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Optimal Production Scheduling in the Dairy Industries Georgios P. Georgiadis, Georgios Kopanos, Antonis Karkaris, Harris Ksafopoulos, and Michael C. Georgiadis Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b05710 • Publication Date (Web): 27 Mar 2019 Downloaded from http://pubs.acs.org on March 28, 2019

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Industrial & Engineering Chemistry Research

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Optimal Production Scheduling in the Dairy Industries

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Georgios P. Georgiadisa,b, Georgios M. Kopanosa, Antonis Karkarisc, Harris

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Ksafopoulosc, Michael C. Georgiadisa,b,*

*

To whom correspondence should be addressed e-mail [email protected]

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aDepartment

of Chemical Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124,

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Greece

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bChemical

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Hellas (CERTH), PO Box 60361, 57001, Thessaloniki, Greece

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cMEVGAL

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Abstract

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This work presents the application of a Mixed-Integer Linear Programming (MILP) model, that

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was previously developed in our group, for the lot-sizing and production scheduling problem in a

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real-life industry. In particular, the case of yoghurt production, a representative food process, in

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a large-scale dairy facility from Greece is studied in detail. The model takes into account all

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constraints typically met in production scheduling (inventory limitations, material balances,

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equipment capacity etc.) having as major objective the minimization of cost. New information

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regarding new orders, order modifications and/or cancellations and deviations of actual to

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planned production is considered by utilizing a rolling horizon algorithm. The suggested solution

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strategy has been successfully integrated with the industrial partner’s system to facilitate the

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decision-making process for the production scheduling of the facility.

Process and Energy Resources Institute (CPERI), Centre for Research and Technology

Dairy Industry, Koufalia 57100, Greece

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Keywords: production scheduling, rescheduling, dairy industry, MILP model, uncertainty

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1. Introduction

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Market trends and competitiveness in food industries result in a complex production process that

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requires greater flexibility and efficient coordination of resources. Production scheduling is the

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major component for the efficient management of production. Scheduling refers to the allocation

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of equipment, utilities and labor resources over a time horizon of interest, in order to execute all

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required processing tasks for the satisfaction of product demands. In practice, all scheduling2 ACS Paragon Plus Environment

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related decisions are mainly derived by managers or operators; thus, the overall performance

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and plant productivity are subject to their experience and understanding of the status of each

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production equipment or line. In order to systematically improve their decisions, computer-aided

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tools can significantly enhance the global production scheduling by proper consideration of the

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involved parameters and the dynamic demand changes1. Although optimization of production

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scheduling is a widely studied topic in the context of process industries2, few works have studied

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real food processing industries3. One of the main reasons is the complex flow process met in food

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industries. A typical food processing facility is a make-and-pack process. In particular, multiple

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batch and/or continuous stages are required to prepare the food-stuff, which is finally packed in

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a form appropriate for the needs of the market. Polon4 studied the scheduling of a sausage

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industry without considering the packaging stage, which often constitutes the main production

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bottleneck. A novel MILP was proposed by Baldo5 for a brewery industry, and heuristics were

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integrated for the investigation of real-life study cases. Xie and Li6 proposed an analytical model

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based on collected data for a single meat shaving and packaging line of a meat company.

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Production scheduling of batch or continuous processes has been a hot topic in the last 30 years.

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Exact methods based on mathematical programming (MILP) have been mostly utilized, since they

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are flexible, rigorous and provide optimal solutions. The main mathematical frameworks for

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production scheduling rely on generic process representations such as the state-task-network7

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and the resource-task-network8. Based on the time representation, the available models in the

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literature can be classified into discrete9,10 and continuous11–15. Floudas and Lin16 and Mendez et

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al.17 provide comprehensive overviews of the available mathematical formulations for

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production scheduling in the process industries. Both representations display specific strengths

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and weaknesses. Discrete-time formulations provide a reference time-grid for all shared

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resources, hence material balances between production stages, inventory and backlog levels, as

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well as availability and consumption of utilities can be monitored and modelled without the

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introduction of any nonlinearities. Moreover, recent studies18 have displayed the supremacy of

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discrete formulations in terms of solution quality, when dealing with large problem instances.

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Nevertheless, discrete formulations lead to very large and often intractable models, especially

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when small discretization of time is required. Continuous-time formulations can potentially 3 ACS Paragon Plus Environment


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alleviate some of the computational problems that arise by discrete-time formulations, since in

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principle fewer binary variables are required, while they can provide more accurate solutions.

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However, they lack efficiency when processing features like modelling, material balances,

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inventories and shared utilities. In order to combine the advantages of both methods, Lee and

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Maravelias19,20 proposed a three-stage method for chemical production scheduling problems.

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The proposed framework showed interesting results as it maintains the advantages of discrete

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formulations, while it addresses their major disadvantage, specifically, the accuracy of the

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provided solutions.

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Dairy manufacturing is a major food industry and a significant component in most economies.

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For example, in 2015 the estimated global market for industrial yoghurt production was about

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67 billion dollars showing a significant increase of production over the last years21. However, the

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use of optimization techniques for the production scheduling problem in a real dairy plant has

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received little attention. This can be mainly attributed to the unique features that differentiate

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dairy from other industries. The high product complexity occurring at the packaging level,

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combined with sequence dependent setup operations, shared resources, multiple identical

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machines and the use of both mixed batch and continuous production modes make production

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scheduling of dairy plants a challenge22. Moreover, tight operating, design and quality constraints

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such as perishability issues23,24 and preferable production sequences25 need to be taken into

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account.

