Optimal payment time and replenishment decisions for retailer’s inventory system under trade credit and carbon emission constraints

Abstract

This study presents a multi-item inventory and pricing model by considering marketing, service activities, trade credit, carbon emissions, and the restrictions of production cost and storage space. In the proposed model, shortages are allowed and demand rate is a power function of service and marketing costs, and selling price. The main objective of this study is to optimize retailer’s payments time, service and marketing expenditure, and replenishment decisions in order to maximize retailer’s total profit and minimize carbon emissions, simultaneously. Model is developed in a fuzzy environment under carbon tax regulation when the length of credit period provided by supplier is less than or equal to the length of time in which no shortage happens. To solve the proposed model, we first transform the original problem into a multi-objective Signomial Geometric Programming (SGP) problem using fuzzy and hybrid parameters, which minimizes both the mean value and the total dispersion value of the objective function. Then a global optimization problem method has been used to solve the SGP problem. Efficiency of this algorithm is tested and compared with multi-objective genetic algorithm, multi-objective genetic algorithm with varying population, and hybrid heuristic algorithm. At the end, several numerical examples and sensitivity analysis are performed to demonstrate the application of the proposed model and solution procedure to obtain managerial insights.

Introduction

Trade credit strategy (or permissible delayed payment) is a broad topic in the business practice, constituting a main source of short-term financing. In order to promote market share, stimulate sales, and decrease on hand inventory level, business provide a period of time during which purchasers may delay their payments without having to pay interest. On the other hand, delay in payments indirectly decrease the cost of purchasing inventory, and therefore motivate purchasers to order more quantities. Therefore, both business and purchasers can use advantages of the trade credit strategy. It is notable that approximately eighty percent of United States firms provide trade credit, and more than eighty percent of business-to-business (B2B) transactions in the United Kingdom are made on credit (Seifert et al. 2013).

In the past few years, global warming and greenhouse effect have drawn more consideration due to extreme climate events. It is a general consensus that the main reason of global warming is the aggregation of carbon dioxide created by human operations (He et al. 2015). To slow down the global warming and greenhouse effect some global environmental conferences were organized (Azadeh et al. 2017a, b). For example, the United Nations Conference on Environment and Development (UNCED) in Rio de Janeiro in 1992 and the Kyoto Protocol in Japan in 1997, which introduced effective mechanisms to control carbon emissions. Carbon tax regulation is one of the main mechanisms in the world to reduce carbon emissions and promote energy saving (Xu et al. 2016). Under this regulation, companies are charged for each unit of their carbon emissions with a constant tax rate. This regulation has been imposed in many countries, such as Australia, Canada, Finland, Denmark, and Sweden (He et al. 2015). Also according to Liu et al. (2015), it is possible to adopt carbon tax policy in China in the years of 2016–2020. In the supply chain, emissions generate from transporting (order placement), production process, storage (holding stock), and other components. Therefore, the environmental legislations affect inventory management decisions at any time, especially for multi-item inventory models due to product portfolio (Azadeh et al. 2017a, b).

In recent years, imprecision in input parameters has received great academic attention in inventory models. The most populated approach to consider uncertainty in inventory parameters is by using fuzzy set theory, as introduced by Zadeh (1965). In general, the parameters may be uncertain in nature in fuzzy-stochastic sense. This variation may occur in two different ways:(1) some parameters may be fuzzy random i.e. they become fuzzy with some probability; (2) some parameters may vary in such a way that a part is fuzzy and another part is random in nature. These parameters are called hybrid parameters. Here we consider the cost parameters as hybrid numbers (more details about hybrid numbers are given in “Appendix 1”).

Literature review

In this section, the literature concerning inventory models under trade credit policy, carbon emissions, and inventory models in fuzzy environments, is reviewed.

