Sustainable Economic Production Quantity Model Considering Greenhouse Gas and Wastewater Emissions

Abstract

Pursuing sustainability in the production inventory model has motivated researchers to reconsider carbon emission costs. In this study, we develop an economic production quantity (EPQ) inventory model to consider greenhouse gas and wastewater emission costs. This study aims to optimize the production quantity, the total costs per production cycle, and the waste emissions. To validate and test the model, data from a pulp and paper mill industry was used. The effectiveness of the model was determined by comparing the costs generated under three (3) scenarios: (1) an EPQ model that considers the costs of greenhouse gas (GHG) emissions; (2) an EPQ model considering shortages; and (3) a multiproduct EPQ model considering shortages. Our proposed sustainable EPQ models can provide decision-makers with insights to visualize how certain factors affect the manufacturing system. Future research can be conducted to consider the product life cycle as well as deteriorating items.

1. Introduction

Driven by urbanization and industrialization, the amount of wastewater increases rapidly. Wastewater emissions result from untreated wastewater discharged into rivers and lakes [1]; this eventually leads to eutrophication and risks human health. Some efforts to address these problems have been made by various researchers, utilizing different processes to reduce water emissions from industrial plants. These methods include both linear and nonlinear models [2,3], pinch analysis [4], genetic algorithms [5], management strategies [6], benchmarking [7], and wastewater treatment [8].

Harmful GHG emissions from manufacturing industries include carbon dioxide (CO₂), sulfur oxide (SOx), nitrogen oxide (NOx), and methane (CH4). These gases are emitted through the combustion, production, and transportation of fossil fuels, as well as many other procedures. A study by Ritchie et al. [9] revealed that the annual GHG emissions would reach 11 million tons and continually increase [10]. However, CH4 emissions associated with landfills, coal mining, and natural gas and petroleum distribution have decreased. Other research has been conducted to address GHG emission problems, focusing on inventory management [11], fuel-saving strategies [12], and transshipment routes [13].

The pollution emission problems may also contribute to the substantial costs of implementing sustainable processes and a cleaner production system [14]. Other researchers linked the problem to the green attitude-behavior gap of consumers [15], which compares customers’ favorable attitudes and purchasing activities with green products. However, implementing sustainability is a dilemma for industries as most consumers are more concerned with product price than green products.

In this study, we developed three scenarios to minimize the total inventory costs of the model that considers sustainability. Numerous researchers have applied the economic production quantity (EPQ) model, considering different types of demand [16], shortage situations [17], and deteriorating rates [18]. Research on sustainable inventory models integrating environmental costs has been conducted by Taleizadeh et al. [17], who considered carbon emission costs and shortage situations. Zadfajar and Gholamian [2] developed an inventory model that considered the emission of SOx, NOx, biological oxygen demand (BOD), and chemical oxygen demand (COD) from the manufacturing process of pulp and paper mills. The model was developed using the exact nonlinear constrained solution method. Another study by Daryanto and Wee [19] developed a sustainable EPQ model considering carbon emissions from production, warehousing, and waste disposal activities, then extended it to a full backorder situation.

This study aims to provide a practical EPQ model that determines the correct production quantity while minimizing the total costs per production cycle and the environmental effect. Our model extends the study by Taleizadeh et al. [17] to consider full backordering and multiproduct situations.

2. Research Gap: Literature Review

2.1. Sustainable Inventory Models

The economic order quantity (EOQ) model was first developed by Harris [20]. The same author extended the work to consider the production environment. Since then, environmental and sustainable issues have received much attention [21]. With the development of technology, environmentally sustainable machines, and treatment processes, producers and buyers are able to reduce costs and expenses [14]. Taleizadeh et al. [17] analyzed four (4) sustainable EPQ models for different shortage situations using a direct accounting approach. Daryanto and Wee [19] applied an optimization method to consider carbon emission costs from production, warehousing, and waste disposal activities. Tiwari et al. [22] considered a single-vendor, single-buyer situation to provide insights on the optimal production quantity while minimizing the total inventory and carbon emission costs. The work of Zadjafar and Gholamian [2], on the other hand, presented a revised EOQ model by adding income from waste sales as well as sulfur, nitrogen, BOD, and COD emission costs. Hovelaque and Bironneau [23] developed an integrated EOQ model to maximize profit while decreasing carbon emissions. Finally, Arslan and Turkay [24] revised the traditional EOQ, considering direct accounting, carbon tax, direct cap, cap and trade, and carbon offsets on carbon emissions.

This study attempts to fill the gaps in past research articles by considering both carbon and wastewater emission costs. The model’s effectiveness is determined by comparing the costs generated under three scenarios. Moreover, the study also presents two kinds of models: single- and multiple-product models. A mathematical optimization process is used to optimize the initial working equation, which considers the factors listed in Table 1. The said process follows the theorem of absolute extrema for partial derivatives, transforming an equation by taking its first derivative. This is explained in detail in the model formulation part. This study used actual data from a pulp and paper mill to minimize inventory and emission costs through environmental economics. The proposed sustainable EPQ model is expected to provide significant insights to relevant industries and could be a good decision-making reference for manufacturing companies.

2.2. Pulp and Paper Mill Industry

Numerous pieces of literature on sustainability for various industries have been published over the years. The pulp and paper mill industry has received widespread attention due to its high energy consumption [25]. The industry generates different pollutants depending on the type of pulping process [8]. Paper production includes three (3) primary activities: pulp-making, paper formation, and finishing. During the pulp-making and paper-formation process, coatings and fillers are added to the pulp that converts into paper. This process produces massive emissions of GHGs, water effluents, and other pollutants [26]. Various types of research on sustainability methods include generating environmentally friendly management strategies [6], treatment processes [8], and the integration of inventory models [2].

Table 1. Summary of Studies on Sustainable Inventory Modeling.

Authors	EPQ/EOQ	Carbon Emission Cost	SOx Emission Cost	NOx Emission Cost	CH4 Emission Cost	Wastewater Cost	Backorder/Shortage	Single or Multiproduct	Constraints	Method
Taleizadeh et al. [17]	EPQ	Yes (Inventory Holding, Obsolescence, Production)	No	No	No	No	Yes (Lost Sale, Full Backordering, Partial Backordering)	Single	No	Direct Accounting
Daryanto and Wee [19]	EPQ	Yes (Inventory Holding, Production, Waste Disposal)	No	No	No	No	Yes (Full Backorder)	Single	No	Optimization
Zadfajar and Gholamian [2]	EOQ	No	Yes	Yes	No	Yes (BOD and COD)	No	Multi	No	Linear Programming
Pasandideh et al. [27]	EPQ	No	No	No	No	No	Yes (Full Backorder)	Multi	Yes (Machine Capacity Constraint, Budget, Service Level)	Optimization
Ghosh et al. [28]	EPQ	No	No	No	No	No	No	Multi	Yes (Warehouse Capacity Constraint)	Lagrange Optimization
Arslan et al. [24]	EOQ	Yes	No	No	No	No	No	Single	No	Optimization
Hovelaque and Birronneau [23]	EOQ	Yes	No	No	No	No	No	Single	No	Optimization
This study	EPQ	Yes (Inventory Holding, Production and Waste Disposal)	Yes	Yes	Yes	Yes (BOD and COD)	Yes (Full Backorder)	Single and Multi	Yes (Warehouse Capacity Constraint)	Mathematical Optimization

Table 1. Summary of Studies on Sustainable Inventory Modeling.

Authors	EPQ/EOQ	Carbon Emission Cost	SOx Emission Cost	NOx Emission Cost	CH4 Emission Cost	Wastewater Cost	Backorder/Shortage	Single or Multiproduct	Constraints	Method
Taleizadeh et al. [17]	EPQ	Yes (Inventory Holding, Obsolescence, Production)	No	No	No	No	Yes (Lost Sale, Full Backordering, Partial Backordering)	Single	No	Direct Accounting
Daryanto and Wee [19]	EPQ	Yes (Inventory Holding, Production, Waste Disposal)	No	No	No	No	Yes (Full Backorder)	Single	No	Optimization
Zadfajar and Gholamian [2]	EOQ	No	Yes	Yes	No	Yes (BOD and COD)	No	Multi	No	Linear Programming
Pasandideh et al. [27]	EPQ	No	No	No	No	No	Yes (Full Backorder)	Multi	Yes (Machine Capacity Constraint, Budget, Service Level)	Optimization
Ghosh et al. [28]	EPQ	No	No	No	No	No	No	Multi	Yes (Warehouse Capacity Constraint)	Lagrange Optimization
Arslan et al. [24]	EOQ	Yes	No	No	No	No	No	Single	No	Optimization
Hovelaque and Birronneau [23]	EOQ	Yes	No	No	No	No	No	Single	No	Optimization
This study	EPQ	Yes (Inventory Holding, Production and Waste Disposal)	Yes	Yes	Yes	Yes (BOD and COD)	Yes (Full Backorder)	Single and Multi	Yes (Warehouse Capacity Constraint)	Mathematical Optimization

2.3. Emissions during the Pulp and Paper Production

The GHG and wastewater emissions of pulp and paper processes are presented in

Table 2. Source of Emissions from a Pulp and Paper Mill Process.

Considered Processes	Emissions
	NOx	SOx	BOD	COD	CO2	CH4
Wood Handling	●	●	●	●
Pulping		●	●	●
Chemical Recovery	●	●
Pulp Washing			●	●
Pulp Bleaching			●	●
Paper Making			●	●
Finished Paper Storage
Wastewater Treatment			●	●
Disposal of Solid Waste (Dry Sludge)						●
Whole Production Process (Electricity)					●

. Wood handling, pulping, and chemical recovery units emit SOx in the process, while NOx emissions are present only in pulping and chemical recovery.

Wastewater emissions can be found in all units except in packaging, while the wastewater treatment unit produces refined water (BOD and COD) and dry sludge. The dry sludge is then disposed of in landfills that produce methane, making air and water pollutants present in the pulp and paper mill industry.

