, and the hybrid variable neighborhood search and linear programming (VNS/LP) model. The mathematical model can solve small-scale instances to optimality, where the obtained solutions are used to benchmark and validate the heuristic solutions of the hybrid VNS/LP model. Real-life large-scale instances cannot be solved to optimality via a mathematical model in a reasonable computational time, and these instances are solved only by the hybrid VNS/LP model.

The remainder of this paper is organized as follows. Section 2 describes the observed P2SMM problem. The mathematical formulation for solving this problem is presented in Section 3, while the hybrid VNS/LP model is presented in Section 4. The test instances (five variants of the small- and large-scale problem setup) and computational results are given in Section 5. Concluding remarks are given in Section 6.

2. Problem Description

In this paper, we observe a production lot-sizing and scheduling problem that can be found at a fruit juice company. Each production line has one dedicated tank with raw material (flavor of a drink) bottled on an automated filling line. A product is defined by a flavor and package size combination (different bottle sizes). Usually, there is more than one production line, where processing times vary for different products. Raw material tanks have minimal required and maximal possible levels, where production lot sizes must be between these two values (the maximum tank capacity is large enough for continuous production of any product in a one-day period). A flavor changeover represents one type of production setup time, which is required for the preparation of raw material in the tank between two consecutive production lots (a setup time is required for mandatory cleaning/disinfection purposes regardless of the type of flavor before or after the changeover). The second production setup time is a package changeover, referring to the time required to change the product’s package (e.g., bottle size) at a filling line. Both flavor and package changeovers are directly dependent on the production sequence (combinations of the flavor and package size on the line production plan). The production plan for each line must define the sequence and quantity of products that will be produced in a given planning horizon. The start time of production and the production duration (schedule and lot size) must be determined by the production plan for each product. The production plan should meet the expected products’ demand, as it is the primary factor driving the production process. In addition, the production plan is made for a period of a few weeks only for those products that are in danger of stockout in that period. The rest of the products have sufficient stock in the planning period (thus, they are not included in the production lot-sizing and scheduling problem), where the stock of these products decreases over time as a result of clients’ orders. Concurrently, this leads to the everyday increase in the available warehouse capacity during the planning period.

Compared to the available literature (Table 2), the proposed approach is unique due to a greater focus on the inventory management and logistics of finished products. This is mainly achieved through problem extensions related to the safety stock, product shelf life and limited in-house warehouse capacity for storing finished products. Besides these additional inventory-related optimization segments, we consider backorders, production setup time and utilization of production capacity. Backorder is required in the case of product shortage, which incurs additional costs. Safety stock is required to reduce the possibility of shortages (in reality, product demand is stochastic). In addition, product shelf life is important for sales because customers may demand a minimum allowed product shelf life (below which they can refuse or return the product). Thus, extremely high stock levels can incur significant quantities of obsolete products due to the higher risk of shelf-life diminishment. For example, customers can refuse products with diminished shelf life, these products can have lower sale price, and eventually must be destroyed in the case of extremely low shelf life. All these shelf-life diminishment consequences can incur some additional cost. Silver [47] published one of the first papers that emphasized the importance of shelf life in production. Since then, many publications on a similar topic of product decay and perishability in production were published [11,12,13,14]. In the problem observed in our research, the production facility has an in-house warehouse with limited capacity for product storage. Excessive quantities that cannot be stored in the in-house warehouse must be outsourced to a 3PL warehouse. The cost of outsourced storage with transportation incurs additional costs (more expensive than the in-house storage cost). The flowchart of the observed problem is presented in Figure 1.

The production setup time is necessary due to flavor and package size changeovers, and represents lost production time that should be minimized. On the other hand, production capacity should be fully utilized in a production plan, with respect to inventory-related side constraints. In general, if there is enough warehouse capacity, each product should be produced in quantities up to the maximum shelf-life stock level. Therefore, we discarded classical total stock minimization from the objective function and introduced maximum desirable stock (regarding shelf life) because it is better to utilize existing warehouse and production capacities than to have somewhat lower total stocks. This is especially important for the production plan in the successive planning horizon, where higher starting stocks (that are in the target stock levels) result in more relaxed production requirements. The described problem setup is not restricted only to fruit juice production but can also be found in other P2SMM production scheduling problems.

3. The Mixed Integer Linear Programming Model

The MILP formulation to solving the production scheduling and lot-sizing problem is based on the modelling approaches introduced by Pressmar [34] with a two-level time structure of micro- and macro-periods for a general dynamic production system; Fleischmann and Meyr [35] with a two-level time structure mathematical formulation for general lot-sizing and scheduling; and Toledo et al. [7] with a two-level time structure mathematical formulation to solving the P2SMM problem in the soft drink production. We have modified and extended the latter formulation to reflect the case study problem with an additional product stock (product shelf-life and warehouse capacity) and production-time-related objective function and constraints. Notations, objective function and constraints of the proposed MILP model are presented in this section. The parameters and variables related to the two stages of the production process are noted with superscript I (for tank stage I) and with superscript II (for bottling stage II).

• Indices:

• j - products planned for production in the planning horizon (j = 1, …, J)
• t - days in the planning horizon (t = 1, …, T)
• m - production lines (m = 1, …, M)
• l - raw material (l = 1, …, L)
• s - micro-period that corresponds to an index of production lot in day t (S_t is a set of micro-periods s in day t)

• Parameters:

• g_j - cost of backorder cost for product j (daily per product unit)
• a_j - cost of safety stock violation for product j (daily per product unit)
• b_j - cost of shelf-life stock violation for product j (daily per product unit)
• c - cost of warehouse capacity overflow for all products (daily per pallet)
• w - cost of lost time from production setup (per time unit)
• u - cost of unused production time (per time unit), (idle time, not used for production or changeover)
• d_jt - demand of product j in day t (in product units)
• r_jl - required quantity of raw material l to produce unit of product j (in L)
• a^II_mj - unit production time for product j in production line m
• K^I_m - capacity of tank m for production line m (in L)
• K^II_mt - available time capacity for production line m in day t
• q^I_lm - minimum quantity of raw material l in tank at production line m required for production (in L)
• α_m - set of eligible products for production line m
• β_m - set of eligible raw materials for production line m (depends on the α_m).
• SS_jt - minimum target safety stock level for product j in day t (in product units)
• SL_jt - maximum target stock level for product j in day t (in product units)
• WH_t - total warehouse capacity in pallet locations in day t (in pallets)
• WH⁺ - daily increase in available warehouse capacity (in pallets) due to the expected sales of products not in the production planning horizon (products in the company’s portfolio that are not in danger of stockout in the time period of T days)
• P_j - number of j product units on one pallet
• s⁰_t - first micro-period s in day t
• ST_ij - production setup time from the product i to j (equals to max{raw material changeover time in tanks, product changeover time on production lines}). We presume that this time has the same value for all changeovers between products, regardless of the production line.

