171
Indonesian Journal of Science & Technology 8.
171-196 Indonesian Journal of Science & Technology Journal homepage: http://ejournal.
edu/index.
php/ijost/ Sustainable Production-Inventory Model with MultiMaterial.
Quality Degradation, and Probabilistic Demand: From Bibliometric Analysis to A Robust Model Dana Marsetiya Utama1,2,*.
Imam Santoso1.
Yusuf Hendrawan1.
Wike A.
Dania1 Universitas Brawijaya.
Malang.
Indonesia University of Muhammadiyah Malang.
Malang.
Indonesia * Correspondence: E-mail: dana@umm.
ABSTRACT
An adequate sustainable production inventory model is expected to represent complex real-life cases involving fuel, emissions, and electricity costs as well as multi-materials, quality degradation, and probabilistic demand.
Therefore, this study was conducted to develop this kind of model to determine the number of raw material shipments .
j ), production cycle time (T), and the number of finished goods delivered .
to maximize the Expected Total Profit (ETP).
The proposed model is based on a bibliometric literature analysis of the sustainable production-inventory problem which is visualized using the VOSviewer.
Moreover, a sophisticated Harris-Hawks Optimization (HHO) algorithm was proposed to solve the problems identified in the sustainable production inventory model optimization.
It is also important to note that three numerical cases were provided to evaluate the performance of the algorithm.
The findings showed that the suggested HHO method outperforms the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) in maximizing ETP and this means it is better for ETP optimization.
It was also discovered from the sensitivity analysis that an increase in the rate of quality degradation .
led to a reduction in both the ETP and T.
A 2023 Tim Pengembang Jurnal UPI
ARTICLE INFO
Article History:
Submitted/Received 12 Sep 2022
First Revised 21 Nov 2022
Accepted 03 Jan 2023
First Available Online 04 Jan 2023
Publication Date 01 Sep 2023
____________________
Keyword:
Harris Hawks Optimization.
Inventory model.
Production.
Sustainable.
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INTRODUCTION
Sustainability is currently an important global issue and has promotes stakeholders to focus on increasing the economic, environmental, and social dimensions (Negri et al.
, 2.
This requires implementing sustainable practices in industries to significantly reduce emissions, conserve resources (Mashud et al.
, 2022.
Taghikhah et al.
, 2.
, and solve social problems.
The concept is being applied widely to supply chain activities and has been proven to be performance (Ho et al.
, 2022.
Shekarian et , 2022.
Wang et al.
, 2.
Meanwhile, it is important to note that procurement, production, and distribution decisions in supply chain systems can affect supply chain performance (Lu et al.
, 2020.
Maulana et al.
This is the reason scholars have attempted to integrate inventory decisions into the supplier and manufacturing levels (Utama et al.
, 2022a.
Utama et al.
, 2022.
, internal manufacturing (Liu et al.
, 2.
, and manufacturing-customer relationship.
One of the problems identified in integrated procurement system which is popularly known as the Production Inventory model (Goyal & Deshmukh, 1992.
Park, 1.
This led to the conduct of several relevant studies to solve this problem through the optimization of only the economic Therefore, there is a need to investigate the environmental and social Different forms of sustainable production inventory models have been proposed with most of the previous studies discovered to have focused on minimizing one indicator of the environmental aspect such as electricity costs (Gautam et al.
, 2.
, fuel consumption (Sarkar et al.
, 2017.
Utama et , 2022a.
Wangsa & Wee, 2.
, and emissions (Jaber et al.
, 2013.
Jauhari et al.
Ullah et al.
, 2.
Others also attempted to combine two indicators such as fuel consumption and emissions in developing a new model (Wangsa, 2017.
Wangsa et al.
, 2.
It was discovered that only Jauhari .
considered fuel, emissions, and electricity costs simultaneously.
Moreover, it is also important to consider quality degradation in this model due to its existence in several industries including pharmaceutical (SilvaAravena et al.
, 2.
, food (Ibrahim et al.
Lee et al.
, 2.
, and agro-industry (Liu et al.
, 2.
Most previous studies assumed that a single finished product requires a single raw material (SRM) (Bhattacharjee & Sen, 2.
This means their models cannot be applied to products requiring multiple raw materials (MRM).
The studies also assumed that product demand is deterministic (Fiorotto et , 2.
without any consideration for stochastic demand.
Advanced metaheuristic procedures have been proposed to optimize production inventory models based on the rapid advances in computer technology.
These include Particle Swarm Optimization (PSO) (Taleizadeh et al.
, 2.
and Genetic Algorithms (GA) (Sadeghi et al.
, 2.
, as well as the integration of the two algorithms (Sadeghi et al.
, 2.
However, no study used the Harris-Hawks Optimization (HHO) algorithm to optimize the sustainable production inventory model It was discovered that Heidari et .
only proposed the HHO algorithm by mimicking Harris HawksAo herd behavior in hunting prey.
The algorithm was reported to have good performance in optimizing scheduling (Utama & Widodo, 2.
, forecasting (Chaudhuri & Alkan, 2.
, energy (Dev et , 2.
, and engineering field (Shehab et , 2.
This means it has the potential to solve the problems associated with the sustainable production inventory model.
Only a few studies considered fuel, emissions, and electricity cost indicators DOI: https://doi.
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It was discovered that none considered the multi-material, quality degradation, and probabilistic demand indicators and this is the primary motivation for this study.
The gaps in sustainable production inventory model research are also evident in presented in section 2.
Furthermore.
HHO advanced algorithm was reported to have the potential to solve the problem of a sustainable production inventory model but it has not been applied for this purpose.
This study also proposes to apply the HHO algorithm in resolving problems associated with the model.
Therefore, the Research Goals (RG) include (RG .
developing a sustainable production inventory model that considers multimaterials, probabilistic demands and (RG .
applying the HHO algorithm to optimize the problems in the model.
This means the practical contributions involved include:
the development of a new model of sustainable production inventory by considering multi-materials, quality degradation, and probabilistic demands.
the application of the HHO algorithm as an optimization tool to solve the problems of sustainable production inventory model.
The structure of this paper is below:
Section 2 provides a literature review and bibliometric analysis of the sustainable production inventory model.
Section 3 describes the system's characteristics, assumptions, notations, and the proposed model on the sustainable production inventory model.
The proposed algorithm for optimizing the sustainable production inventory model is presented in Section 4.