2.4. EPQ in Shortage and Multiproduct Situations

In practice, a shortage happens when demand for a product exceeds available supply; costs from this type of situation will inevitably emerge as a shortage cost. The economic production quantity (EPQ) model is frequently used to determine the optimal production quantity that balances the costs of setting up a production run and the costs of holding inventory. The work of Cárdenas-Barrón [29] extended the traditional EPQ model to shortage situations and utilized an algebraic approach to modify the EPQ model. It covered the multiproduct EPQ model in a vendor-buyer system through a heuristic algorithm. Another study considering shortages by Wee et al. [30] included linear and fixed backorder costs using the optimization method. The optimal inventory policies were analyzed, as well as the schedule for shortages. Wang et al. [31] considered the lot-sizing and joint pricing problem in the inventory model, especially when the setup cost largely exceeds the order cost. They included all unit discount policies and a franchise fee to determine finite production rates in their optimization method. On considering multiproduct, Chiu et al. [32] determined the optimal standard production cycle policy for a multi-item EPQ model with the scenarios of scrap, rework, and multiple deliveries. Their model was obtained through mathematical modeling and analysis. Similarly, Nobil et al. [33] considered a multimachine and multiproduct EPQ problem for an imperfect manufacturing system. In their study, a hybrid genetic algorithm was used to identify the machines to purchase, the items to allocate in each machine, and determine the optimal cycle length. Overall, the EPQ models developed from previous studies served as a guide in creating the sustainable EPQ models presented in this paper that consider GHG and wastewater emission costs, shortages, and multiproduct costs in a real production plant. The effectiveness of the EPQ model was determined by comparing production quantity, cycle length, and total costs generated under the three scenarios.

2.5. Sustainable EPQ Models

Using the mathematical optimization method and modifying the equation from Taleizadeh et al. [17], the three (3) scenarios developed in this study include:

(1) Sustainable EPQ model with CO₂ emission costs from inventory holding and production; SO_X and NO_x emission costs from production; CH₄ emission costs from waste disposal; BOD and COD emission costs during production for the wastewater emission.

(2) Sustainable EPQ model using the exact emission costs in scenario 1, with a shortage.

(3) Sustainable EPQ model using the exact emission costs in scenario 1, with multi-product and shortage situations.

Unlike most relevant researchers who did not consider CO₂, SO_x, NO_x, CH₄, and wastewater emissions (BOD and COD) in their models, this study develops an EPQ model to determine and compare the costs and optimal production quantities of the enumerated scenarios. The study covered a backordering situation and a multiproduct sustainable EPQ model with warehouse capacity constraints.

3. Mathematical Modeling

3.1. Solution Method

Wood is the primary raw material in the pulp and paper industry. A variety of wood is available and used to produce paper. The manufacture of pulp and paper includes seven distinct processes: wood handling, pulping, chemical recovery, pulp washing, pulp bleaching, papermaking, and packaging. Wood handling refers to the conversion of raw wood materials into wood chips. Pulping focuses on producing pulp, a material that is pivotal in papermaking and can be created from an artificial mineral, cellulosic, or wood fibers.

There are various types of pulping processes. In this study, a chemical pulping process was considered where wood logs are chopped to form wood chips that are cooked with chemicals under high pressure. Cooking these wood chips removes the lignin and separates it into cellulose fibers. The fibers are then screened to remove impurities and prevent any discoloration or disintegration of the final product—the paper. Half of the used wood chips are dissolved in a black liquor substance and separated from the pulp during the screening process. The process is performed because the black liquor can be recycled and reused in the future by undergoing a chemical recovery process and because only the wood pulp is needed for the succeeding process.

After pulping comes papermaking, where the wood pulp is diluted a hundred times its original weight. The diluted pulp is then pressed and dried to produce paper sheets. The liquids generated during this process run through the machine and go through a treatment process so that they will become reusable. The last stage is packaging. There are different types of finished paper, and the process depends on the type of paper the manufacturer wants to produce. Packaging generally includes cutting and coating the paper sheets to convert them into paper of a specific grade or color. After packaging, the produced papers are packed in preparation for storage or delivery.

Figure 1 presents a general flowchart of a paper and pulp mill. The wood handling unit involves subprocesses called wood log storage, debarking, chipping, chip storage, and wood chip waste storage. In this unit, wood logs are received from suppliers, and excess logs are stored for future use. One of the notable activities in the process is the cooking or heating of wood chips, as black liquor is produced and screened to separate the pulp from the weak liquor. The weak liquor goes through the chemical recovery unit, where it will be evaporated, clarified, kilned, and causticized to produce white liquor. Furthermore, unnecessary effluents or strong liquor will go through steam generation to produce green liquor that will eventually be converted into white liquor. This white liquor can be reused for pulp manufacturing and pulp bleaching. The processes are thoroughly discussed since they became the blueprint for identifying the processes that emit the five pollution factors and eventually helped formulate the initial equation.

3.1.1. The Sustainable EPQ Model

The model was developed to derive the optimal quantity from production to minimize the total cost and the adverse effect on the environment. The parameters’ sensitivity to change is extensively tested using various test cases. Figure 2 illustrates the traditional EPQ model developed by Bajpai [34] under two stages: production and nonproduction.

His work assumed constant demand and production rates, assuming production is more prominent than demand in the production stage. The inventory then decreases in the nonproduction stage because of demand. Figure 2 shows the production and nonproduction stages as T1 and T2, respectively, and the total cycle length is T. The objective of this model is to optimize production quantity over time.

The main assumptions for the proposed sustainable EPQ models are: (1) a single product; (2) production and demand rates are known and constant; (3) constant periods are assumed; (4) the production process is continuous; (5) shortage is not allowed; (6) wastewater from all manufacturing processes is collected at the end of the production cycle and treated before disposal; (7) the dry sludge collected from the wastewater treatment is not used as biomass by the mill; and (8) the carbon emission of the plant is based on the amount of electricity it uses for producing one ton of paper. The notations used in the model include:

Parameters

D = Demand rate

P = Production rate

C1= Setup cost per cycle (USD/cycle)

C2= Inventory cost per ton of paper (USD/ton)

C3= Production cost per ton of paper (USD/ton)

C4= Carbon emission cost (USD/kg CO2)

C5= fine for NOx emission (USD/mg/lit NOx)

C6= fine for SOx emission (USD/mg/lit SOX)

C7= Water treatment cost per cycle (USD/cycle)

C8= BOD discharge cost (USD/kg BOD)

C9= COD discharge cost (USD/kg COD)

C10= Disposal cost of dry sludge per cycle (USD/cycle)

C11= Methane emission cost (USD/ton CH4)

NOx = NOx per ton of product (kg/ton)

SOx = SOx per ton of product (kg/ton)

UBOD = BOD per treated wastewater (kg/m3)

UCOD = COD per treated wastewater (kg/m3)

USLD = % dry sludge per treated wastewater (ton of dry sludge/m3 wastewater)

UCH4= CH4per m3of dry sludge (ton CH4/m3 dry sludge)

v = Required space per ton of product (m3/ton)

er = Average inventory energy consumption per cubic meter (kWh/m3)

ep = Average production energy consumption per ton of paper (kWh/ton)

ew = Total wastewater output per ton of paper (m3/ton)

Eg = Energy generation standard emission (tonCO2/kWh)

Dependent Variables

TC = Total cost function

Im = Maximum inventory level (tons)

T1= Production and Consumption Period (time unit)

C2e= Average inventory emission cost (USD/ton)

C4e= Production carbon emission cost (USD/ton)

C5e= NOx emission cost (USD/ton)

C6e= SOx emission cost (USD/ton)

C8e= BOD emission cost (USD/ton)

C9e= COD emission cost (USD/ton)

C11e= CH4emission cost (USD/ton)

CS = Setup cost (USD)

CHE = Inventory cost (USD)

CPE = Production cost (USD)

CWW = Wastewater cost (USD)

CWE = Solid waste cost (USD)

Decision Variables

Q = Optimum production size (Traditional)

Q* = Optimum production size (Sustainable)

T = Cycle length

The sustainable EPQ model considered carbon emissions from inventory holding, production, and waste disposal. It also included costs from SO_x and NO_x emissions. Wastewater emissions during production were also accounted for, where BOD and COD were used to measure the cost. The total cost for the proposed sustainable EPQ is:

$$TC\left(Q\right)=C_S+C_{HE}+C_{PE}+C_{WW}+C_{WE}$$

(1)

1. Setup Cost. Cost of preparing the equipment before the start of production based on the traditional EPQ model by Taft [35].

$$C_S=\frac{C_1}{T}=\frac{C_{1}D}{Q}$$

(2)

2. Inventory Cost (Holding Cost and Carbon Emission of Inventory). This includes holding costs from the traditional EPQ and carbon emission costs from storing final products [19]. To prevent pulp damage, heating, ventilation, and air conditioning (HVAC) systems were used. C_2e, in Equation (3), refers to the space needed per ton of paper (v), average warehouse energy consumption per cubic meter (e_r), energy generation standard emission (E_g), and cost of emitting carbon per ton (C₄).

$$C_{2e}=v{e}_r{E}_g{C}_4$$

(3)

In finding the average inventory level [35], the inventory cost function presented in Equation (7) contains the holding cost and carbon emission cost of the warehouse.

$$T_1=\frac{Q}{P}$$

(4)

$$I_m=\left(P-D\right)T_1=\left(P-D\right)\frac{Q}{P}$$

(5)

$$C_{HE}=\left(C_2+C_{2e}\right)\frac{I_m}{2}$$

(6)

$$C_{HE}=(C_2+C_{2e})\frac{Q(P-D)}{2P}$$

(7)

3. Production Costs (CO₂+ NO_x + SO_x). Carbon emission for production is based on the work of Daryanto and Wee [19]. In Equation (8), C_4e is defined by the average energy consumption of production per ton (e_p), energy generation standard emission (E_g), and cost of emitting carbon per ton (C₄).