Inventory under trade credit policy

The area of inventory models with trade credit strategy has received a large amount of attention in the recent years. Dye (2012) presented an EOQ model, in which demand rate is sensitive to selling price and time, under trade credit policies with finite planning horizon. Soni (2013) developed an inventory model to optimize the inventory level and cycle time with price and stock dependent demand under permissible delay in payment with limited space. Chen et al. (2014) combined a partial delay in payment with a permissible delayed payment in an EOQ model including shortages. Ghoreishi et al. (2015) investigated the effects of inflation, credit period and customer returns in an EOQ model for non-instantaneous deteriorating items by considering partial backordering. Sharma (2016) extended the work of Chen et al. (2014) with partial backordering and partial delay in payment based on order quantity.

Nowadays, marketing activities and quality of services play an important role in increasing the sales of the products and consequently achieving more money. Marketing activities (such as advertising and market research) provide the customers with information about fair price, good quality, and longevity of the items. Service means activities and benefits that are considered for purchasers such as customer support and repurchase after sale services. In reality, service quality and marketing activities that are suggested to customers can attract them and consequently increase the demand rate. Therefore, demand rate is proportionate to cost of marketing activities and quality of services. Tsao and Sheen (2012) studied the problem of marketing, pricing, and replenishment with a finite time horizon under permissible delay in payments. In their work, the demand rate is a function of time and selling price, and marketing expenditure is a function of demand rate. Another problem of the marketing expenditure and trade credit policy by EOQ based model including shortages was formulated by Sundara Rajan and Uthayakumar (2017), where the holding cost varies with time and demand rate depends on sales teams’ initiatives.

All the aforementioned articles assumed that buyers pay sellers at the end of the credit period. However, in reality, buyers can delay their payments in case they can obtain more money through investing the money that must pay sellers at the end of credit period by collecting revenues on the sale or use of the products and earning more interest on that revenue than paying rate. Thus, it is advantageous for buyers to defer their payments until the interest earned is smaller than the interest charges. Therefore, determining the best payment time for buyers to complete the payment to sellers in order to make the best use of the overdue amount for minimizing cost (or maximizing profit) through investing that amount elsewhere is an interesting problem. Jamal, Sarker et al. (2000) was the first study that addressed an EOQ model for a deteriorating item to optimize the payment time of the retailer under permissible delay in payments. Song and Cai (2006) identified three small errors in the model of Jamal, Sarker et al. (2000), reformulated the correct model and obtained the optimal payment time for a retailer. Chang et al. (2009) determined the optimal payment time for a retailer under inflation in an EOQ framework. However, none of the above-mentioned studies took the effect of realistic features such as shortages, the quality of service, and marketing activities in finding the retailer’s optimal payment time into consideration.

Most models on trade credit are developed for a single product. Although, in the real world, many suppliers, retailers, and wholesalers’ inventory systems are dealing with multi-items. Researchers that considered the trade credit in multi-item inventory models are Jiangtao et al. (2014), Tiwari et al. (2018), Pakhira et al. (2018a, b).

Inventory under carbon policy

Some researches integrate carbon emission concern into classical production and inventory models. Hammami et al. (2015) formulated a multi-echelon production inventory model with carbon emission consideration and lead-time restrictions under carbon tax policy. They explained how carbon emissions could affect the decisions of manufacturing and inventory systems. Xu et al. (2016) incorporated carbon emissions in a joint pricing and production problem for multi-items under carbon tax and cap-and-trade policies. They optimized the tax rate and production quantities to maximize social welfare. Bozorgi et al. (2014) developed a sustainable EOQ model in which costs and emissions related to holding and transportation were considered as non-linear objective functions. They solved the problem using a set of exact algorithms to determine the optimal order quantity in environmental context. Hovelaque and Bironneau (2015) proposed a novel pricing model by considering carbon emission dependent demand and total carbon emissions, simultaneously. They found two optimal order quantities that maximized the retailer’s total profit, and reduced the amount of carbon emissions under carbon tax and cap-and-trade policy. For the first time, Dye and Yang (2015) investigated the effect of carbon emissions in the joint trade credit and inventory decisions under different environmental policies where demand rate was sensitive to the length of credit period provided for customers. They determined the optimal credit period, cycle time, and ordering quantity under different carbon price and carbon cap. Tsao et al. (2017) built a newsvendor model that takes trade credit, carbon emissions, and product recycling into consideration. Cao and Yu (2018) investigated the interaction of operational decision and financial decision in an emission-dependent supply chain with a capital constrained retailer under stochastic demand and carbon cap-and-trade mechanism. Tiwari et al. (2018a, b) presented a sustainable integrated inventory model for deteriorating items with imperfect quality. Kazemi et al. (2018) formulated a sustainable EOQ model for imperfect quality item considering factors such as human errors in quality inspection and human learning in imperfect quality.