Section 5 provides study data and Section 6 presents results and Finally, this article concludes with conclusions.
LITERATURE REVIEW AND BIBLIOMETRIC
ANALYSIS
Bibliometric Analysis This section presents the bibliometric problem of the sustainable production inventory model.
The keywords used for this search are "Sustainable" and "Production" or "Inventory" and "Model".
Fifty papers were collected from the Scopus database published in 2013-2022.
Figure 1 presents the development of article publications related to the sustainable production inventory model.
This result shows that this topic started to be published in 2013.
This topic increased dramatically from 2020-2022, and 17 papers were published in 2022.
Network Visualization of sustainable production inventory keywords based on VOSviewer is depicted in Figure 2.
This result shows that 6 clusters were identified based on co-occurrence analysis.
The main popularly used keywords are presented in cluster 1 .
ed colo.
In this cluster, the main keywords are sustainable inventory model and its derivatives, such as sustainable economic production quantity (EPQ), controllable carbon emission, deteriorating, green technology, and shortage.
The second cluster .
reen colo.
includes a sustainable integrated inventory model, sustainable supply chain, controllable lead time, sustainable location, defective items, and stock levels that focus on the supply chain network.
The third cluster .
categorizes terms related to the economic order quantity, green inventory model, supply lead time uncertainty, and sustainable order quantity inventory model that focuses on the model for order quantity.
The fourth cluster in yellow is a group of sustainable production inventory model DOI: https://doi.
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Figure 1.
Development of article publications related to sustainable production inventory Figure 2.
Network Visualization of sustainable production inventory keyword.
In this cluster, some related derivative keywords are investment, carbon emission, collaborative investment, preservation technology, and production inventory This cluster shows that the consideration of quality degradation, multiraw materials, and stochastic demand are not discussed in previous studies.
The fifth cluster .
urple colo.
shows the cluster group of sustainable supply chain inventory models with qualities such as imperfect quality, perishable products, maintenance, and unit quantity discount.
The last cluster in light blue is the green inventory management cluster with derivatives such as carbon emission and trade policy.
Finally.
Figure 3 analyzes co-occurrence by all keywords with an overlay The analysis results show that the most used keywords between 2021 and 2022 correspond to the green and yellow colors: carbon emission, sustainable production inventory model, defective items, and trade policy.
Content Analysis and Gaps Based on the bibliometric analysis.
Content Analysis, and Gaps model explained in this section.
Previous studies that have been conducted concerning the problems of the sustainable production inventory model are reviewed with a focus on the integration of production and inventory policies.
It is pertinent to note that the procurement and production subsystems are interconnected in making decisions on raw material procurement and finished goods production.
The model was initially proposed by Goyal .
and GoyalDeshmukh .
to minimize total DOI: https://doi.
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Analysis of representative terms on the subject over time.
Sustainability requires an integrated and collaborative approach in supply chain It is also important to note that sustainable inventory management is one aspect of sustainable supply chain management (Becerra et al.
, 2.
which is critical and recognized as corporate social and environmental responsibility (Pattnaik et al.
, 2.
The previous studies that have been conducted on sustainable production inventory models are summarized in Table It was discovered that most of these studies focused on the complexities of Single Raw Material (SRM).
Single Stage Production (SSP), and Single Product (SP).
This means attention was generally on the development of economic and environmental models consisting of emission cost and fuel usage without consideration for the quality degradation in raw materials.
It was also discovered that they mostly consider customer demand while deterministic and heuristic procedures are the popular methods applied to solve the problem.
There is no present study conducted on the sustainable production inventory model that considers multiple materials, quality degradation, and probabilistic demand.
Therefore, this study was conducted to fill this gap by proposing a new model that considers these indicators.
The HHO algorithm which is classified as a metaheuristic procedure was also proposed to optimize the problems associated with the sustainable production inventory model.
SYSTEM CHARACTERISTICS,
ASSUMPTIONS.
NOTATIONS.
AND
PROPOSED MODEL
System Characteristics The proposed model was designed to address the shortcomings of earlier models.
It can represent complex real cases due to the inclusion of the costs for fuel, emissions, multi-materials, degradation, and probabilistic demand in the model.
Moreover.
Figure 4 shows the sustainable production inventory system procurement, production, and distribution The figure shows the process through which products are produced to meet the stochastic demands of buyers .
using several raw materials .
ordered from It is pertinent to state the producers are required to order raw materials from suppliers yco times for each raw material yc .
coyc ).
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Table 1.
Literature Review on the sustainable production inventory model Complexity-Based Classification Research (Fiorotto et al.
, 2.
(Fang et al.
, 2.
(Budiman & Rau, 2.
(Omar & Zulkipli, 2.
(Karaba & Tan, 2.
(Khara et al.
, 2.
(Shafiee et al.
, 2.
(Jauhari et al.
, 2.
(Jauhari, 2.
(Mashud et al.
, 2.
(Wangsa et al.
, 2.
(Gautam et al.
, 2.
(Bhattacharjee & Sen, 2.
(Mishra et al.
, 2.
(Mashud et al.
, 2.
(De-la-Cruz-Myrquez et al.
, 2.
This research .
SRM
MRM
SSP
MSP
Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Fuel Tax Electricity Demand Characteristic Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Oo Deterministic Deterministic Stochastic Deterministic Deterministic Deterministic Deterministic Stochastic Stochastic Deterministic Stochastic Deterministic Deterministic Deterministic Deterministic Stochastic Stochastic Quality Degradatio Oo Oo Oo Oo Oo Oo Oo Optimization Tools Exact Heuristic Heuristic Exact Metaheuristic Heuristic Hybrid Heuristic Heuristic Heuristic Heuristic Heuristic Heuristic Heuristic Heuristic Heuristic Metaheuristic Where: Single Raw Material (SRM).
Multi Raw Material (MRM).
Single Stage Production (SSP).
Multi-Stage Production (MSP).
Single Product (SP).
Multi Product (MP) DOI: https://doi.
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September 2023 Hal 171-196 Economic 1D Supplier Raw Raw Raw Raw jD Raw Social Enviroment Single Stage Production Buyer Order Frequency Raw material Single finish good mj, n.
T Inventory Inventory Raw Material finish good Quality Manufacturer Stochastic Figure 4.
Characteristics of the sustainable production inventory model system.