$$C_{4e}=e_p{E}_g{C}_4$$

(8)

The cost for emitting NO_x is described by the amount of NO_x emitted per ton of paper (NO_x) produced and the emission cost per NO_x emission (C₅), thus, resulting in the following:

$$C_{5e}={NO}_x{C}_5$$

(9)

The SO_x emission cost function is constructed similarly to the NO_x function; therefore, we obtain the following:

$$C_{6e}={SO}_x{C}_6$$

(10)

The environmental ergonomic costs of SO_x and NO_x are based on the model by Zadjafar and Gholamian [2], where costs are calculated by capping or limiting the number of emissions. However, their equation was modified to fit the requirements of this study. The costs are all related to the quantity produced per cycle, equivalent to the demand rate. Therefore, the production cost function would become:

$$C_{PE}={(C_3+C}_{4e}+C_{5e}+C_{6e})\frac{Q}{T}$$

(11)

$$C_{PE}={(C_3+C}_{4e}+C_{5e}+C_{6e})D$$

(12)

4. Wastewater Cost (Treatment Cost, BOD, and COD). Adapted from the model of Zadjafar and Gholamian [2], the modified equation is determined by the fixed cost of treating the effluent (C₇) and the fine per discharge of the remaining BOD and COD present in the treated effluent, C₈, and C_9, respectively. The total wastewater cost is the total amount of wastewater per ton of paper (e_w), the treatment cost of BOD per treated wastewater (U_BOD), and the penalty for discharging BOD. Therefore, we obtain the following:

$$C_{8e}=e_w{U}_{BOD}{C}_8$$

(13)

The same pattern goes for the COD cost function,

$$C_{9e}=e_w{U}_{COD}{C}_9$$

(14)

The amount of discharged BOD and COD depends on the plant’s wastewater and how much is produced per cycle. Thus, C_8e and C_9e are dependent on D. Combining all the costs into Equation (15), the cost function for wastewater is:

$$C_{WW}=\frac{C_{7}D}{Q}+(C_{8e}+C_{9e})D$$

(15)

5. Solid Waste Disposal Cost (Disposal Cost and CH₄). From the model by Daryanto and Wee [19], the waste disposal cost is a function of fixed costs associated with the disposal itself and variable costs from solid waste emissions. The disposal cost of dry sludge is determined by the fixed cost of waste disposal from the plant to the landfill (C₁₀). The environmental cost is equal to the % of dry sludge per treated wastewater (U_SLD) multiplied by the amount of CH₄ per cubic meter of dry sludge (U_CH4), the variable cost.

$$C_{11e}={e_{w}U}_{SLD}{U}_{CH4}{C}_{11}$$

(16)

$$C_{WE}=\frac{C_{10}}{T}+C_{11e}D$$

(17)

$$C_{WE}=\frac{C_{10}D}{Q}+C_{11e}D$$

(18)

6. Total Cost. Substituting Equations (2), (7), (12), (15) and (18) to Equation (1), the total cost function per unit time is:

$$\begin{gathered} TC\left(Q\right)=\frac{C_{1}D}{Q}+(C_2+C_{2e})\frac{Q^2\left(P-D\right)}{2DP}+{(C_3+C}_{4e}+C_{5e}+C_{6e})D+\frac{C_{7}D}{Q} \\ +(C_{8e}+C_{9e})D+\frac{C_{10}D}{Q}+C_{11e}D \end{gathered}$$

(19)

Taking the first derivative of Equation (19) yields:

$$\frac{dTC}{dQ}=-\frac{C_{1}D}{Q^2}+\frac{\left(C_2+C_{2e}\right)\left(P-D\right)}{DP}-\frac{C_{7}D}{Q^2}-\frac{C_{10}D}{Q^2}$$

(20)

Proof of the convexity of the total cost is given in Appendix A.

Equating Equation (20) to zero at the minimum total cost, Q can be derived. Therefore, the optimal Q* resulting in the optimal solution is:

$$Q^{\ast }=\sqrt{\frac{2DP(C_1+C_7+C_{10})}{(C_2+C_{2e})(P-D)}}$$

(21)

To mathematically illustrate the theory, secondary data from the case studies of Zadjafar and Gholamian [2] and Daryanto and Wee [19] were gathered from the pulp and paper mill industry in Iran. Parameters C₄, C₇, C₁₀, C₁₁, U_SLD, U_CH4, v, e_r, e_p, and E_g are unavailable. Thus, the values were assumed and modified in this study. From the numerical example, fluting paper is chosen to be produced.

D	=	84,000	ton/year		C11	=	65	USD/ton
P	=	0	ton/year		NOx	=	1.12	kg/ton
C1	=	5000	USD/cycle		SOx	=	1.9	kg/ton
C2	=	2.5	USD/ton		UBOD	=	23.5	kg/m3
C3	=	275	USD/ton		UCOD	=	44	kg/m3
C4	=	65	USD/ton CO2		USLD	=	0.5
C5	=	5	USD/kg NOx		UCH4	=	0.24	ton CH4/m3
C6	=	5	USD/kg SOx		v	=	3.71	m3/ton
C7	=	2000	USD/cycle		er	=	1.5	kWh/m3
C8	=	0.02	USD/kg BOD		ep	=	1078	kWh/ton
C9	=	0.02	USD/kg COD		ew	=	3.49	m3/ton
C10	=	500	USD/cycle		Eg	=	0.005	ton CO2/kWh

The production quantity (Q) and the cycle length (T) are computed using the traditional EPQ model to evaluate the proposed model.

$$\begin{gathered} Q=\sqrt{\frac{2DP{C}_1}{C_2\left(P-D\right)}}=\sqrt{\frac{2\left(84000\right)\left(336000\right)\left(5000\right)}{2.5\left(336000-84000\right)}}=21,166.01 tonsT=\frac{Q}{D}=\frac{21166.01}{84000} \\ =0.252\hspace{0.17em}time\hspace{0.17em}unitTC\left(Q\right)=\frac{\left(5000\right)\left(84000\right)}{21166.01}+\left(2.5+0.1809\right)\frac{{21166.01}^2\left(336000-84000\right)}{2\left(84000\right)\left(336000\right)} \end{gathered}$$

$$+\left(275+35.035+5.6+9.5\right)\left(84000\right)+\frac{2000\left(84000\right)}{21166.01}+\left(1.6403+3.0712\right)\left(84000\right)$$

$$+\frac{500\left(84000\right)}{21166.01}+27.222\left(84000\right)=30,044,797.39\hspace{0.17em}USD$$

Calculating the dependent variables from Equations (3), (8), (9), (10), (13), (14) and (16) results in the following values:

C2e = (3.71)(1.5)(0.0005)(65) = 0.1809
C4e = (1078)(0.0005)(65) = 35.035
C5e = (1.12)(5) = 5.6
C6e = (1.9)(5) = 9.5
C8e = (3.49)(23.5)(0.02) = 1.6403
C9e = (3.49)(44)(0.02) = 3.0712
C11e = (3.49)(0.5)(0.24)(65) = 27.222

Therefore, from Equations (21) and (19), the computed values for Q* and TC are:

$$\begin{gathered} Q^{\ast }=\sqrt{\frac{2\left(84000\right)\left(336000\right)\left(5000+2000+500\right)}{\left(2.5+0.1809\right)\left(336000-84000\right)}}=25,033.26\hspace{0.17em}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s} \\ T=\frac{Q\ast }{D}=\frac{25033.26}{84000}=0.298\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\hspace{0.17em}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t} \end{gathered}$$

$$TC\left(Q^{\ast }\right)=\frac{\left(5000\right)\left(84000\right)}{21631.21}+\left(2.5+0.1809\right)\frac{{21631.21}^2\left(336000-84000\right)}{2\left(84000\right)\left(336000\right)}$$

$$+\left(275+35.035+5.6+9.5\right)\left(84000\right)+\frac{2000\left(84000\right)}{21631.21}+\left(1.6403+3.0712\right)\left(84000\right)$$

$$+\frac{500\left(84000\right)}{21631.21}+27.222\left(84000\right)=\mathrm{30,044,087.72} \mathrm{U}\mathrm{S}\mathrm{D}$$

3.1.2. Sensitivity Analysis for the Sustainable EPQ Model

The sensitivity analyses are performed to test the validity of the results from the proposed sustainable EPQ models. The following parameters are considered for all the models: −20%, −10%, +10%, and +20%. The optimal Q* and TC(Q*) are computed.

Table 3. Sensitivity analysis of Q*, T, and TC(Q*).

