Multistage Production Planning in the Dairy Industry: A Mixed-Integer

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Multi-stage Production Planning in the Dairy Industry: A Mixed Integer Linear Programming Approach Bilge Bilgen, and Koray Do#an Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b02247 • Publication Date (Web): 03 Nov 2015 Downloaded from http://pubs.acs.org on November 8, 2015

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Industrial & Engineering Chemistry Research

Paper Title: Multi-stage Production Planning in the Dairy Industry: A Mixed Integer Linear Programming Approach

The name of the first author: Bilge BĐLGEN, PhD (Corresponding Author) Affiliation: Associate Professor at Dokuz Eylul University, Department of Industrial Engineering, Address: Dokuz Eylul University, Department of Industrial Engineering, Tinaztepe Campus, Buca, Izmir, TURKEY E-mail: [email protected] Phone: 90 (232) 3017615, Fax: 90 (232) 3017608

The name of second author: Koray Dogan, PhD Affiliation: Solvoyo Innovation Lab, Chief Executive Officer Address: ARI 2 Teknokent ITU Ayazaga Kampusu, B Blok B1-3, Maslak, Istanbul, Turkey

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Multi-stage Production Planning in the Dairy Industry: A Mixed Integer Programming Approach Bilge Bilgena*, Koray Doganb a

Dokuz Eylul University, Department of Industrial Engineering, Tinaztepe Campus, Buca, 35160 Izmir, Turkey

b

Solvoyo Innovation Lab, ITU Ayazaga Campus, Maslak, Istanbul, Turkey

*Corresponding author [email protected]

Abstract

This paper addresses a production planning problem in a multi-stage production system consisting of continuous processing resources separated by finite capacity storage tanks, stimulated by a particular case study in the dairy industry. The problem is formulated as a mixed integer linear programming (MILP) model that incorporates several distinguishing characteristics of dairy production, such as multi-stage bulk production, shelf life requirements, intermediate storage, setups, resource speeds, limitations on minimum and maximum lot size, and the conservation of flow among various tanks. The objective is to maximize total profit while determining the quantity of intermediate products and SKUs processed on various resources, the assignment of products to various resources and intermediate storage tanks, the quantity of each SKU sold, lost sales of each SKU, and waiting times. The determinations often reveal production bottlenecks. The efficiency of the proposed model is illustrated through its application to the milk production plant of a leading dairy company. Special features of the proposed model are highlighted through several examples. Computational performance of the MILP model is examined in several test scenarios. Keywords: OR in agriculture, production planning, milk production, mixed integer linear programming, dairy industry 2


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1. Introduction The dairy industry is a significant component of many economies, and is a major industry in most developed and developing economies of the world. This industry can be considered as a part of process industry. In a process industry, the production planning problem contains structural features such as multi-purpose production processes, cleaning costs, combined divergent material flows, multi-stage bulk production, multi-component flows, finite intermediate storage, and dedicated and variable tanks.1 Process manufacturing is different from discrete manufacturing in that finished goods are created through a continuous manufacturing process that involves the creation of multiple intermediate products which are converted into tens or hundreds of finished products. Production recipes are very different from a bills of material structure. For example, cleaning procedures are mandatory, more frequent and often more disruptive than equipment maintenance in discrete manufacturing. One result of these differences is that production planning is generally more difficult in process manufacturing plants. The dairy processing industry has a specific set of product and process characteristics such as high number of SKUs, divergent product structures, seasonal product demand, high demand variability, multiple intermediate products feeding many finished goods with limited storage capacity for intermediate products, high consumer driven demand variability, long lead-times for some packaging material, and a high level of complexity in the production process. For production facilities, the planning problems contain the following structural features: multi-product production, setup times, costs, multi-stage production, production bottlenecks and available connections, continuous flow, multi-component flow, finite intermediate storage, dedicated production lines, and shelf life. Such challenges have

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underlined the need for efficient management of the dairy supply chain, which is critical for profitability. The production process in the dairy industry can be divided into two stages: the processing of raw materials into intermediate products and the packaging of end products. These stages must be closely coordinated. Out of a limited number of raw materials (e.g. raw milk) a moderate number of intermediate products (e.g. full-fat milk, diet milk, aroma milk) are produced within the processing step. The high product complexity typically occurs at the packaging level due to different tastes and customer individual packaging forms.2 The challenges associated with demand variability are compounded by the short shelf life of the finished products and relatively long production lead times. There are many challenges regarding managing tanks, complex cleaning rules, and traceability requirements. The production planning problems encountered in the dairy industry are considered intractable due to several practical constraints. The aim is to implement a flexible solution capable of improving key operational performance metrics. The main challenge is to manage all production constraints and still be able to keep total costs low. This paper presents a novel MILP model for the production planning of a multi-stage process within the dairy industry. We consider multi-stage production planning where the objective is to determine when and how much of each intermediate and end product should be produced over a finite planning horizon in order to maximize profit, while considering production, inventory holdings, lost sales, and setup costs. The problem we consider has multiple stages where raw milk or intermediate products serve as an input in the production of one or more intermediate products or end products. The production process uses parallel pasteurizers, sterilizers, and filling and packaging machines. The synchronization of production stages is difficult due to the difference between processing and packaging rates, and the limitations on the intermediate storage. 4


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In summary, the optimization of a production planning problem should simultaneously take into account all of the costs and operating constraints. The objective of this paper is to develop an optimization model for multi-stage production planning in a dairy industry. The remainder of this paper is organized as follows: In Section 2, the literature review is presented. In Section 3, the problem is defined with a discussion of the practical aspects and modeling challenges that represent the distinguishing characteristics of the dairy industry. Section 4 contains a detailed description of the model notation and formulation. The applicability of the developed model is illustrated in Section 5. Concluding remarks and future research directions are proposed in the final section.