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Entrup et. al22 presented three different MILP formulations, for planning and scheduling

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problems in the packing stage of the yoghurt production. A combined discrete-continuous

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representation was employed, while shelf life issues were explicitly considered. Doganis and

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Sarimveis26 presented a MILP model for the production scheduling of a yoghurt packaging facility

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in Greece. The model considered sequence dependent setup times and costs in addition to

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material balances, storage limitations, manpower and equipment capacity. However, their study

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was restricted to one packaging line. The same authors extended their previous work to optimally

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schedule multiple parallel packaging machines over a 5-day horizon to meet the demand for 25

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final products27. All parallel packaging lines shared the same feeding line, therefore simultaneous

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packaging of multiple products was not allowed, thus simplifying the problem at hand. 4 ACS Paragon Plus Environment

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Additionally, possible limitations of the previous stages (fermentation-pasteurization) were

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neglected. Marinelli et al.28 addressed the planning problem in a packing line producing yoghurt

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products, assuming that all setup times and costs are sequence-independent. The authors

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presented a two-stage heuristic to obtain near-optimal solutions for the problem under study.

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Gellert et al.29 investigated the scheduling problem for filling lines in the dairy industry. However,

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their problem only concerned the filling lines and all potential limitations regarding the rest of

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the processing system were completely ignored. Kopanos et al.30 developed a novel

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mathematical framework for the production scheduling and lot-sizing in yoghurt production

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lines. The problem was mainly focused on the packaging stage, while timing and capacity

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constraints of the pasteurization, homogenization and fermentation stage were imposed to

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ensure the feasibility of the extracted schedules. Extending this work, Kopanos et al.31 considered

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renewable resource limitations. Bilgen and Celebi32 proposed a hybrid method, which combines

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MILP modelling and iterative simulation approaches to solve the integrated production

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scheduling and distribution planning problem in yoghurt production. Wari and Zhu33 presented

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an MILP model of the multi-week scheduling problem for an ice cream facility. Another common

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issue in dairy industries is the quality decay of the intermediate and final products. Claasen et

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al.34 developed a model that addresses product decay of inventory due to perishability. Recently,

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Sel et al.35 proposed a chanced-constrained programming model that accounts for the

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uncertainty of lifetime of intermediate products. A thorough review of all quantitative models

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for the production scheduling of dairy plants can be found in Sel and Bilgen36.

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Few attempts have been made to address production scheduling problems of real-life plants,

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thus a gap is noticed between theory and industrial practice37. Kopanos et al.38 studied a real-life

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ice-cream facility, aiming to minimize the production makespan, while fulfilling the weekly

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demand for 24 products. Extending their previous work Kopanos et al.39 introduced valid integer

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cuts, to improve the computational efficiency of their proposed model. Baumann and

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Trautmann40 proposed a hybrid three-stage method for the scheduling optimization of make-

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and-pack processes, by combining an MILP model and a critical path-based improvement

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method. Aguirre et al.41 addressed medium-sized multistage continuous processes with a single

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unit in every stage. They proposed a solution strategy relying on a combination of exact 5 ACS Paragon Plus Environment


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optimization methods, a rolling horizon approach and iterative improvement algorithms. van

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Elzakker et. al42 developed a problem-specific MILP model to address the scheduling problem in

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the Fast Moving Consumer Goods (FMCG) industry and tested the model’s efficiency on an ice

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cream industry scheduling problem. An extremely large optimization problem for the tactical

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planning in FMCG industry has been solved by van Elzakker et.al43. The authors proposed a novel

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MILP model and a SKU-based decomposition algorithm. Bilgen and Dogan44 solved the multi-

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stage production planning problem of a milk industry, taking into account shelf life requirements

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and allowing for intermediate storage between the continuous processes. Recently, Georgiadis

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et al.45 examined a complex real-life canned-fish production facility. They proposed a case-

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specific continuous MILP model to efficiently model both batch and continuous processing stages

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and an order-based decomposition strategy to solve large industrial study cases.

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In real-world situations, production scheduling is affected by numerous sources of uncertainty.

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Many parameters, like processing times, stream quality, yields etc. are inherently uncertain,

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while unforeseen events, such as machine breakdowns, lack of personnel or raw materials, could

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occur. Thus, scheduling imposes an inherently dynamic behaviour that needs to be accounted

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through rescheduling, since a suboptimal production plan can lead to product waste and financial

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losses46,47. The rescheduling approaches can be categorized into deterministic/reactive and

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stochastic/proactive. In the former, a model is solved in an iterative manner whenever there is

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new information in the system48–51, while stochastic rescheduling explicitly models any

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uncertainties52,53. A thorough review of the techniques for scheduling under uncertainty can be

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found in Li and Ierapetritou54. One of the salient features of the dairy industry is that order

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quantities are confirmed or changed just prior to their dispatch. In some industries this could be

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handled by carrying out stocks (make-to-stock). However, this is not the case in the yoghurt

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industry. Since yoghurt is a perishable product, building up inventories will result to compromises

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in quality and freshness. Moreover, the expected profits would decrease, due to reduced selling

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prices and increased costs which are typically required to maintain an inventory (cooling).