Parameter	Results	−20%	−10%	Base Value	+10%	+20%
D	Q*	21,679.44	23,362.45	25,033.26	26,703.94	28,385.05
	T	0.322611	0.309027	0.298015	0.289004	0.281598
	TC(Q*)	24,041,498.87	27,042,918.03	30,044,087.04	33,045,031.85	36,045,772.28
P	Q*	26,146.38	25,510.12	25,033.26	24,662.37	24,365.58
	T	0.311266	0.303692	0.298015	0.293600	0.290066
	TC(Q*)	30,041,944.22	30,043,146.15	30,044,087.04	30,044,843.97	30,045,466.28
C1	Q*	23,304.69	24,184.42	25,033.26	25,854.24	26,649.94
	T	0.277437	0.287910	0.298015	0.307789	0.317261
	TC(Q*)	30,040,611.51	30,042,380.34	30,044,087.04	30,045,737.74	30,047,337.61
C2	Q*	27,754.95	26,289.02	25,033.26	23,941.81	22,981.70
	T	0.330416	0.312965	0.298015	0.285022	0.273592
	TC(Q*)	30,039,151.30	30,041,682.75	30,044,087.04	30,046,381.61	30,048,580.23
C4	Q*	25,203.87	25,118.13	25,033.26	24,949.24	24,866.06
	T	0.300046	0.299025	0.298015	0.297015	0.296025
	TC(Q*)	29,455,158.32	29,749,622.97	30,044,087.04	30,338,550.54	30,633,013.47
C7	Q*	24,356.56	24,697.23	25,033.26	25,364.84	25,692.14
	T	0.289959	0.294015	0.298015	0.301962	0.305859
	TC(Q*)	30,042,726.44	30,043,411.40	30,044,087.04	30,044,753.73	30,045,411.82
C10	Q*	24,865.81	24,949.67	25,033.26	25,116.56	25,199.59
	T	0.296022	0.297020	0.298015	0.299007	0.299995
	TC(Q*)	30,043,750.36	30,043,918.98	30,044,087.04	30,044,254.54	30,044,421.48
v	Q*	25,203.87	25,118.13	25,033.26	24,949.24	24,866.06
	T	0.300046	0.299025	0.298015	0.297015	0.296025
	TC(Q*)	30,043,746.32	30,043,916.97	30,044,087.04	30,044,256.54	30,044,425.47
er	Q*	25,203.87	25,118.13	25,033.26	24,949.24	24,866.06
	T	0.300046	0.299025	0.298015	0.297015	0.296025
	TC(Q*)	30,043,746.32	30,043,916.97	30,044,087.04	30,044,256.54	30,044,425.47
Eg	Q*	25,203.87	25,118.13	25,033.26	24,949.24	24,866.06
	T	0.300046	0.299025	0.298015	0.297015	0.296025
	TC(Q*)	29,455,158.32	29,749,622.97	30,044,087.04	30,338,550.54	30,633,013.47

shows the sensitivity analysis results, while

Table 4. Percent Error for Model 1.

		% ERROR
		−20%	−10%	Base Value	+10%	+20%
D	Q*	−13.40%	−6.67%	0.00%	6.67%	13.39%
	T	8.25%	3.70%	0.00%	−3.02%	−5.51%
	TC(Q*)	−19.98%	−9.99%	0.00%	9.99%	19.98%
P	Q*	4.45%	1.90%	0.00%	−1.48%	−2.67%
	T	4.45%	1.90%	0.00%	−1.48%	−2.67%
	TC(Q*)	−0.01%	0.00%	0.00%	0.00%	0.00%
C1	Q*	−6.91%	−3.39%	0.00%	3.28%	6.46%
	T	−6.91%	−3.39%	0.00%	3.28%	6.46%
	TC(Q*)	−0.01%	−0.01%	0.00%	0.01%	0.01%
C2	Q*	10.87%	5.02%	0.00%	−4.36%	−8.20%
	T	10.87%	5.02%	0.00%	−4.36%	−8.20%
	TC(Q*)	−0.02%	−0.01%	0.00%	0.01%	0.01%
C4	Q*	0.68%	0.34%	0.00%	−0.34%	−0.67%
	T	0.68%	0.34%	0.00%	−0.34%	−0.67%
	TC(Q*)	−1.96%	−0.98%	0.00%	0.98%	1.96%
C7	Q*	−2.70%	−1.34%	0.00%	1.32%	2.63%
	T	−2.70%	−1.34%	0.00%	1.32%	2.63%
	TC(Q*)	0.00%	0.00%	0.00%	0.00%	0.00%
C10	Q*	−0.67%	−0.33%	0.00%	0.33%	0.66%
	T	−0.67%	−0.33%	0.00%	0.33%	0.66%
	TC(Q*)	0.00%	0.00%	0.00%	0.00%	0.00%
v	Q*	0.68%	0.34%	0.00%	−0.34%	−0.67%
	T	0.68%	0.34%	0.00%	−0.34%	−0.67%
	TC(Q*)	0.00%	0.00%	0.00%	0.00%	0.00%
er	Q*	0.68%	0.34%	0.00%	−0.34%	−0.67%
	T	0.68%	0.34%	0.00%	−0.34%	−0.67%
	TC(Q*)	0.00%	0.00%	0.00%	0.00%	0.00%
Eg	Q*	0.68%	0.34%	0.00%	−0.34%	−0.67%
	T	0.68%	0.34%	0.00%	−0.34%	−0.67%
	TC(Q*)	−1.96%	−0.98%	0.00%	0.98%	1.96%

shows the percent error of each parameter from the changes. It can be observed that Q* is highly sensitive to the change in parameters of D, C₂, and C₁, moderately sensitive to P and C₇, and insensitive to C_4, C₁₀, e_r, v, and E_g. TC(Q*), on the other hand, is highly sensitive to D, moderately sensitive to C₄ and E_g, and insensitive to P, C_1, C_2, C_7, C_10, e_r, and v. Finally, T is highly sensitive to the change in D, C_1, and C₂, moderately sensitive to P and C₇, and insensitive to C₄, C₁₀, v, e_r, and E_g.

3.1.3. Sustainable EPQ Model with Shortage

The second model contains the same elements as the previous EPQ model but with an additional shortage backorder scenario.

The traditional EPQ model, shown in Figure 3, extends to a shortage situation and includes additional variables: the time cycles (T), namely T₃ and T₄. The objective is to compute the optimal production time with a shortage (Q) and the optimal cycle length (T). This model has the same assumptions as the previous one, except that it allows for a shortage to illustrate a real-life scenario. Additional notations include:

Parameter

C12= Backordering cost per ton (USD/ton)

Dependent Variables

Ib = Maximum backorder level (ton)

T3 and T4 = Shortage period (time unit)

CB = Backorder cost

Decision Variable

T2 = Consumption Period (time unit)

As seen in Figure 3, the system allows full backordering with two periods; the shortage during downtime production, denoted by T₃, and the shortage during uptime production, denoted by T₄. Therefore, the production cycle is computed as follows:

$$T=T_1+T_2+T_3+T_4$$

(22)

Since periods T₁ and T₃ sum the production period of one cycle, then Q is expressed as:

$$Q=P(T_1+T_4)$$

(23)

The periods T₁, T₃, and T₄ are defined by T and T_2. Period 1 denotes that the production starts at zero inventory level. Inventory increases based on demand and production, which stops until it reaches the maximum inventory level. Mathematically,

$$T_1=\frac{I_m}{P-D}$$

(24)

However, I_m can be solved as:

$$I_m=T_{2}D$$

(25)

Substituting Equation (25) into Equation (24), one has

$$T_1=\frac{T_{2}D}{P-D}$$

(26)

During period T₃, all inventories are consumed, and production is still down while backorders increase. As production starts in the fourth period, T₄ accumulates backorders and demands that the period decreases the inventory levels. Therefore, we obtain the following:

$$P{T}_4=D(T_3+T_4)$$

(27)

Solving T₄ by transferring P to the right-hand side of Equation (27) results in:

$$T_4=\frac{D}{P}(T_3+T_4)$$

(28)

Expressing T₄ and T₃ in terms of T and T₂, and substituting Equations (22) and (24) into Equation (28), one has:

$$T_4=\frac{D}{P}\left(T-T_1-T_2\right)=\frac{D}{P}\left(T-\frac{T_{2}D}{P-D}-T_2\right)$$

(29)

Solving T₃ using the same method as finding T₄ results in the following:

$$T_3=\frac{P-D}{P}(T-\frac{D{T}_2}{P-D}-T_2)$$

(30)

Simplify T₃ by expansion,

$$\begin{gathered} T_3=\frac{P-D}{P}\left(T-\frac{D{T}_2}{P-D}-T_2\right)=\frac{P-D}{P}T-\frac{P-D}{P}\ast \frac{D{T}_2}{P-D}-\frac{P-D}{P}{T}_2 \\ =\frac{P-D}{P}T-\frac{D{T}_2}{P}-\frac{P-D}{P}{T}_2=\frac{PT-DT-D{T}_2-P{T}_2+D{T}_2}{P} \\ =\frac{PT-DT-P{T}_2}{P}=\frac{T\left(P-D\right)-P{T}_2}{P}=\frac{T\left(P-D\right)}{P}-T_2 \end{gathered}$$

(31)

The total cost with C_B as the backordering cost is:

$$TC\left(T,T_2\right)=C_S+C_{HE}+C_{PE}+C_{WW}+C_{WE}+C_B$$

(32)

The inventory cost is derived as follows:

The area of each triangle computes the inventory level of each period. It is then multiplied by the variable costs C₂ and C_2e as follows:

$$C_{HE}=\frac{\left(C_2{+C}_{2e}\right)}{T}\left(\frac{I_m{T}_1}{2}+\frac{I_m{T}_2}{2}\right)$$

(33)

Substitute Equations (25) and (26) and simplifying C_HE, one has:

$$C_{HE}=\frac{\left(C_2{+C}_{2e}\right)}{T}\left(\frac{(P-D){(T_1)}^2}{2}+\frac{D{(T_2)}^2}{2}\right)=\frac{\left(C_2{+C}_{2e}\right)D{(T_2)}^2}{2T}\left(1+\frac{D}{P-D}\right)$$

(34)

The backorder cost is derived as follows:

Shortages occur during periods T₃ and T₄, thus, the area under these periods is the backorder cost. The backorder cost function is formulated as follows:

$$C_B=\frac{C_{12}}{T}\left(\frac{I_b{T}_3}{2}+\frac{I_b{T}_4}{2}\right)$$

(35)

Substitute Equations (29) and (30) to (35), C_B is

$$C_B=\left(\frac{D}{2}{\left(\frac{P-D}{P}\left(T-\frac{D{T}_2}{P-D}-T_2\right)\right)}^2+\frac{P-D}{2}{\left(\frac{D}{P}\left(T-\frac{D{T}_2}{P-D}-T_2\right)\right)}^2\right)$$

(36)