• Decision variables:

• x^II_mjs - production quantity (lot size) of product j on line m in micro-period s (in product units)
• y^II_mjs - binary variable that takes value 1 if product j is scheduled on line m in micro-period s
• z^II_mijs - binary variable that takes value 1 if there is a changeover in micro-period s on line m from product i to product j.
• W_mt - total production setup time for line m in day t
• I^-_jt - backorder of product j in day t
• I⁺_jt - total stock of product j in day t (in product units)
• V^SS_jt - violation of target minimum stock level (SS_jt) for product j in day t (in product units)
• V^SL_jt - violation of target maximum stock level (SL_jt) for product j in day t (in product units)
• V^WH_t - violation of warehouse capacity in day t (in pallets)
• U_mt - unused production time of line m in day t

The four sub-functions of the objective function (1) are cost of backorder, cost of target minimum and maximum stock level violation, cost of outsourced warehousing, cost of production setup time and unused available production time.

Objective function:

$$\begin{array}{ll}\min \to & {\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{g}_j\cdot I_{jt}^{-}}}\\ & +{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T\left(a_j\cdot V_{jt}^{SS}+b_j\cdot V_{jt}^{SL}\right)}}\\ & +{\displaystyle \sum _{t=1}^{T}c\cdot V_t^{WH}}\\ & +{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{t=1}^T\left(w\cdot W_{mt}^{}+u\cdot U_{mt}^{}\right)}}\end{array}$$

(1)

Subject to:

$$I_{j(t-1)}^{+}+{\displaystyle \sum _{m\in {\lambda }_j}{\displaystyle \sum _{s\in S_t}{x}_{mjs}^{II}}}+I_{jt}^{-}=I_{jt}^{+}+I_{j(t-1)}^{-}+d_{jt} j=1,\dots ,J,t=1,\dots ,T$$

(2)

$$U_{mt}^{}\ge K_{mt}^{II}-{\displaystyle \sum _{j\in {\alpha }_m}{\displaystyle \sum _{s\in S_t}{a}_{mj}^{II}\cdot x_{mjs}^{II}}}-W_{mt} m=1,\dots ,M,t=1,\dots ,T$$

(3)

$$r_{jl}\cdot x_{mjs}^{II}\le K_m^I\cdot y_{mjs}^{II} m=1,\dots ,M,l\in {\beta }_m,j=1,\dots ,J,t=1,\dots ,T,s\in S_t$$

(4)

$$r_{jl}\cdot x_{mjs}^{II}\ge q_{lm}^I\cdot y_{mjs}^{II} m=1,\dots ,M,l\in {\beta }_m,j=1,\dots ,J,t=1,\dots ,T,s\in S_t$$

(5)

$$V_{jt}^{SS}\ge I_{jt}^{-}-I_{jt}^{+}+S{S}_{jt} j=1,\dots ,J,t=1,\dots ,T$$

(6)

$$V_{jt}^{SL}\ge I_{jt}^{+}-S{L}_{jt} j=1,\dots ,J,t=1,\dots ,T$$

(7)

$$V_t^{WH}\ge {\displaystyle \sum _{j=1}^J\frac{I_{jt}^{+}}{P_j}}-W{H}_t t=1,\dots ,T$$

(8)

$$W_{mt}={\displaystyle \sum _{i\in {\alpha }_m}{\displaystyle \sum _{j\in {\alpha }_m}{\displaystyle \sum _{s\in S_t}S{T}_{ij}\cdot z_{mijs}^{II}}}} m=1,\dots ,M,t=1,\dots ,T$$

(9)

$$y_{mjs}^{II}\ge \frac{a_{mj}^{II}}{K_{mt}^{II}}{x}_{mjs}^{II} m=1,\dots ,M,j\in {\alpha }_m,t=1,\dots ,T,s\in S_t$$

(10)

$$y_{mj(s-1)}^{II}\ge y_{mjs}^{II} m=1,\dots ,M,j\in {\alpha }_m,t=1,\dots ,T,s\in S_t-\left\{s_t^0\right\}$$

(11)

$$\begin{array}{l}{y}_{mi(s-1)}^{II}+y_{mjs}^{II}-1\le z_{mijs}^{II}\le \frac{y_{mi(s-1)}^{II}+y_{mjs}^{II}}{2}\\ m=1,\dots ,M,i\in {\alpha }_m,j\in {\alpha }_m,t=1,\dots ,T,s\in S_t-\left\{s_t^0\right\}\end{array}$$

(12)

$${\displaystyle \sum _{i\in {\alpha }_m}{\displaystyle \sum _{j\in {\alpha }_m}{z}_{mijs}^{II}}}\le 1 m=1,\dots ,M,t=1,\dots ,T,s\in S_t$$

(13)

$${\displaystyle \sum _{j\in {\alpha }_m}{y}_{mjs}^{II}}\le 1 m=1,\dots ,M,t=1,\dots ,T,s\in S_t$$

(14)

$$y_{mjs}^{II},z_{mijs}^{II}\in \left\{0,1\right\} m=1,\dots ,M,i=1,\dots ,J,j=1,\dots ,J,t=1,\dots ,T,s\in S_t$$

(15)

$$\begin{array}{l}{x}_{mjs}^{II},I_{jt}^{+},I_{jt}^{-},V_{jt}^{SS},V_{jt}^{SL},V_t^{WH},W{H}_t,V_{mt}^{PT},U_{mt}^{},W_{mt}^{}\ge 0\\ m=1,\dots ,M,j=1,\dots ,J,t=1,\dots ,T,s\in S_t\end{array}$$

(16)

Constraint (2) defines the lot-sizing segment that is dependent on demand, stock level and stockout of product j in day t. The unused production time for line m in day t is defined by constraint (3). Constraint (4) limits the maximum quantity of the lot size for product j to the available tank capacity on the production line m. Constraint (5) limits the minimum quantity of the lot size for product j to the minimum required quantity of raw material in the tank. Safety stock and shelf-life stock target level violations are defined by constraints (6) and (7), respectively. Constraint (8) defines the value of warehouse capacity overflow. Constraints (9) define the total production setup time. Constraint (10) defines the binary variable related to the schedule of production (for a production line with a product being produced in a micro-period). In the case of less than the maximum micro-periods (lots) per day used for production, constraint (11) assures that the production plan uses a micro-period from the first to last successively. Constraint (12) defines the existence of changeover on a production line from the product i to j before micro-period s. Constraints (13) and (14) limit the sum of all product changeovers before a micro-period to a single one, and the sum of all products produced in a micro-period to a single one, respectively. Constraints (15) and (16) define the nature of the variables.

The daily increase in available warehouse capacity (due to the expected sales of products that are not in the production planning horizon) is defined a priori for all t > 1 by Equation (17).

$$W{H}_t=W{H}_{t-1}+W{H}^{+}\cdot (t-1)\forall t=2,\dots ,T$$

(17)

Constraints (9)–(15) are related to the scheduling segment of the considered problem, which is solved using the VNS heuristic in the proposed hybrid VNS/LP model, described in the next section.