Fuzzy inventory models

During the past years, a vast number of studies have been published to integrate inventory models and fuzzy set theories. Panda et al. (2008) proposed a multi-item EOQ model with hybrid cost parameters under fuzzy/fuzzy-stochastic constraints. A fuzzy inventory model with shortages was presented by Samadi et al. (2013). In their model, demand rate is a function of selling price, marketing and service expenditure and unit cost is a function of order quantity.

Shekarian et al. (2016a, b) formulated two fuzzy models for a reverse inventory problem with learning effect, where the collection rate of the recoverable products from customers and the demand rate of the serviceable products were presented as fuzzy numbers. Shekarian et al. (2016a, b) developed a fuzzy EOQ model for imperfect quality items under different holding costs. For the first time, a comprehensive and systematic review of fuzzy inventory models was provided by Shekarian et al. (2017). Pramanik et al. (2017) developed an inventory model under trade credit policy in fuzzy and rough environments. Pakhira et al. (2018a, b) introduced a fuzzy multi-item two-level supply chain under promotional cost sharing and two-level price discount, assuming that the demand of products depends on the amount of cash discount offered by the retailer to the customers and by promotional effort. Dey (2017) investigated an integrated production-inventory model for imperfect quality items when demand rate is a fuzzy random variable.

A comparison of the mentioned papers is illustrated in Table 1. From the Table 1, some of the major shortcomings of previous papers in the formulation of inventory models can be summarized as follows:

Table 1 Brief review of mentioned studies
  • Most inventory models with trade credit have failed to consider the issue of carbon emissions.

  • Most previous studies on carbon emissions have failed to consider trade credit financing.

  • No inventory model with carbon emissions and trade credit is developed in an uncertain environment.

  • Very few studies on trade credit considered the issue of how retailers determine the best time to complete their payments to suppliers, so that their total profit is maximized.

  • No inventory model with trade credit and carbon emissions considered the service expenditure dependent demand.

Incorporating all the phenomena mentioned above, we have developed a fuzzy sustainable inventory model for multi-items with marketing expenditure, service expenditure, and selling price dependent demand under permissible delay in payment, partial backordering, carbon tax regulation, and restriction on total available budget to purchasing inventory and space area. The supplier offers a fixed credit period to the retailer, which is less than the length of time in which no inventory shortage occurs. At the end of credit period, the retailer has two choices to pay the supplier: (1) he/she can pay off at the end of credit period or, (2) he/she can pay off after credit period by investing the money until the interest earned is smaller than interest payable for the supplier. Therefore, the payment time is higher than or equal to the credit period and less than or equal to the length of time in which no inventory shortage occurs. The proposed model measures the amount of carbon emissions from warehouse and transport operations, and also makes a rational tradeoff among the environmental and economic objectives. In order to deal with uncertainty, fuzzy theory and hybrid numbers are used for available resources, and some cost parameters, respectively. Incorporating these realistic features to the supplier-retailer inventory system, the proposed problem is modeled as a fuzzy geometric programming problem of profit maximization and solved using a global optimization approach (Xu 2014) to determine the optimal payment time, marketing expenditure, service expenditure, selling price, and optimal inventory replenishment decisions. Efficiency of this algorithm is tested and compared with three proposed algorithms of Jana and Das (2017): multi-objective genetic algorithm (MOGA), multi-objective genetic algorithm with varying population (MOGAVP), and hybrid heuristic algorithm (HA).