Substituting, in terms of T and T_2, using the expressions from the first model and those of Equations (2), (12), (15) and (18), into (32), one has:

$$\begin{gathered} TC(T,T_2)=\frac{C_1}{T}+\frac{1}{2}\frac{\left(C_2+C_{2e}\right)D{T_2}^2}{T}\left(1+\frac{D}{P-D}\right)+\left(C_3+C_{4e}+C_{5e}+C_{6e}\right)D+\frac{C_7}{T} \\ +\left(C_{8e}+C_{9e}\right)D+\frac{C_{10}}{T}+C_{11e}D \\ +\frac{C_{12}}{T}\left(\frac{D}{2}{\left(\frac{P-D}{P}\left(T-\frac{D{T}_2}{P-D}-T_2\right)\right)}^2+\frac{P-D}{2}{\left(\frac{D}{P}\left(T-\frac{D{T}_2}{P-D}-T_2\right)\right)}^2\right) \end{gathered}$$

(37)

To solve for T and T₂, take the first derivative using Maple15, which results in the following:

$$\frac{\partial TC(T,T_2)}{\partial T_2}=\frac{\left(T_{2}P\left(C_2+C_{2e}\right)+C_{12}D(TD-TP+T_{2}P\right))}{T(P-D)}$$

(38)

$$\frac{\partial TC(T,T_2)}{\partial T}=\frac{\left(2C_1\left(PD-P^2\right)-D{T_2}^2{P}^2{(C}_{2e}+C_2)+2C_{10}(PD-P^2\right)+C_{12}D(T^2{P}^2+T^2{D}^2-2D{T}^{2}P-T^2{P}^2)}{2P{T}^2(P-D)}$$

(39)

By taking the second derivative, the convexity property of the total cost function can be proven.

$$\frac{{\partial }^{2}TC}{\partial {T_2}^2}=\frac{\left(P{C}_2+P{C}_{2e}+C_{12}P\right)D}{T(P-D)}$$

(40)

$$\frac{{\partial }^{2}TC}{\partial T^2}=\frac{(2C_1\left(P-D\right)-D{T_2}^{2}P\left(C_2+C_{2e}\right)+2C_7\left(P-D\right){+2C}_{10}\left(P-D\right)+C_{12}D{T}^{2}P)}{T^3\left(P-D\right)}$$

(41)

Taking the derivative of Equation (37) concerning T and T₂ results in the following equation:

$$\frac{\partial TC}{\partial T\partial T_2}=\frac{{(C}_2+C_{2e}+C_{12})T_{2}PD}{T^2(-P+D)}$$

(42)

Proof for convexity of the total cost is given in Appendix B.

By equating Equations (38) and (39) to zero, T₂ and T can be derived.

$$T_2=\frac{C_{12}T(P-D)}{P(C_2+C_{2e}+C_{12})}$$

(43)

$$T=\frac{\sqrt{DP(2C_1\left(P-D\right)+D{T}^{2}P\left(C_2+C_{2e}+C_{12}\right)+C_7\left(P-D\right)+C_{10}(P-D)}}{D(P-D)}$$

(44)

Substituting Equation (44) into (43), we have T₂ as follows:

$$T_2=\frac{\sqrt{2DP{C}_{12}(P-D)(C_2+C_{2e})(C_2+C_{2e}+C_{12})\left(C_{10}+C_7+C_1\right)}}{DP(C_2+C_{2e})(C_2+C_{2e}+C_{12})}$$

(45)

Substituting Equation (45) into (44), we have:

$$T=\frac{1}{C_{12}D(P-D)}\ast \sqrt{\frac{2C_{12}DP(P-D)(C_2+C_{2e}+C_{12})(C_{10}+C_7+C_1)}{C_2+C_{2e}}}$$

(46)

Substituting Equation (45) into (26), T₁ becomes:

$$T_1=\frac{\sqrt{2PD{C}_{12}(P-D)(C_2+C_{2e})(C_2+C_{2e}+C_{12})\left(C_{10}+C_7+C_1\right)}}{P(P-D)(C_2+C_{2e})(C_2+C_{2e}+C_{12})}$$

(47)

Substituting Equations (45) and (46) into (29) yields:

$$T_4=\sqrt{\frac{2D(C_{10}+C_7+C_1)(C_2+C_{2e})}{C_{12}P(P-D)(C_2+C_{2e}+C_{12})}}$$

(48)

To derive Q, substitute Equations (47) and (48) into (23).

$$Q=\sqrt{\frac{2D\left(C_{10}+C_7+C_1\right)}{\left(C_2+C_{2e}\right)\left(1-\frac{D}{P}\right)}}\ast \sqrt{\frac{C_2+C_{2e}+C_{12}}{C_{12}}}$$

(49)

The same data and assumptions are used to test the model numerically. Calculating the dependent variables from (3), (8), (9), (10), (13), (14) and (16) results in the following values:

C2e = (3.71)(1.5)(0.0005)(65) = 0.1809
C4e = (1078)(0.0005)(65) = 35.035
C5e = (1.12)(5) = 5.6
C6e = (1.9)(5) = 9.5
C8e = (3.49)(23.5)(0.02) = 1.6403
C9e = (3.49)(44)(0.02) = 3.0712
C11e = (3.49)(0.5)(0.24)(65) = 27.222

Therefore, Q is:

$$Q=\sqrt{\frac{2D\left(C_{10}+C_7+C_1\right)}{\left(C_2+C_{2e}\right)\left(1-\frac{D}{P}\right)}}\ast \sqrt{\frac{C_2+C_{2e}+C_{12}}{C_{12}}}$$

$$Q=\sqrt{\frac{2\ast 84000\ast \left(500+2000+5000\right)}{\left(2.5+0.1809\right)\left(1-\frac{84000}{336000}\right)}}\ast \sqrt{\frac{2.5+0.1809+50}{50}}=25,695.60\hspace{0.17em}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}$$

$$\begin{gathered} T=\frac{1}{50\ast 84000\left(252000\right)}\ast \sqrt{\frac{2\ast 50\ast 84000\ast 336000\left(252000\right)\left(2.5+0.1809+50\right)\left(7500\right)}{2.5+0.1809}} \\ =0.306\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\hspace{0.17em}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t} \end{gathered}$$

$$T_2=\frac{\left(50\right)\left(0.3590003\right)\left(336000-84000\right)}{336000\left(2.5+0.1809+50\right)}=0.218\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\hspace{0.17em}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}$$

$$\begin{gathered} TC\left(T,T_2\right)=16345.20925+23270.27205+27311340+402304.08+2288282.52+23270.27205 \\ =30,042,789.62\hspace{0.17em}\mathrm{U}\mathrm{S}\mathrm{D} \end{gathered}$$

3.1.4. Sensitivity Analysis for Sustainable EPQ Model with Shortage

By varying the parameters, Q*, T, T₂, and TC (T, T₂) are computed as follows:

It can be observed from

Table 5. Sensitivity analysis of Q*, T, $T_2$, and TC (T, T₂).

Parameter	Results	−20%	−10%	Base Value	+10%	+20%
D	Q*	22,253.04	23,980.59	25,695.60	27,410.49	29,136.07
	T	0.33	0.32	0.31	0.30	0.29
	T2	0.25	0.23	0.22	0.20	0.19
	TC(T,T2)	24,040,300.38	27,041,666.84	30,042,789.62	33,043,693.98	36,044,399.23
P	Q*	26,838.18	26,185.09	25,695.60	25,314.91	25,010.26
	T	0.32	0.31	0.31	0.30	0.30
	T2	0.21	0.21	0.22	0.22	0.22
	TC(T,T2)	30,040,702.04	30,041,873.00	30,042,789.62	30,043,527.04	30,044,133.31
C1	Q*	23,921.30	24,824.31	25,695.60	26,538.30	27,355.06
	T	0.28	0.30	0.31	0.32	0.33
	T2	0.20	0.21	0.22	0.22	0.23
	TC(T,T2)	30,039,403.69	30,041,126.91	30,042,789.62	30,044,397.78	30,045,956.42
C2	Q*	28,353.79	26,920.49	25,695.60	24,633.52	23,701.45
	T	0.34	0.32	0.31	0.29	0.28
	T2	0.24	0.23	0.22	0.21	0.20
	TC(T,T2)	30,038,192.50	30,040,558.50	30,042,789.62	30,044,903.82	30,046,915.31
C4	Q*	25,861.85	25,778.29	25,695.60	25,613.76	25,532.75
	T	0.31	0.31	0.31	0.30	0.30
	T2	0.22	0.22	0.22	0.22	0.22
	TC(T,T2)	29,453,886.42	29,748,338.33	30,042,789.62	30,337,240.32	30,631,690.40
C7	Q*	25,001.00	25,350.68	25,695.60	26,035.96	26,371.92
	T	0.30	0.30	0.31	0.31	0.31
	T2	0.21	0.21	0.22	0.22	0.22
	TC(T,T2)	30,041,768.40	30,042,439.98	30,042,789.62	30,043,756.04	30,044,401.26
C10	Q*	25,523.72	25,609.81	25,695.60	25,781.11	25,866.34
	T	0.30	0.30	0.31	0.30	0.31
	T2	0.21	0.22	0.22	0.22	0.22
	TC(T,T2)	30,042,772.30	30,042,937.62	30,042,789.62	30,043,266.62	30,043,430.30
v	Q*	25,861.85	25,778.29	25,695.60	25,613.76	25,532.75
	T	0.31	0.31	0.31	0.30	0.30
	T2	0.22	0.22	0.22	0.22	0.22
	TC(T,T2)	30,042,474.42	30,042,632.33	30,042,789.62	30,042,946.32	30,043,102.40
er	Q*	25,861.85	25,778.29	25,695.60	25,613.76	25,532.75
	T	0.31	0.31	0.31	0.30	0.30
	T2	0.22	0.22	0.22	0.22	0.22
	TC(T,T2)	30,042,474.42	30,042,632.33	30,042,789.62	30,042,946.32	30,043,102.40
Eg	Q*	25,861.85	25,778.29	25,695.60	25,613.76	25,532.75
	T	0.31	0.31	0.31	0.30	0.30
	T2	0.219309249	0.218525644	0.217749874	0.216981809	0.216221321
	TC(T,T2)	29453886.42	29748338.33	30042789.62	30337240.32	30631690.4
C12	Q*	25,858.54	25,768.15	25,695.60	25,636.10	25,586.40
	T	0.31	0.30	0.31	0.30	0.30
	T2	0.21	0.22	0.22	0.22	0.22
	TC(T,T2)	30,042,787.51	30,042,961.70	30,042,789.62	30,043,218.42	30,043,315.72

that Q* is highly sensitive to the change in parameters D, C₂, and C₁, moderately sensitive to P and C₇, and insensitive to C₄, C_10, v, e_r, E_g, and C₁₂. On the other hand, T is highly sensitive to D, C₁, and C₂, moderately sensitive to P, C_7, and C₁₀, and insensitive to C₄, v, e_r, E_g, and C₁₂. Moreover, T₂ is highly sensitive to D, C_2, and C₁, moderately sensitive to P, C_7, C_10, and C₁₂, and insensitive to C₄, v, e_r, and E_g. Finally, TC (T, T₂) is only moderately sensitive to E_g.