2. Literature Review The optimization problems inherent to the dairy industry have motivated the research community to develop efficient planning and scheduling approaches that have had a significant economic impact. Despite its importance in practical settings, in the late 1990s only few researchers addressed the production planning problem in dairy industry. Pioneering work was performed by Smith-Daniels and Ritzman3, Claassen and van Beek4, van Dam et al.5, and Nakhla6. Smith-Daniels and Ritzman3 developed a general lot sizing model for process industries and applied their method to a situation representative of a food processing facility. Claassen and van Beek4 developed an approach to solve a planning and scheduling problem for a bottleneck in the packaging facilities of the cheese production division of a large dairy company. van Dam et al.5 investigated and compared the origins of the increasing complexity of the scheduling problems of several companies in the process industry producing dairy products, pharmaceutical products, tobacco, paint, chocolate products, and foods. Nakhla6 emphasized the need for flexibility for operations scheduling in the dairy industry, and proposed a rule-based approach for scheduling packaging lines in a dairy 5



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industry. Foulds and Wilson7 proposed two heuristic algorithms for a variation of the generalized assignment problem arising in the New Zeland dairy industry. Then Basnet et al.8 described an exact algorithm to solve the scheduling and sequencing problems in the same industry. In recent years there have also been papers that focus on side aspects of production scheduling, like the environmental effects of production tasks and workforce scheduling. Berlin et al.9 studied a heuristic to arrange products to minimize the environmental impact of yoghurt products in their life cycle. Until recently, the use of optimization-based techniques for the production planning problem in the dairy industry has received little attention in the operations research literature. Lütke Entrup et al.10 presented three different MILP formulations for scheduling in the fresh food industry. The shelf life of the products has been explicitly considered. To guarantee the compactness and computability of the models, a block planning approach developed by Guenther and Neuhaus11 is chosen. However, the MILP models focus on the flavouring and packaging stages. Thus, operations involving the processing and storage of products are neglected. Doganis and Sarimveis12 proposed a model that aims for the optimal production scheduling in a single yoghurt production line. The model takes into account all the standard constraints encountered in production scheduling (material balances, inventory limitations’, and machinery capacity). However, the model is limited to a single production line. In another study13 they presented a methodology for optimum scheduling of yoghurt packaging lines that consists of multiple parallel machines. The methodology incorporates features that allow it to tackle industry-specific problems, such as multiple intermediate due dates, job mixing and splitting, product specific machine speed, minimum and maximum lot size, and sequencedependent changeover times and costs. The model does not incorporate multi-stage production decisions, and ignores some industry-specific characteristics, such as shelf life. Doganis and Sarimveis14 studied the same model, and extended it to include the product shelf 6


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life. The extended model considers shelf life restrictions and optimizes the balance between the cost factors and the profit-contributing aspects of minimizing the time between the production and delivery of products to retailers. For a packaging company producing yoghurt, Marinelli et al.15 proposed a solution for the capacitated lot sizing and scheduling problems with parallel machines and shared buffers. A two-stage heuristic methodology based on decomposition of the problem into lot sizing and scheduling problems has been developed. This approach assumes that the production rate is fixed by a single bottleneck stage, and that setup times and costs are sequence-independent. Gellert et al.16 investigated the planning for filling lines, taking into account the new aspect of a flexible scheduling environment regarding cleaning and sterilization. Their problem limits itself to concerns about the scheduling and sequencing for filling lines, and ignores all potential limitations regarding the rest of the processing system. Kopanos et al.17 studied the lot sizing and scheduling problem in a multi-product yoghurt production plant. A mixed discrete/continuous time MILP model was proposed. The problem under question is mainly focused on the packaging stage, whereas timing and capacity constraints are imposed with respect to the pasteurization, homogenization, and fermentation stage. Sequence dependent setup times and costs are explicitly taken into account and optimized by the proposed framework. However, the scheduling problem they consider only involves the packaging stage. Kopanos et al.18 presented a novel MILP formulation and solution strategy to address the challenging production scheduling problems in multi-product, multi-stage food industries. The main features of the proposed approach rely on the integrated production stages, and the inclusion of strong valid integer cuts, favouring shorter computational times. Kopanos et al.19 presented a MILP framework for the resource constrained production planning problem in a semi-continuous food process, similar to the

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dairy industry. Quantitative, as well as qualitative, optimization goals were included in the proposed model. Renewable resource limitations are appropriately taken into account. All of the above mentioned works are related to the single stage production systems in the dairy industry. In a paper by Kopanos et al.20 the MILP model developed by Kopanos et al.18 is further enhanced by introducing new sets of tightening constraints in order to improve computational efficiency for industrial size scheduling problems in food industries. Both papers consider the production scheduling problem in a real world, multi-stage, food processing industry with the limited shelf life of intermediate mixes in the aging stage. Kopanos et al.21 presented a novel MILP framework based on a hybrid discrete/continuous time representation for the simultaneous detailed production and distribution planning problem of the multi-site, multi-product, semi-continuous food processing industry. The novelty of the proposed mathematical formulation rests upon the integration of the different modelling approaches and the detailed production and distribution operations. Amorim et al.22 developed multi-objective, mixed integer programming (MIP) models to deal with simultaneous lot sizing and the scheduling of perishable products for different strategic scenarios: pure make-to-order, hybrid

make-to-order, and make-to-stock

environments. A hybrid genetic algorithm is developed to solve both models, and various problem instances are tested based on the dairy plant described by Kopanos et al.17. The main contribution of this paper is the multi-objective framework. They differentiate between costs and freshness. Amorim et al.23 addressed the integrated production and distribution planning of perishable products in a multi-objective framework. They formulated models for the case where perishable goods have a fixed and loose shelf life. In a recent study Amorim et al.24 assessed the suitability of financial risk measures for mitigating crucial risks in the production planning of perishable foods including dairy products. Bilgen and Celebi25 addressed the short-term production scheduling and distribution planning problem within the dairy industry. 8