The rest of this study is organized as follows. Notations and necessary assumptions are given in Sect. 2. In Sect. 3, we formulate sustainable EOQ model with permissible delay in payment under carbon tax policy. Moreover, the proposed problem is solved using global optimization of fuzzy geometric programming problem, MOGA, MOGAVP, and HA in Sect. 4. Several numerical examples and sensitivity analysis are done to demonstrate the applicability of the proposed model and solution procedure in order to obtain managerial insights in Sects. 5 and 6. At the end, conclusions and directions for future research are described in Sect. 7.

Notation and assumptions

The proposed mathematical model is formulated using the following notations and assumptions:

Notations

Indices
 i Sets of product types; \(i \in \left\{1,2,3, \ldots ,n \right\}\)
Crisp parameters
 \(I_{e}\) Interest earned rate ($/year)
 \(I_{p}\) Interest charged rate ($/year)
 \(\gamma_{i}\) The percentage of shortages that will be backordered for each item \(i\); \(\gamma_{i} \in \left[ {0,1} \right]\)
 \(f_{i}\) Storage space of item \(i\)
 \(C_{i}\) Unit purchasing cost of an item ($/unit)
 \(M\) Time- allowed delay in payment (credit period)
 \(s\) Carbon tax rate per unit
 \(a\) The amount of carbon emissions per order
 \(H\) The amount of carbon emissions per unit of inventory quantity
Hybrid parameters
 \(\underline{{\tilde{A}}}\) Hybrid ordering cost ($/order)
 \(\underline{{\tilde{\pi }}}_{i}\) Hybrid backordering cost($/unit/year)
 \(\underline{{\tilde{g}}}_{i}\) Hybrid goodwill loss for unit lost sales
 \(\underline{{\tilde{h}}}_{i}\) Hybrid holding cost ($/unit/year)
Fuzzy parameters
 \(\tilde{B}\) Total available production cost
 \(\tilde{X}\) Total available storage space
Decision variables
 \(P\) The portion of demand that will be satisfied from warehouse, \(P \in \left[ {0,1} \right]\)
 \(T\) The length of an inventory cycle time
 \(S_{i}\) The unit selling price
 \(E_{i}\) Service expenditure per unit
 \(G_{i}\) Marketing expenditure per unit
 \(F\) Payment period
Independent decision variables
 \(\lambda_{i}\) Demand rate of item \(i\) per year
 \(Q_{i}\) Order quantity
 \(B_{i}\) The level of shortages
 \({\text{CO}}_{2}\) The amount of carbon emissions
Other variables
 \(SR_{i}\) Average sales revenue per year
 \(Cm_{i}\) Average marketing expenditure per year
 \(Ce_{i}\) Average service expenditure per year
 \(Ch_{i}\) Average holding cost per year
 \(Cp_{i}\) Average purchasing cost per year
 \(Co_{i}\) Average ordering cost per year
 \(Cb_{i}\) Average backordering cost per year
 \(Gl_{i}\) Average goodwill losses cost per year
 \(Cc_{i}\) Average capital cost per year
 \(Ie_{i}\) Average interest earned per year
 \(Z\) Average total profit

Assumptions

  1. 1.

    The discussion and analysis in this study are limited to a supply chain including a single supplier and a single retailer.

  2. 2.

    The demand rate of item \(i,\lambda_{i}\) is a function of marketing and service expenditure, and selling price, that is \(\lambda_{i} = V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} ,\) where \(V_{i} ,\delta_{i} ,\chi_{i}\) are positive and constant,\(\alpha_{i} > 1\), and \(i = 1,2,3, \ldots ,n.\) Pictorial representation of demand function is shown in Fig. 1.

    Fig. 1
    figure1

    Pictorial representation of demand

  3. 3.

    Shortages are allowed and as combination of lost sales and backorders.

  4. 4.

    Replenishment rate is instantaneous and lead time is zero.

  5. 5.

    There is a limitation on the total production cost and total available space area with Triangular Fuzzy Number (TFN).

  6. 6.

    For each item, ordering cost, holding cost, and shortage costs \(\left( {\underline{{\tilde{A}}} ,\underline{{\tilde{h}}}_{i} ,\underline{{\tilde{\pi }}}_{i} ,\underline{{\tilde{g}}}_{i} } \right)\) are considered as hybrid numbers.