Table 6. Percent Error for Model 2.

		% ERROR
		−20%	−10%	Base Value	+10%	+20%
D	t	−13.40%	−6.67%	0.00%	6.67%	13.39%
	T	8.25%	3.70%	0.00%	−3.02%	−5.51%
	T2	15.47%	7.15%	0.00%	−6.26%	−11.81%
	TC(T,T2)	−19.98%	−9.99%	0.00%	9.99%	19.98%
P	Q*	4.45%	1.90%	0.00%	−1.48%	−2.67%
	T	4.45%	1.90%	0.00%	−1.48%	−2.67%
	T2	−4.26%	−1.87%	0.00%	1.50%	2.74%
	TC(T,T2)	−0.01%	0.00%	0.00%	0.00%	0.00%
C1	Q*	−6.91%	−3.39%	0.00%	3.28%	6.46%
	T	−6.91%	−3.39%	0.00%	3.28%	6.46%
	T2	−6.91%	−3.39%	0.00%	3.28%	6.46%
	TC(T,T2)	−0.01%	−0.01%	0.00%	0.01%	0.01%
C2	Q*	10.34%	4.77%	0.00%	−4.13%	−7.76%
	T	10.34%	4.77%	0.00%	−4.13%	−7.76%
	T2	11.40%	5.27%	0.00%	−4.59%	−8.63%
	TC(T,T2)	−0.02%	−0.01%	0.00%	0.01%	0.01%
C4	Q*	0.65%	0.32%	0.00%	−0.32%	−0.63%
	T	0.65%	0.32%	0.00%	−0.32%	−0.63%
	T2	0.72%	0.36%	0.00%	−0.35%	−0.70%
	TC(T,T2)	−1.96%	−0.98%	0.00%	0.98%	1.96%
C7	Q*	−2.70%	−1.34%	0.00%	1.32%	2.63%
	T	−3.32%	−1.97%	0.00%	0.68%	1.98%
	T2	−3.39%	−2.03%	0.00%	0.61%	1.91%
	TC(T,T2)	0.00%	0.00%	0.00%	0.00%	0.01%
C10	Q*	−0.67%	−0.33%	0.00%	0.33%	0.66%
	T	−1.30%	−0.97%	0.00%	−0.30%	0.03%
	T2	−1.37%	−1.03%	0.00%	−0.37%	−0.04%
	TC(T,T2)	0.00%	0.00%	0.00%	0.00%	0.00%
v	Q*	0.65%	0.32%	0.00%	−0.32%	−0.63%
	T	0.65%	0.32%	0.00%	−0.32%	−0.63%
	T2	0.72%	0.36%	0.00%	−0.35%	−0.70%
	TC(T,T2)	0.00%	0.00%	0.00%	0.00%	0.00%
er	Q*	0.65%	0.32%	0.00%	−0.32%	−0.63%
	T	0.65%	0.32%	0.00%	−0.32%	−0.63%
	T2	0.72%	0.36%	0.00%	−0.35%	−0.70%
	TC(T,T2)	0.00%	0.00%	0.00%	0.00%	0.00%
Eg	Q*	0.65%	0.32%	0.00%	−0.32%	−0.63%
	T	0.65%	0.32%	0.00%	−0.32%	−0.63%
	T2	0.72%	0.36%	0.00%	−0.35%	−0.70%
	TC(T,T2)	−1.96%	−0.98%	0.00%	0.98%	1.96%
C12	Q*	0.63%	0.28%	0.00%	−0.23%	−0.42%
	T	0.00%	−0.35%	0.00%	−0.87%	−1.06%
	T2	−1.34%	−0.99%	0.00%	−0.47%	−0.27%
	TC(T,T2)	0.00%	0.00%	0.00%	0.00%	0.00%

shows the percent error for Model 2.

3.1.5. Multiproduct Sustainable EPQ Model with Shortage

The multiproduct sustainable EPQ model considers warehouse capacity constraints. The goal is to find the optimal quantity to produce in a cycle with multiple products while minimizing total cost and environmental problems.

Figure 4 shows a multiproduct EPQ model considering shortages and parameter variations. During T_1i and T_4i, the inventory level increases due to production. T_2i and T_3i are the periods where production is down and inventory for each item has decreased to shortage level I_bi.

The assumptions used in this model are: (1) n-type of products; (2) there is a production period during which all products can be produced; (3) a single machine is used for all n-type of products; (4) a single and continuous production line; (5) all parameters are deterministic; (6) shortages are allowed; (7) all products have perfect quality; (8) there is a warehouse capacity constraint; and (9) there is no cost for holding waste that is to be disposed of. Additional notations used in the model include:

Parameters i = Product type index (i = a, b, …)

Di = Demand rate for item i (ton/year)

Pi = Production rate for item i (ton/year)

C2i=Inventory cost per item i (USD/ton)

C3i= Production cost per item i (USD/ton)

C4i= Carbon emission cost (USD/kg CO2)

C5i= fine for NOx emission (USD/mg/lit NOx)

C6i= fine for SOx emission (USD/mg/lit SOX)

C7i= Water treatment cost per item i (USD/cycle)

C8i= BOD discharge cost (USD/kg BOD)

C9i= COD discharge cost (USD/kg COD)

C10i= Disposal cost of dry sludge per item i (USD/cycle)

C11i= Methane emission cost (USD/ton CH4)

C12i= Backordering cost per ton (USD/ton)

NOxi = Amount of NOx per item i (kg/ton)

SOxi = Amount of SOx per item i (kg/ton)

UBODi = Amount of BOD per treated wastewater per item i (kg/m3)

UCODi = Amount of COD per treated wastewater per item i (kg/m3)

USLDi = Percentage of dry sludge per treated wastewater per item i (ton of dry sludge/m3 wastewater)

UCH4i= Amount of CH4per m3of dry sludge per item i (ton CH4/m3 dry sludge)

vi = required space per ton of product per item i (m3/ton)

epi = Average production energy consumption per item i (kWh/ton)

ewi = Total wastewater output per item i (m3/ton)

F = Total space available in the warehouse (m3)

Dependent Variables

Imi = Maximum inventory level for item i (ton) Ibi = Maximum backorder level of item i (ton) Ti = Length of one cycle (time unit)

T1iand T2i= Production and Consumption Period (time unit)

T3iand T4i= Shortage period (time unit)

C2ei= average inventory emission cost (USD/ton)

C4ei= production carbon emission cost (USD/ton)

C5ei= NOx emission cost (USD/ton)

C6ei= SOx emission cost (USD/ton)

C8ei= BOD emission cost (USD/ton)

C9ei= COD emission cost (USD/ton)

C11ei= CH4emission cost (USD/ton)

CSi = Setup cost

CHEi = Inventory cost

CPEi = Production cost

CWWi = Wastewater cost

CWEi = Solid waste cost

CBi = Backorder cost

Decision Variables

Qi = Optimum production size (Traditional)

Qi* = Optimum production size (Sustainable)

T2i = Consumption period (time unit) λ = Lagrange multiplier

The production period for each cycle is the sum of all the production times allotted to each type of product; this is the same explanation for periods T_2i, T_3i, and T_4i. Following Pasandideh et al. [27], each period per product was defined as:

$$T_{1i}=\frac{I_{mi}}{P_i-D_i}=\frac{T_{2i}{D}_i}{P_i-D_i}Q=\sqrt{\frac{2D\left(C_{10}+C_7+C_1\right)}{\left(C_2+C_{2e}\right)\left(1-\frac{D}{P}\right)}}\ast \sqrt{\frac{C_2+C_{2e}+C_{12}}{C_{12}}}$$

(50)

$$T_{4i}=\frac{D_i}{P_i}\left(T_{3i}+T_{4i}\right)=\frac{D_i}{P_i}\left(T_i-T_{1i}-T_{2i}\right)=\frac{D_i}{P_i}\left(T_i-\frac{T_{2i}{D}_i}{P_i-D_i}-T_{2i}\right)$$

(51)

$$T_{3i}=\frac{P_i-D_i}{P_i}\left(T_i-\frac{D_i{T}_{2i}}{P_i-D_i}-T_{2i}\right)=\frac{T_i\left(P_i-D_i\right)}{P_i}-T_{2i}$$

(52)

$$Q=P(T_1+T_4)$$

(53)