The system has different requirements for each raw material to produce a finished This is indicated by the fact that each raw material yc has a requirement coefficient yuI .
uIyc ) to produce a food product.
The system also considers the degradation of the quality of the raw materials over time to ensure appropriate optimization of those in the warehouse inventory.
Moreover, the system requires that the producers determine the production cycle .
cN), the finished goods to be sent to buyers in n times as well as each raw material to be ordered m times .
coyc ).
The goods are produced at a production rate .
cE) which is more than the buyer demand .
Assumptions and Notations The assumptions made in developing the mathematical models to represent the problem are stated as follows:
Demand for finished goods is probabilistic based on the normal .
The finished good production rate exceeds the product demand rate .
cE > y.
This is to ensure all the demands are .
Raw material yc has the highest quality .
cEycoycaycuyc ) when it arrives in the warehouse for manufacturing.
Each raw material is adequate to meet production requirements.
None of the raw materials is also expected to expire during the planning period because the producers have complete control over the procurement process.
There is no shortage of raw materials because suppliers can meet demands.
The buyer's request for a shortage of finished goods is permitted.
The quantity of raw materials ordered is not limited by vehicle capacity.
Vehicle capacity does not limit the number of finished good shipments.
The notation used in this model includes:
Index yc : index of raw materials yc = 1 A ycAyc Parameters ycE : production rate ya : finished good demand ycAyc : number of raw materials yuIyc : the requirement of a finished good on yc raw materials yc0yc : order quantity of raw materials j yc1 : finished good delivery quantity ycoyc : rate of degradation quality of the yc-th raw materials per unit of time ycEycoycaycuyc : maximum quality on yc-th raw DOI: https://doi.
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ycEycoycnycuyc : minimum quality on the yc-th raw ycEyc .
: quality remaining of the yc-th raw materials in period yc OIycEyc .
: quality degradation of the jth raw material until period yc yuaycoycaycuyc : duration yc-th raw materials can be ycaycoycuycycyc : loss costs quality of the yc-th raw ycaycoycuycycycy : loss sales cost of finished goods : finished goods selling price : purchasing cost of the yc-th raw : processing cost of the finished ya0 yc : ordering cost of the yc-th raw Trj : transportation costs for the yc-th raw material ycNycyycn : finished good transportation costs aryc : fixed cost of transportation on the yc-th raw material : fixed costs of transportation of finished goods : distance of the yc-th raw material : distance between the producer and the buyer yaycEyaycj : kilometers per litre for the yc-th raw material procurement unload yaycEyayc j : kilometers per litre for the yc-th raw material procurement full load KPLp : kilometers per litre of unloading for delivery of finished goods KPLp O : kilometers per litre for the shipment of finished goods with a full load r j : fuel cost for the shipment of the ycth raw material p : fuel cost used to deliver the finished good yuUyc : emissions for 1 litre of fuel in the yc-th raw material shipment yuUycy : emissions for 1 litre of fuel at product delivery yuUyco yuUycyco yuUycnyco yuU0yc At Cyc Cr yc Cp Cfp Cmp Cmi Ciryc yunyc yunycy yunycn yun0yc yayun ya yua ycIycI yaya yceyc .
a ) yayc .
a ) ycI ya0yc ya1 ya0yc ya1 : emissions for the production of each unit of product : emissions for production setups : emissions for finished goods : emissions for the yc-th raw material : emission tax per kg : fixed social cost of procurement the yc-th raw material per horizon : social cost of procurement the yc-th raw material per order : social costs of delivering the finished good : social fixed costs of manufacturing : the social cost of manufacturing once produced : social costs of inventory finished : social cost of the yc-th raw material : energy required for the production setup : energy required to produce each : energy required for storage of the finished good : energy required for the inventory of the yc-th raw material : energy tariff per kWh : safety factor : standard deviation demands : safety stock : estimated cost of loss sales of finished goods : probability density function of the normal distribution : cumulative distribution function of the normal distribution : setup costs for processing the finished good : inventory costs for the yc-th raw : finished good inventory costs : average inventory for the yc-th raw : average finished good inventory DOI: https://doi.
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September 2023 Hal 171-196 yayc : total costs due to degradation of the quality of the yc-th raw materials ycNya0yc : total cost of the yc-th raw materials procurement system yaycNya1 : expected total cost of the finished ETP : expected total profit in the sustainable production Inventory Decision Variable ycoyc : frequency shipment the yc-th raw : production cycle time : delivery frequency of finished The Proposed Model of Sustainable Production Inventory The proposed model for sustainable production inventory problems associated with multi-raw materials and quality degradation is discussed in this section.
It is important to note that the quality degradation of each raw material yc was used to calculate the costs incurred by the company due to the reduction in quality.
Therefore, a kinetic model function was used in this study to formulate the degradation of raw material quality in the inventory system (Rong et al.
, 2.
It was assumed that the entire supply of raw materials yc .
c0yc ) was used for only production purposes .
cE) during the procurement cycle .
cNycy /ycoyc ).
It was also assumed at the beginning of the filling cycle that the raw material quality level yc is the maximum level .
cEycoycaycuyc ).
Moreover, the degradation rate formula at time yc or ycE.
is shown in Equation .
and the quality loss for raw material j for production at time t is presented in Equation .
The raw material quality degradation was estimated by determining the maximum quality of yc-th .
cEycoycaycuyc ), achieving minimum quality level .
cEycoycnycuyc ), and the yc-th maximum duration .
uaycoycaycuyc ).
Furthermore.
Equation .
indicates the model of the decline rate of raw material yc quality in each period yc.
The linear relationship between the quality degradation j from period 0 .
cEycoycaycuyc ) to yc is also modeled in Equation .
The total cost quality reduction yc .
coyc , ycN)) during period yc is indicated in Equation .
Figure 5 shows the system profile of the sustainable production inventory model designed for the problems investigated.
was discovered that there are two levels of inventory including the finished products and raw materials yc.
For finished products, the raw materials are processed in the amount of j P with a production time of Tp to meet the demand of buyers .
Where j indicates the yc-th raw material needed to have a finished product.
Moreover, the producers are required to ensure the production rate .
cE) is greater than demand .
and the rate of raw material yc needed for production is yuIyc ycE.
It is important to note that the proposed model estimates the number of finished products during the production cycle .
cN) to meet demand based on yc1 = yaycN with Tp = yaycN/ycE.