4. The Hybrid VNS/LP Model

The VNS metaheuristic approach was developed by Mladenović and Hansen [48], based on the idea of the variable neighborhood algorithm published by Mladenović [49]. Hansen and Mladenović [50] published a review of VNS applications to various combinatorial and global optimization problems. Since the first published paper addressing the VNS approach, different variants have been developed and applied to numerous problems with complex combinatorial backgrounds [51]. In VNS, the solution is searched for via a systematic change in neighborhoods within a search algorithm organized into three general steps: the initial solution construction, the local search and the shaking procedure.

The hybrid VNS/LP model is based on the relaxation of the MILP model by removing all integer scheduling-related constraints (scheduling part of the considered production problem). Relaxed mathematical formulation becomes an LP model that can solve real-life production lot-sizing problems promptly. The VNS model is used to obtain solutions for the production scheduling part of the problem. The LP model takes these production scheduling solutions as an input to solve production lot-sizing and returns the value of the overall scheduling and lot-sizing objective function. More precisely, the VNS model is the platform of the hybrid approach where the LP model is used to obtain a value of the objective function for different solutions found via an initial construction heuristic, and local search and shaking procedures. The outline of the hybrid VNS/LP model is given in Algorithm 1, with maximum CPU (central processing unit) time as the VNS termination criteria (the LP model is applied as part of lines 1, 2 and 8).

Algorithm 1. The outline of the proposed hybrid VNS/LP model.
Termination criteria = False until the maximum allowed CPU time
1:	Construct Initial solution
2:	Apply Local search procedure on Initial solution to obtain Current best solution
3:	While Termination criteria == False:
4:	For Sh_intensity in SoI:  #Set of intensities
5:	For Shaking pass in range(0, Tnp):  #Total number of passes
6:	Apply Shaking(Sh_intensity) move on Current best solution to obtain New solution
7:	Apply Local search procedure on New solution
8:	If New solution is better than Current best solution:
9:	Current best solution ← New solution
10:	Break to line 3

For the initial solution construction, we developed the best-reduced insertion (BRI) heuristic presented in Algorithm 2. Starting from an empty solution, the next best insertion of product j that has some approximated benefit (incurs lower total costs) is added to the current solution. To obtain the best insertion, we use the insertion benefit list per product that has three parts: first day t’ with stockout for product j, backorder cost for that day, total cost of backorder and safety stock violation. The N best (first) insertions are evaluated via the LP model. The list of benefits is sorted in the following manner: ascending by day t’, descending by backorder cost for day t’, and descending by the total cost of backorder and safety stock violation.

Algorithm 2. The initial solution construction.
Solution ← empty set of sequences and lot-sizes
Improvement ← True
1:	While Improvement:
2:	Best_new_solution ← Solution
3:	For j in J: Define Insertion_benefit_list
4:	Sort Insertion_benefit_list by highest approx. benefit
5:	benefit = 0
6:	For ins in N best insertions from Insertion_benefit_list:
8:	t’, j ← ins
9:	For t in [t’−Δ, t’+Δ]:  #evaluate Δ days before and after t’
10:	For m in M:
11:	New_solution ← insertion for j,t,m i n Solution  #using LP model
12:	If cost(Solution) − cost(New_solution) > benefit:
13:	benefit = cost(Solution) − cost(New_solution)
14:	Best_new_solution ← New_solution
15:	If benefit > 0:
16:	Solution ← Best_new_solution
17:	Else: Improvement = False

The core of the VNS model is based on the neighborhoods in which local search and shaking procedures try to find a better solution. We present seven different neighborhoods in Figure 2, for a demonstrative example with two production lines, two macro-periods (days), three bottle sizes and three flavors (nine different products):

(a) Insertion	- of a new product on line m in day t
(b) Removal	- of a product j on line m in day t
(c) Change product	- for line m in day t, removal of a product j and insertion of a new product j*
(d) Reallocate 2	- reallocate product j on line m from day t1 to day t2
(e) Swap 2	- for line m, swap product j1 from day t1 with product j2 from day t2
(f) Reallocate 3	- reallocate product j in day t from line m1 to line m2
(g) Swap 3	- for day t, swap product j1 from line m1 with product j2 from line m2

The best sequence of products on a production line within each day t (during a search of a neighborhood) is defined in such a manner to incur the lowest changeover time. For example, in the Swap 2 move in Figure 2, the sequence of products is changed in the new solution and has a lower changeover time. In addition, each neighborhood move is focused on changing the schedule part of the problem and needs to be imported to the LP model to obtain new lot-sizing. Therefore, the lot size for all products in a production plan (for all days and bottling lines) can be affected by a change in production structure on only one day (e.g., Insertion on Figure 2 has an impact on both days, although it was conducted only on day 1). The change in production on a single day can have an influence on the production in the entire planning horizon.

To solve the considered problem in a reasonable computational time, we used a reduced VNS variant with first improvement local search. The classical reduced VNS model [51] does not use the local search procedure. Instead, neighborhoods are randomly searched via the shaking procedure only. In our approach, the local search procedure is used, where each possible move can be evaluated with probability P_LS, as presented in Algorithm 3. Additionally, the order of neighborhoods is shuffled at the start of the local search procedure for the purpose of search diversification.

Algorithm 3. The local search procedure.
Input Current_best_solution
Randomize order of neighborhoods in List_of_neighborhoods
Improvement ← True
1:	While Improvement:
2:	Improvement = False
3:	For neighborhood in List_of_neighborhoods:
4:	For move in neighborhood: #iterate trough all possible search moves
5:	Generate random_number from [0.0, 1.0]
6:	If random_number < PLS:
7:	New_solution ← Current_best_solution & move
8:	If New_solution is better than Current_best_solution:
9:	Current_best_solution ← New_solution
10:	Improvement = True

The shaking procedure is used to move the current best solution to some random point in the solution space, from where a local search procedure is applied, with the main goal of overcoming the local optimum trap. Neighborhoods from the local search procedure, with additional Remove_all neighborhood (line 4 in Algorithm 4), are used in the shaking procedure to randomly shake the current solution. The shaking procedure is structured to begin with minor adjustments, and then to gradually intensify the magnitude of those changes as it progresses towards completion. The shaking procedure is presented in Algorithm 4. With the increase in shaking intensity, more neighborhoods are used (0.0 < R_S1 < R_S2 < R_S3 < R_S4 < 1.0).