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They present an efficient hybrid solution methodology based on a MILP formulation and simulation approach. This solution obtains optimal production schedules, and a distribution plan via mathematical modeling, while incorporating uncertainties in the execution of these plans via simulation. Baumann and Trautmann26 developed an efficient, continuous, time precedence-based MILP model considering all technological constraints for a make-and-pack production process. The proposed model accounts for sequence dependent changeover times, multipurpose storage units with limited capacities, quarantine times, batch splitting, partial equipment connectivity, and transfer times. They also embed new sets of symmetry-breaking constraints. In a more recent work, Baumann and Trautmann27 presented a novel hybrid method for the short-term scheduling of make-and-pack production processes van Elzakker et al.28 developed a new MILP scheduling model and algorithm for scheduling in the Fast-moving Consumer Goods (FMCG) industry. The model is demonstrated on an ice cream scheduling problem. In a very recent work van Elzakker et al.29 propose an MILP model to address the optimization of the tactical planning for the FMCG industry. To solve these extremely large problems, they proposed an algorithm based on SKU decomposition. Sel et al. (2015)30 address a dairy industry problem on integrated planning and scheduling of set yoghurt production. The integrated planning and scheduling problem is divided into two distinct sub-problems. The sub-problems are solved by the decomposition heuristic. A hybrid MILP/Constraint Programming is proposed. Interested

readers

are

referred to Sel and Bilgen (2015)31 for the extensive review and discussion on quantitative models within the dairy industry. Table 1 classifies the literature and the relation of the proposed research to the existing literature. The dairy industry is an area that is attracting a growing interest in the recent years. Our problem differs from these studies in several ways. First, based on observed practices, we 9



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consider all stages of the production planning problem. Second, the products are allowed to be produced on multiple production lines, which more accurately captures the intricacies of real world production. Third, tank capacities, process times, flow rates, and a host of other operational constraints are taken into consideration. The overview of the relevant literature reveals that most research efforts towards process planning in the dairy industry concentrate on the packaging stage, and do not integrate the rest of the production systems into planning. Capacity limitations in the intermediate storage are often neglected. There is an established need for an optimization model in process industries that recognizes interrelationships between production capacity, lot sizes, and the sequence of multi-stage production. The first attempt to solve multi-stage scheduling problems in the ice cream industry was made by Bongers and Bakers32. To the best of our knowledge, of all the previously published literature, the models developed by Kopanos et al.18 Kopanos et al.20, and Baumann and Trautmann26, which are the multi-stage extension of ice cream production, are the closest to the problem studied in this paper. This research develops a MILP model for production planning which features a variety of distinguishing dairy industry characteristics. The model to be presented has the flexibility to be applied to a wide variety of production environments.

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Table 1. Literature Summary

Sequencing Constraints

Conservation of Flow

Timing Constraints

Storage Constraints

Objective Function

Time Representation

CF

C

MILP

Yoghurt

CF

C

MILP

Yoghurt

CF

C

MILP

Yoghurt

F

H

C

MILP

Yoghurt

Doganis and Sarimveis

P+PF

Doganis and Sarimveis14

P+PF

Kopanos et al.

17

PF

Kopanos et al.

18

PF

Kopanos et al. 19

P+PF

Kopanos et al. 20

PF

Kopanos et al. 21

P+PF

22

P

Baumann and Trautmann26

P

Bilgen and Celebi25

P

Packaging

Proposed Research

Dairy

P C

P Product PF Product Family F Flavor H Hybrid CF Continuous Flow MB Multi Bucket

Yoghurt

Two-stage Heuristics

H CF

MILP Block Planning

13

Application Industry

Doganis and Sarimveis

van Elzakker et al.

Solution Methodology

P

12

28

Model Characteristics

Distribution

OF+constraints

Make and pack

PF

All stages

Shelf Life

Marinelli et al.15

Product/ Product Family

Lütke Entrup et al.10

Amorim et al.

Problem Focus

Problem Characteristics Sequence dependent setup

Reviewed Literature

Lot sizing Scheduling

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


OF+constraints

constraints

OF+constraints

OF+constraints

constraints

C

MILP with valid cuts

Food+Dairy

C

MILP with valid cuts

Ice cream

H

C

H

C

MB

MO

CF

Makespan

CF

P

Hybrid MILP+Simulation

Yoghurt

Mix +

STN

Makespan

MILP heuristic algorithm

Ice cream

CF

P

MILP

Milk

H H

MILP with tightening, symmetry breaking constraints MILP with tightening, constraints MILP Block Planning Hybrid MTO/MTS MILP symmetry breaking constraints

constraints

Ice cream Dairy Yoghurt Make and pack

P Profit C Cost MO Multi-objective STN State Task Network

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3. Problem Definition The dairy industry tends to operate under intense competition. The production planning problem studied in this paper is based on a real case study in a leading dairy company. In the dairy industry, the milk undergoes a series of transformations linked by the same overall sequences. The individual products are made through a complex, multi-stage process, including raw milk collection and preparation, pasteurization, sterilization, and filling and packaging. Extra care must be taken to ensure high standards of sanitation, control of allergens, batch traceability, and maximum product freshness. The main goal of the company is to implement a solution capable of improving key performance metrics, such as resource utilization and cost minimization. The plant has three stages with parallel, non-identical, batch or semi-continuous processing units: i.

Processing of raw milk into intermediate products (pasteurization).

ii.

Storage of intermediate product and further processing (sterilization).

iii.

Filling and packaging of end products.