The finished products are also poured in batches .
c1 ) and sent to sales with the delivery frequency of n times.
This makes it possible to estimate the cycle of finished product orders by sales using yc1 /ya.
For the raw material inventory, producers obtain raw materials from suppliers with size yc0yc and yc0yc procurement cycles yuI ya.
These materials are yc subsequently sent to the producers with a delivery frequency of ycoyc times.
The demand for finished products .
in this problem is stochastic based on the normal distribution and this means it can be estimated using the mean ya.
cN) and the standard deviation yuaOoycN during the period ycN.
Moreover, the average inventory was estimated by calculating the average ycNperiod inventory added with the safety stock (Jauhari et al.
, 2021.
Jauhari et al.
, 2.
The safety stock formula in inventory is indicated in Equation .
It is also possible for finished products to experience a loss of DOI: https://doi.
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sales due to stochastic demand.
These lost sales are estimated in period ycN using Equations .
while the inventory for the finished products .
a1 ) and raw materials yc .
a0yc ) is modelled in Equations .
The proposed transportation model assumes that the vehicle departs to pick up raw materials with an empty load.
Therefore, the model to procure raw materials from suppliers is presented in Equation .
and the model to ship finished products is formulated in Equation .
The costs related to raw material management are calculated in Equation .
while the expected total cost of the finished product system .
cNya1 .
cu, ycN)) is presented in Equation .
Moreover, the formula to determine the total revenue .
cNycI.
cu, ycN)) in the system is indicated in Equation .
The Mixed-Integer Nonlinear Programming equation presented in Equation .
is designed to predict the total revenue of the system under study with the constraints identified in Equations .
The Expected Total Profit (ETP) of the model is shown in Equation .
with certain constraints required to be satisfied during First, the production level needs to meet all the demands in Equation .
Second, the production cycle requirement in Equation .
needs to be greater than 0.
Third, the constraint in Equation .
ensures the delivery frequency of raw materials yc and finished products need to be an integer that is greater than 1.
It is important to note that profit maximization was conducted through the simultaneous determination of the optimal decision variables including ycoyc , ycu, and ycN.
ycEyc .
= ycEycoycaycuyc Oe ycoyc yc OIycEyc .
= ycEycoycaycuyc Oe ycEyc .
ycoyc = ycEycoycaycuyc OeycEycoycnycuyc OIycEyc .
= ycoyc yc ycoyc yuIyc ycE coyc , ycN) = ycaycoycuycycyc yuIycyaycN ycoyc yuIycycE yuuycEyc .
ycIycI = yayuaOoycN yaya = yuaOoycNyue.
a ) = .
a ) Oe ya.
Oe yayc .
a )]) .
yaycN ya ya1 = 2ycu .
cE .
Oe yc.
cu Oe .
) yayuaOoycN .
yuIyc ya2 ycN ya0yc = 2yco yuI ycE yc yc yccycy ycNycyycn = .
ycy Cp ) yccycy DycN yaycEyaycy OeKPLpO ycu yaycEyaycy .
cy yuUycy O At ) Trj = .
ryc Cryc ) O .
cy yuU O At ) yuIyc yaycN yaycEyaycj OeyaycEyayc Oj ycoyc yaycEyaycj j yuUyc O At ) .
j yuUyc O At ) .
ycNya0 .
coyc , ycN) = Ocyc=1 .
ca0yc yuIyc ya Cyc .
a0yc ycoyc Tr.
ycN .
a0yc Ciryc yun0yc O yayun yuU0yc O At )ya0yc ycaycoycuycycyc ycoyc yuIyc ycE yuIycyaycN ycoyc yuIyc ycE yuuycEyc .
yaycNya1 .
cu, ycN) = .
ca1 yunycy O yayun yuUyco O At ) O .
cI Cmp yunyc Oyayun yuUycyco OAt ) ycNycyycn Oycu ya Cfp ycN ycN (Cmi yunycn O yayun yuUycnyco O At ya1 )ya1 yaya*ycaycoycuycycycy .
ycNycI.
cu, ycN) = ycaycycaycoyce ya .
yaycNycE.
coyc , ycN, yc.
= ycNycI .
cu, ycN) Oe .
cNya0 .
coyc , ycN) ycNya1 .
cu, ycN)) .
ycE Ou ya.
T > 0.
ycoyc , .
Ou 1.
Integer DOI: https://doi.
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System profile of the sustainable production inventory model.
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PROPOSED ALGORITHM
An HHO algorithm was proposed to optimize the objective function of the model problem through the application of the decision variables presented in Section 3.
It is important to note that the number of decision variables can be calculated to solve the problem identified using ycAyc 2.
This means the number is determined based on the number of raw materials used to manufacture a product.
Heidari et al.
introduced the HHO algorithm with two main behaviors which include exploration and exploitation as shown in Algorithm 1.
The exploration phase involves applying the Harris Hawks behavior to detect rabbit prey as expressed in Equation .
represents the current position from Harris Hawks while ycU.
indicates its position in the next iteration.
The rabbit's position is denoted by ycUycycaycaycaycnyc .
while yc1 , yc2 , yc3 , and yc4 are all random numbers in the range .
Moreover, the upper and lower limit variables are denoted as ycOyaA and yayaA, ycUycycaycuycc .
also simulates the Harris Hawks selected randomly from the current population while Equation .
is used to calculate the average position of the Harris Hawks .
cUyco .
ycUycn .
calculates the location of each Harris Hawks in the current iteration and ycA is the total number of hawks.
It was observed that the prey's energy .
decreases during the transition from exploration to exploitation as shown in Equation .
The notation shows that 2ya0 represents the rabbit's initial energy and ya denotes the energy released by the prey depending on the maximum number of iterations .
cN).
It is pertinent to note that the Harris Hawks is exploring and experiencing ycU
STUDY DATA AND PROCEDURES
The data used to conduct the experiments include numerical examples from three different cases involving small (Case .
, moderate (Case .
, and large numbers of raw material variations (Case .
It is important to state that Case 1 focuses on the production problem requiring two raw materials.
Case 2 involves five raw materials, and Case 3 uses ten raw .
Oeyc1 .
cUycycaycuycc.
Oe2yc2ycU.
| ycycaycaycaycnyc .
OeycUyco .
)Oeyc3 .
ayaA yc4 .
cOyaAOeyayaA)) ycU.