Algorithm 4. The shaking procedure.
Input Current_best_solution; Sh_intensity
1:	For change in range(1, Sh_intensity+1):
2:	Apply a random Removal, Insertion, and Change Product
3:	If change/Max_intensity > RS1:
4:	Remove all j from one random t of one random m
5:	If change/Max_intensity > RS2:
6:	Apply a random Reallocate 2, and Swap 2
7:	If change/Max_intensity > RS3:
8:	Apply a random Reallocate 3, and Swap 3
9:	If change/Max_intensity > RS4:
10:	Apply one random change from each neighborhood used in shaking

The LP model takes the following scheduling input parameters from the heuristic approach:

• ns_mjt - number of production lots for product j on line m in day t
• W_mt - production setup time for line m in day t

Decision variable:

• u^II_mjt - lot size for product j on line m in day t. From u^II_mjt, other decision variables are derived and used in objective function (18) to obtain the values of five sub-objectives.

In LP objective function (18), we added a fifth segment to evaluate the feasibility of the scheduling segment (regarding the sum of setup times and production times for minimal quantities of all allocated products): the cost of available production time violation (a high value of coefficient Ω will penalize this violation so that any unfeasible scheduling input from the heuristic solution will not be taken as current best in the neighborhood search). Additional variables and their coefficient in this segment of the objective function are:

• V^PT_mt - the violation of production time of line m in day t
• Ω - big number (coefficient used to eliminate solutions with a production time violation in the process of finding the best solution in the proposed hybrid approach)

Solution feasibility is not an issue in the MILP formulation because the solution will always be feasible. Instead, the MILP solution can have a higher or lower value than the objective function (for example, if production capacity is scarce, stockouts can occur which can lead to an increase in total costs). More precisely, the MILP model simultaneously solves both production lot-sizing and scheduling segments of the problem, while the LP model takes the scheduling solution from VNS and solves the lot-sizing segment. In the latter case, heuristic scheduling can incur infeasible lot-sizing (due to the minimum lot size constraints and changeover times). To overcome this possible infeasibility, the fifth segment of objective function is added.

The LP objective function:

$$\begin{array}{ll}\min \to & {\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{g}_j\cdot I_{jt}^{-}}}\\ & +{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T\left(a_j\cdot V_{jt}^{SS}+b_j\cdot V_{jt}^{SL}\right)}}\\ & +{\displaystyle \sum _{t=1}^{T}c\cdot V_t^{WH}}\\ & +{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{t=1}^T\left(w\cdot W_{mt}^{}+u\cdot U_{mt}^{}\right)}}\\ & +{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{t=1}^T\Omega \cdot V_{mt}^{PT}}}\end{array}$$

(18)

In addition to constraints (6)–(9) from the MILP model, the LP objective function is also subject to

$$V_{mt}^{PT}\ge {\displaystyle \sum _{j\in {\alpha }_m}{a}_{mj}^{II}\cdot u_{mjt}^{II}}+W_{mt}-K_{mt}^{II} m=1,\dots ,M,t=1,\dots ,T$$

(19)

$$I_{j(t-1)}^{+}+{\displaystyle \sum _{m\in {\lambda }_j}{u}_{mjt}^{II}}+I_{jt}^{-}=I_{jt}^{+}+I_{j(t-1)}^{-}+d_{jt} j=1,\dots ,J,t=1,\dots ,T$$

(20)

$$U_{mt}^{}\ge K_{mt}^{II}-{\displaystyle \sum _{j\in {\alpha }_m}{a}_{mj}^{II}\cdot u_{mjt}^{II}}-W_{mt} m=1,\dots ,M,t=1,\dots ,T$$

(21)

$$r_{jl}\cdot u_{mjt}^{II}\le K_m^I\cdot n{s}_{mjt} m=1,\dots ,M,l\in {\beta }_m,j=1,\dots ,J,t=1,\dots ,T$$

(22)

$$r_{jl}\cdot u_{mjt}^{II}\ge q_{lm}^I\cdot n{s}_{mjt} m=1,\dots ,M,l\in {\beta }_m,j=1,\dots ,J,t=1,\dots ,T$$

(23)

$$\begin{array}{l}{I}_{jt}^{+},I_{jt}^{-},u_{mjt}^{II},V_{jt}^{SS},V_{jt}^{SL},V_t^{WH},W{H}_t,V_{mt}^{PT},U_{mt}^{}\ge 0\\ m=1,\dots ,M,j=1,\dots ,J,t=1,\dots ,T\end{array}$$

(24)

Constraint (19) specifies the violation of total available time allotted per line. Constraint (20) defines the lot-sizing segment that is dependent on demand, stock level and stockout of product j in day t. Constraint (21) defines unused production time per line. Constraint (22) ensures that the production lot size does not surpass the tank capacity on the line. Constraint (23) defines the minimum production lot size based on the minimum required quantity of raw material that needs to be filled in the tank. Constraint (24) defines the nature of the variables.

5. Test Instances and Computational Results

To test the proposed approach, we generated five types of instances of small- and large-scale problem dimensions. Small-scale instances were solved with both the MILP and hybrid VNS/LP model, while large-scale instances could not be solved by the MILP model in a reasonable computational time due to its combinatorial complexity (even some of the small-scale instances could not be solved to optimality by the MILP model in the 1 h of available computational time). All instances had M = 2 production lines to fill different fruit juice flavors into PET packages (0.33, 0.5 and 1.5 L).

The parameters for small- and large-scale instances were:

Small-scale instances   

- K^II_mt = 8 h; J = 10 products; L = 6 raw materials; T = 3 days; 3 micro-periods per day; WH₀ = 1000 pallets

Large-scale instances   

- K^II_mt = 20 h; J = 30 products; L = 20 raw materials; T = 10 days; 5 micro-periods per day; WH₀ = 3000 pallets

The detailed input parameters of all instances are available in the Supplementary Materials. In the following text, we outline only the key parameter values. The capacity of tanks was 350,000 L, while the minimal quantity that needed to be filled in tanks in order to start the production was 17,500 L. The daily cost of total warehouse capacity overflow was c = 5 EUR/pallet; the unit cost of lost time from production setup was w = 4 €/min; the unit cost of unused production time was u = 3 EUR/min. The daily stock-related cost per product unit of a backorder, safety stock violation and shelf-life stock violation were between EUR [0.10–0.40], [0.05–0.20], and [0.02–0.08], respectively. Daily demand for products djt was between [3000–24,000] units, while the minimum desirable safety stock level was equal to 5*djt, and the maximum target stock level based on shelf-life was equal to 60*djt. The production setup time (changeover) from product i to j was between [30–240] min.

In addition, we generated five variants of problem setup to further analyze the proposed models (10 randomly generated instances per each variant):

• - basic variant with two production lines (both can bottle all products); input parameters for one small-scale instance are presented in

  Table 3. An example of a test instance (small-scale instance 1 of basic variant A).