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Additionally, there is a constant pressure to maximize the shelf life of products. For example, a

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customer will probably prefer a yoghurt that is 2 weeks prior to its expiration time, rather than a

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yoghurt that expires in the upcoming days, leading to reduction of sales and product waste. Since 6 ACS Paragon Plus Environment

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the total shelf life of a yoghurt product is limited (30-54 days depending on the type of yoghurt),

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it is important to narrow down the time between production and on-shelf availability as much as

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possible. Consequently, yoghurt, as any fresh food industry, operates in a make-to-order fashion,

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rather than a make-to-stock, leading to reduced production flexibility. As a result the production

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facility may not be able to handle demand variations. This may lead to increased production

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costs, reduced productivity and considerable product waste. Therefore, rescheduling techniques

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and production agility is crucial for the efficient operation of dairy plants. However, integration

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of uncertainty in the dairy industry is still in its infancy55–57.

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Only in recent years practical cases of optimization-based scheduling have been presented, while

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very few real case studies have addressed the issue of uncertainty3. Furthermore, most of the

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previous studies do not illustrate the practical integration of optimization-based models and

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approaches in the decision-making infrastructure of an industrial facility. In this work, the

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industrial application of an optimization-based technique for the production scheduling problem

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of a large scale yoghurt production facility is addressed. A real-life industrial case study is

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optimally solved, using an MILP framework proposed in our previous work58. All realistic

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operational and design constraints, such as, mass balances, equipment capacity, inventory

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limitations and sequence-dependent setup times have been taken into account, while valid

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industry specific cuts and aggregation techniques (product families) are utilized to reduce the

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underlying computational cost. An extension of the MILP model through the implementation of

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a rolling horizon framework is proposed to efficiently handle new information (order

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cancellations, new orders, change in order quantities) and improve the production flexibility of

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the examined yoghurt facility. Finally, the proposed MILP framework and solution strategy are

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integrated with the ERP tool of the company to develop a computer-aided decision-making tool

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in order to assist the production engineers to generate near-optimal weekly schedules.

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The remainder of this paper is organized as follows: In section 2, the industrial process of yoghurt

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production is described, while in section 3 the problem in hand is formally defined. Section 4

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presents the mathematical model and the proposed solution strategy for the consideration of

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new information. Finally, in section 5 several industrial study cases are solved to illustrate the

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efficiency and applicability of the suggested approach. 7 ACS Paragon Plus Environment


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2. Yoghurt production process

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Yoghurts are dairy products obtained from the lactic acid fermentation of the bacteria species

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Streptococcus thermophilus and Lactobacillus delbrueckii. While all yoghurt products originate

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from the fermentation process of milk, the final products can have various sensory properties

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which rely on: a) the composition of milk, that depends on the source (e.g. cow or goat milk,

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conventional or organic) and the fat content; b) the addition of other ingredients, such as fruits,

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nuts, sweeteners, stabilizers or emulsifiers; and c) the technology employed during milk

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pretreatment and yoghurt posttreatment. The main types of yoghurt products are set and stirred

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yoghurt, however a multitude of new recipes have been created to address customer needs,

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leading to a broad diversification of yoghurt products and a significant increase in the

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consumption of yoghurt. Yoghurt products can vary on the texture (set, stirred,

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concentrated/strained, frozen, dried), flavor (natural, with added fruits, nuts or honey,

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sweetened or flavored) and nutritional content (fat and lactose content). Yoghurt production is

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a complex procedure that requires multiple processing steps (figure 1). A brief description of

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these stages follows.

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Fig. 1. Schematic diagram of yoghurt production 8 ACS Paragon Plus Environment

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Collection of milk

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Initially, milk is collected in a daily basis from dairy farms. It should be noted that the chemical

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composition of milk (e.g. content of water, fat, protein or lactose) varies from day to day even

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within the same breed. This is due to multiple factors such as the season of year, the stage of

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lactation, the age and the nutrition of the animal. In this stage, fresh milk contains contaminants

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from the udder of the animal, hair, straws, leaves and soil. These are removed, usually with cloth

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or centrifugal filters, to ensure product safety and quality. Finally, road tankers equipped with

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cooled containers are used to transfer the fresh milk to the plant under refrigeration (at about

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5oC).

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Milk pretreatment

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In this step, the fresh milk that has been delivered to the dairy plant is prepared in order to obtain

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the mix to be fermented. First, standardization of the fresh milk in terms of fat and protein

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content takes place. Fat standardization is done by removal of fat from milk by centrifugation at

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a high temperature (close to 55oC), followed by reincorporation of cream to get the desired fat

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content. With the addition of milk powder or milk proteins and replacers like whey or caseinates,

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increased protein-content is achieved. This process improves the yoghurt’s texture, while it

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reduces syneresis. For fruit or flavored yoghurt, sweetening agents are added after fat and

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protein standardization. In some countries the use of thickeners and stabilizers (gelatin,

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carrageenan, starch etc.) is allowed to improve the yoghurt texture. Following standardization,

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the milk is heated at around 90oC for 3-7 minutes to remove any spoilage microorganisms.

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Afterwards, homogenization takes place at high pressure (20 or 25Mpa) and a temperature of

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around 70oC. Homogenization is compulsory for yoghurt quality as it (i) reduces the size of fat

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globules, (ii) increases gel texture and (iii) reduces syneresis.

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Fermentation process

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The heat-treated milk is cooled down to 42oC and is inoculated with starter cultures

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(Streptococcus thermophilus and Lactobacillus delbrueckii) to initiate the lactic-acid fermentation

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of milk. The next production step depends on the type of yoghurt produced. Stirred-type

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yoghurts are transferred to the fermentation tanks, while set-type are directly packaged 9 ACS Paragon Plus Environment


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(fermentation in cups). The fermentation time may vary due to numerous reasons like starter

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composition, temperature and final product type.

