# Provider Fairness and Beyond-Accuracy Trade-offs in Recommender Systems

Saeedeh Karimi  University of ZanjanIran<EMAIL_ADDRESS>,  Hossein
A. Rahmani  University College LondonUK<EMAIL_ADDRESS>,
Mohammadmehdi Naghiaei  University of Southern CaliforniaUS<EMAIL_ADDRESS>and  Leila Safari  University of ZanjanIran<EMAIL_ADDRESS>
###### Abstract.

Recommender systems, while transformative in online user experiences, have
raised concerns over potential provider-side fairness issues. These systems
may inadvertently favor popular items, thereby marginalizing less popular ones
and compromising provider fairness. While previous research has recognized
provider-side fairness issues, the investigation into how these biases affect
beyond-accuracy aspects of recommendation systems—such as diversity, novelty,
coverage, and serendipity—has been less emphasized. In this paper, we address
this gap by introducing a simple yet effective post-processing re-ranking
model that prioritizes provider fairness, while simultaneously maintaining
user relevance and recommendation quality. We then conduct an in-depth
evaluation of the model’s impact on various aspects of recommendation quality
across multiple datasets. Specifically, we apply the post-processing algorithm
to four distinct recommendation models across four varied domain datasets,
assessing the improvement in each metric, encompassing both accuracy and
beyond-accuracy aspects. This comprehensive analysis allows us to gauge the
effectiveness of our approach in mitigating provider biases. Our findings
underscore the effectiveness of the adopted method in improving provider
fairness and recommendation quality. They also provide valuable insights into
the trade-offs involved in achieving fairness in recommender systems,
contributing to a more nuanced understanding of this complex issue.

Provider Fairness, Recommendation, Beyond-Accuracy

††copyright: none††conference: 17th ACM Conference on Recommender Systems;
Sep. 18–22, 2023; Singapore, Singapore††ccs: Information systems Recommender
systems

##  1\. Introduction and Context

The rapid proliferation of digital content and the abundance of information
available on the internet have made recommendation systems indispensable tools
for users to discover relevant and engaging items. These systems are widely
employed across various domains such as e-commerce, entertainment, and news,
among others. Despite their widespread adoption, recommendation systems often
suffer from popularity bias, a phenomenon where popular items are
disproportionately recommended at the expense of less popular or long-tail
items (Rahmani et al., 2022c, b). This leads to a skewed exposure of items and
potentially unfair treatment of providers, particularly those offering niche
or less popular content. Additionally, popularity bias may adversely affect
the diversity, novelty, and serendipity of recommendations, thereby limiting
users’ experiences and the discovery of new content. In some cases, this bias
could reinforce echo chambers or marginalize certain content providers,
leading to broader societal implications (Geschke et al., 2019; Naghiaei et
al., 2022b; Rahmani et al., 2022a).

Previous research in the field of recommendation systems has primarily focused
on improving accuracy, with little attention paid to the trade-offs between
provider fairness and other essential dimensions of recommendation quality
(Zhao et al., 2022; Naghiaei et al., 2022a; Rahmani et al., 2022d). Some
existing approaches to mitigate popularity bias include re-sampling techniques
(Li and Vasconcelos, 2019; Naghiaei et al., 2022d) and diversity-aware
algorithms (Schelenz, 2021). However, these methods often overlook the complex
interplay between provider fairness and other important aspects of
recommendation quality, such as diversity, serendipity, and novelty. This
leaves a significant gap in our understanding of the broader impact of
fairness-aware algorithms.

In contrast, our research takes a comprehensive approach to examine the
intricate and complex trade-offs that come into play when measures to enhance
fairness in recommendation systems are implemented. We introduce a method,
inspired by existing post-processing techniques (Li et al., 2021), that is
uniquely designed to promote fairness among providers without significantly
sacrificing accuracy. Post-processing methods are particularly advantageous as
they strike a balance between provider fairness, user relevance, and beyond-
accuracy performance (Naghiaei et al., 2022c). Furthermore, these methods are
recommendation model agnostic, enhancing their suitability for real-world
applications. Furthermore, we delve into an in-depth investigation of the
consequences and trade-offs between provider fairness and recommendation
quality across four recommendation algorithm baselines and four datasets in
different domains. To thoroughly investigate the trade-offs between provider
fairness and recommendation quality, we pose several research questions that
guide our study:

  1. RQ1

How does the proposed post-processing re-ranking optimization framework affect
the exposure of long-tail items and overall provider fairness in
recommendation systems?

  2. RQ2

What are the consequences of improving provider fairness on other important
aspects of recommendation quality, such as diversity, serendipity, and
novelty?

  3. RQ3

How do the trade-offs between provider fairness and recommendation quality
manifest across different recommendation algorithms and real-world datasets?

Our experiments offer compelling evidence of the effectiveness of our proposed
framework. It not only improves the exposure of long-tail items and enhances
provider fairness, but also preserves other important aspects of
recommendation quality such as diversity, serendipity, and novelty. The
subsequent sections present a detailed discussion of our methodology,
experiments, and findings.

##  2\. Proposed Provider Fairness Model

In this section, we propose a simple yet effective model to improve fairness
among providers to analyse the effect of mitigating provider bias on beyond-
accuracy metrics.

Provider Fairness. In recommendation systems, provider fairness can be
addressed by mitigating the popularity bias, where popular items are
disproportionately recommended at the expense of less popular or long-tail
items. Let $U=\\{u_{1},u_{2},\dots,u_{m}\\}$ denote the set of users and
$I=\\{i_{1},i_{2},\dots,i_{n}\\}$ represent the set of items in the
recommendation system, $m$ and $n$ are the number of users and items,
respectively. We divide the items into two groups, popular (or short-head) and
non-popular (or long-tail) items. To measure the average difference in the
chosen popularity measure between popular and non-popular items across all
users, we introduce binary decision variables $Y_{j}$, where $Y_{j}=1$ if item
$j$ is a popular item, and $Y_{j}=0$ if item $j$ is a non-popular item. We
define the provider fairness metric $F$ as follows:

(1)
$F=\frac{1}{m}\sum_{i=1}^{m}\left(\sum_{j=1}^{n}Y_{j}*X_{ij}-\sum_{j=1}^{n}(1-Y_{j})*X_{ij}\right)$

The provider fairness metric, $F$, captures the difference in exposure between
popular and non-popular items in the recommendation system. By considering
this metric, our goal is to balance the exposure of items from both groups,
ensuring that long-tail items receive adequate visibility while still
delivering relevant recommendations to users. An $F$ of $0$ indicates equal
exposure to long-tail and short-head items.

Fairness-aware Algorithm. We define a framework that generates fairness-aware
recommendation lists by applying a re-ranking method to the output of a
baseline recommendation model. Our fairness-aware algorithm is further
illustrated by the Algorithm 1, which provides a step-by-step implementation
of the re-ranking process. The algorithm takes as input the score matrix $R$,
the number of items to recommend $K$, the binary popularity matrix $Y$, and
the hyperparameter $\lambda$. The score matrix $R$ is generated by baseline
recommendation models and has dimensions $m\times n$, where $R_{ij}$
represents the predicted score or preference of user $i$ for item $j$. We
introduce binary decision variables $X_{ij}$, where $X_{ij}=1$ if item $j$ is
selected in the re-ranked list for user $i$, and $X_{ij}=0$ otherwise.
Furthermore, to incorporate the provider fairness aspect into the re-ranking
process, we introduce a hyperparameter lambda ($\lambda$) that controls the
trade-off between recommendation score and provider fairness. In the integer
programming model, we aim to maximize the sum of scores while minimizing the
difference in the chosen popularity measure between popular and non-popular
items, controlled by the hyperparameter $\lambda$. The ReRank algorithm
formulates and solves the integer programming problem to generate the re-
ranked recommendation matrix $X$.

Algorithm 1 Re-ranking Recommendations with Provider Fairness

1:procedure ReRank($R$, $K$, $Y$, $\lambda$)

2: Initialize $m\times n$ binary matrix $X$

3: Input: Score matrix $R\in\mathbb{R}^{m\times n}$, number of items to
recommend $K$, binary popularity matrix $Y\in\\{0,1\\}^{n}$, hyperparameter
$\lambda$

4: Output: Re-ranked recommendation matrix $X\in\\{0,1\\}^{m\times n}$

5: Define the objective function as:
$Z(X)=\sum_{i=1}^{m}\sum_{j=1}^{n}(R_{ij}*X_{ij})-\lambda*F$

6: Define the constraints as:

  * •

$\sum_{j=1}^{n}X_{ij}=K,\quad\forall i\in\\{1,\dots,m\\}$

  * •

$X_{ij}\in\\{0,1\\},\quad\forall i\in\\{1,\dots,m\\},\forall
j\in\\{1,\dots,n\\}$

  * •

$Y_{ij}\in\\{0,1\\},\quad\forall j\in\\{1,\dots,n\\}$

7: Formulate the integer programming problem: $\max_{X}Z(X)$ subject to the
constraints

8: Solve the integer programming problem using a solver (e.g., Gurobi or
CPLEX)

9: Return the re-ranked recommendation matrix $X$

10:end procedure

The constraints of the model ensure that for each user $i$, exactly $K$ items
are selected in the re-ranked list (i.e., $X_{ij}=1$ for $K$ items). Integer
programming is an NP-hard problem, which means that the runtime complexity
increases exponentially with the size of the problem. However, modern integer
programming solvers such as Gurobi111https://www.gurobi.com/ or
CPLEX222https://www.ibm.com/products/ilog-cplex-optimization-studio can
efficiently handle problems of moderate size, making them suitable for solving
the re-ranking problem in this context. In this formulation, the
hyperparameter $\lambda$ allows for controlling the trade-off between
maximizing the sum of recommendation scores and improving provider fairness.
When $\lambda$ is set to a high value, the optimization model places more
emphasis on provider fairness, while lower values of $\lambda$ prioritize
recommendation scores. When $\lambda=0$, the re-ranked list will be the same
as the initial top-$N$ recommendations, as no emphasis is placed on provider
fairness. By tuning $\lambda$, practitioners can balance the need for accurate
recommendations with the importance of giving exposure to less popular items,
thus creating a more diverse and fair recommendation experience for users and
content providers.