  Input Data	Values
  WHt	= [1000, 1170, 1340]
  Pj	= [1584, 1584, 1584, 1296, 1296, 1296, 1296, 504, 504, 504]
  gj	= [0.28, 0.3, 0.36, 0.14, 0.34, 0.24, 0.12, 0.2, 0.1, 0.34]
  aj	= [0.14, 0.15, 0.18, 0.07, 0.17, 0.12, 0.06, 0.1, 0.05, 0.17]
  bj	= [0.056, 0.06, 0.072, 0.028, 0.068, 0.048, 0.024, 0.04, 0.02, 0.068]
  djt  (for all t)	= [19,019, 18,880, 14,933, 19,405, 12,030, 4559, 19,633, 23,867, 18,545, 5237]
  SSjt  (for all t)	= [95,095, 94,400, 74,665, 97,025, 60,150, 22,795, 98,165, 119,335, 92,725, 26,185]
  SLjt  (for all t)	= [1,141,140, 1,132,800, 895,980, 1,164,300, 721,800, 273,540, 1,177,980, 1,432,020, 1,112,700, 314,220]
  I+j0	= [149,679, 147,075, 105,277, 120,505, 74,826, 35,469, 78,532, 90,813, 130,742, 34,668]
  rjl  [l = 1]	= [0.33,  -,  -,  -,  -,  -,  -,  -,  1.5,  - ]
  rjl  [l = 2]	= [-,  -,  -,  -,  -,  0.5,  -,  -,  -,  - ]
  rjl  [l = 3]	= [-,  0.33,  -,  0.5,  -,  -,  -,  -,  -,  - ]
  rjl  [l = 4]	= [-,  -,  -,  -,  -,  -,  0.5,  -,  -,  - ]
  rjl  [l = 5]	= [-,  -,  -,  -,  -,  -,  -,  -,  -,  1.5 ]
  rjl  [l = 6]	= [-,  -,  0.33,  -,  0.5,  -,  -,  1.5,  -,  - ]
  STij  [i = 1]	= [30,  240,  120,  240,  120,  240,  240,  240,  60,  120 ]
  STij [i = 2]	= [120,  30,  120,  240,  240,  60,  120,  240,  240, 120 ]
  STij [i = 3]	= [120,  120,  30,  120,  120,  120,  60,  120,  120,  120 ]
  STij [i = 4]	= [120,  240, 240,  30,  120,  60,  120,  120,  120,  60 ]
  STij [i = 5]	= [120,  120,  120,  120,  30,  120,  60,  120,  240,  120 ]
  STij [i = 6]	= [240,  120,  120,  60,  120,  30,  30,  240,  120,  120 ]
  STij [i = 7]	= [120,  240,  240,  60,  120,  30,  30,  120,  240,  240 ]
  STij [i = 8]	= [120, 120,  120,  120,  60,  240,  120,  30, 120,  120 ]
  STij [i = 9]	= [240, 240,  120,  240,  120,  240,  240,  120,  30,  120 ]
  STij [i = 10]	= [120,  60,  240,  120,  120,  240,  120,  60,  120,  30 ]
  aIImj [m = 1]	= [0.0036, 0.0036, 0.0036, 0.0036, 0.0036, 0.0036, 0.0036, 0.005, 0.005, 0.005]
  aIImj [m = 2]	= [0.0036, 0.0036, 0.0036, 0.0036, 0.0036, 0.0036, 0.0036, 0.005, 0.005, 0.005]

• - increased production-time-related costs by 25% (setup time and unused time costs)
• - smaller starting warehouse capacity by 25%; increase in outsourced warehousing cost by 25%
• - increased inventory-related costs by 25% (backorder, minimum safety stock and shelf-life stock violations)
• - increased product demand (including minimum and maximum stock level) by 25%

The hybrid VNS/LP model had the following tune-up parameter values:

• Initial solution (BRI) - N = 2 and Δ = 1 for small-scale instances, and N = 3 and Δ = 2 for large-scale instances
• Local search - P_SL = 0.5
• Shaking - SoI = [1,2,3,4,5,6,7,8,9,10]; R_S1 = 0.4, R_S2 = 0.5, R_S3 = 0.6, R_S4 = 0.7; Tnp = 1 and Tnp = 2 for small and large-scale instances, respectively
• Termination criterion - max CPU time was set to 60 s and 14,400 s for small- and large-scale instances, respectively

The models were implemented using Python and API CPLEX on a PC with Intel^® Core™ i5-3470 CPU @ 3.20 GHz with 8 GB RAM.

5.1. Small-Scale Instance Computational Results

To solve small-scale instances with the LP model, we set the maximum computational time limit to 3600 s. The hybrid VNS/LP model solved 20 iterations of each small-scale instance. The results of the small-scale instances obtained from the hybrid VNS/LP model are presented in

Table 4. Results obtained from the MILP and hybrid VNS/LP models for the small-scale instances.