The multiproduct EPQ model has the same assumptions. An additional expression, ∑, is used to represent all the products. Therefore, the total cost function is derived as follows:

$$\begin{array}{ll}TC\left(T_i,T_{2i}\right)=& {\displaystyle \sum _{i=1}^n}\frac{C_{1i}}{T_i}+{\displaystyle \sum _{i=1}^n}\frac{1}{2}\frac{\left(C_{2i}+C_{2ei}\right)D_i{T_{2i}}^2}{T_i}\left(1+\frac{D_i}{P_i-D_i}\right)\\ & +{\displaystyle \sum _{i=1}^n}\left(C_{3i}+C_{4ei}+C_{5ei}+C_{6ei}\right)D_i+{\displaystyle \sum _{i=1}^n}\left(\frac{C_{7i}}{T_i}+\left(C_{8ei}+C_{9ei}\right)D_i\right)\\ & +{\displaystyle \sum _{i=1}^n}\left(\frac{C_{10i}}{T_i}+C_{11ei}{D}_i\right)\\ & +{\displaystyle \sum _{i=1}^n}\frac{C_{12i}}{T_i}(\frac{D_i}{2}{\left(\frac{P_i-D_i}{P_i}\left(T_i-\frac{D_i{T}_{2i}}{P_i-D_i}-T_{2i}\right)\right)}^2\\ & +\frac{P_i-D_i}{2}{\left(\frac{D_i}{P_i}\left(T_i-\frac{D_i{T}_{2i}}{P_i-D_i}-T_{2i}\right)\right)}^2)\end{array}$$

(54)

Many researchers compute this problem by solving T and T_2i. However, in real situations, resources such as capacity or space may be limited. In Figure 4 the warehouse space is affected by the production and consumption rate in period T_1i. The maximum production should be less than or equal to the total available space (F). Therefore, the expression would be:

$$v_i{P}_i{T}_{1i}\le F$$

(55)

Substituting Equation (50) into (55) results in Equation (56), with the constraint F_.

$$v_i{P}_i\frac{T_{2i}{D}_i}{(P_i-D_i)}\le F$$

(56)

The constraint in Equation (56) is defined by the function g (T_i, T_2i).

$$g\left(T_i,T_{2i}\right)\to v_i{P}_i\frac{T_{2i}{D}_i}{\left(P_i-D_i\right)}\le F$$

(57)

Since the right-hand side F value is greater than the left-hand side value, the constraint is the active Lagrangian function; L, is used [28] to solve this. The Lagrange multiplier method minimizes the exact values of variables T_i and T_2i. The objective function, TC, can be solved.

Transforming the objective function from Equation (54) with the Lagrange multiplier, λ, the function in Equation (58) becomes L (T_i, T_2i, λ).

$$L\left(T_i,T_{2i},\lambda \right)=TC\left(T_i,T_{2i}\right)+\lambda \left[F-g(T_i,T_{2i})\right]$$

(58)

$$\begin{array}{ll}L\left(T_i,T_{2i},\lambda \right)=& {\displaystyle {\displaystyle \sum _{i=1}^n}}\frac{C_{1i}}{T_i}+{\displaystyle \sum _{i=1}^n}\frac{1}{2}\frac{\left(C_{2i}+C_{2ei}\right)D_i{T_{2i}}^2}{T_i}\left(1+\frac{D_i}{P_i-D_i}\right)+{\displaystyle \sum _{i=1}^n}\left(C_{3i}+C_{4ei}+C_{5ei}+C_{6ei}\right)D_i\\ & +{\displaystyle \sum _{i=1}^n}\left(\frac{C_{7i}}{T_i}+\left(C_{8ei}+C_{9ei}\right)D_i\right)+{\displaystyle \sum _{i=1}^n}\left(\frac{C_{10i}}{T_i}+C_{11ei}{D}_i\right)\\ & +{\displaystyle \sum _{i=1}^n}\frac{C_{12i}}{T_i}\left(\frac{D_i}{2}{\left(\frac{P_i-D_i}{P_i}\left(T_i-\frac{D_i{T}_{2i}}{P_i-D_i}-T_{2i}\right)\right)}^2+\frac{P_i-D_i}{2}{\left(\frac{D_i}{P_i}\left(T_i-\frac{D_i{T}_{2i}}{P_i-D_i}-T_{2i}\right)\right)}^2\right)\\ & +\lambda {\displaystyle \sum _{i=1}^n}\left(F-v_i{P}_i\frac{T_{2i}{D}_i}{(P_i-D_i)}\right)\end{array}$$

(59)

Using the saddle point theorem developed in Kuhn and Tucker [36] the following condition must be satisfied to solve the decision variables:

$$\frac{\partial L\left(T_i,T_{2i},\lambda \right)}{\partial T_i}=0;T_i\ge 0;T_i\left(\frac{\partial L\left(T_i,T_{2i},\lambda \right)}{\partial T_i}\right)=0$$

(60)

$$\frac{\partial L(T_i,T_{2i},\lambda )}{\partial T_{2i}}=0;T_{2i}\ge 0;T_{2i}\left(\frac{\partial L\left(T_i,T_{2i},\lambda \right)}{\partial T_{2i}}\right)=0$$

(61)

$$\frac{\partial L(T_i,T_{2i},\lambda )}{\partial \lambda }=0;\lambda \ge 0;\lambda \left(\frac{\partial L\left(T_i,T_{2i},\lambda \right)}{\partial \lambda }\right)=0$$

(62)

Using Maple 15 and the conditions in Equation (60)–(62), the following equations are derived:

$$T_i=\frac{\sqrt{C_{12i}{D}_i{P}_i(2(C_{1i}+C_{7i}+C_{10i})(P_i-D_i)+\left(C_{2i}+C_{2ei}+C_{12i}\right){(D_i{T}_{2i}}^2{P}_i))}}{C_{12i}{D}_i(P_i-D_i)}$$

(63)

$$\lambda =\frac{\left(C_{2i}+C_{2ei}\right)\left(T_{2i}{P}_i\right)+C_{12i}(T_i{P}_i-T_i{D}_i-T_{2i}{P}_i)}{v_i{P}_i{T}_i}$$

(64)

$$T_{2i}=\frac{F(P_i-D_i)}{v_i{P}_i{D}_i}$$

(65)

By solving the three unknown variables, the total cost function can now be solved.

A case study with two items in the system—fluting paper (Product a) and applying newsprint paper (Product b)—illustrates the theory. Data for the two products are as follows:

Da	=	84,000	ton/year		Db	=	40,250	ton/year
Pa	=	336,000	ton/year		Pb	=	16,100	ton/year
C1a	=	5000	USD/cycle		C1b	=	5000	USD/cycle
C2a	=	2.5	USD/ton		C2b	=	2.5	USD/ton
C3a	=	275	USD/ton		C3b	=	395	USD/ton
C4a	=	65	USD/ton CO2		C4b	=	65	USD/ton CO2
C5a	=	5	USD/kg NOx		C5b	=	5	USD/kg NOx
C6a	=	5	USD/kg SOx		C6b	=	5	USD/kg SOx
C7a	=	2000	USD/cycle		C7b	=	2000	USD/cycle
C8a	=	0.02	USD/kg BOD		C8b	=	0.02	USD/kg BOD
C9a	=	0.02	USD/kg COD		C9b	=	0.02	USD/kg COD
C10a	=	500	USD/cycle		C10b	=	500	USD/cycle
C11a	=	65	USD/ton CH4		C11b	=	65	USD/ton CH4
C12a	=	50	USD/ton		C12b	=	50	USD/ton
NOxa	=	1.12	kg/ton		NOxb	=	0.4	kg/ton
SOxa	=	1.9	kg/ton		SOxb	=	0.34	kg/ton
UBODa	=	23.5	kg/m3		UBODb	=	38.5	kg/m3
UCODa	=	44	kg/m3		UCODb	=	65	kg/m3
USLDa	=	0.5			USLDb	=	0.5
UCH4a	=	0.24	ton CH4/m3 sld		UCH4b	=	0.24	ton CH4/m3 sld
va	=	3.71	m3/ton		vb	=	3.71	m3/ton
era	=	1.5	kWh/m3		erb	=	1.5	kWh/m3
epa	=	1078	kWh/ton		epb	=	1457	kWh/ton
ewa	=	3.49	m3/ton		ewb	=	3.49	m3/ton
Ega	=	0.0005	ton CO2/kWh		Egb	=	0.0005	ton CO2/kWh
F	=	185000	m3

Using Equations (63)–(65), the total cost function is calculated as 49,450,942.27 USD.

The procedures on how to obtain Q* for the multiproduct sustainable EPQ model with shortage are elaborated in the following statements:

• Compute T_2i (where i = product a, product b) using Equation (65).
• Determine T_i (where i = product a, product b) using Equation (63).
• Find T_1i (where i = products a, product b) by inserting the values of the solved T_2i into Equation (50).
• Using Equation (51), compute T_4i.
• By inserting the values of T_4i and T_1i into Equation (53), determine Q*.

Following the procedure of computing for Q*, the following values are obtained:

$$T_{1a}=\frac{{(T}_{2a})(D_a)}{\left(P_a-D_a\right)}=\frac{0.44523\ast 84000}{336000-84000}=0.148\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\hspace{0.17em}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}$$

$$T_{1b}=\frac{{(T}_{2b})(D_b)}{(P_b-D_b)}=\frac{0.92917\ast 40250}{161000-40250}=0.310\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\hspace{0.17em}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}$$

$$\begin{gathered} T_{4a}=\frac{D_a}{P_a}\left(T_a-\frac{{(T}_{2a}\ast D_a)}{\left(P_a-D_a\right)}-T_{2a}\right)=\left(\frac{84000}{336000}\right)\left(0.61324-\frac{0.44523\ast 84000}{336000-84000}-0.44523\right) \\ =0.005\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\hspace{0.17em}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t} \end{gathered}$$

$$\begin{gathered} T_{4b}=\frac{D_b}{P_b}\left(T_b-\frac{{(T}_{2b}\ast D_b)}{\left(P_b-D_b\right)}-T_{2b}\right)=\left(\frac{40250}{161000}\right)\left(1.27557-\frac{0.44523\ast 40250}{161000-40250}-0.92917\right) \\ =0.009\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\hspace{0.17em}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t} \end{gathered}$$

$${Q_a}^{\ast }=P_a\left(T_{1a}+T_{4a}\right)=336000\left(0.14841+0.00490\right)=51,512.16\hspace{0.17em}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}$$

$${Q_b}^{\ast }=P_b\left(T_{1b}+T_{4b}\right)=161000\left(0.30972+0.00917\right)=51,341.69\hspace{0.17em}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}$$

3.1.6. Sensitivity Analysis for a Multiproduct Sustainable EPQ Model with Shortages

The sensitivity analysis for Q*, T, T₂, and TC (T, T₂) for both products is performed using the same parameters. The changes in parameters affect the results of T₂, T, λ, and the total cost (TC). These effects are shown in

Table 7. Sensitivity analysis of T₂, T, and λ for Product a.