The units within a stage are non-identical in the sense that their suitability for processing products and their processing rates are different. The pasteurization, sterilization, filling and packaging stages constitute the bottleneck of a dairy production facility mainly due to the low packaging rates compared to the flow rates of previous stages, and the batch size nature of the process itself. The production of intermediate and end products must be tightly synchronized. The production process must take into account tank capacities, conversion of process time, flow rates, and a host of other operational constraints. Intermediate products are produced in batches that must be kept within the maximum tank capacities. Cleaning rules on tanks and resources can be extremely complex. Shelf life adds another layer of complexity to the

12


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production process. The production process under consideration has some distinctive characteristics that require careful consideration in the context of production planning. The primary production stages are highlighted in Figure 1.

OP1 - pasteurizers PAS1, PAS2 30ton/hr, 25ton/hr

raw milk tanks 80t

OP2 – processed milk tanks, 2x80ton, 2x100ton capacity

PMT

OP3 - sterilizers SPR1, SPR2, and SPR3 22ton/hr, 20ton/hr, and 18ton/hr flow

OP4 – aseptic tanks SPR1AT1, SPR2AT1, and SPR3AT1 25ton, 22ton, and 10ton capacity

OP5 – sterilized filling & packaging lines

Figure 1. Production planning process in the dairy industry

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The production process in the dairy industry consists of: raw milk tanks, pasteurizers, processed milk tanks, sterilizers, aseptic tanks, and filling and packaging lines. Raw milk is subject to a number of preparatory treatments before it is processed. As the raw milk components vary significantly, it is first necessary to standardize the raw milk in order to meet the compositional standards for the final products. Then the milk undergoes a series of transformations linked by the same overall sequence. The storage tanks are not capable of storing all intermediate product types. The flavoring and packaging step has to cope with a high number of product variants because of the variety of consumer tastes and packaging materials.10 The raw milk goes through processing lines where the milk is pasteurized. Then it is standardized in processed milk tanks before it goes through processing lines where it is sterilized. Then it is stored into the aseptic tanks. Finally, in the packaging stage, sterilized milk products are packed in different sizes according to the customer preferences. The key decision variables of this process are: the quantity of intermediate products; SKUs processed on resources in each time period; waiting times as raw milk and intermediate products on resources; quantity of SKUs sold; quantity of SKUs held in inventory; lost sales of SKUs; and time consumed by resources. The objective is to maximize the total profits based on sales revenue from the products against the costs, such as setup, inventory holdings, lost sales, and production costs. This study has specifically focused on a number of industry specific characteristics (e.g. shelf life, intermediate storage, multi-stage processing, setup times, and costs). The stages are linked by intermediate storage tanks with limited capacities. To produce and package products, different operations have to be performed for which a set of parallel processing units (mixing tanks and packing lines) is available on each stage. Several technological constraints have to be respected: each resource cannot process more than one product at a time, but various products can share a common production resource and various resources can produce 14


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the same product in parallel. The physical configuration consists of various production resources, and tanks to transfer dairy products between production resources. A production resource is initially adjusted to produce dairy products of a given flavor in a given pack size. To produce a diary product of another flavor and pack size, it is necessary to stop the production and make all the necessary adjustments for the production of another product with a different flavor and pack size. Optimizing a production plan should take into account all of the different costs and operating constraints. The main assumptions of the proposed model include the following: • The processing structure consists of multiple processing stages arranged in series, each one containing several resources operating in parallel. •

A set of products should be processed by following a predefined sequence of processing stages with processing units working in parallel.

•

Production process in each stage involves identical, uniform, or unrelated parallel machines of the same or of different resources. A production resource cannot process all products.

•

Every intermediate tank (e.g. processed milk tank, and aseptic tank) has a maximum capacity.

•

Production speed of all resources is given.

•

Production, inventory holdings, lost sales, and setup costs are covered in the objective function.

•

All model parameters are deterministic.

•

The time horizon is taken as approximately one month.

•

All resources, including filling and packaging lines, must be cleaned at the end of each day.

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The orders to be processed follow the same processing sequence, though, at the final stage, they are packed at only dedicated resources. The proposed MILP approach is based on continuous time domain representation and accounts for waiting times between different production resources. The objective is to determine the feasible capacity with optimal production quantities based on the availability of materials and raw milk.

4. Model Formulation This study is mainly concerned with analyzing the production planning decisions in the multi- stage process that might be encountered in a flow-shop production system. In the proposed model formulation, the problem’s constraints have been grouped according to the type of decision (timing, capacity of resources, flow conservation, minimum and maximum lot sizes, resources speeds, and demand). The optimization model finds the best possible solution that maximizes the total net profit, involving total sales, total production, inventory, lost sales, and setup costs, while respecting all operational constraints and regulations.

Indices, Sets, Parameters H

Raw milk

I, J

Set of intermediate products

K

Set of SKUs

R

Set of production resources (pasteurization, sterilization, filling and packaging machines)

RPAS

Set of pasteurization resources 16


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RSTER

Set of sterilization resources

RFP

Set of filling and packaging machines

T

Set of time periods (days)

Parameters RCap r

Capacity of production resource r (ton)

PTCapi

Pasteurization tank capacity (ton)

STCap j

Sterilization tank capacity (ton)

pt r

Unit production speed at resource r (ton/hour)

d kt

Demand of product ( SKUs) k at time period t. (ton)

δ rt

Setup time on resource r in period t (hour)

prk

Unit sales price for SKU k

SLk

Shelf life of SKU k

ICostk

Unit Inventory Cost for SKU k

PPCost rPAS Unit Production Cost at Pasteurization resource

SPCost rSTER Unit Production Cost at Sterilization resource FPCost rFP Unit Production Cost at filling and packaging resource 17



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LSCostk

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Unit Lost Sale Cost of SKU k