##  3\. Experimental Setup

This section briefly describes the datasets, baseline models, and evaluation
metrics. To foster the reproducibility of our experiments333We release our
codes and dataset for the reproducibility and future work at
https://github.com/rahmanidashti/BeyondAccProvider, we implemented and
evaluated all the recommendations with the open-source Python-based
recommendation toolkit Cornac (Salah et al., 2020).

###  3.1. Datasets

To evaluate our proposed approach, we use four publicly available datasets
from different domains and the details of these datasets are given in Table 1.
We divided the dataset into three subsets: a training set (70%), a validation
set (10%), and a test set (20%), which were used for all model training and
evaluation. Additionally, we classified items into two groups: popular or
short-head items, representing the top 20% of items with the most
interactions, while the remaining items were classified as non-popular or
long-tail or items.

Table 1. Statistics of the datasets: $\left|U\right|$ is the number of users,
$\left|I\right|$ is the number of items, $\left|P\right|$ is the number of
interactions, $\left|SI\right|$ is the number of popular (short-head) items,
$\left|LI\right|$ is the number of non-popular (long-tail) items.

Dataset $\left|U\right|$ $\left|I\right|$ $\left|P\right|$
$\frac{\left|P\right|}{\left|U\right|}$
$\frac{\left|P\right|}{\left|I\right|}$ %Sparsity Domain $\left|SI\right|$
$\left|LI\right|$ Epinions 2,677 2,060 103,567 38.6 50.2 98.12% Opinion 412
1,648 BookCrossing 1,136 1,019 20,522 18.0 20.1 98.22% Book 203 816 Gowalla
1,130 1,189 66,245 58.6 55.7 95.06% POI 237 952 Last.fm 1,797 1,507 62,376
34.7 41.3 97.69% Music 301 1,206

###  3.2. Baselines

To analyse the effectiveness of our provider fairness method, we compared its
performance to that of two traditional models (WMF (De Pauw et al., 2022; Tran
et al., 2018) and PF (Mnih and Salakhutdinov, 2007)) and two neural network-
based recommendation models (NeuMF (Xue et al., 2017; Zou et al., 2020) and
VAECF (Yalcin, [n. d.])). In addition to assessing accuracy, our target is to
examine the method’s ability to meet other important beyond-accuracy criteria,
such as diversity, novelty, and serendipity. By including these beyond-
accuracy aspects in our evaluation, we were able to provide a more
comprehensive analysis of providing provider fairness compared to the
traditional and neural network-based models.

###  3.3. Evaluation Metric

While accuracy is a crucial metric to assess the performance of recommender
systems, it is vital to recognize that it is not the sole indicator of a
system’s effectiveness. Other essential factors to consider while evaluating a
recommender system’s performance include diversity, novelty, and exposure.
Therefore, to assess the performance of recommender systems and the proposed
provider bias strategy, we rely on accuracy-based metrics including Precision
(Arif et al., 2022), Recall (Arif et al., 2022; Cremonesi et al., 2010; Ashok
et al., 2009), NDCG (Han and Yamana, 2017), and beyond-accuracy metrics,
namely, Diversity (Han and Yamana, 2017; Faria, 2023), Novelty (Kaminskas and
Bridge, 2016), Coverage (Ge et al., 2010), Serendipity (Ge et al., 2010),
Personalisation (Ge et al., 2010), and Item Exposure (Naghiaei et al., 2022c).

A diverse and novel recommendation system can help users discover new and
unexpected items that they may not have considered otherwise. Additionally,
serendipity measures the system’s ability to offer unique and pleasant
surprises to users beyond their usual preferences.

##  4\. Result

Table 2. The recommendation performance of our re-ranking method and
corresponding baselines on Epinion and BookCrossing datasets. The evaluation
metrics here are calculated based on the top-10 predictions in the test set.
Our best results are highlighted in bold. RelShort and Rellong values denote
the number of relevant recommended short-head and long-tail items. Fairness-
unaware algorithm is shown with the notation $N$ and our proposed re-ranking
method is denoted with $P$.

Model Type Accuracy Beyond-Accuracy Item Exposure NDCG Pre Rec Nov. Div. Cov.
Per. Ser. Short. RelShort Long. RelLong Epinion PF N 0.0321 0.0312 0.0445
4.9602 0.9153 0.5073 0.9601 0.9388 22,169 8,637 4,601 3,595 PF P 0.0323 0.0314
0.0448 4.9915 0.9161 0.5112 0.9613 0.9391 8,316 7,768 18,454 4,774 WMF N
0.0235 0.0225 0.0317 4.9484 0.9274 0.2913 0.9366 0.9441 25,225 6,566 1,545
1,276 WMF P 0.0225 0.0215 0.0296 5.1762 0.9319 0.3587 0.9457 0.9467 20,846
6,316 5,924 2,110 NeuMF N 0.0451 0.0406 0.0574 3.8923 0.8708 0.1214 0.7714
0.9257 26,533 6,276 237 291 NeuMF P 0.0442 0.0392 0.0556 3.9874 0.8743 0.1340
0.7853 0.9267 24,443 6,128 2,327 591 VAECF N 0.0444 0.0410 0.0597 4.3809
0.8794 0.2893 0.9038 0.9266 24,707 7,998 2,063 1,850 VAECF P 0.0445 0.0410
0.0600 4.4043 0.8804 0.2947 0.9052 0.9269 15,676 7,522 11,094 2,622
BookCrossing PF N 0.0108 0.0106 0.0276 5.8712 0.9620 0.8940 0.9780 0.9716
5,858 1,554 5,502 1,715 PF P 0.0111 0.0107 0.0271 5.9188 0.9624 0.8989 0.9794
0.9721 337 794 11,023 1,952 WMF N 0.0062 0.0059 0.0158 6.6007 0.9733 0.9715
0.9802 0.9770 1,968 1,323 9,392 1,998 WMF P 0.0062 0.006 0.0161 6.6111 0.9735
0.9725 0.9801 0.9771 0 0 11,360 2,061 NeuMF N 0.0165 0.0147 0.0380 4.5143
0.9422 0.0343 0.2026 0.9671 11,360 411 0 0 NeuMF P 0.0165 0.0147 0.0380 4.5143
0.9422 0.0343 0.2026 0.9671 11,360 411 0 0 VAECF N 0.0211 0.0189 0.0473 5.1439
0.9440 0.4004 0.9165 0.9630 9,287 1,446 2,073 648 VAECF P 0.0190 0.0177 0.0475
5.5079 0.9430 0.4524 0.9338 0.9647 5,672 1,339 5,688 826

In this section, we provide a thorough examination of the beyond-accuracy
metric in the fairness-aware recommendation performance in comparison to
fairness-unaware baseline models. Our analysis seeks to understand the trade-
offs among different item groups in terms of beyond-accuracy objectives.

###  4.1. Fairness-aware Algorithm Effectiveness.

Trends in Tables 2 and 3 show that the fairness-aware algorithm effectively
increases the exposure of long-tail items while reducing the prominence of
short-head items. This shift allows a broader range of items to gain exposure,
providing a more equitable recommendation environment for providers. The
redistribution of item exposure has several implications. First, it benefits
providers by offering a fairer distribution of exposure, enabling long-tail
items to compete with more popular items and potentially gain traction among
users. For instance, as you see in Table 2 for Epinion the total number of
short-head and long-tail recommended items using PF baseline has changed from
$(22169,4601)$ to $(8316,18454)$, respectively. Second, it contributes to the
overall diversity and novelty of the recommended items, enhancing the user
experience and encouraging them to explore new content while keeping the
relevant popular items. For example, for the case of PF and on Epinion dataset
in Table 2, one can see improvement along all beyond-accuracy metrics.
Furthermore, the number of relevant short-head and long-tail recommended items
(denoted as Relshort and Rellong) changes from $(8637,3595)$ to $(7768,4774)$
indicating the model capability of choosing relevant popular items while
giving more exposure to relevant long-tail items. Similar trends can be
observed in other baselines and datasets. This equitable distribution aligns
with the goals of our fairness-aware algorithm, as it seeks to strike a
balance between promoting fairness and maintaining the quality of
recommendations.

Table 3. The recommendation performance of our re-ranking method and
corresponding baselines on Gowalla and Last.fm datasets. The evaluation
metrics here are calculated based on the top-10 predictions in the test set.
Our best results are highlighted in bold. RelShort and Rellong values denote
the number of relevant recommended short-head and long-tail items. Fairness-
unaware algorithm is shown with the notation $N$ and our proposed re-ranking
method is denoted with $P$.