= { ycycaycuycc .
cU exploitation when ya0 Ou1.
The four strategies associated with Harris Hawks during the exploitation phase include soft besiege, hard besiege, soft besiege with progressive rapid dives, and hard besiege with progressive rapid dives.
The soft besiege behavior occurs when yc Ou 0.
5 and .
Ou 0.
5 as indicated in Equations .
Moreover, the OIX .
shows the difference between the position vector of the rabbit and the current location in yaycyceyc iteration with a value of ya = 2 .
Oe yc5 ) while yc5 describes the random numbers in the range .
The hard besiege strategy occurs when yc Ou 0 and .
< 0 as modeled in Equation .
The soft besiege with progressive rapid dives occurs when yc < 0 and .
Ou 0 as presented in Equations .
It is important to note that the levy flight function is denoted as yaya, a random vector with size 1 ycu ya is represented by ycI, and the problem dimensions are described as ya.
The yaya function can be estimated using Equation .
where yu is a constant of 1.
while yc and yuO are random values in the range .
Hard besiege with progressive rapid dives occurs when yc < 0.
5 and .
< 5 as modeled in Equation .
Meanwhile.
Y' and Z' values can be estimated using Equations .
ycUyco .
= ycA OcycA ycn=1 ycUycn .
ycOu0.
yc<0.
DOI: https://doi.
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p- ISSN 2528-1410 e- ISSN 2527-8045 183 | Indonesian Journal of Science & Technology.
Volume 8 Issue 2.
September 2023 Hal 171-196 ycU ycU.
= { ycycaycuycc .
cU .
Oeyc1 .
cUycycaycuycc.
Oe2yc2ycU.
| ycycaycaycaycnyc .
OeycUyco .
)Oeyc3 .
ayaA yc4 .
cOyaAOeyayaA)) ycUyco .
= ycA OcycA ycn=1 ycUycn .
ycOu0.
yc<0.
yaycyceyc ya = 2ya0 .
Oe ycN ) .
= OIycU.
Oe ya .
aycUycycaycaycaycnyc .
Oe ycU.
| .
OIycU.
= ycUycycaycaycaycnyc .
Oe ycU .
= ycUycycaycaycaycnyc .
Oe ya |OIycU.
| .
ycU = ycUycycaycaycaycnyc .
Oe ya .
aycUycycaycaycaycnyc .
Oe ycU.
| .
ycs = ycU ycI ycu yaya .
a ) .
cu ) = 0.
01 ycu ycycuyua .
u , yua=( yuUyu ) yu 1 yu yuOe1 e( )x x 2( e.
)x sin( .
ycU ycnyce ya .
cU) < ya.
) ycU.
= { ycs ycnyce ya.
< ya.
) .
ycUA ycnyce ya .
cUA) < ya.
) ycU.
= { ycsA ycnyce ya.
csA) < ya.
) .
Algorithm 1.
Pseudo-code of HHO algorithm Inputs: The population size ycA and the maximum number of iterations Outputs: The location of the rabbit and its Atness value Initialize the random population Xi.
= 1,2,.
,N)
topping condition is not me.
do Calculate the Atness values of hawks = ycU O Set Xrabbit as the location of the rabbit .
est locatio.
ach hawk (X.
) do Update the initial energy E0 and jump strength J (E0=2rand()-1.
J=2.
-rand()) Update the E using Eq.
if (|E|Ou .
hase of exploratio.
Update the location using Eq.
if (|E| < .
hase of exploitatio.
Ou0.
5 and |E|Ou 0.
then (Soft besiege Update the location vector using Eq.
else if .
Ou0.
5 and |E| < 0.
then (Hard besiege Update the location vector using Eq.
else if .
<0.
5 and |E|Ou 0.
then (Soft besiege with progressive rapid dives Update the location using Eq.
else if .
<0.
5 and |E| < 0.
then (Hard besiege with progressive rapid dives Update the location using Eq.
Evaluate the rabbit position Update rabbit position .
cU O ) if there is a better solution for the population yei=yei ya Return Xrabbit DOI: https://doi.
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| 184
Data for case 1 are ycE=10,500, ya=8,500, ycAyc =2, yuI1 = 4, yuI2 =2, yco1 =0.
1, yco2 =0.
ycaycoycuycyc1 =2,000, ycaycoycuycyc2 =1,000, ycaycoycuycycycy =140,000, yaycycaycoyce =140,000, yca01 =12,000, yca02 =15,000, yca1 = 125, ya01 =125, ya02 =100, ar1 =100, ar2 =100, ap =700, dr1 =4, dr2 =6, dp =10, yaycEyayc1 =20, yaycEyayc2 =20, yaycEyayc O1=18, yaycEyayc O 2 =18.
O KPLp =20.
KPLp =18, r1 =1,000, r1 =1,000, p =1,000, yuU1 =1.
5, yuU2 =1.
2, yuUycy =1.
75, yuUyco =5, yuUycyco =50, yuUycnyco =1.
25, yuU01 =1, yuU01 =1.
At =30.
C1 =25.
C2 =25.
Cr1 =25.
Cr 2=25.
Cp =50.
Cfp =1500.
Cmp= 70,000.
Cmi =250.
Cir1 =15.
Cir2 =15, yunyc =550, yunycy =15, yunycn =1, yun01 =0.
yun02 =0.
05, yayun =1,444, ya=1.
645, yua=125, ycI=45,000, ya01 =250, ya02 =155, ya1 =300.
Data for case 2 is presented as follows:
ycE=9,500, ya=7,400, ycAyc =5, yuI1 =2, yuI2 =0.
1, yuI3 =0.
25, yuI4 =0.
1, yuI5 =1, yco1 =0.
yco2 =0.
01, yco3 =0.
01, yco4 =0, yco5=0, ycaycoycuycyc1 =200, ycaycoycuycyc2 =100, ycaycoycuycyc3 =200, ycaycoycuycyc4 =0, ycaycoycuycyc5 =0, ycaycoycuycycycy =80,000, ycaycycaycoyce=75,000, yca01 =8,000, yca02 =1,300, yca03 =3,400, yca04 =500, yca05 =500, yca1=100, ya01 =50, ya02 =50, ya03 =50, ya04 =50, ya05 =50, ar1 =20, ar2 =20, ar3 =20, ar4 =5, ar5 =10, ap =5, dr1 =5, dr2 =2, dr3 =5, dr4 =7, dr5 =1, dp =5, yaycEyayc1 =20, yaycEyayc2 =20, yaycEyayc3 =20, yaycEyayc4 =20, yaycEyayc5 =2, yaycEyayc O1=19, yaycEyayc O 2 =19, yaycEyayc O 3 =19.
yaycEyayc 4 =19, yaycEyayc 5 =19.