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Posttreatment and packaging

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The fermentation process is stopped with fast cooling when the targeted pH of the yoghurt (4.8

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- 4.5) is achieved. At this point flavoring, filling and packaging take place. For fruit yoghurts,

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aromas, sweeteners, fruits or nuts can be added to stirred yoghurts. For all other yoghurts this

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stage is skipped. Multiple parallel lines perform the filling of yoghurt in cups, as well as labeling

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and packaging of the cups in the final desired form (single cups, 6-packs etc.). The equipment

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needs to be cleaned and/or sterilized between the packaging of any two different products.

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Finally, the packaged products are stored at a temperature below 10oC, which is maintained

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during transportation and commercialization. Maintaining the product at this low temperature

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limits postacidification of yoghurt and ensures the safety of the final product. The interested

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reader can find more detailed information regarding yoghurt manufacturing in Varnam and

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Sutherland59 and Corrieu and Béal21.

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3. Problem Statement

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This work was initiated by an effort to address the lot-sizing, scheduling and rescheduling

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problem at a yoghurt production line of a real multiproduct dairy plant. In a real plant, the lot-

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sizing and scheduling decisions are usually made heuristically, based on the experience of the

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operators. This is a very challenging task, especially in the food industry due to the existence of

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sequence-dependent changeovers. The operators are sequentially solving first the lot-sizing

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problem, without taking into account the changeovers, and then the scheduling problem. Very

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often changeovers are not correctly accounted, therefore infeasible schedules may be generated.

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As a result, the procedure must be re-initiated and even after many iterations a near-optimal

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schedule is rarely achieved. The utilization of an exact mathematical method that incorporates

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all industry-specific parameters, allows the integration of the lot-sizing and scheduling problem

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that leads to solutions otherwise unachievable.

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In this study, the production process of MEVGAL Ltd., one of the largest dairy industries in Greece,

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that deals with over 150 different products of set or stirred yoghurt, is considered. Both yoghurt 10 ACS Paragon Plus Environment

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types are processed in a similar manner. The only difference lies in the fermentation process.

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More specifically, set yoghurt is initially packed and then fermented in the final retail container,

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while stirred yoghurt is fermented before the packaging stage, which is fulfilled by 6 parallel lines.

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The short-term scheduling horizon of the plant is 5 days (Monday to Friday), whereas the

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production time is 24 hours. The plant remains closed during weekends. Each product may have

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multiple intermediate due dates within the scheduling horizon. Demand for all orders must be

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met on time, so tardiness is not allowed. Orders that are not met on time are considered as lost

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sales and they are represented in the total production costs. Prior to their distribution, the final

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products are stored in refrigerated conditions, thus earliness, despite being allowed is not

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preferred due to the imposed holding costs. Rescheduling needs to be addressed, since

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suboptimal schedules after demand disruptions lead to product waste, decreased productivity

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and increased operating costs, while orders may not be satisfied. The yoghurt production facility

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(see figure 2) is a multistage multiproduct mixed batch-continuous plant with multiple parallel

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units. However, the packaging stage was identified as the main bottleneck. The timing and

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capacity constraints along with switchover times are imposed with respect to the batch stage

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(fermentation) to ensure the generation of feasible production plans and optimal production

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scheduling in the packaging stage. This approach immensely reduces the computational

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complexity of the problem without sacrificing the quality of the provided solution.

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Fig. 2. Yoghurt production line layout31



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All data are assumed to be deterministic, while resources-related constraints, such as manpower,

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steam, water, electricity etc., are not taken into account, since they are irrelevant to the industrial

3

case under consideration. Shelf-life issues are incorporated in the model implicitly. The

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consideration of holding costs in the objective function favour a just-in-time production. As a

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result, the products’ shelf life is maximized from a production scheduling point-of-view.

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The problem under consideration can be formally defined in terms of the following items:

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(i)

A known scheduling horizon T divided into a set of time periods n∈ N.

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(ii)

The set of packaging lines j ∈ J, with daily availability of ωjn hours and a specific setup cost νjn.

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(iii)

The set of products p ∈ P to be processed within the scheduling horizon, with

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specific demand (ζpn), production (ηpn) and inventory (ξpn) costs, initial (  0p ) and

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safety stock (  pmin ), processing rates (ρpj), minimum (  pmin, j , n ) and maximum (  pmax , j ,n )

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processing runs. Moreover, the cost of failing to meet each products safety stock

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(υp) and the lost sales cost (ψpn) is given. Jp specifies the packaging lines that can

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process product p.

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(iv)

Products p are categorized into families f ∈ F. Subset Pf denotes the products

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belonging to family f, while Fj are the available lines j to process family f.

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Accordingly, Jf defines the families that can be processed by line j. Each family f

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has a given operating (𝜃𝑓𝑗𝑛) and setup cost (χfjn).

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(v)

Each family f is associated to a fermentation recipe r ∈ R, utilizing the subset Rf.

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max Each recipe has a minimum ( rmin , n )and maximum (  r , n ) daily production capacity

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and a minimum preparation time (τr).

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(vi)

A changeover task is required in a packaging line whenever the production is

24

changed between two different families. Each changeover operation requires a

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specific time (γff΄j ) and cost φff΄jn.

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(vii)

Forbidden sequences between families are predefined.

The key decisions to be determined are: (i)

The allocation of product families into packaging lines (Yfjn). 12 ACS Paragon Plus Environment

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(ii)

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The amount of product p produced in line j (Qpjn), the inventory levels of each product (Ipn) as well as the lost sales (Lpn).