Variables r xhit

Quantity of raw milk h pasteurized on resource r ∈ RPAST and sent to processed milk tank as product i in time period t. (novel variable structure is developed: i carries information regarding both the intermediate product, and the tank that has been stored, for example i represents 81500002001_PST01, the amount of pasteurized full-fat milk (81500002001) and stored on processed milk tank PST01).

yijtr

Quantity of intermediate product i sterilized on resource r and sent to aseptic tank as product j in time period t. (novel variable structure is developed: similarly, j carries information regarding both the intermediate product and the tank; for example j represents 81500002001_SPR03AT01, the amount of sterilized full-fat and stored aseptic tank SPR03AT01).

z rjkt

Quantity of intermediate product j filled and packaged on resource r as demanded product (SKU) k in time period t.

r whit

Waiting time for raw milk h pasteurized on resource r sent to processed milk tank as product i in time period t.

wtijtr

Waiting time for product i sterilized on resource r sent to aseptic tank as product j in time period t.

qkt

Quantity of SKU k sold in time period t

18


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I kt

Inventory of SKU k in time period t

LSkt

Lost sale of SKU k in time period t

r X hit

1, if raw milk h is pasteurized on resource r and sent to processed milk tank as product i in time period t

Yijtr

1, if intermediate product i is sterilized on resource r and sent to aseptic tank as product j in time period t

Z rjkt

1, if intermediate product j is filled and packed on resource r as product (SKU) k in time period t

tconrt

Time consumed by resource r in time period t

Maximize Total Profit

∑∑ pr q k

k

kt

-

[

t

∑∑ ∑ ∑ PPCost x r

h

i

r∈PAS

r hit

t

+ ∑∑ i

∑ ∑ SPCost

j r∈STER

r

t

+ ∑∑ ICostk I kt + ∑∑ LSCostk LSkt ] k

t

k

yijtr + ∑∑ ∑∑ FPCost r z rjkt j

k r∈FP t

(1)

t

Subject to Supply Capacity

∑ ∑x i

r hit

≤ Supplyht

∀h, t

r∈PAS

(2) 19



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

Page 20 of 37

Tank capacity Processed milk tank capacity

∑∑

r xhit ≤ PTCapi

∀i, t

(3)

∀j, t

(4)

h r∈PAS

Aseptic tank capacity

∑∑ i

yijtr ≤ STCap j

r∈STER

Relationship between production resources

∑ ∑ (x

r hit

/ pt r ) + ∑

h r∈PAS

∑ ∑(y i

∑w

≥∑

r hit

h r∈PAS

r ijt

/ pt r ) + ∑

r∈STER

i

∑(y

r ijt

/ pt r )

∀i, ∀t

(5)

∀j , ∀t

(6)

j r∈STER

∑ wt

r ijt

r∈STER

≥ ∑ ∑ ( z rjkt / pt r ) k

r∈FP

Minimum and maximum lot sizes

∑

r r r MinLot. X hit ≤ ∑ xhit ≤ ∑ MaxLot. X hit

h

∑

h

MinLot.Yijtr ≤ ∑ yijtr ≤ ∑ MaxLot.Yijtr

i

∑

i

(7)

∀j, r ∈ RSTER , t

(8)

∀k , r ∈ RFP , t

(9)

i

MinLot .Z rjkt ≤ ∑ z rjkt ≤ ∑ MaxLot .Z rjkt

j

∀i, r ∈ RPAS , t

h

j

j

Flow of Conservation Constraints Flow of conservation at processed milk tanks

∑∑ h r∈PAS

r xhit ≥∑

∑y

r ijt

∀i, t

j r∈STER

(10) 20


Page 21 of 37

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


Flow of conservation at aseptic tanks

∑∑ i

yijtr ≥ ∑ ∑ z rjkt

r∈STER

∀j, t

(11)

k r∈FP

Time Availability constraint r tcon1rt = δ rt ∑∑ X hit h

i

tcon 2 rt = δ rt ∑∑ Yijtr i

j

tcon 3rt = δ rt ∑∑ Z rjkt j

k

∑∑ x h

ptr

∑∑ +

(12)

r ∈ STER, t

(13)

r ∈ FP, t

(14)

∑∑

r hit

i

j

r ∈ PAS , t

r + tcon1rt + ∑∑ whit + h

z

i

i

yijtr

j

ptr

+ tcon 2 rt + ∑∑ wijtr i

j

r jkt

k

ptr

+ tcon3rt ≤ 23

∀t, ∀r ∈ RPAS , r ∈ RSTER , r ∈ RFP (15)

Demand constraints

∑ ∑z

r jkt

+ I kt −1 − I kt = qkt

∀k, t

j r∈FP

(16)

t + SL

I kt ≤

∑q

ks

∀k, t

(17)

s = t +1

qkt ≤ Dkt

∀k, t

(18)

Dkt − qkt ≤ LSkt

∀k, t

(19)