KPLp =20.
O KPLp =18, r1 =1,000, r 2 =1,000, r 3 =1,000, r 4 =1,000, r5 =1,000, p =1,000, yuU1 =0.
yuU2 =0.
yuU3 =0.
yuU4 =0.
001, yuU5 =0.
001, yuUycy =0.
0015, yuUyco=5, yuUycyco = 1,000, yuUycnyco =1.
25, yuU01 =0.
yuU02 =0.
yuU03 =0.
5, yuU04 =0.
2, yuU05 =1.
At =30.
C1 =10.
C2 =10.
C3 =10.
C4=5.
C5 =5.
Cr1 =10.
Cr2 =10.
Cr 3 =10.
Cr4 =50.
Cr 5 =25.
Cp =10.
Cfp =1,000.
Cmp=500,000.
Cmi =100.
Cir1 =25,Cir 2 =25.
Cir3 =25.
Cir4 =125.
Cir 5=100, yunyc =1,000, yunycy =20, yunycn =1, yun01 =0.
yun02 =0.
25, yun03=0.
52, yun04 =25, yun05 =15, yayun =1,444, ya=1.
645, yua=100, ycI=9,000,000, ya01 =150, ya02 =175, ya03 =150, ya04 =850, ya05 =750, ya1 =750.
Furthermore, data for case 3 are ycE=9,500, ya=7,400, ycAyc =10, yuI1 =2, yuI2 =0.
yuI3 =0.
25, yuI4 =0.
1, yuI5 =1, yuI6 =2, yuI7 =0.
yuI8 =0.
25, yuI9 =0.
1, yuI10=1, yco1 =0.
025, yco2 =0.
yco3 =0.
01, yco4 =0, yco5 =0, yco6 =0.
025, yco7 =0.
yco8 =0.
01, yco9 =0, yco10=0, ycaycoycuycyc1 =200, ycaycoycuycyc2 =100, ycaycoycuycyc3 =200, ycaycoycuycyc4 =0, ycaycoycuycyc5 =0, ycaycoycuycyc6 =200, ycaycoycuycyc7 =100, ycaycoycuycyc8 =200, ycaycoycuycyc9 =0, ycaycoycuycyc10 =0, ycaycoycuycycycy =150,000, ycaycycaycoyce =120,000, yca01 =8,000, yca02 =1,300, yca03 =3,400, yca04 =500, yca05 =500, yca06 =8,000, yca07 =1,300, yca08 =3,400, yca09 =500, yca010 =500, yca1=100, ya01 =50, ya02 =50, ya03 =50, ya04 =50, ya05 =50, ya06 =50, ya07 =50, ya08 =50, ya09 =50, ya010 =50, ar1 =20, ar2 =20, ar3 =20, ar4 =5, ar5 =10, ar6 =20, ar7 =20, ar8 =20, ar9 =5, ar10=10, ap =5, dr1 =5, dr2 =2, dr3 =5, dr4 =7, dr5 =1, dr6 =5, dr7 =2, dr8 =5, dr9 =7, dr10=1, dp =5, yaycEyayc1 =20, yaycEyayc2 =20, yaycEyayc3 =20, yaycEyayc4 =20, yaycEyayc5=20, yaycEyayc6 =20, yaycEyayc7 =20, yaycEyayc8 =20, yaycEyayc9 =20, yaycEyayc10=20, yaycEyayc O1=19.
yaycEyayc 2 =19, yaycEyayc 3 =19, yaycEyayc O 4 =19, yaycEyayc O 5 =19, yaycEyayc O 6 =19, yaycEyayc O 7 =19, yaycEyayc O 8 =19, yaycEyayc O 9=19, yaycEyayc O10=19.
KPLp =20.
KPLp O=18, r1 =1,000, r 2 =1,000, r 3 =1,000, r 4 =1,000, r 5 =1,000, r 6 =1,000, r 7 =1,000, r 8 =1,000, r 9 =1,000, r10=1,000, p =1,000, yuU1 =0.
yuU2 =0.
2, yuU3 =0.
02, yuU4 =0.
001, yuU5 =0.
yuU6 =0.
yuU7 =0.
yuU8 =0.
yuU9 =0.
yuU10=0.
001, yuUycy =0.
0015, yuUyco =5, yuUycyco = 1,000, yuUycnyco =1.
25, yuU01 =0.
yuU02 =0.
5, yuU03 =0.
yuU04 =0.
2, yuU05 =1, yuU06 =0.
yuU07 =0.
yuU08 =0.
5, yuU09 =0.
2, yuU010=1.
At =30.
C1 =10.
C2 =10.
C3 =10.
C4 =50.
C5 =25.
C6 =10.
C7 =10.
C8 =10.
C9 =50.
C10=25.
Cr1 =10.
Cr2 =10.
Cr 3 =10.
Cr 4 =50.
Cr5 =25.
Cr6 =10.
Cr 7=10.
Cr 8 =10.
Cr 9 =50.
Cr10 =25.
Cp =10.
Cfp =1,000.
Cmp=500,000.
Cmi =100.
Cir1=25.
Cir2 =25.
Cir3 =25.
Cir4 =125.
Cir 5 =100.
Cir6 =25.
Cir7 =25.
Cir8 =25.
Cir 9=125.
Cir10 =100, yunyc =1,000, yunycy =20, yunycn =1, yun01 =0.
5, yun02 =0.
yun03 =0.
yun04 =0.
yun05 =1, yun06 =0.
yun07 =0.
yun08 =0.
yun09=0.
yun010 =1, yayun =1,444, ya=1.
645, yua= 100, ycI=9,000,000, ya01 =150, ya02 =175, ya03 =150, ya04 =850, ya05 =750, ya06 =150, ya07 =175, ya8 =150, ya09 =850, ya010 =750, ya1 =750.
DOI: https://doi.
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p- ISSN 2528-1410 e- ISSN 2527-8045 185 | Indonesian Journal of Science & Technology.
Volume 8 Issue 2.