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(iii)

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The processing (Tfjn) and completion time (Cfjn) for the production of each product family.

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(iv)

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The sequencing between product families in each packaging line (Xff΄jn).

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Additionally, any alterations in demand e.g. order cancellations/modifications or new orders,

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need to be considered through rescheduling actions, to ensure the optimal operation of the

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plant.

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4. Mathematical formulation

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4.1 Conceptual model design

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Since scheduling problems are NP-hard in nature, real-life applications require significant

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computational effort. Especially, in scheduling problems of complex industries, like dairy, MILP

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formulations can easily result to intractable problems. Fortunately, one of the main

14

characteristics of a typical dairy industry is that a large number of final products originate from

15

few recipes, which share similar processing characteristics. Therefore, they can be grouped into

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product families60. This categorization alleviates the problem’s computational burden, since the

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optimization procedure focuses on the product families and not on the individual products. Thus,

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the total number of variables is vastly decreased, without violating any constraint and without

19

sacrificing the quality of the provided solution. Products are categorized in the same family if i)

20

they follow the same recipe, ii) have the same processing rate in the packaging stage and iii) no

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changeover operation (cleaning, change in the setup of equipment) between them is required.

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Additionally, product family succession can be defined a priori. A natural sequence of products

23

based on their characteristics (e.g. fat content, taste, color) frequently exists in dairy plants, so

24

that unnecessary cleaning tasks are avoided. Thus, all product families are placed in a succession

25

with a ranking from the first to the last to be produced. The model is derived using a MILP-based

26

framework introduced in our previous work30, which is general enough and has been able to

27

address different industrial cases in the past. The production horizon is divided into production



Page 14 of 43

1

periods (days) whose material balances are modeled with a discrete time representation. Within

2

each production day an immediate-precedence continuous-time representation is adopted.

3

4.2 MILP model.

4

The constraints are categorized based on the type of decision they subject to (allocation, timing

5

etc.). For simplicity, Latin letters are used for indices, sets and variables, while parameters are

6

represented using Greek characters.

7

Lot-sizing constraints

8

Constraint (1) sets an upper and lower bound to the produced amount of product p in packaging

9

line j in time period n. Binary variable Y p , j ,n defines whether a product p is packaged in unit j in

10

time period n.

11

min max  pjn  Y pjn  Q pjn   pjn  Y pjn

12

Definition of product family processing time

13

Since no changeover is required between products of the same family (𝑝 ∈ 𝑃𝑓), all timing and

14

sequencing constraints are deployed just for the product families. Nevertheless, setup times for

15

the packaging lines based on the processed product (𝛿𝑝𝑗) may exist. Hence, the total family

16

processing times can be defined as:

17

T fjn 

 Q pjn

  

pPf



pj

   pj  Y pjn   

p, j  J p , n

f , j  J f , n

(1)

(2)

18

Family allocation constraints

19

A family of products f is set in a unit j in time period n, if at least on product 𝑝 ∈ 𝑃𝑓, belonging

20

to that family is being processed in that unit and at the specific time period (constraint (3)).

21

Constraint (4) forces the binary variable to zero when no product of this family is produced.

22

Y fjn  Y pjn

23

Y fjn 

Y

pPf

f , n, j  J f , p  Pf

(3)

f , n, j  J f

(4)

pjn


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1

The following constraint ensures that packaging line j is utilized in time period n (𝑖.𝑒. 𝑉𝑗𝑛 = 1), if

2

at least one family f is processed in line j in time period n.

3

V jn  Y fjn

4

Family timing and sequencing constraints

5

The binary variables 𝑋𝑓𝑓′𝑗𝑛 are introduced to express the sequence between two families 𝑓 and

6

𝑓′ in a packaging line 𝑗 in time period 𝑛. Constraint sets (6) and (7) denote that when a product

7

family f is allocated to line j at period n, at most one product family 𝑓′ is processed right after

8

and/or before it.

9

(5)



X f ' fjn  Y fjn

f , n, j  J f

(6)



X ff ' jn  Y fjn

f , n, j  J f

(7)

f '  f , f 'F j

10

f , n, j  J f

f ' f , f 'F j

11

Obviously, the total number of active sequencing variables 𝑋𝑓𝑓′𝑗𝑛 plus the utilization binary

12

variable of line j, 𝑉𝑗𝑛, must be equal to the number of product families processed in line j in time

13

period n:

14

 

f F j f ' f , f 'F j

X ff ' jn  V jn 

Y

f F j

fjn

j , n

(8)

15

Constraint (9) forces a product family f’ that follows another product family f on a packaging line

16

j (i.e. Xff’jn=1) to start after the completion time of family f plus the required changeover time

17

between them.

18

C fjn   ff ' j  C f ' jn  T f ' jn  M  1  X ff ' jn 

f , f '  f , n, j  ( J f  J f ' )

(9)

19

The following constraints reduce the model’s search space and alleviate the computational

20

complexity of the problem. In particular, constraint sets (10) and (11) impose bounds on the

21

starting and completion times of every product family. Specifically, the starting time of processing

22

a family f, (𝐶𝑓𝑗𝑛 ― 𝑇𝑓𝑗𝑛) must be larger than the daily setup time of the facility 𝛼𝑗𝑛, plus the

23

minimum required time for the preparation of each recipe 𝜏𝑟, plus any changeovers between

24

family f and f’. Two supplementary terms are also implemented. Parameter 𝜊𝑗𝑛 represents any 15 ACS Paragon Plus Environment


Page 16 of 43

1

additional required time in line j due to possible maintenance or other technical issues, while 𝜀𝑟𝑛

2

denotes the time for the preparation of a batch of recipe r. The production of a batch of recipe r

3

can only start if all components of the recipe are available. Moreover, constraint (11) ensures

4

that the scheduling horizon is not violated. Specifically, it guarantees that the completion time of

5

every packaging process will be less than the daily horizon (24h) minus the shutdown time

6

required for the unit.