21



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Page 22 of 37

The profit of the plant is equal to the sales revenue minus operating costs, including processing, setup, inventory, and lost sales. Constraint (2) guarantees that total quantity of raw materials sent to various processed milk tanks as pasteurized intermediate product i is bounded by the available raw milk supply amount. Constraint (3) indicates that the total quantity of raw milk h pasteurized at different resources and sent to processed milk tanks as a pasteurized intermediate product i in time period t must not exceed the available tank capacity. Constraint (4) guarantees that the quantity of various pasteurized intermediate products sterilized at various resources and sent to aseptic tanks as a sterilized intermediate product j should not exceed the available tank capacity. The constraints (3) and (4) also allow each tank to contain only one kind of mixture. Constraint (5) ensures that total processing time of pasteurization resources should be greater than the total processing time of sterilization resources. Constraint (6) ensures that total processing time of sterilization resources should be greater than the total processing time of filling and packaging resources. There should be a waiting time if the speed of the upstream resources exceeds the speed of the downstream resources. Lower and upper bounds on the intermediate products and SKUs are imposed by Constraints (7), (8), and (9). Constraint (7) poses minimum and maximum lot sizes for individual products on pasteurization resources. Constraints (8) and (9) can be written as sterilization resources, and filling and packaging resources. Constraints (10) and (11) correspond to the conservation of flow constraints for each pasteurized intermediate product at each processed milk tank and period, and for each sterilized intermediate product at each aseptic tank, in each time period, respectively. Constraint (10) enforces the amount of raw milk pasteurized at various resources and sent to each processed milk tank as a pasteurized product, in which i should be greater than the total amount of products sent from that tank for sterilization at various resources and sent to various aseptic tanks as sterilized intermediate products. Constraint (11) enforces the amount of intermediate products sterilized 22


Page 23 of 37

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at various resources and sent to aseptic tank as a sterilized product, in which j should be greater than the amount of products sent from that tank for filling and packaging at various resources as various SKUs. Constraints (12), (13), and (14) equal the time consumed by pasteurization, sterilization, and filling and packaging resources, respectively. Constraint (15) limits the sum of the processing times, waiting times, and setup time on resources to the total available processing time. Constraint (16) ensures demand satisfaction for each SKU in each time period. Constraint (17) is the perishability constraint, where SL denotes the maximum number of periods that a product can be stored. This constraint ensures that the inventory in any period can not exceed the amount of SKUs sold in the next SL periods. Constraint (18) ensures the sales of each product at each customer during each time period should not exceed its corresponding demand. Constraint (19) guarantees that the sales of each SKU during each day should not exceed its corresponding demand, while the unsatisfied amount is lost; lost sales amount is greater than the demand minus the sales of each SKU at each day.

5. Computational Results 5.1 Illustrative Example 1 To illustrate the application of the proposed formulation, we consider an example from a real dairy supply chain. The data originate from the processing plant of a dairy company. However, the data are slightly modified because of confidentially concerns. The production process consists three intermediate products, 16 SKUs, two pasteurizers, three sterilizers, and six filling and packaging machines. The model allows for storage between processing steps. After the termination of a pasteurization operation the intermediate products are pumped to one or more processed milk tanks. Similarly, after the termination sterilization operation, sterilized milk is pumped to aseptic tanks. There are four processed milk tanks and three aseptic tanks. Processed milk tank capacities are 80, 80, 100, 100 tons, while aseptic tank 23



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

Page 24 of 37

capacities are 25, 22, 10 tons. In total, 16 different SKUs can be produced and packed in various machines from four different intermediate products, namely full-fat, half-fat, diet milk, and aroma milk. The illustrative example reflects the actual production environment addressed in this paper. Production rates and setup time parameters for different resources are presented in Table 2.

Table 2. Resource Parameters Resource (pasteurizers) PAS01 PAS02 Resource (sterilizers) SPR01 SPR02 SPR03 Resource (filling & packaging) SFM01 SFM04 SFM02 SFM06 SFM07

Resource_Speed (ton/h) 30 25 Resource_Speed (ton/h) 22 20 18

Setup_Time (h) 0,075 0,075 Setup_Time (h) 0,08 0,075 0,075

Resource_Speed (ton/h) 10 10 10 6 6

Setup_Time (h) 0,025 0,045 0,03 0,035 0,025

Numerical experiments are performed to test the significance and the practical applicability of the proposed model. The MILP formulation described in the previous section was implemented and solved using the modeling language IBM ILOG CPLEX Optimization Studio 12.5 on an Intel Core i7-Q720 with 1.6 GHz processor and 8 GB RAM. The main research question will be, “How strong is the influence of the problem’s size? e.g. the number of SKUs, the number of periods?” To answer this question, different problem sizes are defined, differing mainly by the number of periods. Model statistics for the examples with 16 SKUs, and 5, 10, 15, 20-day planning horizons are shown in Table 3.

24


Page 25 of 37

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


Table 3. Model Statistics with 16SKUs and 5, 10, 15, and 20-day planning horizons Model Statistics Time horizon Number of constraints The number of variables The number of binary variables Others Total Profit (in thousands) CPU (h:m:s:ms)

Instances 5-day 4712

10-day 7657

15-day 10602

20-day 13547

6865

11155

15445

19735

4840 2025

7865 3290

10890 4555

13915 5820

3247,636 6185,475 9088,3 11355,501 00:00:59:08 00:08:12:24 00:10:24:62 00:22:18:64

For illustrative purposes we report the results of the Case Problem with 16 SKUs and a 15-day planning horizon. Daily demand data for the case problem are provided in Figure 2. Note that no demand is assigned to the first three days so that the inventory can be accumulated. Since a short-term planning horizon of three weeks is considered, seasonal demand variations are not essential. Total demand data for the case problem are provided in Figure 3. There are three types of demand frequency elements. For the high frequency demand element, one demand element is assigned to each day. For medium and low demand frequency elements, demand elements are assigned randomly. Each SKU has a shelf life of five days. The unit lost sales cost of each product is assumed to be half of its initial price at the market.