Model Type Accuracy Beyond-Accuracy Item Exposure NDCG Pre Rec Nov. Div. Cov.
Per. Ser. Short. RelShort Long. RelLong Gowalla PF N 0.0592 0.0558 0.0568
4.0587 0.8462 0.6098 0.9615 0.8730 7,939 4,884 3,361 3,233 PF P 0.0592 0.0559
0.0571 4.0868 0.8467 0.6165 0.9631 0.8733 2,085 4,192 9,215 4,375 WMF N 0.0338
0.0347 0.0368 4.5012 0.8862 0.4617 0.9563 0.8937 6,526 3,777 4,774 2,173 WMF P
0.0339 0.0346 0.0365 4.5163 0.8871 0.4626 0.9567 0.8942 40 386 1,1260 2,618
NeuMF N 0.0563 0.0528 0.0536 3.3409 0.8127 0.1463 0.8815 0.8649 10,447 3,451
853 577 NeuMF P 0.0509 0.0485 0.0470 3.6512 0.8093 0.1615 0.8901 0.8657 7,361
3,309 3,939 877 VAECF N 0.0652 0.0625 0.0673 3.8219 0.8092 0.4079 0.9543
0.8579 8,025 4,360 3,275 2,724 VAECF P 0.0569 0.0548 0.0560 4.3186 0.8145
0.4407 0.9552 0.8682 3,299 3,739 8,001 3,304 Last.fm PF N 0.0372 0.0350 0.0469
5.0905 0.9201 0.6682 0.9775 0.9213 12,480 6,978 5,490 2,967 PF P 0.0371 0.0350
0.0470 5.1089 0.9206 0.6709 0.9779 0.9215 10,724 6,939 7,246 3,133 WMF N
0.0319 0.0301 0.0434 5.5358 0.9294 0.7804 0.9723 0.9257 8,000 6,267 9,970
3,272 WMF P 0.0279 0.0262 0.0380 5.8797 0.9420 0.7963 0.9725 0.9328 0 0 17,970
3,688 NeuMF N 0.0415 0.0390 0.0519 3.7975 0.8647 0.0916 0.8802 0.9034 17,863
4,866 107 67 NeuMF P 0.0387 0.0366 0.0488 3.8812 0.8694 0.0962 0.8843 0.9049
16,665 4,835 1,305 184 VAECF N 0.0555 0.0514 0.0725 4.6164 0.8769 0.4280
0.9661 0.8985 13,998 6,746 3,972 1,920 VAECF P 0.0556 0.0517 0.0732 4.6635
0.8794 0.4326 0.9669 0.8996 6,786 6,354 11,184 2,406

###  4.2. Beyond-accuracy Metrics Influence.

According to Fig. 1, the implementation of our proposed algorithm has led to a
consistent upward trend in beyond-accuracy metrics across all models and
datasets. For example, as can be seen in Table 2, we observe notable
improvements in novelty, diversity, coverage, and serendipity for all models
in Epinion. For instance, the WMF model’s novelty increased from 4.94 to 5.17,
while diversity in the VAECF model rose from 0.87 to 0.88. Similar positive
trends are observed in the BookCrossing dataset. Overall, the novelty has the
most improvement in the Last.fm dataset, while diversity and coverage have the
best improvement in the Epinion dataset. This can be due to the underlying
data characteristics of the datasets. However, among the models, the coverage
in the NeuMF model has relatively low values, although it has increased in the
fairness-aware model, its result is insignificant compared to other models.
The personalisation and serendipity also have fewer changes compared to
novelty and diversity, but they improved more in Epinion.

###  4.3. Dataset and Baseline Model Dependency.

Our analysis reveals dependencies on both the dataset and the baseline model
used when implementing the fairness-aware algorithm. The BookCrossing dataset,
for example, exhibited more significant improvements across various criteria
than other datasets, such as Gowalla. This dataset dependency can be
attributed to the unique data structure and characteristics within each
dataset, which can affect the outcomes of the research and alter the values
obtained from the implementation of the models. On the other hand, the
baseline model also plays a crucial role in the effectiveness of the fairness-
aware algorithm. The VAECF model, in particular, exhibits the most substantial
enhancement in beyond-accuracy criteria due to its ability to capture complex
and non-linear relationships between user preferences and item features
through its variational autoencoder framework. Conversely, the NeuMF model
shows comparatively less improvement in beyond-accuracy criteria, possibly due
to its reliance on a combination of matrix factorization and multi-layer
perceptron techniques. In conclusion, the performance differences among models
can be ascribed to their distinct underlying mechanisms, which consider user
preferences, item features, and intricate interactions. Nonetheless, applying
the fairness-aware reranking to all models consistently improved beyond-
accuracy metrics, with the extent of this improvement varying notably among
the models.

(a) NDCG

(b) Pre

(c) Rec

(d) Nov

(e) Div

(f) Cov

(g) Per

(h) Ser

Figure 1. Comparative performance of fairness-unaware and fairness-aware
models across key evaluation metrics. Each box represent the variation among
all the datasets.

##  5\. Discussion

In this section we summarize the answers we found to the research questions we
listed in Section 1.

###  5.1. RQ1: Impact on exposure of long-tail items and provider Fairness.

Our findings demonstrate that the proposed post-processing re-ranking
optimization framework effectively increased the exposure of long-tail items,
resulting in a more diverse and equitable distribution of recommendations. On
average, the number of recommended and relevant recommended long-tail items
for fairness-unaware and fairness-aware models are $(3565.8,2116.6)$ and
$(8765,2573.1)$, respectively. This enhancement in exposure subsequently
contributed to improving overall provider fairness.

###  5.2. RQ2: Consequences of improving provider fairness on other aspects of
recommendation quality.

By improving provider fairness, we observed positive effects on other aspects
of recommendation quality, such as diversity, serendipity, and novelty. This
suggests that promoting fairness can enhance the user experience by offering
more diverse and unexpected recommendations, catering to a wider range of user
interests and preferences. However, it is important to strike a balance
between increasing fairness and maintaining the quality of recommendations.
Overemphasizing fairness could potentially result in recommendations that do
not align with users’ interests or needs, leading to dissatisfaction and
reduced system efficiency. Thus, it is crucial to exercise moderation when
suggesting unpopular items, ensuring that popular items are still included in
the recommendations.

###  5.3. RQ3: Trade-offs between provider fairness and recommendation quality
across different algorithms and datasets.

Our analysis indicates that the trade-offs between provider fairness and
recommendation quality manifest differently across various recommendation
algorithms and real-world datasets. The nature of the dataset, data structure,
and data characteristics can affect the performance of fairness-aware
algorithms and influence the balance between fairness and recommendation
quality. The choice of recommendation algorithm also plays a role in managing
this trade-off. While some algorithms may perform better in certain datasets,
they may not yield the same results in others. As such, it is essential to
consider the specific context and requirements of a recommendation system when
selecting and implementing a fairness-aware algorithm.

##  6\. Conclusion

In conclusion, this study presents a fairness-aware re-ranking algorithm
designed to mitigate biases in recommendation systems, specifically addressing
the overemphasis on popular items. By incorporating a comprehensive set of
beyond-accuracy evaluation metrics, including novelty, diversity, coverage,
and serendipity, we thoroughly analyze the impact of our fairness-aware
approach on these metrics and their implications for item providers. The
results demonstrate that our fairness-aware approach has a positive impact on
these beyond-accuracy metrics, with only a minor reduction in recommendation
accuracy. This indicates that the overall effectiveness of the system is not
significantly compromised when provider is introduced in this way.

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title section subsection subsubsection descriptionlabel paragraph

# A Two-Phase Method for Production Planning and Machine Speed Optimization
Problem

Mustafa Kaan Topaloğlu  Department of Industrial Engineering, İstanbul Bilgi
University, İstanbul, Turkey  Banu Kabakulak  Department of Industrial
Engineering, İstanbul Bilgi University, İstanbul, Turkey

###### Abstract

Textile industry is becoming a highly competitive area with the increase in
demand for textile products. Since expanding the production capacity is not
always feasible, optimizing the existing system is more practical. In
particular, we consider a felt production system of a textile factory
operating in Turkey in this study. We aim to minimize the production costs by
optimizing machine operating speeds as well as building an efficient
production lot sizing plan within the planning horizon. In this direction, we
propose the Lot Sizing and Machine Speed (LSMS) nonlinear model to determine
the optimal unit processing times and production quantities while minimizing
the work-in-process and end item inventories by changing the machine operating
speeds dynamically according to the demands. Since LSMS nonlinear optimization
problem is NP-hard, we design a Two-Phase heuristic which iteratively
processes a linear programming model by utilizing a commercial solver at each
phase. We intensively test our Two-Phase heuristic via randomly generated
demand, planning horizon and machine-hour capacity scenarios. Our
computational experiments show that the introduced Two-Phase heuristic can
find the local-optimal results in acceptable amount of time.

Keywords: Nonlinear mathematical modelling, Two-Phase heuristic, lot sizing,
machine speed optimization

††*Corresponding author: Banu Kabakulak, e-mail<EMAIL_ADDRESS>
##  1 Introduction and Literature Review

Industries are trying to adapt themselves to the increase in demand in their
products, which is caused by the increase in the population worldwide. Today,
the textile industry, which is used in automotive, ready-made clothing,
packaging, construction, and many similar areas, has become a competitive
industry due to the increasing demands in the areas where it is used. The fact
that most textile factories are formerly established and do not have major
production systems, the competition possibilities of these factories are
limited.

The felt production system examined in this study is optimized by minimizing
the total operating cost, additionally making a production plan via optimizing
unit processing times and the production amounts of the machines. At the end
of the optimization, it is envisioned that competition level of the production
system will increase while keeping efficiency level of the production system
high, as well as increasing the total production capacity of the system.

Production plan plays a crucial role in every factory and in the production
system. Success and profitability of the production system depends on a good
plan. There have been many researches in developing and generalizing the idea
of the production plan with many variables. To achieve the least possible cost
in a given planning horizon whilst satisfying all the requirements of the
system, [17] examine a production schedule of a system and shape up a
production plan by constructing different scenarios with requirements.
Modigliani and Hohn form production intervals throughout the planning horizon
and show that the optimal production plan consists of series of optimal
production intervals. In later years, [9] generalize on Modigliani and Hohn’s
work and provide a computational algorithm called Generalized Modigliani and
Hohn (GMH) Algorithm. [4] hypothesized the idea of deterministic production
plan with non-convex cost and capacity constraints, with finite number of
periods and without varying capacity from period to period. While providing
two scenarios where backlogging allowed and not allowed, [4] use dynamic
programming to generate the shortest path route to find optimal solutions.