		The MILp Model						The Hybrid VNS/LP Model				Difference		CV [%]
								Total Costs [€]				[%]
Inst.		Total Costs [€]	VSS Cost [€]	VWH Cost [€]	W Cost [€]	U Cost [€]	CPU Time [s]	Avg	Stdev	Min	Max	Avg	Min
A	1	1369.83	120.36	289.47	960	0.00	1408	1408.16	91.33	1369.83	1633.10	2.80	0.00	6.49
	2	667.49	67.49	0.00	600	0.00	1567	667.49	0.00	667.49	667.49	0.00	0.00	0.00
	3	528.78	0.00	48.78	480	0.00	908	532.83	9.64	528.78	555.78	0.77	0.00	1.81
	4	1048.51	325.93	122.58	600	0.00	359	1050.23	5.14	1048.51	1065.65	0.16	0.00	0.49
	5	969.63	85.70	306.05	480	97.88	239	969.63	0.00	969.63	969.63	0.00	0.00	0.00
	6	1528.63	292.90	515.73	720	0.00	485	1599.36	88.55	1528.63	1728.34	4.63	0.00	5.54
	7	960.00	0.00	0.00	960	0.00	2095	960.00	0.00	960.00	960.00	0.00	0.00	0.00
	8	840.00	0.00	0.00	840	0.00	914	840.00	0.00	840.00	840.00	0.00	0.00	0.00
	9	1481.00	161.00	0.00	1320	0.00	393	1498.61	42.20	1481.00	1659.06	1.19	0.00	2.82
	10	1181.35	0.00	461.35	720	0.00	1199	1203.18	29.87	1181.35	1291.65	1.85	0.00	2.48
Avg		1057.52	105.34	174.40	768	9.79	957	1072.95				1.14	0.00	1.96
B	1	1609.83	120.36	289.47	1200	0.00	558	1624.26	43.27	1609.83	1754.05	0.90	0.00	2.66
	2	817.49	67.49	0.00	750	0.00	1384	817.49	0.00	817.49	817.49	0.00	0.00	0.00
	3	645.78	195.78	0.00	450	0.00	1228	645.78	0.00	645.78	645.78	0.00	0.00	0.00
	4	1155.65	583.07	122.58	450	0.00	96	1157.80	9.34	1155.65	1198.51	0.19	0.00	0.81
	5	1114.10	85.70	306.05	600	122.35	181	1117.85	11.25	1114.10	1151.60	0.34	0.00	1.01
	6	1708.63	292.90	515.73	900	0.00	141	1734.12	45.73	1708.63	1846.41	1.49	0.00	2.64
	7	1182.38	282.38	0.00	900	0.00	1559	1182.38	0.00	1182.38	1182.38	0.00	0.00	0.00
	8	1050.00	0.00	0.00	1050	0.00	1550	1050.00	0.00	1050.00	1050.00	0.00	0.00	0.00
	9	1809.06	459.06	0.00	1350	0.00	343	1809.25	0.58	1809.06	1811.00	0.01	0.00	0.03
	10	1361.35	0.00	461.35	900	0.00	753	1371.09	20.77	1361.35	1455.33	0.72	0.00	1.51
Avg		1245.43	208.67	169.52	855	12.24	779	1251.00				0.37	0.00	0.87
C	1	4379.81	120.36	2966.43	960	333.02	441	4383.53	11.72	4379.81	4428.39	0.08	0.00	0.27
	2	1936.90	67.49	447.89	960	461.52	488	1954.75	22.24	1936.90	2012.46	0.92	0.00	1.14
	3	2828.32	195.78	823.99	480	1328.55	29	2846.33	25.23	2828.32	2910.82	0.64	0.00	0.89
	4	3280.86	183.04	2017.82	1080	0.00	803	3462.76	92.09	3280.86	3526.79	5.54	0.00	2.66
	5	3931.60	85.70	2358.46	840	647.43	247	3938.30	13.49	3931.60	3968.58	0.17	0.00	0.34
	6	4238.69	165.54	2633.15	1440	0.00	1496	4253.76	19.35	4238.69	4324.26	0.36	0.00	0.45
	7	1771.65	0.00	305.22	960	506.42	249	1804.67	32.83	1771.65	1868.40	1.86	0.00	1.82
	8	2315.39	0.00	765.71	960	589.68	75	2356.88	24.97	2315.39	2400.54	1.79	0.00	1.06
	9	3716.21	179.43	1976.78	1560	0.00	1145	3741.05	47.20	3716.21	3851.89	0.67	0.00	1.26
	10	4031.51	339.48	2871.01	600	221.03	* 3600	4042.13	14.61	4031.51	4070.74	0.26	0.00	0.36
Avg		3243.09	133.68	1716.65	984	408.77	857	3278.42				1.23	0.00	1.03
D	1	1399.92	150.45	289.47	960	0.00	811	1399.92	0.00	1399.92	1399.92	0.00	0.00	0.00
	2	684.36	84.36	0.00	600	0.00	1445	684.36	0.00	684.36	684.36	0.00	0.00	0.00
	3	528.78	0.00	48.78	480	0.00	881	539.46	25.43	528.78	600.00	2.02	0.00	4.71
	4	1130.00	407.41	122.58	600	0.00	195	1138.14	24.43	1130.00	1211.42	0.72	0.00	2.15
	5	991.05	107.13	306.05	480	97.88	295	1005.41	47.49	991.05	1197.91	1.45	0.00	4.72
	6	1601.85	366.13	515.73	720	0.00	321	1669.91	101.30	1601.85	1923.60	4.25	0.00	6.07
	7	960.00	0.00	0.00	960	0.00	1631	960.00	0.00	960.00	960.00	0.00	0.00	0.00
	8	840.00	0.00	0.00	840	0.00	1061	840.00	0.00	840.00	840.00	0.00	0.00	0.00
	9	1521.25	201.25	0.00	1320	0.00	364	1547.77	53.03	1521.25	1653.83	1.74	0.00	3.43
	10	1181.35	0.00	461.35	720	0.00	999	1212.67	47.83	1181.35	1285.73	2.65	0.00	3.94
Avg		1083.86	131.67	174.40	768	9.79	800	1099.76				1.28	0.00	2.50
E	1	17,025.49	15,006.24	339.24	1680	0.00	* 3600	17,436.19	422.54	17,025.49	18,133.88	2.41	0.00	2.42
	2	2752.42	952.42	0.00	1800	0.00	140	2943.54	235.49	2752.42	3298.03	6.94	0.00	8.00
	3	1210.25	130.25	0.00	1080	0.00	70	1260.68	52.68	1210.25	1450.25	4.17	0.00	4.18
	4	16,508.71	15,188.71	0.00	1320	0.00	972	16,623.57	216.28	16,508.71	17,264.36	0.70	0.00	1.30
	5	14,725.33	13,661.05	104.28	960	0.00	* 3600	14,999.34	262.63	14,725.33	15,358.59	1.86	0.00	1.75
	6	24,296.25	22,769.38	206.88	1320	0.00	* 3600	25,053.78	268.60	24,484.38	25,449.34	3.12	0.77	1.07
	7	3176.07	1736.07	0.00	1440	0.00	184	3205.17	89.92	3176.07	3535.16	0.92	0.00	2.81
	8	4057.90	2137.90	0.00	1920	0.00	257	4442.31	378.22	4057.90	5421.86	9.47	0.00	8.51
	9	30,405.42	29,085.42	0.00	1320	0.00	1689	30,473.46	255.90	30,405.42	31,573.86	0.22	0.00	0.84
	10	17,134.78	15,381.21	73.58	1680	0.00	* 3600	17,578.61	393.72	17,134.78	18,405.39	2.59	0.00	2.24
Avg		13,129.26	11,604.86	72.40	1452	0.00	1771	13,401.67				3.24	0.08	3.31
Total Avg		3951.83	2436.85	461.47	965.40	88.12	1033	4220.76				1.45	0.02	1.93

* CPLEX time limit parameter was set to 3600 s.

. The hybrid VNS/LP model was able to find the same optimal solution as the MILP model at least once (out of 20 solving iterations), except for one instance (for instance, the E-6 heuristic minimum solution gap to MILP solution was 0.77%). On average, for all 50 small-scale instances and 20 iterations, the hybrid approach returns solutions that have 1.45% higher total costs than MILP solutions. At the same time, the computational time for heuristics was on average less than 6% of the MILP computational time required to obtain the solution. The highest average difference of 3.24% was for variant E with increased demand (due to increased combinatorial complexity), while the lowest average difference of 0.37% was for variant B with an increased cost of production setup and unused time. The combinatorial complexity of variant E is also evident from the required computational time to obtain a solution with the MILP model, which is considerably higher than in the case of other variants (for four instances of variant E, the MILP model was not able to confirm the optimality of the solution within 3600 s). The increased demand in variant E leads to zero unused production time and increased changeover time, and considerably lower stock than safety-stock level (solutions with an increased risk of understock, although without any backordering). In variant C, with a lower starting level of available warehouse capacity, the cost of warehouse capacity overflow and the cost of unused production time are significantly increased (in this case, it is better to produce less than to have the increased cost for 3PL warehousing). There is no significant difference between variants A, B and D (basic, increased setup and unused time, increased inventory costs), except an almost double safety-stock violation cost in variant B. Backordering and shelf-life violation was not a part of any small-scale instance solution (the models were able to obtain solutions without any backordering or violation of shelf-life). The average coefficient of variation (CV) for all 50 small-scale instances of the hybrid approach solutions was 1.93%.