7

C fjn  T fjn  ( jn  max  jn ,  rn    r )  Y fjn 

8

C fjn  ( jn   jn )  Y fjn

9

Batch recipe stage



f  f ', f 'Fj

 ff ' jn  X ff ' jn ,

f , n, j  J f , r  R f

f , n, j  J f

(10) (11)

10

To obtain feasible production schedules, the batch stages (fermentation and pasteurization)

11

must be included to the mathematical model. Constraint (12) states that the total amount of

12

products p coming from the same fermentation recipe r must be within allowed limits based on

13

the production capacity of the pasteurization/fermentation stage. Furthermore, a fermentation

14

recipe is produced in period n, if at least one product family of that recipe is packaged at the

15

same period.

16

rnmin Wrn 

17

Wrn 

Y

jJ f

 Q

pPr jJ p

fjn

pjn

 rnmax  Wrn

r , n

r , n, f  Fr

(12) (13)

18

Inventory constraints

19

The inventory balance is guaranteed by constraints (14) and (15). In particular constraint (14)

20

describes the special case of n=1, where the inventory of product p, Ipn, is the summation of the

21

initial inventory 𝜎0𝑝 and the produced amount, minus the product demand ζpn. Additionally,

22

variables Lpn and LSpn are incorporated, representing any lost sales of product p in period n and

23

the amount that is left in order to meet the safety stock of the product p in every period n.

24

Constraint (15) corresponds to those cases, when n>1. Here, the initial inventory is omitted, while

25

the inventory level of the previous period Ipn-1 is taken into account. 16 ACS Paragon Plus Environment

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1 2

Ipn   0p   Qpjn   pn  LS pn  Lpn

(14)

p, j , n : n  1

jJp

3

Ipn  Ipn  1   Qpjn   pn  LS pn  Lpn

(15)

p, j , n : n  1

jJp

4

Constraint (16) imposes a lower bound in the product inventory. In particular, the inventory at

5

the end of every production day must be larger than the safety stock 𝜎𝑚𝑖𝑛 minus any amount 𝑝

6

that is not met LSpn.

7

I pn   pmin  LS pn

8

Objective function

9

The objective is to minimize the total operating cost including numerous factors such as (i)

10

production costs, (ii) unit operating costs, (iii) inventory costs, (iv) unit utilization costs, (v) family

11

setup costs, (vi) changeover costs, (vii) lost sales costs and (viii) costs of not meeting the safety

12

stock as portrayed below:

p, n

(16)

min(   pjnQ pjn    fjnT fjn    pn I pn   jnV jn 13

p

jJ p



n

f

j

n

   fjnY fjn   

f F j

j

n

f

p



n

j

n

  ff ' jn X ff ' jn   p Lpn   p LS pn )

f ' f j(J f  J f ' ) n

p

n

(17)

p

14

4.3 Implementation of a rolling horizon framework

15

A common issue in dairy plants is the volatility of product orders as they can be confirmed or

16

changed just prior to their dispatch. An optimization method that does not consider these

17

demand fluctuations, may result in suboptimal or even infeasible production schedules. In such

18

cases a reactive scheduling approach can be applied utilizing the aforementioned MILP model in

19

the context of a rolling horizon algorithm, to ensure that an optimal schedule is generated after

20

the arrival of new information.

21

The implementation of the algorithm requires the introduction of two new subsets Tp and Tc. Tp

22

corresponds to the prediction horizon, which is the time horizon considered by the optimization

23

model at each iteration of the algorithm. In that time horizon order quantities are assumed to be 17 ACS Paragon Plus Environment


Page 18 of 43

1

known with some certainty. A typical prediction horizon is weekly; thus, it includes five time

2

periods (|Tp|=5). This also applies for the case of MEVGAL dairy industries. The second subset,

3

Tc, is the control horizon, which includes the time periods in which the decisions provided by the

4

optimization problem are applied. A usual control horizon includes one time period (|Tc|=1).

5

Notice that the initial state of the plant in a given prediction horizon Tp,h is equal to the final state

6

of the plant in the previous control horizon Tc,h-1. At each discrete time instant (end of production

7

day) the optimization model receives new information regarding the demand and the state of

8

the plant (see figure 3).

9 10

Fig. 3. Rescheduling via a rolling horizon approach

11

Assume an example where the complete scheduling horizon is |T|=7 with time periods

12

T={n1,…,n7}, |Tp|=5 with time periods Tp={n1,…,n5} and |Tc|=1. This means that the optimization

13

problem will consider time periods n1 through n5, but the solution (lot-sizes, unit allocation,

14

timing and sequencing of tasks) will be implemented only for n1, so Tc={n1}. In the next iteration,

15

subsets Tp and Tc are updated so that Tp= {n2,…,n6} and Tc={n2}. This procedure continues until a

16

production schedule for the complete horizon T is extracted. Thus, in the rolling horizon approach

17

the prediction horizon is moving forward in steps of Tc time periods. 18 ACS Paragon Plus Environment

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1

This rescheduling strategy is implemented in our case study in order to handle any changes in

2

demand. Figure 4 illustrates the proposed rescheduling algorithm. Initially, the problem

3

parameters of the plant (e.g. inventories) and the necessary rolling horizon related parameters

4

are defined. Additionally, the number of required iterations to optimize the complete scheduling

5

horizon (total) is calculated. Then, the uncertain parameters (product orders) are updated. Thus,

6

order modifications, cancellations and new orders can optimally be dealt with. Afterwards, the

7

proposed MILP is used to solve the scheduling problem for the prediction horizon and the optimal

8

decisions are fixed for the control horizon. At the end of every iteration the state of the plant and

9

the horizon subsets are updated. The algorithm is terminated, when an optimal schedule for the

10

complete time horizon is generated.