To design good production parameters, such as lead time and lot size, in a
hierarchical production plan environment, [6] propose a frame-work which is
optimized by simulation-based optimization methods. [15] construct a
linearized model for molded pulp packing industries production plan with the
objective of minimizing total set up cost and inventory costs, to achieve this
planning horizon is splitted into periods, and periods divided into slots. In
their research, [18] develop a dynamic programming model for minimizing the
cost of manufacturing and remanufacturing system while minimizing inventory of
the system and the backlog. The system in [18] has finite number of stages
within the planning horizon. With the proposed deterministic and stochastic
dynamic programming model, [18] produce policies how much to produce at each
stage. In their work, [18] show that the optimal production plan achieved by
developing optimal policies for stages throughout the planning horizon. In
work of [14], goal programming model is developed and for the multi-objective
production plan problem, a Linear Psychical Programming is proposed. [22]
generate two hybrid evolutionary methods named as gradient-based SPEA2 and
gradient-based NSGA_II, to optimize a nonlinear multi-objective production
plan in mineral processing. In creation of production plan for continuous
steel casting, [24] propose a hybrid optimization model which minimizes the
queues, starting time of the operations and conflicts of tasks by combining
genetic algorithm and parallel-backward inferring algorithm, that works
consecutively solve the model. To evaluate the model, [24] create a simulation
model based on cellular automata (CA). [11] propose a hybrid cell evaluated
genetic algorithm (CEGA) which merges genetic algorithm with fractional
factorial design for creating a simulation based optimization method for a
remanufacturing system’s production plan in a stochastic environment with the
objective of maximizing profit. With the proposed method, CEGA is more
effective method to search global optimum than other simulation methods. In a
very sensitive production system where the raw materials holding cost is high
and raw materials have different processes to make different products, such as
sawmill system, [13] propose multi period production plan model (M3PM) with
the objective of maximizing the total net revenue. Proposed M3PM method
consists of two segments: first one is Linear Programming (LP) algorithm named
as resource coordinator which creates optimal levels of activities that meets
the constraints, second one is a simulation model which constructs columns
(activities) by using Dynamic Programming (DP) for the LP model.

Not all production systems are discrete. [3] propose methods for the
continuous-time production systems, where demands are satisfied continuously
as well as raw materials arrives to the system successively. To generate a
solving procedure for this problem, [3] discuss that planning horizon should
be breakdown into a periods and production rates of the machines should be
allowed to change to reduce the cost within the period, called at _switching
times_ , not at the end of the period. The method, named as two-step iterative
procedure, consists of two stages, at first stage production rates are
optimized, at second stage switching times within the periods are optimized.
Objective of this method is that minimizing the generated cost from backorder
and inventory keeping within the production system. Main difference between
our study and the problem in [3] is that, our system is a discrete system with
allowed work-in-process (WIP) accumulation, not a continuous one nor WIP
disallowed. Therefore there is no need of changing the machine speed within a
period. Additionally, our mathematical model is a nonlinear programming (NLP)
formulation due to the machine speed constraints, whereas [3] have a
nonlinearity because of objective coefficients.

Production planning contains a lot-sizing problem (LSP), that is deciding how
many products to produce in each period. Although in our production system
there is a cutting operation, our study does not include the cutting stock
problem (CSP) since the aim of our study to minimize cost while changing the
unit processing times, thus we only consider the LSP within our production
plan. In [8], both LSP and CSP problems are considered for production planning
in a furniture industry. [8] say that, treating LSP and CSP problems
separately, increases the global cost, hence propose a Lagrangian-based
heuristic to combine and optimize the lot-cut problem (LCP) while generating
lower cost.

One form of production planning is aggregated production plan (APP) which
includes laying off, hiring and overtime. By combining dynamic programming and
metaheuristic methods such as bat and bee algorithms, [12] proposes an a
method named as _Two-Phase_ Approximation Method. The method first uses bat
and bee algorithms to find feasible solutions for the APP problem, then this
dynamic programming method choose the optimal algorithm for the problem. For
finding an optimal APP, [10] come up with a stochastic robust optimization
model with a nonlinear objective. To solve the problem as a linear problem,
[10] introduce two deviational decision variables to the objective function.
Offered robust optimization method is used later on forming of multi-site
production plan for a company within the unclear environment with the
objective of minimizing total cost. [21] find out that particle swarm
optimization (PSO) is insufficient method for optimizing an APP with mixed
integer linear programming model and propose a modified PSO (MPSO) which has
sub-particles in contrast to the standard PSO. In measuring the efficiency of
MPSO, [21] use Genetic Algorithm (GA) and standard PSO and show that
modification provides better results in accuracy, converge time and
reliability. In creating an APP in a continuous-time environment [5] introduce
a hybrid method for finding a global optimum which combines GA and simulated-
annealing (SA) named as GA-SA. [5] optimize the continuous-time APP with the
objective of minimizing the total cost while considering randomly generated
production rates and planning horizons within the APP. In their work, [5] use
GA, SA and hybrid GA-SA separately and conclude that proposed hybrid method
works best among all three methods. [2] use a modified and specific variant of
PSO named as linear reduction of inertia weight to solve the multi-period,
multi-product integer linear programming APP problem with possibilistic
environment (PE) and the objective of minimizing total cost. To solve the
uncertainty which comes from PE, [2] propose fuzzy triangular membership
function. With a case study, proposed PE-PSO method showed better qualities
than GA and fuzzy based GA.

[16] set up a linear multi-period, multi-objective APP for energy saving with
some uncertain parameters and objectives of minimizing energy consumption,
operating cost and carbon emission while considering production, inventory and
backlog. Proposed solution method first uses goal attainment technique and
then utilizes robust optimization method with aggregate constraint approach.
Another approach for forming a multi-objective multi-period APP with
uncertainty according to seasons, mixed integer nonlinear programming (MINLP)
model of [7] with objectives of maximizing the customer satisfaction and
minimizing the total cost. [7] create four seasons with different uncertain
demands and design the model such that it can optimize every season
separately. To deal with the uncertainty, robust optimization is introduced to
the model. After clarifying the uncertainty, goal programming with the
algorithms multi-objective invasive weed optimization algorithm and non-
dominated sorting genetic algorithm are introduced to solve the model. After
the optimization with applying Taguchi Method, the efficiency is improved by
finding the best set of parameters of the proposed model.

Key processes like mining, drilling or, in general extracting raw materials,
happens outside of the factory environment. Hence, not all production plans
have to be in a factory environment. In organizing a production plan in an
open-pit mine which consists of set of blocks with different attributes, [20]
discuss a new metaheuristics which derives from Ant Colony Optimization to
maximizing the total profit. [20] show that the proposed metaheuristics are
Max-Min Ant System and the Ant Colony System can solve the open-pit mine
problem with any type of nonlinear constraints and objective function.
Establishing a production plan for petroleum refineries, [19] propose a
extended model that is a multiperiod MINLP problem with uncertainty which is
derived from the Pinto and Moro’s model. The objective function is maximizing
the profit while taking inventory levels, operation and raw materials costs
into consideration and solved by using MINLP algorithms. In a uncertain
environment with significant changes in market demands, companies may use
reconfigurable manufacturing system (RMS) to adapt to changes. To utilize RMS
effectively [1] proposes a MINLP model with stochastic order environment.The
RMS model optimizes the production tasks within the planning horizon as well
as optimizing the batch sizes to minimize the inventory. The RMS model
objective is maximizing profit while considering production costs, inventory
costs and changeover costs from single-task to multiple-task production.
Proposed solving procedure is finds optimal/near-optimal output using GA. [23]
introduce a MINLP model with the objective of maximizing the overall profit
for ethylene plant while considering inventory, processing operations,
materials, energy consumption and energy utilization which is more efficient
than the base model. [23] create two scenarios with same planning horizons and
different number of periods and noted that as the number of periods increases
(as the system becomes continuous time), the quality of the solution and
profit increases, the size of the model increases, thus increases the
computational time.

In this study, for developing and optimizing the felt production system, we
proposed a NLP production model with the objective of minimizing the total
production cost which consists of processing cost, inventory holding cost,
transportation cost of the inventories and energy consumption of the machines.
We aim to minimize the objective function with lot sizing which will be
achieved by rearranging the machine speeds (unit processing times) at each
period according to the demand. To solve the proposed mathematical model in
polynomial time we developed a heuristic named as _Two-Phase_ method to get a
near-optimal solution for the proposed mathematical model. We tested the
system under three different scenarios generated with three key parameters
which are planning horizon of the production, maximum inventory levels and
plant capacity.

The contributions of this research to the literature can be summarized as
follows:

  * •

We propose a NLP model for the felt production system which creates a
production plan by changing machine speeds at the beginning of each period
within the planning horizon.

  * •

We develop a linear programming based _Two-Phase_ heuristic, which works at
polynomial time to get a near-optimal solution for the nonlinear model.

  * •

We test the performance of _Two-Phase_ heuristic on randomly generated
instances. The results indicate that _Two-Phase_ can find near optimal
production plan and machine speeds for real size instances in acceptable time.

##  2 Problem Definition

In this study, we focus on the machine speed optimization and lot sizing
problem in an existing felt production facility. The proposed production plan
aims to minimize the total cost by decreasing the inventory levels, deciding
what to produce and when to produce by dynamically changing the machine speeds
(unit processing times of the machines) within the planning horizon $T$.
Planning horizon $T$ is divided into periods $t$ to create a efficient plan
for deciding the unit processing times of the machines as well as satisfying
demands of the considered periods.

In the felt production system, there are three different machines which are
named as Production Line 1 (PL1, $m=1$), Production Line 2 (PL2, $m=2$) and
Cutting Machine (CM, $m=3$). In PL1, there are three operations, these are
mixing, carding and needling. These mechanical operations turn the raw
material into the compact felt products. At the end of the PL1, the felt is
rolled into a cylinder. In PL2, compact felt products go through chemical
process by chemicalization operation and free conveyor dryer machine. Similar
to PL1 chemical felts are rolled into a cylinder at the end of the PL2. CM is
cutting the felt product into the plaque shapes. According to these machines,
there are four different types of products with respect to their production
line and machine routing. These are: Non-Chemical Cylinder, Non-Chemical
Plaque, Chemical Cylinder and Chemical Plaque.