September 2023 Hal 171-196 The procedures conducted in this study with HHO used different population variations and iterations.
Each case was optimized using three population variations which include the small, medium, and large A total of 100 populations and 100 iterations (Pop 100 x Iter .
were used as small variations, 250 populations and 250 iterations (Pop 250 x Iter .
for medium, as well as 500 populations and 500 iterations (Pop 500 x Iter .
for large.
Each experiment was run 30 times, thereby leading to 90 trials for each case and a total of 270 trials for the three cases.
The quality of the solution provided by the proposed algorithm was benchmarked using ETP and computation time with the GA and PSO algorithms.
The parameters used to compare the algorithms were large populations and iterations which include Pop 500 x Iter 500.
In the GA algorithm, a crossover probability of 0.
8 and mutation of 8 was used while an inertia weight of 0.
was applied in the PSO algorithm.
It is important to state that all the algorithms were decoded on MATLAB R2018a on Windows 10 AMD A8 with x64-64 4GB RAM.
Moreover, the ANOVA test was used to determine the quality of the solution based on ETP and the computation time to compare the proposed algorithm with the GA and PSO algorithms.
A sensitivity analysis was also conducted to examine the effect of changing variables on decision variables and the expectation of total profit.
It was applied to Case 1 using the quality degradation rate .
, the standard deviation of demand .
, and the safety factor .
as variables.
Each variable was changed with 10 different data and the results were recorded.
RESULTS AND DISCUSSION
Expected Total Profit Optimization Using HHO which involves incorporating costs of fuel, multi-materials, quality degradation, and probabilistic It was applied to the aforementioned three cases.
The ETP optimization using HHO based on trial variations is summarized in Table 1.
The experimental results showed that the experimental variations in Cases 1 and 2 are small (Pop 100 x Iter .
, medium (Pop 250 x Iter .
, and large (Pop 500 x Iter .
, and they all have the same solution.
This means the problems associated with a small or medium number of raw materials produced the same ETP without any difference based on population variations and iterations.
However, the problems associated with a large number of raw materials in Case 3 showed that only the trials of medium variations and large variations produced similar and better ETP solutions compared to the population variation experiment and small iteration.
This means the optimal solution for Case 3 was found in the population experiment as well as the medium and large iterations.
Computation Time on Problem-Solving with HHO The results of the computation time required to solve the problems using HHO are presented in Table 2 based on variations in trials and cases.
It was discovered that an increase in the population and iterations led to an increment in the computation time needed to solve the HHO algorithm The time was observed to reduce for smaller populations and iterations.
The results from each case showed that the problems associated with a larger quantity of raw materials as indicated in Cases 1-3 necessitate an increase in computation (ETP) The proposed model was developed based on the complex real-world situation DOI: https://doi.
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Table 2.
Computation time to solve problems using HHO (Secon.
Cases Case 1
Case 2
Case 3
Results Average Standard deviation Minimum
Maximum
Average Standard deviation Minimum
Maximum
Average Standard deviation Minimum
Maximum
Pop 100 x Iter 100
Pop 250 x Iter 250
2,010
1,869
2,130
Pop 500 x Iter 500
1,605
1,557
1,679
3,141
2,998
3,305
7,615
7,183
7,883
Table 3.
Results of expected total profit optimization using HHO.
Cases Case 1
Case 2
Case 3
Results Average Standard deviation Minimum
Maximum
Average Standard deviation Minimum
Maximum
Average Standard deviation Minimum
Maximum
Pop 100 x Iter 100
164,137,878
164,137,878
164,137,878
14,444,202
14,444,202
14,444,202
76,290,250
167,177
75,999,721
76,387,543
Tables 2 and 3 showed that the problems in Case 1 or 2 can be solved by varying population trials and small iterations (Pop 100 x Iter .
This is reasonable because the small population and iteration experiments produced solutions considered to be as good as those classified as medium and large.
They also have faster computation times than the other variations and iterations.
Medium population and iteration variations were also recommended to solve the problems in Case 3 because they produced similar ETP solutions with large variations and better than small variations.
However, large variations require more computation time.
Pop 250 x Iter 250
164,137,878
164,137,878
164,137,878
14,444,202
14,444,202
14,444,202
76,387,543
76,387,543
76,387,543
Pop 500 x Iter 500
164,137,878
164,137,878
164,137,878
14,444,202
14,444,202
14,444,202
76,387,543
76,387,543
76,387,543
Algorithm Comparison ETP and computation time for each algorithm were compared and presented in the Boxplot.
The results for PSO and GA algorithms are listed in Tables A7 and A8 respectively in Appendix A.
Moreover.
Figures 6-8 show a Boxplot of the ETP results for each algorithm in Cases 1Ae3.
The solution provided to Cases 1 and 2 by the proposed HHO algorithm was observed to be as good as the PSO algorithm.
However, the solution provided in Case 3 was found to be better.
These findings were further supported by the ANOVA test conducted on yaycNycE as shown in Tables 4 and 5 where the variance of the yaycNycE value was found to be different .
ig<0.
It was discovered that HHO and DOI: https://doi.
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Volume 8 Issue 2.
September 2023 Hal 171-196 PSO produced the same solution .
ig>0.
in Cases 1 and 2 as indicated in Table 5.
However.
HHO performed better than PSO in Case 3 as evidenced by a sig value <0.
The yaycNycE comparison results between HHO and GA also showed that the proposed model is superior in all cases.
Figures 9-11 show a Boxplot comparison of Cases 1-3 in terms of computation time and the PSO algorithm was observed to have outperformed the proposed HHO and GA algorithms.
This was supported by the findings of the ANOVA test in Tables 4 and 5 that the variance values of HHO.
PSO, and GA algorithms differ.
The computation time was discovered to be significantly different as indicated by the sig value <0.
Meanwhile, the HHO algorithm produced a better yaycNycE than PSO despite having a longer computation time.
Figure 6.
Boxplot of ETP results for each algorithm in Case 1.
Figure 7.
Boxplot of ETP results for each algorithm in Case 2.
Figure 8.
Boxplot of ETP results for each algorithm in Case 3.
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Table 4.
The results of the ANOVA for the expected total profit (ETP) test and computation time in each Case.
Tests ETP Computation Time Anova Nilai F Sig Nilai F Sig Case 1 Case 2 Case 3 Table 5.