  7. 7.

    All products have a same cycle length to avoid frequent orders of the products, which would increase the ordering cost for the retailer.

  8. 8.

    In the presented supply chain, the retailer purchases the items in each cycle under the trade credit strategy provided by the supplier. It means the supplier gives a full credit period of \(M\) years to the retailer. During the credit period \(M\), the retailer sells the products and collects the sale revenue and obtains interest at a rate \(I_{e}\). At the end of the credit period, the retailer has two ways when pays the supplier: he/she can decide to pay off either at the time \(M\) or any time between \(M\) and \(PT\), the length of time in which no inventory shortage occurs, \(\left( {i.e.\,\,M \le F \le PT} \right)\).

  9. 9.

    According to Hovelaque and Bironneau (2015), warehouse operations and transportation create the longer carbon emissions. Therefore, we calculated the total carbon emission function based on the storage amount and the frequency of delivery.

Model formulation

In the start of each inventory cycle, the retailer orders \({\text{Q}}_{i}\) units of item \(i\) and obtains a credit period of \(M\) years from its supplier. At the end of credit period of \(M\), the retailer can decide to pay off either at the time \(M\) or at the time \(F\)\(\left( {M \le F \le PT} \right)\). The behavior of the considered inventory system along with both payment time \(F\) and credit period \(M\) are shown in Fig. 2. According to Fig. 2 the retailer’s order quantity of item \(i\) is obtained as:

Fig. 2
figure2

Inventory diagram

$${\text{Q}}_{i} = \lambda_{i} PT + \gamma_{i} \lambda_{i} (1 - P)T = \lambda_{i} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right) = T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }}$$
(1)

The following are components of the retailer’s total annual profit for item \(i\):

The retailer sells the products during time \(\left[ {0,PT} \right]\) and accumulates all sells revenue to pay back the total purchasing cost at time \(F\). Since before \(F\), the retailer uses all accumulated sells revenue to pay off the supplier, the retailer can only create profit for his/her own after he/she pays back the total purchasing cost at the time \(F\). Therefore, the sales revenue of item \(i\) after \(F\) per cycle is equal to {the unit selling price \(\times\) (order quantity of item \(i\)—the amount sold during time \(F\))}. So, the sales revenue after \(F\) per year for item \(i\) is:

$$SR_{i} = \frac{{S_{i} \left( {Q_{i} - \lambda_{i} F} \right)}}{T}$$
(2)

The marketing cost per year for item \(i\) is:

$$Cm_{i} = \frac{{G_{i} Q_{i} }}{T}$$
(3)

The service expenditure per year for item \(i\) is:

$$Cs_{i} = \frac{{E_{i} Q_{i} }}{T}$$
(4)

The holding cost per year for item \(i\) is:

$$Ch_{i} = \frac{1}{T}\left( {\underline{{\tilde{h}}}_{i} \frac{{\lambda_{i} PT \times PT}}{2}} \right) = 0.5\underline{{\tilde{h}}}_{i} \lambda_{i} P^{2} T\,$$
(5)

where \(\underline{{\tilde{h}}}_{i} = \left( {h_{1} ,h_{2} ,h_{3} } \right)( + )^{\prime } \left( {\mu_{{h_{i} }} ,\sigma_{{h_{i} }}^{2} } \right)\)

The purchasing cost per year for item \(i\) is:

$$Cp_{i} = \frac{{C_{i} Q_{i} }}{T}$$
(6)

The ordering cost per year for item \(i\) is:

$$Co = \frac{{\underline{{\tilde{A}}} }}{T}\,$$
(7)

where \(\underline{{\tilde{A}}} = \left( {A_{1} ,A_{2} ,A_{3} } \right)( + )^{\prime } \left( {\mu_{A} ,\sigma_{A}^{2} } \right).\)