  * •

_Non-Chemical Cylinder ( $i=1$): _Raw material goes through only PL1 and
becomes an end item.

  * •

_Non-Chemical Plaque ( $i=2$): _Raw material goes through PL1 and CM
respectively and becomes an end item.

  * •

_Chemical Cylinder ( $i=3$): _Raw material goes through PL1 and PL2
respectively and becomes an end item.

  * •

_Chemical Plaque ( $i=4$): _Raw material goes through PL1, PL2 and CM
respectively and becomes an end item.

Raw material processing in the system always starts in PL1. Production can not
start in PL2 nor in CM, since the inputs to these machines inputs are basic
compact felts which are only produced in PL1. Machine routing sequence defines
the product types, thus it is not possible to change the production routing.
For example it is not possible to produce a Chemical Plaque with a sequence of
PL1-CM-PL2, since PL2 takes basic compact felts as a cylinder unit, not as a
plaque unit. This makes the production system a flow-shop. In Figure 1, the
layout of the production system can be seen.

Figure 1: Facility layout for felt production

$\mathbf{A}=\begin{bmatrix}1\ 0\ 0\\\ 1\ 0\ 1\\\ 1\ 1\ 0\\\ 1\ 1\ 1\\\
\end{bmatrix}\ \ \ \mathbf{P}=\begin{bmatrix}1\ 0\ 0\\\ 1\ 3\ 0\\\ 1\ 2\ 0\\\
1\ 2\ 3\\\ \end{bmatrix}$

(a) (b)

Figure 2: (a) Production route matrix $\mathbf{A}_{im}$, (b) production
sequence matrix $\mathbf{P}_{i}$ of product $i$

Figure 2a shows product-machine assignment as $\mathbf{A}$ matrix. For
example, chemical cylinders ($i=3$) are processed in PL1 ($m=1$) and PL2
($m=2$), In the $\mathbf{A}$ matrix this is shown as 1 in the corresponding
$(i,m)$ cells, i.e., $\mathbf{A}_{31}$ $\mathbf{A}_{32}$ 1 and
$\mathbf{A}_{33}$ is 0, since the chemical cylinder ($i=3$) does not go
through the cutting operation. Whereas in Figure 2b, shows the precedence
relationship $\mathbf{P}$ of the products in particular, $\mathbf{P}_{i}$
represents the sequence of the machines that product $i$ visits respectively.

For computational simplicity, the machine speeds are converted to the unit
processing time of a machine, which simply shows the time it takes to produces
one unit of a product in the unit of minute per unit. The unit processing time
on machine $m$ is limited by the lower bound $v^{min}_{m}$ and the upper bound
$v^{max}_{m}$. The unit processing time and the machine speed are inversely
proportional. That is as the unit processing time decreases the machine speed
increases. High machine speeds consume more energy and the energy cost of the
system increase in parallel. Hence, we assume that all machines are operating
at $v^{min}_{m}$ and aim to optimize the machine speeds to decrease the energy
consumption. Cost of energy consumed by the machine $m$ in one minute is
represented as $r_{m}$. Total energy consumption of the machine $m$ in a
production cycle is the multiplication of $r_{m}$ with the unit processing
time. Daily production capacity of the machines is the same for every machine
and expressed as $m_{t}$.

In every period $t$, there is a demand for each product type ($d_{it}$) that
must be satisfied throughout the planning horizon $T$. At the end of the
periods, it is possible to keep products as end item in the warehouse with the
cost of $h_{i}$ for satisfying the demand in the following periods.
Alternatively, it is practical to keep work-in-process (WIP) inventory at the
warehouse with the cost of ($w_{i}$) between the periods. In the case of
inventory accumulation, forklifts transport the products to the warehouse from
the factory level, generating the inventory transportation (IT) cost
($\bar{t}_{i}$).

Table 1: List of the parameters

Parameters
---
$\mathbf{I}$ |  set of products | $r_{m}$ |  operating cost of machine $m$
$I$ |  number of products | $\bar{t}_{i}$ |  IT cost of product $i$
$\mathbf{T}$ |  set of periods | $c_{m}$ |  VAO made by machine $m$
$T$ |  number of periods (planning horizon) | $h_{i}$ |  end item IH cost of product $i$
$\mathbf{M}$ |  set of machines | $w_{i}$ |  WIP holding cost of product $i$
$M$ |  number of machines | $d_{it}$ |  demand of product $i$ in period $t$
$\mathbf{A}_{im}$ |  production route matrix | $m_{t}$ |  plant capacity of the factory in period $t$
$\mathbf{P}_{i}$ |  production sequence matrix | $u^{max}$ | WIP inventory capacity
$v^{min}_{m},v^{max}_{m}$ |  lower and upper bounds on the | $s^{max}$ | end item inventory capacity
|  unit processing time of machine $m$ | $L$ | a unit of product in meters

For formulation simplicity, we assume that production cost of a product is
distributed into the machines. In other words, production cost of the products
consist of all the value added operations (VAO) that made by that machine. For
example, in PL2 there is a chemicalization and drying processes to compact
felt. With these processes, basic compact felt is turned to insulated felt.
These operations as a whole adds value to the product. Raw materials that is
consumed and the machining parts that is used in these processes are shown as
$c_{m}$, VAOs that happened on the product $i$ by machine $m$. By this
adjustment, machines and their operations generate cost, not the product type
as a whole.

The proposed production plan aims to minimize the total cost by decreasing the
inventory levels, deciding what to produce when to produce all by dynamically
changing the machine speeds (unit processing times of the machines) at the
start of the each period $t$, throughout the planning horizon $T$. We aim to
achieve just-in-time (JIT) production at the end of the production.

This paper is organized at follows: Section 3 explains the followed solution
method in detail. Section 3.1 explains the LSMS model and Section 3.2
describes the developed _Two-Phase_ Heuristic. Section 4 shows computational
results of the _Two-Phase_ Heuristic performance according to the generated
scenarios. The behaviour of the system according to the unit processing time
and demand is analyzed explained at Section 4.1, the total production and
inventory explained in Section 4.2. Section 5 concludes the paper together
with some comments on the future work.

##  3 Solution Method

We first propose a NLP formulation for the LSMS problem in Section 3.1. The
problem complexity of the LSMS problem is discussed in the same section. Then
to produce near-optimal solution in polynomial time, our _Two-Phase_ Heuristic
is introduced in Section 3.2.

The model we proposed is NLP which will be discussed in Section 3.1. With the
aim of solving the model in a polynomial time, we created a heuristic named as
_Two-Phase_ which will be explained in Section 3.2.

###  3.1 Mathematical Formulation

We proposed an NLP model with the objective of minimizing the total cost while
taking into account unit processing time, plant capacity and inventory
constraints of the felt production system. In this production plan we allowed
to change the machine speeds, in other words, unit processing times of the
machines by overloading the machines in a period within the acceptable
boundaries. Decreasing the unit processing time of the machine (increasing the
machine speed) from its original time generates a penalty which corresponds to
the energy consumption of the machine. By this idea, we proposed our
mathematical model with consideration of changing the unit processing times of
the machines at the start of each period.

Table 2: List of decision variables

Decision Variables
---
$v_{mt}$ |  unit processing time on machine $m$ | $y_{imt}$ |  amount of production of product $i$
|  in period $t$ |  |  on machine $m$ in period $t$
$s_{it}$ |  inventory level of product $i$ | $u_{it}$ |  WIP inventory level of product $i$
|  at the end of period $t$ |  |  at the end of period $t$

Objective function (1) consists of summing all cost which are production,
inventory holding and transportation cost then subtracting machine speed
penalty within a period until the end of the planning horizon. Normally, we
want to add the machine speed penalty to the objective. But we have $v_{mt}$
unit processing time as decision variable which is inversely proportional to
the machine speed. Hence, in the objective machine speed penalty appears with
a minus sign. With this modification of the penalty, we ensure that the unit
processing times of the machines change according to the demand requirements
of the periods.

Lot Sizing and Machine Speed (LSMS) Model:

$\displaystyle\min z=$ $\displaystyle\sum_{t\in T}c_{m}y_{imt}+\sum_{i\in
I}\left\\{s_{it}(h_{i}+\bar{t}_{i})+u_{it}(w_{i}+\bar{t}_{i})\right\\}-\sum_{m\in
M}r_{m}v_{mt}$ (1) $\displaystyle\sum_{i\in I}y_{imt}v_{mt}\leq m_{t}\hskip
91.04881pt\forall m\in M,\quad\forall t\in T$ (2) $\displaystyle
y_{imt}\leq\left(\sum_{t\in T}d_{it}\right)\mathbf{A}_{im}\hskip
79.66771pt\forall i\in I\quad\forall t\in T$ (3) $\displaystyle\sum_{t\in
T}y_{i1t}=\sum_{t\in T}y_{i2t}\hskip 102.42992pt\text{ for }\hskip
4.2679pti\in\\{3,4\\}$ (4) $\displaystyle
s_{it-1}+y_{im^{*}t}=d_{it}+s_{it}\hskip 76.82243pt\forall i\in I,\hskip
4.2679pt\forall t\in T$ (5) $\displaystyle
u_{2t-1}+y_{21t}=y_{23t}+u_{2t}\hskip 93.89409pt\forall t\in T$ (6)
$\displaystyle u_{4t-1}+y_{42t}=y_{43t}+u_{4t}\hskip 93.89409pt\forall t\in T$
(7) $\displaystyle v^{min}_{m}\leq v_{mt}\leq v^{max}_{m}\hskip
91.04881pt\forall m\in M,\quad\forall t\in T$ (8) $\displaystyle\sum_{i\in
I}s_{it}\leq s^{max}\hskip 142.26378pt\forall t\in T$ (9)
$\displaystyle\sum_{i\in I}u_{it}\leq u^{max}\hskip 142.26378pt\forall t\in T$
(10) $\displaystyle u_{it}=0\hskip 145.10905pt\forall t\in T,\text{ for
}\hskip 4.2679pti\in\\{1,3\\}$ (11)

In constraints (2) the total production time in a given machine must not
exceed the available time $m_{t}$. Constraint (2) are nonlinear due to the
multiplication of the decision variables $y_{imt}$ and $v_{mt}$. This makes
the LSMS formulation a nonlinear model.