11

Fig. 4. Rolling horizon algorithm

12 13

5. Case studies

14

In this section numerous real-life production scheduling problems are studied considering the

15

complex yoghurt production line of MEVGAL. The facility under consideration utilizes six parallel

16

packaging lines (J1-J6) to process a total of 158 final yoghurt products, both set and stirred (P119 ACS Paragon Plus Environment


Page 20 of 43

1

P158), which are categorized into 44 families (F1-F44) and originate from 28 distinct recipes (R1-

2

R28). The plant is operating in periods of five days from Monday (n1) to Friday (n5), therefore a

3

weekly scheduling horizon is implemented.

4

The production targets of the packaging stage, e.g. the demand quantities and due dates are

5

based on the product orders that arrive by the end of the previous week, as well as on forecasts.

6

Real data has been slightly changed in these studies due to confidentiality issues, while cost data

7

cannot be shared due to industry privacy issues. The main data for products and product families

8

such as inventories, packaging rates, availability of units, allocation of products into families and

9

recipes, changeover and setup times can be found in the supporting information.

10

The MILP model was developed using the GAMS 24.9 interface and all considered problem

11

instances were solved to optimality in an Intel Core i7 @3.4Gz with 16GB RAM using CPLEX 12.0

12

(Brooke et al.61). An optimality gap of 1% has been used to ensure a global optimal solution.

13

5.1 Weekly scheduling

14

In the first study case the proposed model is used to optimize the weekly schedule of the MEVGAL

15

yoghurt facility. All demand-related data is deterministic and no modifications in the order

16

quantities occur. The demand for this scenario can be found in the supporting information (table

17

S10) and corresponds to real industrial data provided by MEVGAL. The quantities of the product

18

orders have been slightly changed due to confidentiality issues. The optimization problem

19

consists of 18128 equations, 15489 continuous variables and 6075 binary variables. The optimal

20

solution corresponds to a total cost of 484439 RMU† and it was reached in 290s. Figure 5

21

illustrates the proposed optimal weekly schedule. Table 1 provides the production quantities for

22

every family in each production day, while table 2 reports the inventory levels for each family at

23

the end of each day.

†

Relative Money Units


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1 2

Fig. 5. Gantt chart of weekly schedule

3

Table 1. Production schedule Family

Line

F1 F1 F1 F2 F3 F3 F3 F4 F5 F5 F6 F6 F6 F6 F7 F8 F9 F10 F11 F12 F13 F14 F16 F17

j1 j2 j5 j1 j1 j3 j5 j2 j2 j5 j1 j2 j3 j5 j1 j2 j1 j1 j1 j1 j4 j4 j1 j1

Monday 5364 0 0 4144 3969 0 0 0 0 0 0 0 0 0 0 0 0 7852 0 0 0 0 0 0

Tuesday 3321 0 0 0 7488 543 552 2500 281 553 2151 2493 1759 3940 3083 0 3379 0 2991 0 0 0 0 0

Day Wednesday 3378 0 3046 0 10077 0 1650 0 0 0 2265 0 0 0 0 0 0 0 0 0 0 0 0 0

Thursday 3447 3078 0 2500 11024 0 3893 0 0 0 1310 0 0 9431 0 500 0 0 0 3743 0 1210 0 0

Friday 0 0 0 0 0 0 0 0 0 0 5744 0 7405 3000 0 0 0 12367 10854 0 1500 0 1723 777



F19 F19 F20 F21 F22 F23 F23 F24 F25 F26 F27 F28 F29 F29 F30 F31 F31 F32 F32 F33 F34 F35 F35 F36 F37 F38 F39 F41 F41 F42 F44

j2 j4 j2 j2 j4 j1 j2 j2 j2 j2 j2 j2 j1 j2 j2 j1 j2 j1 j2 j2 j2 j1 j2 j6 j2 j2 j2 j5 j6 j1 j1

0 0 0 0 0 2584 0 415 408 413 593 406 3113 0 235 0 0 0 0 0 1989 1110 235 0 0 0 0 9563 6557 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3734 0 0 0 0 0 2500 1000 4738 301 0 3395 0 0

0 0 0 0 3918 0 2227 0 0 0 0 0 0 0 0 3726 2060 512 4049 1902 0 0 0 0 0 0 0 13417 0 235 0

Page 22 of 43

350 650 2342 1000 0 0 3430 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3113 0 0 0 0 0 0 0 0 500

0 0 0 0 1500 0 0 0 0 0 0 0 0 1500 0 0 0 0 1500 1500 0 0 5920 0 0 0 0 0 0 0 0

Thursday 1450 2500 2601 2053 0 14650 2488 500

Friday 1450 2500 2601 2053 0 10492 1000 373

1

Table 2. Inventory levels

2 Family F1 F2 F3 F4 F5 F6 F7 F8

Monday 6393 0 6186 1051 117 29224 598 0

Tuesday 5383 0 3166 2053 0 26575 2488 0

Day Wednesday 1508 0 3081 2053 0 25274 2488 0


Page 23 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


F9 F10 F11 F12 F13 F14 F15 F16 F18 F19 F20 F21 F22 F23 F24 F25 F26 F27 F28 F29 F30 F31 F32 F33 F34 F35 F36 F37 F38 F39 F40 F41 F42 F43 F44