The backordering cost per year for item \(i\) is:

$$Cb_{i} = \frac{1}{T}\left( {\underline{{\tilde{\pi }}}_{i} \frac{{\lambda_{i} T\gamma_{i} \left( {1 - P} \right) \times T\left( {1 - P} \right)}}{2}} \right) = 0.5\underline{{\tilde{\pi }}}_{i} \gamma_{i} \lambda_{i} T\left( {1 - P} \right)^{2}$$
(8)

where \(\underline{{\tilde{\pi }}}_{i} = \left( {\pi_{1} ,\pi_{2} ,\pi_{3} } \right)( + )^{\prime } \left( {\mu_{{\pi_{i} }} ,\sigma_{{\pi_{i} }}^{2} } \right).\)

The lost sale cost per year for item \(i\) is:

$$Gl_{i} = \underline{{\tilde{g}}}_{i} \left( {1 - \gamma_{i} } \right)\left( {1 - P} \right)\lambda_{i}$$
(9)

where \(\underline{{\tilde{g}}}_{i} = \left( {g_{1} ,g_{2} ,g_{3} } \right)( + )^{\prime } \left( {\mu_{{g_{i} }} ,\sigma_{{g_{i} }}^{2} } \right)\).

There are many different ways to calculate the interest earned (Sarker et al. 2000; Jamal et al. 2000; Song and Cai 2006). Here we calculated the interest earned based on the article of Song and Cai (2006). The retailer collects revenue and obtains interest from backordered demand and new demand during time \(\left[ {0,PT} \right]\). At the beginning of cycle, \(\gamma_{i} \lambda_{i} T\left( {1 - P} \right)\) units of the received orders are consumed for satisfying the backordering demand. Therefore, total annual interest earned after \(F\) for item \(i\) is obtained as follows:

$$Ie_{i} = \frac{1}{T}\left( {\frac{{\left( {S_{i} I_{e} \lambda_{i} \left( {PT - F} \right)^{2} } \right)}}{2} + \left( {S_{i} I_{e} \gamma_{i} \lambda_{i} T\left( {1 - P} \right)\left( {PT - F} \right)} \right)} \right)$$
(10)

The interest payable per year is composed of the cost of current inventory at time \(t\), profit of amount sold and the interest earned from the sales revenue during \(\left[ {0,M} \right]\), and margin profit of the sold amount after due date \(M\)(the same method has been applied in Jamal et al. 2000). So the interest charge per year for inventory in stock can be obtained as:

$$\begin{aligned} Cc_{i} & = \frac{1}{T}\left( {I_{p} \int_{0}^{F - M} {\left( {\left( {C_{i} Q_{i} - S_{i} \lambda_{i} M - \left( {\frac{{\left( {S_{i} I_{e} \lambda_{i} M^{2} } \right)}}{2} + \left( {S_{i} I_{e} \gamma_{i} \lambda_{i} T\left( {1 - P} \right)M} \right)} \right)} \right) - S_{i} \lambda_{i} t} \right)dt} } \right) \\ & = I_{p} \left( {C_{i} Q_{i} \,\left( {F - M\,\,} \right) - 0.5S_{i} \lambda_{i} \left( {F^{2} - M^{2} } \right) - S_{i} I_{e} \lambda_{i} M\left( {0.5M + \gamma_{i} T\left( {1 - P} \right)} \right)\left( {F - M} \right)} \right)T^{ - 1} \\ \end{aligned}$$
(11)

The amount carbon emissions (in tons) depends on the storing and delivery items:

$$CO_{2} = \sum\limits_{i = 1}^{n} {\left( {\frac{a}{T} + H\frac{{\lambda_{i} P^{2} T}}{2}} \right)}$$
(12)

Therefore, the retailer’s total annual profit with considering carbon emissions under carbon tax policy can be calculated by the formula

$$Z = \sum\limits_{i = 1}^{n} {\left( {SR_{i} - Cm_{i} - Cs_{i} - Ch_{i} - Cp_{i} - Cb_{i} - Gl_{i} - Cc_{i} + Ie_{i} } \right)} - Co\, - \left( {s \times CO_{2} } \right)$$