In the product-machine $\mathbf{A}$ matrix, cells can only take binary values
indicating the presence of the production. The value of $\mathbf{A}_{32}$ is
one, meaning that there is a production of the product $i=3$ in the machine
$m=2$. The value of $\mathbf{A}_{33}$ is zero, showing that there is no
production of the product $i=3$ in the machine $m=3$. By multiplying the total
demand with $\mathbf{A}$ matrix in constraint (3), we ensure that only the
feasible product types at given machines can be produced as well as allowing
machines to produce more products than that periods demand to keep inventory
if necessary.

######  Lemma 1.

The LSMS model is a non-convex nonlinear model.

Proof. In order to have a convex mathematical model, the objective function
and the equality constraints should be linear and the less-than-or-equal-to
constraints should be convex. The constraints (2) include decision variables
$y_{imt}$ and $v{mt}$ multiplied. That is, these constraints are not convex
which results that the LSMS model to be non-convex. $\square$

One of the main assumptions of the production system is that there will be no
WIP products from $m=1$ to $m=2$. Idea behind this assumption is that products
that unloaded from $m=1$ can be loaded on $m=2$ via crane system which is
embedded between that two production lines. Constraint (4) assure that, total
chemical type $(i=3,4)$ production in $m=1$ (PL1) is equal to total chemical
type production in $m=2$ (PL2). Constraints (4) show that the chemical
products can not start their production on $m=2$ unless they have been
produced in $m=1$ first. At the same time, with constraint (4), system does
not generate any WIP products between $m=1$ and $m=2$.

Some periods may be overloaded with demand, which results in exceeding the
daily capacity of the production system. To reduce this overload on the
production system before these periods it is possible keep the excessive
production as end item inventory. Constraints (5) are the end item inventory
balancing constraints for each product type at the end of a period. All
products become end product when they finish their machine routing in their
sequence matrix $\mathbf{P}$. To guarantee keeping only end item products as
end item inventory, the decision variable $y_{imt}$ is written as
$y_{im^{*}t}$, where $m^{*}$ is the last machine in the production sequence
matrix $\mathbf{P}_{i}$.

It is possible to keep WIP inventory which only consist of plaque type
$(i=2,4)$ of products on machine $m=1$ to $m=2$ and $m=3$. Constraints (6) and
(7) show that the WIP inventory balances at the end of each period.
Constraints (11) ensure that no WIP inventories is generated between the
machines $m=1$ and $m=2$.

Each machine in the production system has their unit processing time
independent from the product type that they produce, and their unit processing
time bounds, which can be seen from Constraints (8), representing the lower
and upper bounds of the unit processing time of the machines.

Constraints (9) and (10) show the total allowed end and WIP item inventories
in a period, respectively. Since it is assumed that cylinder type and plaque
type of products have the same volume, there is no need to separate product
types in the summation.

###  3.2 Two-Phase Heuristic

As stated earlier, constraints (2) make the LSMS model a nonlinear problem due
to the multiplication of the decision variables, $y_{imt}$ and $v_{mt}$. By
using the divide-and-conquer idea we developed the _Two-Phase_ heuristic,
which divides the nonlinear problem into two linear subproblems (SP) and
solves them repeatedly.

In the implementation of the _Two-Phase_ heuristic, we divide the LSMS model
into two subproblems namely Subproblem 1 (SP1) and Subproblem 2 (SP2). Each SP
has the same objective function, minimizing the cost. In particular SP1 has
$y_{imt},s_{mt},u_{mt}$ as decision variables and takes the initial unit
processing times of the machines $v_{mt}$ as parameters in the production
system. Similarly, SP2 takes $v_{mt},s_{mt},u_{mt}$ as decision variables, and
keeps the production amount of the machines $y_{imt}$ as parameters. Each
model has constraints that are subjected to their decision variables.
Mathematical formulation of SP1 can be seen as follows.

SP1:

$\displaystyle\min z=$ $\displaystyle\sum_{t\in
T}c_{m}y_{imt}+\left(\sum_{i\in
I}s_{it}(h_{i}+\bar{t}_{i})+u_{it}(w_{i}+\bar{t}_{i})\right)-\sum_{m\in
M}r_{m}\hat{v}_{mt}$ (12) $\displaystyle\sum_{i\in I}y_{imt}\hat{v}_{mt}\leq
m_{t}\hskip 99.58464pt\forall m\in M,\quad\forall t\in T$ (13)
$\displaystyle(\ref{flow})-(\ref{wipbalance2}),(\ref{maxend})\text{ and
}(\ref{maxwip})$

SP1’s main difference can be seen above. LSMS model is modified according to
the $y_{imt},s_{mt},u_{mt}$ decision variables. $\hat{v}_{mt}$ is introduced
as a parameter which holds the machine unit processing time which is obtained
from SP2’s solution or can be given as an initial machine unit processing time
in the first execution. With this alteration, now SP1 is a linear model since
constraints (13) are no more nonlinear.

SP2:

$\displaystyle\min z=$ $\displaystyle\sum_{t\in
T}c_{m}\hat{y}_{imt}+\left(\sum_{i\in
I}s_{it}(h_{i}+\bar{t}_{i})+u_{it}(w_{i}+\bar{t}_{i})\right)-\sum_{m\in
M}r_{m}v_{mt}$ (14) $\displaystyle\sum_{i\in I}\hat{y}_{imt}v_{mt}\leq
m_{t}\hskip 99.58464pt\forall m\in M,\quad\forall t\in T$ (15) $\displaystyle
s_{it-1}+\hat{y}_{im^{*}t}=d_{it}+s_{it}\hskip 82.51299pt\forall i\in
I,\quad\forall t\in T$ (16) $\displaystyle
u_{2t-1}+\hat{y}_{21t}=\hat{y}_{23t}+u_{2t}\hskip 105.27519pt\forall t\in T$
(17) $\displaystyle u_{4t-1}+\hat{y}_{42t}=\hat{y}_{43t}+u_{4t}\hskip
105.27519pt\forall t\in T$ (18) $\displaystyle(\ref{bounds})-(\ref{maxwip})$

The SP2, LSMS model is modified according to the $v_{mt},s_{mt},u_{mt}$
decision variables. $\hat{y}_{imt}$ is introduced as a parameter which holds
the production amounts which is obtained from the solution of SP1. As in SP1,
SP2 is a linear model since constraints (15) are linear.

To solve the LSMS model, subproblems need to communicate with each other. To
solve SP1, we give initial unit processing times of the machines ($v_{mt}$) as
parameters. With this initial processing times, SP1 is executed and feasible
values for the $y_{imt},s_{mt},u_{mt}$ decision variables are found. As
explained earlier, from these values $y_{imt}$ is given to the SP2 as a
parameter. Then, SP2 is executed and feasible values for the
$v_{mt},s_{mt},u_{mt}$ decision variables are obtained. One cycle of _Two-
Phase_ consists of solving SP1 & SP2 once. At the end of the each cycle, the
objective function values of the SP1 & SP2 compared with each other. If the
objective function values are the same, the algorithm compares
$s_{mt},u_{mt},y_{imt},v_{mt}$ decision variable values from current and
previous cycles. This comparison will continue as long as there is no
difference in the decision variables and the objective functions. Figure 3
visualizes the cycle of the _Two-Phase_ heuristic. Pseudocode of the _Two-
Phase_ heuristic is provided below (see Algorithm 1).

Algorithm 1: (Two-Phase) Input: An instance of $LSMS$ model 0\.
$loop\leftarrow true$ 1\. $\hat{v}_{mt}$ $\leftarrow$ $v^{min}$ for all $m,t$
2\. While $loop\leftarrow true$ 3\. Initialize condition. 4\. Save previous
$y_{imt},v_{mt}$ . 5\. Run SP1 with $\hat{v}_{mt}$, for $\mathbf{I,M,T}$ sets,
find current $y_{imt},s_{it},u_{it}$ 6\. Save $\hat{y}_{imt}$ as ${y}_{imt}$
7\. Run SP2 with $\hat{y}_{imt}$, for $\mathbf{I,M,T}$ sets, find current
$v_{mt},s_{it},u_{it}$ 8\. $\hat{v}_{mt}$ $\leftarrow$ ${v}_{mt}$ 9\. If there
is change in $y_{imt},v_{mt},s_{it},u_{it}$ Then 10\. $condition\leftarrow
true$ 11\. Else $condition\leftarrow false$ 12\. End If 13\. If
$condition\leftarrow false$ Then Go to Step 2. 14\. Else If
$condition\leftarrow true$ Then $loop\leftarrow false$, Go to Step 2 15\. End
If 16\. End While Output: Near-optimal solution of the LSMS model
$y_{imt},v_{mt},s_{it},u_{it}$ values with objective value $Z$

If $T_{max}$ is the maximum iteration limit in the _Two-Phase_ heuristic, then
its time complexity can be given as $\mathcal{O}$((SP1 \+ SP2)$T_{max}$) where
$\mathcal{O}$(SP1) and $\mathcal{O}$(SP2) are the time complexity of SP1 and
SP2, respectively. Note that $\mathcal{O}$(SP1) and $\mathcal{O}$(SP2) are
polynomial time complexity since SP1 and SP2 are linear programming problems

Figure 3: Visualization of the _Two-Phase_ Heuristic

##  4 Computational Results

Computational parameters gathered by examining the production system as well
as making a market research for the production costs. The system is analyzed
for three different planning horizon levels, i.e. $T$ = 10, 20 and 30 periods.
For each $T$, 10 random demand instances are generated. For better
understanding the effects of the parameters on the objective function and the
production system, two key parameters $m_{t}$ and $(s^{max}_{i},u^{max}_{i})$,
which effect the decision variables the most, are chosen and created three
different scenarios for each parameter. These scenarios can be explained as
Low, Medium and High level instances, which are in total of 270 instances.
List of computational parameters can be seen in Table 2.