The results of the comparison of expected total profit (ETP) and computation time
for each algorithm in each Case
Tests ETP
Computation Time
Comparing HHO-GA
HHO-PSO
PSO-GA
HHO-GA
HHO-PSO
PSO-GA
Sig Case 1 Sig Case 2 Sig Case 3 Figure 9.
Boxplot of computation time for each algorithm in Case 1.
Figure 10.
Boxplot of computation time for each algorithm in Case 2.
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Volume 8 Issue 2.
September 2023 Hal 171-196 Figure 11.
Boxplot of computation time for each algorithm in Case 3.
Sensitivity Analysis The results of the sensitivity analysis conducted on the effect of changes in the rate of quality degradation .
, the standard deviation of demand .
, and the safety factor .
on the time of production cycle .
cN) and ETP are explained.
Figure 12 depicts the effect of yco changes on ycN and yaycNycE and it was discovered that an increase in the rate of quality degradation .
led to an increment in yaycNycE and ycN, and vice versa.
Meanwhile, a change in the rate of decline in quality .
did not affect the decision variables including the frequency of ordering raw materials .
coyc ) and delivery of finished products .
as indicated by the value of 1 for both.
The results further showed that an increase in the degradation rate .
caused an increment in the frequency of ordering raw materials in one horizon.
This is reasonable because an increase in the rate of quality degradation .
is expected to cause a reduction in the raw material inventory yc because of the increase in the frequency with which raw materials are ordered .
coyc ) and vice versa.
Figure 13 shows the effects of changes in the standard deviation of demand .
on T and ETP.
The findings showed that an increase in the standard deviation of demand .
led to an increment in ETP and T, and vice versa.
Meanwhile, the change in demand standard deviation .
has no effect on the decision variables associated with ordering raw materials .
coyc ) and shipping finished products .
The results also showed that a reduction in the standard deviation of demand .
led to a decrease in demand uncertainty which caused an ETP and T, and vice versa.
This is reasonable because demand uncertainty usually increases with the standard deviation of demand .
A high uncertainty can cause decision-makers to increase safety stock and reduce T, thereby leading to high inventory costs and lower ETP, and vice It was discovered in Figure 14 that an increase in the safety factor .
also led to an increment in yaycNycE and T, and vice versa.
Meanwhile, the changes in the safety factor (K) did not affect the frequency of ordering raw materials .
coyc ) and delivering finished products .
The findings also indicated that an increase in the safety factor .
increased the average finished product inventory, thereby, leading to a reduction in finished product lost sales .
and an enhancement in ETP and T.
Meanwhile, a decrease in safety factor .
caused a reduction in the average finished product inventory and this led to an increase in yaya.
As a result.
ETP and T fell.
This is considered reasonable because an increase in ya enhances the risk yaya DOI: https://doi.
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Figure 12.
The effect of changing k on T and ETP.
Figure 13.
The effect of changing E on T and ETP.
Figure 14.
Effect of changes in K to T and ETP.
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Volume 8 Issue 2.
September 2023 Hal 171-196 Managerial Insight The proposed model can be implemented in companies with a linear decline in raw material quality such as the agro, food, and Its implementation can assist managers and decision-makers determine production decisions, raw material procurement, and finished product delivery.
Moreover, they can also benefit significantly from the findings related to ETP.
This study proposes the HHO procedure for optimizing the problem of the sustainable production inventory model.
The proposed algorithm outperforms the GA and PSO algorithms.
The findings suggested that managers and decision-makers use a population of 100 and iterations of 100 to solve problems involving raw materials numbers 2 (Case smal.
and 5 (Case To solve problems with ten raw materials (Large Cas.
, 250 populations and 250 iterations are recommended.
The proposed algorithm can increase the company's ETP.
It was also indicated that the degradation of raw materials quality affects the company's ETP.
This is observed from the fact that low-quality degradation can improve ETP.
Therefore, managers and decision-makers are required to consider several factors such as humidity, temperature, and storage time.
It has been indicated that perishable raw materials are extremely sensitive to changes in temperature and humidity (Mahmood et al.
This means proper management needs to be implemented in the storage areas to slow the decline in quality .
There is also the need for strict and effective inventory management procedures such as the principle of a First-In First-Out (FIFO) inventory system.
This method is useful in dealing with quality degradation issues caused by first processing first-come, firstserved raw materials.
It also has the potential to reduce warehouse storage time.
The study also showed that an increase in the standard deviation of demand .
reduced yaycNycE.
This means managers and decision-makers need to effectively manage demand at the sales level through Collaborative Planning.
Forecasting, and Replenishment (CPFR).
CPFR is a method of demand planning and fulfillment that improves the efficiency of manufacturing and supply chain businesses (Danese, 2.
(Panahifar et al.
, 2.
It also can assist producers to obtain reliable demand data (Alptekin et al.
, 2.
An increase in the safety factor .
was also observed to have the ability to raise yaycNycE.
Therefore, managers and decisionmakers need to decide whether to use a high safety factor .
when demand is uncertain to enhance the average inventory, but this can reduce the risk of yaya and yaycNycE.
CONCLUSION
This study proposed a sustainable production inventory model to maximize ETP with due consideration for fuel cost, emissions cost, electricity cost, multimaterials, probabilistic demand which represent complex real-life cases.
This is to ensure the limitations of previous models are resolved in the proposed model.
Moreover, a new HHO procedure was also proposed to optimize the problems associated with the sustainable production inventory model.
The findings showed that the proposed HHO algorithm was able to optimize the sustainable production inventory model It also outperformed the GA and PSO algorithms in ETP but has a slower computation time than PSO.
The sensitivity analysis conducted also presented significant results such as the reduction in yaycNycE and time of production cycle .
cN) due to the increase in the quality degradation rate.
A similar trend was also recorded with the standard deviation of demand .
while an increase in the safety DOI: https://doi.
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was observed to have led to an increment in yaycNycE and ycN.
The proposed study model has limitations that can be addressed in future These consideration of certain factors such as item production in the development of a new model in the future.
The model also assumed the manufacturing process to be flawless with no product In reality, errors in the manufacturing process can result in product Therefore, further studies can be developed by considering the presence of defective items.
There is also the need to account for the uncertainty of delivery lead time because the model designed in this study only considered demand even though the uncertainty for delivery lead time is more common in reality.
It is recommended that the model is developed with due consideration for the uncertainty of the delivery lead time in future studies.
REFERENCES