25



400 81520213001 350

81520210001 81520218010

300

81520218001 81520212004

250

81520212003 81520212002

200

81520212001 81520201003

150

81520212011 81520201001

100

81520203002 81520212021

50

81520207001 81520202002

0 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17

81520208202

Figure 2. Daily demand data for each SKU

81520213001

81520210001

81520218010

81520218001

81520212004

81520212003

81520212002

81520212001

81520201003

81520212011

81520201001

81520203002

81520212021

81520207001

81520202002

900 800 700 600 500 400 300 200 100 0 81520208202

Demand

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

Page 26 of 37

Figure 2. Total demand data for each SKU Figure 4 presents the objective function breakdown as well as the contribution of each cost term in the total cost ( -in thousands). 26


Page 27 of 37

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


Production Cost Lost Sale Cost Inventory Cost Revenue Total Profit 0

2000 4000 6000 8000 10000 12000 14000 16000

Total Profit Revenue Objective Function Components

9088.3

14731

Inventory Cost 365.2

Lost Sale Production Cost Cost 3449.5

1827.8

Figure 4. Objective Function Breakdown for the case with 16-SKU, 15-day

The profiles of quantity of SKUs sold, inventory levels, lost sales for each planning period, and each SKU are shown in Figures 5, 6, and 7, respectively.

27



80 70 Quantity of SKUs sold (ton)

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

Page 28 of 37

60 50 40 30 20 10 0

81520201001

0 0

1 0

2 0

3 60

4 76

5 60

6 44

7 2

8 23

9 8

10 52

11 20

12 51

13 29

14 5

15 18

16 52

17 44

81520201003

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

81520202002

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

8

81520203002

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

81520207001

0

0

0

0

0

39

0

0

0

0

0

0

0

0

0

0

0

0

81520208202

0

0

0

38

0

0

39

0

0

0

0

0

41

39

43

0

0

0

81520210001

0

0

0

0

0

0

0

58

29

44

0

0

0

0

44

34

0

0

81520212001

0

0

0

0

0

0

0

0

0

39

45

0

46

0

0

35

0

35

81520212002

0

0

0

30

0

0

0

0

0

0

0

0

22

0

0

0

31

0

81520212003

0

0

0

0

0

0

20

0

20

0

0

0

0

0

0

0

0

33

81520212004

0

0

0

0

0

22

0

0

20

0

0

0

30

0

0

0

0

0

81520212011

0

0

0

26

0

22

0

0

0

30

0

0

0

0

0

0

0

0

81520212021

0

0

0

0

0

0

0

0

18

0

0

0

0

24

0

29

0

0

81520213001

0

0

0

0

0

25

0

0

0

0

0

32

0

0

27

0

0

0

81520218001

0

0

0

0

0

0

28

0

0

0

0

0

0

0

0

0

34

0.1484

81520218010

0

0

0

0

40

0

24

0

19

0

0

0

0

0

0

0

0

0

Days

Figure 5. The quantity of SKUs sold

28


Page 29 of 37

120

Inventory Level (ton)

100 80 60 40 20 0

0

1

4

5

6

7

8

9

10 11 12 13 14 15 16 17

81520201001

0

52 104 96 60

2

3

0

2

0

0

0

0

81520207001

0

0

0

0

12

0

0

0

0

0

81520210001

0

0

0

0

0

0

6

0

0

0

0

81520212001

0

0

0

0

0

0

0

17 17

0

0

81520212002

0

0

5

0

0

0

0

0

0

0

0

81520212003

0

0

0

0

0

0

0

0

0

0

0

81520212004

0

0

0

0

0

0

0

20

0

0

81520218001

0

0

0

0

10 18

0

0

0

0

81520218010

0

0

0

0

2

0

5

0

0

0

2

0

1

0

0

0

0

0

0

24

0

0

0

0

16

0

0

28

3

3

0

0

0

0

13 13

0

0

0

0

0

0

13 13

0

0

30

0

0

0

0

0

0

0

0

0

0

0 0.15 0

0

0

0

0

0

0 0

0

0

Figure 6. Inventory level for each SKU

100 90 80 70 Lost Sales (ton)

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


60 50 40 30 20 10 0

0 81520201001 0 81520201003 0

1 0

2 0

3 0

4 0

5 0

6 0

7 8 9 10 11 12 13 14 15 16 17 50 26 43 24 47 0 27 38 22 3 0

0

0

45 53 48 60 82 57 59 61 61 52 73 53 49 59 47

81520202002 0 81520203002 0

0

0

41 86 55 64 51 59 44 73 57 49 59 55 59 48 36

0

0

61 82 53 39 73 51 45 59 64 39 57 53 60 45 55

81520207001 0 81520218001 0

0

0

0

0

0

0

0

0

45

0

52

0

0

27

0

49

0

0

0

0

0

0

2

0

0

0

0

0

0

0

0

0

0

22

81520218010 0

0

0

0

0

0

8

0

0

0

0

0

0

0

0

0

0

0

Figure 7. Lost Sales for SKUs 29



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

Page 30 of 37

The highest amount of SKU sold is observed in period three. Since SKUs 81520201001, 81520201003, 81520202002, and 81520203002 are made from full-fat milk, only SKU 81520201001 was produced by accumulating sufficient inventory. The resource and tank capacities could not be sufficient to produce other SKUs, so their demands are lost. Furthermore, inventory has been observed for some SKUs. Due to the shelf life restriction, sufficient inventory could not be held to satisfy the demands for these SKUs. Generally, inventory levels are maintained at low levels during the planning horizon, with an exception in periods two and three. Relatively high inventory levels of 104, 96 tons are detected due to the high demand requirements of the first periods. If full demand satisfaction is not imposed, lost sales might be observed. Lost sales are observed for those SKUs in time periods from day three to day 17. Most of the SKUs that have low frequency demand elements (e.g, 81520212002, 81520212003, 81520218010) are satisfied. A small amount of lost sales occurred for only two SKUs; 81520218001, 81520218010 in time periods six and 17. A large quantity of lost sales was observed for the SKUs which have a high demand frequency, namely 81520201001, 81520201003, 81520202002, and 81520203002.