Table 3: List of the computational parameters

$c_{m}$ | [2400, 5400, 1000]
---|---
$\bar{t}_{i}$ | [100, 120, 120, 140]
$h_{i}$ | [300, 150, 300, 150]
$w_{i}$ | [0, 50, 0, 50]
$r_{m}$ | [1.16, 3.09, 0]
$T$ | 10, 20, 30
$L$ | 400 meter
$m_{t}$ | 630, 720, 810 minutes
$(s^{max}_{i},u^{max}_{i})$ | (6,3), (12,6), (18, 9)
$D_{T}$ | (80,120), (180,250), (290,350)
$(I,M)$ | (4,3)

The felt company provided us a demand ratio for each product type and gave us
a minimum and maximum amount of demand that can be requested in a given
planning horizon $T$. For planning horizons $T=10$, $T=20$ and $T=30$ total
demand bounds are (80, 120), (180, 250) and (290, 350) respectively. From
these total demands:

  * •

16% of the total demand corresponds to demand of the product $i=1$.

  * •

4% of the total demand corresponds to demand of the product $i=2$.

  * •

64% of the total demand corresponds to demand of the product $i=3$.

  * •

16% of the total demand corresponds to demand of the product $i=4$.

With the data provided, demands are generated and distributed on the periods
within the given planning horizon randomly using uniform distribution. VAOs of
the machines, $c_{m}$, represents an array of single instance. In this
instance, $c_{1}$ is 2400, $c_{2}$ is 5400 and $c_{3}$ is 1000. Similar to
$c_{m}$, $\bar{t}_{i}$, $h_{i}$, $w_{i}$ and $r_{m}$ represents an array of
single instances which changes with the corresponding index values. These
computational parameter values are generated by analysis of the production
system.

A unit of product corresponds to a 400 meters of felt. It is assumed that all
product types (plaque and cylinder) takes the same amount of space in the
inventory, at the same time, holding cost of cylinder type products is higher
since it requires more manpower and better allocation in the warehouse. In
Table 3, lower and upper bounds for unit processing time of the machines can
be seen.

Table 4: Unit Processing Time Bounds of Machines

$(v^{min}_{1},v^{max}_{1})$ | (50, 80) min/unit
---|---
$(v^{min}_{2},v^{max}_{2})$ | (22.2, 26.6) min/unit
$(v^{min}_{3},v^{max}_{3})$ | (80, 80) min/unit

The computations are conducted on a computer with 2.80GHZ Intel Core i7-7700HQ
processor working on Windows 10 with 16GB RAM. All codes are written with Java
language, on Eclipse IDE. For generating the optimal solution of the
subproblems we utilized IBM CPLEX 12.10. From the outputs only average values
reported. Since all demands are satisfied throughout the planning horizon,
$\bar{y}_{imt}$ would be equal to $\bar{d}_{it}$ and would be the same from
instance to instance, therefore no $y_{imt}$ value is reported. We reported
only $\bar{v}_{1t}$, the average unit processing time of PL1, in the Tables 4
and 5. Since $m=1$ is the critical production line with producing all the
product types. At the same time, in all instance outputs, ${v}_{2t}$ value is
constant and equal to 26.6 min/unit and ${v}_{3t}$ is a constant and equal to
80 min/unit.

The reason behind that only $\bar{v}_{1t}$ is reported in the table is that,
$m=1$ is the critical production line with producing all the product types, as
well as in 270 instance outputs, ${v}_{2t}$ value is constant and equal to
26.6 min/unit and ${v}_{3t}$ is a constant and equal to 80 min/unit.

For a given $T$ planning period, as the available working hours $m_{t}$ in a
day get higher, the demand can be satisfied with a lower average machine $m=1$
speed, i.e., $\bar{v}_{1t}$ processing times increase, while accumulating
fewer WIP and end-item inventories.

For all $T$ levels, at $m_{t}$(_low_) working hours allowing more WIP and end-
item inventories, i.e., $s_{it}$ and $u_{it}$ get higher, decreases the end-
item inventory $s_{it}$ amounts and increases WIP inventory $u_{it}$ level.
This is since WIP inventory carrying cost is smaller than the one for end-item
inventory.

Table 5: Performance of the Two-Phase Heuristic

|  |  |  | Two-Phase |
---|---|---|---|---|---
T | $m_{t}$ |  ($s^{max},u^{max}$) | $\bar{z}$ | CPU (ms) | # iter | $\bar{s}$ |  $\bar{u}$ | $\bar{v}_{1t}$
|  | low | 737,425.85 | 165.84 | 2 | 0.15 | 0.76 | 59.30
| low | med | 737,319.85 | 164.2 | 2 | 0.04 | 0.87 | 59.30
|  | high | 737,319.85 | 163.22 | 2 | 0.04 | 0.87 | 59.30
|  | low | 735,703.11 | 167.58 | 2 | 0.01 | 0.09 | 66.15
10 | med | med | 735,703.11 | 164.12 | 2 | 0.01 | 0.09 | 66.15
|  | high | 735,324.26 | 158.92 | 2 | 0 | 0.03 | 71.70
|  | low | 735,324.26 | 160.30 | 2 | 0 | 0.03 | 71.70
| high | med | 735,324.26 | 161.80 | 2 | 0 | 0.03 | 71.70
|  | high | 735,324.26 | 158.92 | 2 | 0 | 0.03 | 71.70
|  | low | 1.450,956.60 | 185 | 2 | 0.07 | 0.47 | 60.67
| low | med | 1,453,682.87 | 183.2 | 2 | 0.04 | 0.51 | 60.67
|  | high | 1,456,684.13 | 182.52 | 2 | 0.03 | 0.51 | 60.67
|  | low | 1,451,398.82 | 183.58 | 2.1 | 0 | 0.05 | 67.28
20 | med | med | 1,451,398.82 | 180.84 | 2 | 0 | 0.05 | 67.28
|  | high | 1,451,398.82 | 181.88 | 2 | 0 | 0.05 | 67.28
|  | low | 1,450,799.37 | 183.5 | 2 | 0 | 0.01 | 72.11
| high | med | 1,450,799.37 | 181.88 | 2 | 0 | 0.01 | 72.11
|  | high | 1,450,799.37 | 180.48 | 2 | 0 | 0.01 | 72.11
|  | low | 2,150,772.64 | 214.2 | 2.3 | 0.17 | 0.50 | 61.21
| low | med | 2,158,481.07 | 200.86 | 2 | 0.04 | 0.55 | 59.95
|  | high | 2,158,432.18 | 201.78 | 2 | 0.03 | 0.56 | 59.95
|  | low | 2,154,607.06 | 200.2 | 2 | 0 | 0.07 | 66.01
30 | med | med | 2,154,607.06 | 202.28 | 2 | 0 | 0.07 | 66.01
|  | high | 2,154,607.06 | 199.56 | 2 | 0 | 0.07 | 66.01
|  | low | 2,153,550.94 | 200.88 | 2 | 0 | 0 | 71.22
| high | med | 2,153,550.94 | 198.1 | 2 | 0 | 0 | 71.22
|  | high | 2,145,411.34 | 197.62 | 2 | 0 | 0 | 72.79

There is no significant increase in the computation times as the instance gets
larger (for the parameters $T,m_{t},s^{max},u^{max}$ from _low_ to _high_
level). For each instance, all the machines start with their highest
production speed, i.e., the smallest $v_{mt}$ value, at the beginning of the
algorithm. Under this setting, we observe that the average number of
iterations to converge is 2 for almost all instances.

As $T$ planning horizon gets longer, the total demand for the items increases
in the instances as well. Then, we observe an increase in the average
objective function value $z$ with the increase in $T$ as expected.

We observe the best objective function $z$ values, i.e., the smallest
production costs, for the $m_{t}$(_high_) levels and inv(_high_) bounds. This
is since we decrease the WIP and end-item inventories by higher average
processing times $\bar{v}_{1t}$ and the operating costs improve with the
increased machine speed and the lower inventories in the system.

The computational experiments do not show any change in the unit processing
times of machines $m=2$ (PL2) with 26.6 min/unit, and $m=3$ (CM) with 80
min/unit. Hence, in Table 5 we only reported the machine $m=1$ unit processing
times.

In Table 5, the average outputs of the 270 scenario executions are summarized.
As can be seen, as the $m_{t}$ increases, ($s^{max},u^{max}$) level does not
effect the system in terms of objective function value, end item and WIP
inventory values and the unit processing time of the PL1. Whereas, in the low
levels of $m_{t}$, ($s^{max},u^{max}$) is the key factor in changes of
variables. In all cases, best results can be found in high levels of $m_{t}$.

For each $T$, a random demand instance, which is the most overloaded instance
with demand, is chosen to visualize its outputs for each scenario. In each
graph, columns represents the $m_{t}$ levels, rows represent the
($s^{max},u^{max}$) levels.

In each $m_{t}$ level there is a maximum number of products that can be
produced in a period which can be found by dividing the $m_{t}$ level value by
the $v^{min}_{m}$ value. In this case, as we report only $v_{1t}$, this value
can be found as $m_{t}$ divided by $v^{min}_{1}$. To sum up, for every $m_{t}$
level, the maximum number of products that can be produced by $m=1$ is 12.6
unit, 14.4 unit and 16.2 unit, respectively. These bounds can be surpassed
only if there is an accumulated WIP inventory from the earlier periods.