Although variant E has increased combinatorial complexity compared to other variants, it does not mean that other variants are easily solvable for the case of large-scale dimensions. In addition, combinatorial complexity is to some extent dependent on the input values. This is noticeable from the results in Table 4, where the CPU time to solve small-scale instances by the MILP model can vary significantly for different instances of the same variant (in total, this time is between 29 and 3600 s).

5.2. Large-Scale Instance Computational Results

To solve large-scale instances, we set the maximum computational time limit to 4 h. The hybrid VNS/LP model solved 10 iterations of each large-scale instance. Large-scale instances have a significantly increased complexity compared to small-scale instances. The reason for this is the increased values of several input parameters: from 8 h to 20 h of total available production time, from 10 to 30 different products, from 6 to 20 raw materials, from 3 to 10 working days of the planning horizon, and from 3 to 5 micro-periods per day. As a consequence, large-scale instances could not be solved by the MILP model in a reasonable computational time (e.g., one working day).

The results of the large-scale instances obtained from the hybrid VNS/LP model are presented in

Table 5. Results obtained from the hybrid VNS/LP models for the large-scale instances.

		Total Costs [€]				Partial Costs [€]					Diff. of Avg to Min [%]	CV [%]
Inst.		Avg	Stdev	Min	Max	I- Cost	VSS Cost	VWH Cost	W Cost	U Cost
A	1	8569.46	257.72	8222.75	9058.43	0.00	286.57	416.89	7836.00	30.00	4.22	3.01
	2	5562.91	167.29	5308.48	5872.97	0.00	2.85	180.06	5340.00	40.00	4.79	3.01
	3	3723.06	222.43	3295.54	4051.39	0.00	75.72	141.35	3336.00	170.00	12.97	5.97
	4	11,584.90	353.00	11,017.14	12,142.37	0.00	1710.94	333.96	9540.00	0.00	5.15	3.05
	5	7337.13	175.28	7008.97	7548.19	0.00	121.36	449.77	6756.00	10.00	4.68	2.39
	6	5750.46	207.06	5290.54	6133.85	0.00	104.95	109.51	5496.00	40.00	8.69	3.60
	7	17,026.30	726.85	15,507.13	17,884.30	0.00	4160.51	25.25	12,828.00	12.55	9.80	4.27
	8	12,899.19	603.95	12,141.20	14,280.53	0.00	2537.55	533.65	9768.00	60.00	6.24	4.68
	9	4891.94	119.27	4724.31	5056.31	0.00	79.20	404.75	4308.00	100.00	3.55	2.44
	10	5517.82	141.32	5257.68	5793.42	0.00	91.87	413.95	4992.00	20.00	4.95	2.56
Avg		8286.32				0.00	917.15	300.91	7020.00	48.25	6.50	3.50
B	1	10,445.15	303.24	10,166.58	11,022.20	0.00	355.31	444.84	9570.00	75.00	2.74	2.90
	2	7020.43	241.83	6417.38	7272.28	0.00	75.78	247.15	6660.00	37.50	9.40	3.44
	3	4600.79	232.70	4116.47	4846.52	0.00	39.47	198.82	4200.00	162.50	11.77	5.06
	4	13,685.41	423.88	12,876.41	14,190.08	0.00	1841.26	329.14	11,490.00	25.00	6.28	3.10
	5	8994.01	312.88	8576.09	9547.43	0.00	191.16	607.85	8145.00	50.00	4.87	3.48
	6	7072.46	266.40	6668.70	7591.75	0.00	166.83	118.14	6750.00	37.50	6.05	3.77
	7	20,278.84	733.16	18,690.54	21,453.71	0.00	4585.71	98.08	15,570.00	25.05	8.50	3.62
	8	15,131.46	793.79	14,110.40	16,188.26	0.00	2691.35	515.12	11,775.00	150.00	7.24	5.25
	9	5902.46	217.50	5565.09	6224.74	0.00	57.37	412.59	5370.00	62.50	6.06	3.68
	10	6600.75	200.88	6214.07	6820.52	0.00	212.83	432.92	5955.00	0.00	6.22	3.04
Avg		9973.18				0.00	1021.71	340.46	8548.50	62.50	6.91	3.73
C	1	16,503.16	208.03	16,181.29	16,844.74	0.00	198.11	7456.64	8508.00	340.42	1.99	1.26
	2	14,540.44	101.44	14,393.41	14,734.91	0.00	6.24	6536.71	6480.00	1517.49	1.02	0.70
	3	14,702.20	157.71	14,480.81	14,983.58	0.00	41.30	5032.90	5028.00	4599.99	1.53	1.07
	4	18,544.93	362.15	17,871.28	19,053.99	0.00	1366.06	7352.78	9768.00	58.09	3.77	1.95
	5	15,835.51	161.04	15,617.86	16,102.73	0.00	45.94	7472.74	7932.00	384.83	1.39	1.02
	6	14,239.51	147.36	14,002.33	14,572.45	0.00	91.24	6544.67	6996.00	607.61	1.69	1.03
	7	23,310.24	532.25	22,641.69	24,372.19	0.00	3645.16	6546.02	13,104.00	15.06	2.95	2.28
	8	19,776.42	515.07	19,014.67	20,765.34	0.00	1634.42	8103.11	9936.00	102.89	4.01	2.60
	9	14,819.08	112.71	14,696.65	15,018.73	0.00	50.20	6219.75	5724.00	2825.13	0.83	0.76
	10	15,374.67	105.05	15,163.76	15,517.90	0.00	25.92	6453.97	6456.00	2438.79	1.39	0.68
Avg		16,764.62				0.00	710.46	6771.93	7993.20	1289.03	2.06	1.34
D	1	8898.14	243.74	8487.62	9378.16	0.00	475.52	436.62	7956.00	30.00	4.84	2.74
	2	5677.32	183.94	5388.61	5993.16	0.00	8.25	297.07	5292.00	80.00	5.36	3.24
	3	3731.76	191.42	3479.28	4167.96	0.00	15.41	174.36	3432.00	110.00	7.26	5.13
	4	11,746.16	488.83	10,669.84	12,540.02	0.00	1801.26	200.90	9744.00	0.00	10.09	4.16
	5	7350.83	270.33	6904.37	7771.92	0.00	69.00	481.83	6780.00	20.00	6.47	3.68
	6	5871.09	277.50	5530.59	6374.95	0.00	85.99	99.09	5616.00	70.00	6.16	4.73
	7	18,072.68	683.84	16,761.50	19,012.81	0.00	4674.28	70.87	13,320.00	7.53	7.82	3.78
	8	13,465.46	509.28	12,763.76	14,332.99	0.00	2788.97	508.49	10,128.00	40.00	5.50	3.78
	9	4833.67	127.59	4652.95	5072.43	0.00	44.41	383.26	4296.00	110.00	3.88	2.64
	10	5515.51	167.47	5228.57	5788.41	0.00	68.73	448.78	4968.00	30.00	5.49	3.04
Avg		8516.26				0.00	1003.18	310.13	7153.20	49.75	6.29	3.69
E	1	74,451.91	3042.82	68,311.96	78,167.86	0.00	57,070.45	929.46	16,452.00	0.00	8.99	4.09
	2	9091.98	293.36	8635.51	9619.43	0.00	1383.89	120.10	7548.00	40.00	5.29	3.23
	3	181,091.59	3394.45	174,701.68	186,434.78	100.94	166,737.44	237.20	14,016.00	0.00	3.66	1.87
	4	50,024.48	2552.62	46,430.19	54,421.92	0.00	33,691.72	372.76	15,960.00	0.00	7.74	5.10
	5	27,144.15	1398.61	25,016.19	29,554.75	0.00	12,296.67	3.48	14,844.00	0.00	8.51	5.15
	6	249,303.88	2660.60	243,337.80	252,563.56	228.42	235,556.86	162.59	13,356.00	0.00	2.45	1.07
	7	170,383.97	4061.77	163,621.85	178,858.52	75.48	156,550.33	174.17	13,584.00	0.00	4.13	2.38
	8	11,747.09	356.91	11,116.08	12,503.61	0.00	1724.75	266.35	9756.00	0.00	5.68	3.04
	9	13,475.52	988.26	12,136.74	15,640.31	0.00	2461.13	444.38	10,560.00	10.00	11.03	7.33
	10	22,786.11	1240.46	21,508.19	25,336.75	0.00	9419.67	106.44	13,260.00	0.00	5.94	5.44
Avg		80,950.07				40.48	67,689.29	281.69	12,933.60	5.00	6.34	3.87
Total Avg		24,898.09				8.10	14,268.36	1601.03	8729.70	290.91	5.62	3.23