In summation and simplification, the following results are obtained:

$$\begin{aligned} Z & = \sum\limits_{i = 1}^{n} {\left\{ {V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} } \right.} \left( {\left( {\gamma_{i} - \gamma_{i} I_{e} I_{p} M^{2} } \right)S_{i} } \right. + \left( {1 - \gamma_{i} } \right)PS_{i} - \left( {1 - 0.5I_{e} I_{p} M^{2} } \right)S_{i} FT^{ - 1} + \left( {0.5 - \gamma_{i} } \right)I_{e} S_{i} P^{2} T \\ & \quad - \;\beta_{i} \left( {G_{i} + E_{i} } \right) - \left( {1 - \gamma_{i} } \right)P\left( {G_{i} + E_{i} } \right) - 0.5\left( {\underline{{\tilde{h}}}_{i} + \underline{{\tilde{\pi }}}_{i} \gamma_{i} + sH} \right)P^{2} T - \left( {C_{i} - \underline{{\tilde{g}}}_{i} - C_{i} I_{p} M} \right)\left( {1 - \gamma_{i} } \right)P \\ & \quad + \;\underline{{\tilde{\pi }}}_{i} \gamma_{i} PT - \left( {C_{i} \gamma_{i} + \left( {1 - \gamma_{i} } \right)\underline{{\tilde{g}}}_{i} - C_{i} \gamma_{i} I_{p} M} \right) + 0.5\left( {I_{e} + I_{p} } \right)S_{i} F^{2} T^{ - 1} - \left( {1 - \gamma_{i} } \right)I_{e} S_{i} FP - C_{i} \gamma_{i} I_{p} F \\ & \left. {\quad - \,\,C_{i} I_{p} \left( {1 - \gamma_{i} } \right)PF - 0.5I_{p} M^{2} \left( {1 + 0.5I_{e} M} \right)S_{i} T^{ - 1} + \gamma_{i} I_{e} S_{i} PT - \gamma_{i} I_{e} \left( {1 - I_{p} M} \right)SF - 0.5\gamma_{i} \underline{{\tilde{\pi }}}_{i} T} \right)\left. {} \right\} \\ & \quad - \;\left( {\underline{{\tilde{A}}} + sa} \right)T^{ - 1} \\ \end{aligned}$$
(13)

where \(\underline{{\tilde{h}}}_{i} = \left( {h_{1} ,h_{2} ,h_{3} } \right)( + )^{\prime } \left( {\mu_{{h_{i} }} ,\sigma_{{h_{i} }}^{2} } \right)\), \(\underline{{\tilde{\pi }}}_{i} = \left( {\pi_{1} ,\pi_{2} ,\pi_{3} } \right)( + )^{\prime } \left( {\mu_{{\pi_{i} }} ,\sigma_{{\pi_{i} }}^{2} } \right)\)\(\underline{{\tilde{g}}}_{i} = \left( {g_{1} ,g_{2} ,g_{3} } \right)( + )^{\prime } \left( {\mu_{{g_{i} }} ,\sigma_{{g_{i} }}^{2} } \right)\), \(\underline{{\tilde{A}}} = \left( {A_{1} ,A_{2} ,A_{3} } \right)( + )^{\prime } \left( {\mu_{A} ,\sigma_{A}^{2} } \right)\), and \(i = 1,2,3, \ldots ,n.\)

As explained above, we consider some limitations on accessible resources in a fuzzy environment as follows:

  1. a.

    Production cost of all items is limited to \(\tilde{B}\) and calculated as follows

    $$\sum\limits_{i = 1}^{n} {\left( {C_{i} \lambda_{i} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} \le \tilde{B};\quad \tilde{B} = \left( {B_{1} ,B_{2} ,B_{3} } \right)$$
    (14)
  2. b.