5.2 Illustrative Example 2 Another research question which should be answered by these numerical tests is, “How strong is the influence of the tank capacities on the economical and computational performance of the model formulation?” The model has been solved by using the small planning horizon case with seven days so that the model could be solved in a reasonable computation time. Figure 8 displays the results of the objective function breakdowns under different tank capacity levels ( -in thousands). Increasing tank capacities by 20% creates a remarkable improvement in both total costs and total profit. When both processed milk tank and aseptic tank capacities are increased by 20%, the profit amounts to a 7% increase valued 30


Page 31 of 37

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


at 3494. The additional capacity makes the change in reducing the total cost, as well as leading to a reduction in the inventory cost. 6000 5000 4000 3000 2000 1000 0 Profit

ICost

LSCost

PCost

SalesRev enue

3247.636

251.25

870.87

816.02

5185.8

Increase in Tank Capacity 3494.601 %20

189.23

775.23

917.98

5377

Base Tank Capacity

Figure 8. Objective function breakdown under different tank In addition to decreasing lost sales costs and increasing revenue and total profit, the additional capacity changes the waiting time on resources, as well as leading to a reduction in inventory. When additional capacity is incorporated into the model, the amount of SKUs sold and the production costs are increased. A further research question addresses the impact of the duration of shelf life on inventory levels. Figure 9 displays the inventory level under different shelf life durations. Shelf life durations are taken as 4-day and 1-day, respectively. The impact of shelf life can be easily seen through the changes in inventory level. Short shelf life limits the inventory level. Perishability lowers the inventory level, causing a higher lost sales cost. The relatively short shelf-life of milk at the retail level implies that there is product that reaches its expiration and thus removed from the market supply. Figure 10 displays the results of the objective function breakdowns under different shelf life durations ( -in thousands). It is clear that when shelf life duration is increased, the 31



objective function results in slightly higher revenue. The influence of shelf-life on profit is considerable greater.

Inventory of SKS kin time period t

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

Page 32 of 37

100 90 80 70 60 50 40 30 20 10 0 81520201001 (SL=4)

0 0

1 38

2 95

3 47

4 29

5 37

6 0

81520201001 (SL=1) 81520203002 (SL=4)

0

0

57

9

0

8

0

0

19

19

0

0

0

0

81520203002 (SL=1)

0

0

0

0

0

0

0

81520212001 (SL=4)

0

0

0

22

0

0

0

81520212001 (SL=1)

0

0

0

22

0

0

0

81520212004 (SL=4)

0

0

0

0

25

0

0

81520212004 (SL=1)

0

0

0

0

25

0

0

81520212021 (SL=4)

0

0

0

0

6

0

0

81520212021 (SL=1)

0

0

0

0

6

0

0

81520218001 (SL=4)

0

0

0

11

0

0

0

81520218001 (SL=1)

0

0

0

11

0

0

0

Days

Figure 9. Inventory levels under different shelf life durations 6000 5000 4000 3000 2000 1000 0

Profit

ICost

LSCost

PCost

SalesRevenue

SL=1

3183.236

92.745

967.12

750.17

4993.3

SL=4

3247.636

251.25

870.87

816.02

5185.8

Figure 10. Objective function breakdown under different shelf life

32


Page 33 of 37

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6. Conclusion This study introduces a novel model formulation for the continuous process, multi-stage, production planning problem that arises in the dairy industry. The multi-stage production planning problem addresses the issue of determining the production quantities of various intermediate products, and SKUs appearing in consecutive stages in a dairy production facility over a planning horizon. The problem incorporates production characteristics particular to the dairy industry, including shelf lives, production speed, intermediate storage, multi-stage processing, and the availability of resources at each stage. Although this research is illustrated with an application in dairy industry, it encompasses the main characteristics of many food production systems. Therefore, the proposed model can easily be adapted to other production systems with similar divergent product flows. There are several possible extensions to our research. It would also be interesting to extend the model proposed here to incorporate risk management within the dairy industry. It is likely that another interesting extension would be to investigate the development of rolling horizon based heuristics, because of the exponential growth in the computational effort. Acknowledgments The authors gratefully acknowledge the Editor and anonymous reviewers for their insightful suggestions to improve the manuscript during the review process.

References (1) Kallrath, J. (2002). Planning and scheduling in the process industry. OR Spectrum, 24, 219–250

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(2) Fündeling C.-U. and Trautmann, N. “Scheduling of make and pack plants: a case study,” in 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering, W. Marquardt and C. Pantelides, Eds. Elsevier B.V., 2006, pp. 1551– 1556. (3) Smith-Daniels, V. L., Ritzman, L. P., A model for lot sizing and sequencing in process industries.

International Journal of Production Research, 1988, 26, 647-674. (4) Claassen, G.D.H. and van Beek, P., Planning and scheduling packaging lines in food industry,

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(5) van Dam, P., Gaalman, G., and Sierksma, G., Scheduling of packaging lines in the process industry: An empirical investigation, International Journal of Production Economics, 1993, 30, 579589. (6) Nakhla M., Production control in the food processing industry. International Journal of

Operations & Production Management, 1995, 15, 73–88. (7) Foulds, L.R., Wilson, J.M. A variation of the generalized assignment problem arising in the New Zealand dairy industry. Annals of Operations Research, 1997, 69, 105-114. (8) Basnet, C., Foulds, L.R., Wilson, J.M. An exact algorithm for a milk tanker scheduling and sequencing problem. Annals of Operations Research, 1999, 86, 559 – 568. (9) Berlin, J., Sonesson, U., and Tillman, A. M., A life cycle based method to minimise enviromental impact of dairy production through product sequencing. Journal of Cleaner Production, 2007, 15, 347. (10) Lütke Entrup, M., Günther, H. O., Van Beek, P., Grunow, M., & Seiler, T. Mixed-Integer Linear Programming

approaches

to

shelf-life-integrated

planning

and

scheduling

in

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Multistage Production Planning in the Dairy Industry: A Mixed-Integer

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