1762 232 5438 68 7 916 960 2432 8 0 0 0 58 2633 415 408 413 593 437 82 1091 348 23 426 1989 3421 4255 212 1230 21 5580 21611 11992 539 5

3470 232 1773 68 7 916 960 2432 8 0 0 0 58 546 204 115 132 283 0 82 988 348 2442 426 1989 3421 1900 502 1000 92 5580 16707 11992 539 5

3470 232 1773 68 7 916 960 2432 8 0 0 0 0 546 204 115 132 283 0 82 988 1566 0 113 1000 3421 1900 502 1000 92 5580 18471 12227 539 5

3470 232 1773 3811 7 300 25 2432 8 1000 1000 1000 0 3448 204 115 132 283 0 82 988 1566 0 113 1000 1393 1900 502 1000 92 5580 10031 12227 24 199

1000 7576 5000 0 1507 300 25 688 8 1000 1000 1000 1500 1518 0 0 0 0 0 1582 244 1566 1500 1613 1000 1543 1900 502 1000 92 5580 10031 7299 24 199

1 2

Figure 6 illustrates the distribution of the various cost terms. The cost associated with the

3

required raw materials (production cost) is as expected the largest term (93%) of the total cost.

4

It should be noted that the lost sales are practically zero, so a nearly full demand satisfaction is

5

achieved, while the total cost is minimized. In particular, the demand for just two orders, product

6

P138 on Wednesday (17kg lost) and product P139 on Friday (1kg lost), is not fully met. Figure 7 23 ACS Paragon Plus Environment


Page 24 of 43

1

displays the total time spent for packaging, sequence dependent and sequence independent

2

setups in each line and day. Furthermore, figure 8 illustrates the quantity of every yoghurt recipe

3

being processed daily in each packaging line. From figures 5, 6 and 8 it is evident that the available

4

packaging lines are disproportionally utilized. This is expected, since lines j1 and j2 are able to

5

process much more product families compared to the other lines. In particular, lines j1 and j2 can

6

process 29 and 30 families accordingly, while lines j4 - j6 can process only 6 families and line j3

7

only 3. Therefore, large idle times are reported in lines j3 to j6. This is also enforced by table 3,

8

which reports the number of families being processed by each packaging line on each day.

9 10

Production cost

93

Inventory cost Family setup cost Unit operating cost Unit fixed cost 0.1

0.6 0.1 0.4

0.4 1.6

3.6

Lost sales cost Missed safety stock penalty

11 12

Changeovers cost

Fig. 6. Cost distribution


Page 25 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

Fig. 7. Packaging line utilization

2

3

Fig. 8. Recipe production

4 5

Table 3. Number of families processed

6

Line j1

Monday 4

Tuesday 7

Day Wednesday 6

Thursday 7

Friday 5 25

ACS Paragon Plus Environment


J2 J3 J4 J5 J6

8

6 2

1 1

3 2

Page 26 of 43

4

6

1 3

2 2

4 1 2 1

1 2

A great reduction in total costs is achieved compared to the schedule used by MEVGAL. The

3

optimal solution proposed by our model accomplishes an improvement of around 20% in the

4

total cost. In particular, the greatest reduction is noticed in the lost sales (~80%) and in the missed

5

safety stock penalty (~50%), while the inventory costs are also significantly reduced (~25%). A

6

small improvement is also achieved in the changeover costs (~5%). An overview of the reduction

7

accomplished in the various cost terms of the production can be found in Table 4. Due to

8

confidentiality issues exact numbers cannot be reported, therefore normalized values are used.

9 10 11

Table 4. Comparison of real (MEVGAL) and optimized (MILP) schedule Cost term

MEVGAL schedule

MILP-generated schedule

Lost sales

1

0.2

Missed safety stock

1

0.5

Inventory costs

1

0.75

Changeover costs

1

0.95

Total costs

1

0.8

12 13

To further illustrate the efficiency of our method, 12 additional study cases have been solved to

14

optimality. Table 5 reports a summary of the results. In every subcase a randomized demand

15

using the normal distribution function of GAMS is generated. In cases A.I-A.III the product order

16

quantities have been perturbed by ±10%. Similarly, a perturbation of ±20% and ±30% has been 26 ACS Paragon Plus Environment

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1

considered in cases B.I-B.III and C.I-C.III. Finally, cases D, E and F study the uniform increase in

2

demand by 10%, 20% and 30% accordingly. The cumulative results show that in all cases the

3

optimal solution has been reached in less than 10 minutes. The only case where the time limit of

4

600 seconds has been reached was F (30% increase), where the integrality gap was 2.49%.

5

Moreover, the lost sales are miniscule in every examined demand distribution. In the worst case

6

(B.I) the unfulfilled demand is just 0.06% of the total. Consequently, customers remain satisfied,

7

while the total costs of the facility remain minimal, as it is imposed by the objective function. Table 5: Scalability analysis

8

Study Case

Demand (kg)

Nodes

CPU (s)

Lost sales (kg)

Objective (€)

Gap (%)

A.I

300015

6515

287

9

494908

Optimal Production Scheduling in the Dairy Industries | Industrial

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