There are two types of figures which is used to summarize Table 4. One type is
showing the relationship of $\bar{v}_{1t}$ (Unit Processing Time) with $d_{t}$
(Demand) according to scenarios. In this type red line plot represent the
$\bar{v}_{1t}$ value from period to period and its values are shown at the
secondary y-axis; bars represents the total demand within a period and its
values are shown at the primary y-axis. The other figure type shows the
relationship of $y_{imt}$ (Total Production) with $s_{it},u_{imt}$ (Inventory)
According to Scenarios. Each figure type has three rows and three columns.
Rows shows the $s^{max},u^{max}$ (inventory) scenarios as _inv(low)_ ,
_inv(med)_ and _inv(high)_ ; columns shows the $m_{t}$ (daily capacity)
scenarios as _mt(low)_ , _mt(med)_ and _mt(high)_. In total there are nine
graphs in a figure which corresponds to intersecting scenarios. Each types of
figures are used to summarize each $T$ planning horizon. With total of six
figures Table 4 is visualized. From all planning horizons, an instance is
chosen with most variability from inventory levels to $v_{mt}$ and $y_{imt}$
levels.

###  4.1 The Effects of Unit Processing Time and Demand

In the Figure 4, $v_{1t}$ (unit processing time of $m=1$) can be seen with the
total demand of $d_{t}$. The graph shows how $v_{1t}$ reacts to a change in
demand according to scenarios. As can be seen in the Figure 3, inventory
levels do not effect the $v_{1t}$ value. At _mt(low)_ level from $t=3$ to
$t=10$, $v_{1t}$ value is constant as 50 min/unit, that means that $m=1$ works
at its capacity and consuming more energy to satisfy the demand. But at
_mt(med)_ , $m=1$ can be adjusted with respect to change in demands. As
expected, while $d_{t}$ gets lower, $v_{1t}$ increases ($m=1$ speed decreases)
thus consuming less energy; as $d_{t}$ gets higher, $v_{1t}$ decreases ($m=1$
speed increases) thus consuming more energy. At _mt(high)_ , at relatively low
$d_{t}$ periods (relatively low at _mt(high)_ $v_{1t}$ value generally close
to its upper bound 80 min/unit where energy consumption is low, just like the
other $m_{t}$ levels, as the $d_{t}$ increases and $v_{1t}$ decreases.

Figure 4: Relationship of unit processing time with demand according to the
scenarios (T=10)

The same trend continues as the planning horizon becomes $T=20$ . As can be
seen in the Figure 5 just like the $T=10$, only $m_{t}$ levels affects the
$v_{1t}$ values. In all cases, as the $d_{t}$ increases, $v_{1t}$ tends to
decrease, $m=1$ consumes more energy. Although _mt(med)_ and _mt(high)_ are
almost similar in some periods, best scenario, as expected, is _mt(high)_.

Figure 5: Relationship of unit processing time with demand according to the
scenarios (T=20)

In Figure 6, just like the $T=10$ and $T=20$, as $d_{t}$ increases, $v_{1t}$
decreases. Similar to lower planning horizons, only $m_{t}$ level effects
$v_{1t}$.

As we look at the general case of relationship of $v_{1t}$ (unit processing
time of $m=1$) with the $d_{t}$ (total demand at period $t$), there is a
correlation between them. In all cases regarding the $m_{t}$ and
$s^{max},u^{max}$ levels, as the demand increases unit processing time of
$m=1$ tends to decrease, i.e., PL1 starts to work faster and consumes more
energy to satisfy the given demand. At all $T$ planning horizons,
$m_{t}$(_high_) generates the best results and $s^{max},u^{max}$ does not
effect $v_{1t}$.

Figure 6: Relationship of unit processing time with demand according to the
scenarios (T=30)

###  4.2 The Effects of Total Production and Inventory

In Figure 7, total production within a period and WIP and end-item inventory
levels can be seen. At _mt(low)_ and ($s^{max},u^{max}$) levels, accumulation
of end item inventory can be seen since there is no more room for the WIP
inventory. At _inv(medium)_ and _inv(high)_ levels of ($s^{max},u^{max}$),
there is no end item inventory accumulation. Instead of keeping end item
inventory, system decides to keep inventory as a WIP. This shows that keeping
WIP inventory is more preferable than keeping end item inventory, also it
shows that it is possible to find a feasible solution while keeping end item
inventory at low levels of ($s^{max},u^{max}$). As the $m_{t}$ level increases
to _medium_ , all inventory levels decrease significantly, thus it becomes
possible to satisfy the demand with very-low to zero inventory keeping. At the
level of _mt(high)_ , it is possible to accomplish just-in-time (JIT)
production with zero inventory accumulation on WIP and end-item.

Figure 7: Relationship of total production with inventory according to the
scenarios (T=10)

WIP inventory accumulation does not effect the $y_{imt}$ (total production),
it only effects the $y_{imt}$ value only if the system is using the
accumulated WIP inventory. For an example, at the start of the accumulation of
the WIP inventory, system may be producing 12 units of products, but from
these 12 units, 10 of them will become finished goods and 2 of them stored in
the inventory as WIP inventory, waiting to be cut. In the graphs, only
finished goods are shown as $y_{imt}$. But, as the system starts to use
accumulated WIP inventory, WIP products are turned into finished goods thus
effects the $y_{imt}$ value. By using WIP inventory, system can exceed its
total production capacity in a period. As it comes to the keeping end item
inventory, they must be turned into finished goods to become $s_{it}$ (end
item inventory).

Figure 8: Relationship of total production with inventory according to the
scenarios (T=20)

Unlike $T=10$, when planning horizon increases to $T=20$, it may not be
possible to satisfy the demand with zero inventory as shown in Figure 8.
Although ($s^{max},u^{max}$) inventory levels increase, the system decides to
keep some level of end item inventory together with WIP inventory. As a
result, even if keeping WIP inventory is more profitable, it can be essential
to keep end item as well to satisfy the demand in a period without a backlog.
Only at _mt(high)_ level there can not be found any end item inventory. In
every case, WIP inventory accumulation is unavoidable, but to avoid end item
accumulation _mt(high)_ must be chosen.

Figure 9: Relationship of total production with inventory according to the
scenarios (T=30)

As can be seen in Figure 9, similar to $T=10$ and $T=20$, at _mt(low)_ level,
($s^{max},u^{max}$) inventory level play a key role in determining the amount
of inventory type kept in a period. As the ($s^{max},u^{max}$) level
increases, just like at lower planning horizons, system decreases the amount
of accumulated end item inventory and keeps high levels of WIP inventory. At
$m_{t}$(_med_) system only keeps low amounts of WIP inventory and no end item
inventory. Similar to $T=10$, system can start to satisfy the demand with no
inventory, thus making a JIT production the level is _mt(high)_.

As we look at the general case of relationship of $y_{imt}$ (total production)
with $s_{it}$ and $u_{it}$ (end-item & WIP), we can conclude that only at
_mt(low)_ level ($s^{max},u^{max}$) affects the system. In all cases, system
prefers to keep WIP inventory rather than end item inventory although, in some
periods, it becomes necessary to keep end item inventory as well to satisfy
the demand throughout the planning horizon. Main parameter that affects the
system is $m_{t}$ since as the $m_{t}$ level increases, amount of kept
inventory significantly decreases. It can be seen that, in some instances, at
$m_{t}$(_high_), it might be possible to do JIT production.

###  4.3 The Effects of Daily Capacity and Planning Horizon

Figure 10: Average Processing Times of $m=1$ With Respect to $m_{t}$ Levels

At Figure 10, the trend of average unit processing time of $m=1$ ,i.e.,
$\bar{v}_{1t}$ can be seen. In all planning horizons, $\bar{v}_{1t}$ tends to
increase from level to level as expected. At every $m_{t}$ scenario, best
planning horizon where the average energy consumption is the lowest, i.e., the
highest $\bar{v}_{1t}$ changes. At _mt(low)_ , _mt(med)_ and _mt(high)_ levels
the smallest average energy consumption ($\bar{v}_{1t}$) can be found in
$T=20$, $T=10$ and $T=30$ planning horizon levels respectively.

As can be seen, although lowest average energy consumption happens at
_mt(high)_ , this figure gives an insight about which $T$ planning horizon to
choose at given $m_{t}$ level.

##  5 Conclusions

In this study we focused on the machine speed and lot sizing problem in a felt
production system. We proposed a nonlinear programming (NLP) mathematical
model named as lot sizing and machine speed (LSMS) model with the purpose of
making a production plan, by optimizing the production amounts, decreasing the
inventory levels and by changing the machine speeds which effect the unit
processing times of the machines. By changing the machine speeds, we have been
able to make a lot sizing, thus creating a production plan which can result in
a production system with zero inventory accumulation. To overcome the
nonlinearity in the LSMS, we developed a heuristic called _Two-Phase_. By
implementing the developed _Two-Phase_ heuristic, we solved the NLP model in a
polynomial time and obtain a near-optimal solution for the non-convex LSMS.
After achieving a feasible solution for the LSMS, we generated scenarios
according to the planning horizon, the inventory levels of the inventory types
and the total production capacity of the system in a period. We executed the
LSMS model in accordance with the generated scenarios. Average outputs and
their behaviours of the LSMS, with respect to scenarios, are visualized.
Observations showed that, in some scenarios, it is possible to satisfy the
demand with no inventory, thus just-in-time production is achievable.

In this work, we generate a _Two-Phase_ heuristic for sequentially flowing
felt semi-products in a simple three machine flowshop. As a future work, one
can consider more complicated flow shop in the production environment with
multiple product types.

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