. The average coefficient of variation for all 50 large-scale instances of the hybrid approach solutions is 3.23%, while the average total cost of heuristic solutions has a 5.62% higher total cost than the best obtained heuristic solution. In the case of large-scale instances, variant C, with a lower starting level of available warehouse capacity, has a significantly lower coefficient of variation and “difference to average” value than other variants. Only variant E, with an increased demand for large-scale instances, has some instances in which backordering was a part of the solution.

An example of the convergence of large-scale solutions obtained from the hybrid VNS/LP model, for instance 1 (each line represents an iteration) in all five variants of the problem setup, is presented in Figure 3. The proposed approach has difficulty obtaining high convergence for the case of variant E in the available computational time of 4 h. For other variants, the proposed approach has a significantly better convergence of solutions.

5.3. The Effectiveness of Neighborhoods Used in the Local Search Procedure

The effectiveness of neighborhoods used in the local search procedure is presented in Figure 4, and is presented as the average number of found improvements per each variant of the problem setup. Because of the neighborhoods’ order randomization in each local search run, all neighborhoods had an equal chance to find improvement. The Insertion and Change product neighborhoods have shown the best performance in both small- and large-scale instances of all variants, which were followed by the Swap 2 (especially for the large-scale instances), Removal and Swap 3 neighborhoods. The details regarding the overall effectiveness of the proposed hybrid VNS/LP model, per key model phases (value of the objective function and CPU time) are presented in

Table 6. Overview of proposed hybrid VNS/LP model performance: average total costs and CPU time per key model phases.

Instances		No Production		BRI Insertion		Local Search Improvement of BRI		Final VNS/LP Solution
		Avg. Total Cost [€]	Avg. CPU Time [s]	Avg. Total Cost [€]	Avg. CPU Time [s]	Avg. Total Cost [€]	Avg. CPU Time [s]	Avg. Total Cost [€]	Avg. CPU Time [s]
Small-scale	A	54,948.6	0	1769.7	1.0	1277.2	2.8	1072.9	60
	B	57,108.6	0	2092.2	1.0	1453.2	2.8	1251.0	60
	C	55,132.4	0	4038.6	1.0	3447.3	3.1	3278.4	60
	D	66,525.8	0	1846.6	1.1	1353.3	3.0	1099.8	60
	E	125,319.5	0	24,739.7	0.8	15,462.8	3.5	13,401.7	60
Avg		71,807.0	0	6897.4	1.0	4598.8	3.1	4020.8	60
Large-scale	A	2,535,157.5	0	91,090.8	39.4	11,660.4	566.3	8286.3	14,400
	B	2,553,157.5	0	93,794.3	39.6	13,901.3	538.3	9973.2	14,400
	C	2,536,945.2	0	111,658.9	42.0	18,461.8	668.5	16,764.6	14,400
	D	3,150,946.9	0	112,268.2	38.9	12,969.7	531.3	8516.3	14,400
	E	4,295,280.8	0	857,068.2	25.6	92,433.1	911.7	80,950.1	14,400
Avg		3,014,297.6	0	253,176.1	37.1	29,885.3	643.2	24,898.1	14,400

.

6. Concluding Remarks

In this paper, we considered the lot-sizing and scheduling problem found in a fruit juice company. To reflect an additional real-life environment, we developed additional objective function segments and constraints for the inventory management of finished products (regarding the shelf-life concept and limited warehouse capacity). The proposed approach can serve as a framework for a decision maker to have an overview of the impact of different important input parameters (production time costs, warehouse capacity and costs, inventory related costs and production demand) on the total cost of the production process, and accordingly can help managers to make better decisions regarding production lot-sizing and scheduling in changing conditions.

To solve this novel problem formulation, we developed the MILP and hybrid VNS/LP models. For 50 small-scale instances, the hybrid VNS/LP model was able to find the same solution as the MILP model, except for one instance where the best heuristic solution had 0.77% higher total costs. The MILP model was not able to solve large-scale instances to optimality in a reasonable computational time. The hybrid VNS/LP model was able to solve 50 large-scale instances for different variants of the problem setup (regarding production setup and idle costs, warehouse capacity, demand intensity, and inventory costs). The biggest challenge for the hybrid VNS/LP model was to obtain high convergence for the set of large-scale instances with increased demand, due to the higher combinatorial complexity of instances. For this case, a longer computational time is required in order to obtain better solutions. The production plan is usually made in one working day for a period of more than a week; thus, it is easily achievable to extend the maximum computational time used to solve the large-scale instances beyond the current 4 h. Based on the analysis of neighborhoods used in the local search procedure, the most promising neighborhoods were Insertion and Change product, which were followed by Swap 2, Removal and Swap 3, respectively. Interesting future research directions could be the development of faster search techniques that can use the adaptive neighborhood search principle (based on the neighborhoods’ effectiveness analyzed in this study), and the parallelization concept of the heuristic approach using multicore processors (e.g., parallelization of neighborhoods in local search).