Parameter	Results	−20%	−10%	Base Value	+10%	+20%
D	λ	0.23243	0.21670	0.20154	0.18613	0.17075
	T	0.76533	0.68082	0.61324	0.55797	0.51195
	T2	0.59363	0.51118	0.44523	0.39126	0.34629
P	λ	0.17931	0.19166	0.20154	0.20963	0.21637
	T	0.61359	0.61338	0.61324	0.61312	0.61303
	T2	0.40812	0.42873	0.44523	0.45872	0.46996
C1	λ	0.21023	0.20590	0.20154	0.19719	0.19285
	T	0.61272	0.61298	0.61324	0.61349	0.61375
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C2	λ	0.15188	0.17674	0.20154	0.22629	0.25097
	T	0.61036	0.61180	0.61324	0.61467	0.61610
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C2e	λ	0.19796	0.19975	0.20154	0.20334	0.20513
	T	0.61303	0.61313	0.61324	0.61334	0.61344
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C3	λ	0.20154	0.20154	0.20154	0.20154	0.20154
	T	0.61324	0.61324	0.61324	0.61324	0.61324
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C4e	λ	0.20154	0.20154	0.20154	0.20154	0.20154
	T	0.61324	0.61324	0.61324	0.61324	0.61324
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C5e	λ	0.20154	0.20154	0.20154	0.20154	0.20154
	T	0.61324	0.61324	0.61324	0.61324	0.61324
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C6e	λ	0.20154	0.20154	0.20154	0.20154	0.20154
	T	0.61324	0.61324	0.61324	0.61324	0.61324
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C7	λ	0.20503	0.20328	0.20154	0.19980	0.19806
	T	0.61303	0.61313	0.61324	0.61334	0.61344
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C8e	λ	0.20154	0.20154	0.20154	0.20154	0.20154
	T	0.61324	0.61324	0.61324	0.61324	0.61324
	T2	0.44522	0.44522	0.44523	0.44522	0.44522
C9e	λ	0.20154	0.20154	0.20154	0.20154	0.20154
	T	0.61324	0.61324	0.61324	0.61324	0.61324
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C10	λ	0.20241	0.20198	0.20154	0.20111	0.20067
	T	0.61318	0.61321	0.61324	0.61326	0.61329
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
C11e	λ	0.20154	0.20154	0.20154	0.20154	0.20154
	T	0.61324	0.61324	0.61324	0.61324	0.61324
	T2	0.44522	0.44522	0.44523	0.44522	0.44522
C12	λ	0.20125	0.20141	0.20154	0.20165	0.20174
	T	0.61804	0.61537	0.61324	0.61148	0.61001
	T2	0.44523	0.44523	0.44523	0.44523	0.44523
F	λ	0.16502	0.18627	0.20154	0.21289	0.22155
	T	0.49233	0.55273	0.61324	0.67382	0.73446
	T2	0.35618	0.40070	0.44523	0.48975	0.53427
v	λ	0.28140	0.23774	0.20154	0.17078	0.14412
	T	0.76479	0.68055	0.61324	0.55823	0.51245
	T2	0.55653	0.49469	0.44523	0.40475	0.37102

.

Table 8. Percent Error for Model 3—Product a.

		% ERROR
		−20%	−10%	Base Value	+10%	+20%
D	λ	15.33%	7.52%	0.00%	−7.65%	−15.28%
	T	24.80%	11.02%	0.00%	−9.01%	−16.52%
	T2	33.33%	14.81%	0.00%	−12.12%	−22.22%
P	λ	−11.03%	−4.90%	0.00%	4.01%	7.36%
	T	0.06%	0.02%	0.00%	−0.02%	−0.03%
	T2	−8.34%	−3.71%	0.00%	3.03%	5.55%
C1	λ	4.31%	2.16%	0.00%	−2.16%	−4.31%
	T	−0.08%	−0.04%	0.00%	0.04%	0.08%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C2	λ	−24.64%	−12.31%	0.00%	12.28%	24.53%
	T	−0.47%	−0.23%	0.00%	0.23%	0.47%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C2e	λ	−1.78%	−0.89%	0.00%	0.89%	1.78%
	T	−0.03%	−0.02%	0.00%	0.02%	0.03%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C3	λ	0.00%	0.00%	0.00%	0.00%	0.00%
	T	0.00%	0.00%	0.00%	0.00%	0.00%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C4e	λ	0.00%	0.00%	0.00%	0.00%	0.00%
	T	0.00%	0.00%	0.00%	0.00%	0.00%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C5e	λ	0.00%	0.00%	0.00%	0.00%	0.00%
	T	0.00%	0.00%	0.00%	0.00%	0.00%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C6e	λ	0.00%	0.00%	0.00%	0.00%	0.00%
	T	0.00%	0.00%	0.00%	0.00%	0.00%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C7	λ	1.73%	0.86%	0.00%	−0.86%	−1.73%
	T	−0.03%	−0.02%	0.00%	0.02%	0.03%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C8e	λ	0.00%	0.00%	0.00%	0.00%	0.00%
	T	0.00%	0.00%	0.00%	0.00%	0.00%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C9e	λ	0.00%	0.00%	0.00%	0.00%	0.00%
	T	0.00%	0.00%	0.00%	0.00%	0.00%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C10	λ	0.43%	0.22%	0.00%	−0.21%	−0.43%
	T	−0.01%	0.00%	0.00%	0.00%	0.01%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C11e	λ	0.00%	0.00%	0.00%	0.00%	0.00%
	T	0.00%	0.00%	0.00%	0.00%	0.00%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
C12	λ	−0.14%	−0.06%	0.00%	0.05%	0.10%
	T	0.78%	0.35%	0.00%	−0.29%	−0.53%
	T2	0.00%	0.00%	0.00%	0.00%	0.00%
F	λ	−18.12%	−7.58%	0.00%	5.63%	9.93%
	T	−19.72%	−9.87%	0.00%	9.88%	19.77%
	T2	−20.00%	−10.00%	0.00%	10.00%	20.00%
v	λ	39.62%	17.96%	0.00%	−15.26%	−28.49%
	T	24.71%	10.98%	0.00%	−8.97%	−16.44%
	T2	25.00%	11.11%	0.00%	−9.09%	−16.67%

provides the percent error of each product type parameter change.

For Product a, it can be observed that the variable λ is highly sensitive to D, P, C₂, F, and v, moderately sensitive to C₁, C_2e, and C₇, and insensitive to C₁₀, C_12, and all emission costs excluding C_2e. Variable T, on the other hand, is highly sensitive to D, F, and v and insensitive to all emission costs, including C₁, C₂, C₇, C₁₀, and C_12. Finally, variable T₂ is highly sensitive to D, F, and v, moderately sensitive to P, and insensitive to the remaining parameters.

4. Comparative Results of Q, T, and TC

Table 9. Summary of Results for Q, T, and TC.

	Q		T		TC
Model 1	25,033		0.30		30,044,087 USD
% Difference from Traditional	18.27%		18.27%		−0.0024%
Model 2	25,695.60		0.305900		30,042,790 USD
% Difference from Traditional	21.40%		21.40%		−0.0067%
Model 3	Qa = 51,512	Qb = 51,342	Ta = 0.61	Tb = 1.28	49,450,942 USD
Traditional	21,166		0.25		30,044,797 USD

summarizes the computed Q, T, and TC for the three (3) models. Such results are compared to the Q, T, and TC of the traditional EPQ model.

The values of Q and T in the proposed Models 1 and 2 have significantly increased by 18.27% and 21.40%, respectively, compared to the traditional model. It can be observed that the total cost (TC) for both Models 1 and 2 has decreased by 0.0024% and 0.0067%, respectively, since the % difference for TC is negative. The integrated results demonstrate that the company could produce more products for a longer time and benefit from a minor reduction in total cost. Furthermore, allowing a shortage in a production system may be a good strategy since it results in a minimum cost (Model 2). Our study optimizes production quantity, cycle length, and total cost and reduces GHG and wastewater emissions.

The results for Model 3, however, cannot be compared with the previous two models due to different assumptions. However, it can be a reference for multiproduct manufacturers where sustainability and warehouse capacity constraints are considered.

The results show that the total costs (TC) of the first two models have decreased compared to the traditional EPQ model. The optimal production quantity, Q, and the cycle length, T, on the other hand, have increased compared to the traditional one. This indicates that the plant might need to reduce production time. Moreover, the sensitivity analysis shows that incorporating GHG emission costs in Models 1 and 2 affects the values of Q, T, and TC.

5. Conclusions

The main goal of this research is to utilize the sustainable EPQ model to determine the optimal production quantity while minimizing GHG and wastewater emissions. To validate the proposed EPQ models in this study, data from pulp and paper mill industry waters were considered. The model’s effectiveness was determined by comparing the production quantity, cycle length, and total costs generated under three scenarios. The first scenario in Model 1 considers a sustainable EPQ model without a shortage. The second scenario in Model 2 considers shortage, and the third scenario in Model 3 considers multiproduct production systems and warehouse capacity constraints. Our proposed models provide significant insights for industrial managers to determine the most economical production quantity and environmentally friendly operations.

The results of this study show the importance of controlling GHG and wastewater emissions. Increasing carbon and wastewater emission taxes is one way to reduce gas emissions and wastewater emissions. This research illustrates the application of our model using pulp and paper mill industry data. The result could help decision-makers of manufacturers, pulp and paper mill industry, and the government to implement policies conducive to the business and environment. Future research can be conducted to consider the product life cycle as well as deteriorating items.