    Total space area occupied with all items is restricted to \(\tilde{X}\) and calculated as:

    $$\sum\limits_{i = 1}^{n} {\left( {f_{i} \lambda_{i} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} \le \tilde{X};\quad \tilde{X} = \left( {X_{1} ,X_{2} ,X_{3} } \right)$$
    (15)

Under fuzzy recourse limitations, hybrid cost parameters, trade credit policy, and carbon tax regulation the objective of this study is to obtain the order quantity \(\left( {Q_{i} } \right)\), selling price \(\left( {S_{i} } \right)\), marketing expenditure \(\left( {G_{i} } \right)\), service expenditure \(\left( {E_{i} } \right)\), replenishment time \(\left( T \right)\), and payment time \(\left( F \right)\) to maximize the average retailer’s profit \(\left( Z \right)\) and minimize carbon emissions. So, the mathematical model of the sustainable inventory problem is:

$$\begin{aligned} & \hbox{max} \,\,\,\,\,Z \\ & {\text{s}} . {\text{t}}\quad M \le F \le PT \\ & \quad \quad {\text{Constrains }}\left( {14} \right){-}\left( {15} \right). \\ \end{aligned}$$
(16)

In what follows, two solution approaches introduced by Panda et al. (2008) and Xu (2014) are combined to solve the above model.

Solution method

The proposed problem has been formulated as a Fuzzy Non-Linear Programming (FNLP) problem with hybrid and fuzzy numbers. An effective technique to solve these types of NLP problems is Geometric Programming (GP) method (Samadi et al. 2013). GP method has very useful computational and theoretical properties to solve complex optimization problems in different fields such as engineering, management, science, etc. This technique was extended rapidly by researchers, especially engineering designers. Signomial Geometric Programming (SGP) problem was the first extension of GP problems (Passy and Wilde 1967). SGP problems are categorized in class of non-convex optimization problems that are an inherently intractable NP-hard problem (Xu 2014). SGP problems are well used for solving inventory models (Mandal et al. 2006; Panda and Maiti 2009; Samadi et al. 2013, Aliabadi et al. 2017). In this technique degree of difficulty (DDFootnote 1) has an important role. When \(DD \le 2\), many researchers have applied dual geometric programming for solving inventory models. But if \(DD \ge 3\), solving inventory models will be difficult. In the last decades, many methods have been presented for solving SGP problems. For instance, branch-and-bound based global optimization method by using the exponential variable transformation and convex underestimation (Floudas 2013), MM algorithms (Lange and Zhou 2014), and Range reduction techniques (Lin and Tsai 2012). Panda et al. (2008) presented a multi- item fuzzy EOQ model with hybrid and fuzzy parameters. To solve their problem, they first use some transformations to convert FNLP problem into a SGP problem, then they solve the SGP problem using Geometric Programming (GP) approach. Xu (2014) presented global optimization method to solve SGP problems. Since our proposed problem has been modelled as a FNLP problem, we apply a solution procedure that is a combination of these two solution approaches. According to Panda et al. (2008), we first transform the problem into a Multi-Objective Fuzzy Inventory (MOFI) problem in order to minimize the mean and the total dispersion value of the objective function. Then, a surprise function is defined for the restrictions to transform MOFI problem into a multi-objective deterministic SGP problem. Since SGP problems are non-convex and NP-hard problems, so in next step, some transformation and new variables are used for converting SGP problem. The converted SGP problem is transformed into a series of standard GP problems that are nonlinear convex problems and can be efficiently solved (Xu 2014). To check the efficiency of the proposed algorithm, MOGA, MOGAVP, and HA of Jana and Das (2017) are also employed.

Transforming multi-item fuzzy inventory model into a constrained SGP problem

The Eq. (16) is transformed into the following problem:

$$\hbox{min} \,\,\,\,f\left( I \right) = - Z$$
(17)
$$\begin{aligned} & {\text{s}} . {\text{t}} .\quad M \le F \le PT \\ &\quad\quad I = \left( {S_{i} ,G_{i} ,E_{i} ,F,P,T} \right)\quad {\text{and}}\;{\text{Constrains}}\left( {14} \right){-}\left( {15} \right). \\ \end{aligned}$$
(18)

Note the objective function introduced in the Eq. (17) is the reciprocal of the profit.

According to the hybrid numbers theory as explained by Panda et al. (2008) the objective function of model (17) is reduced to:

$$\hbox{min} \,\,\,\,\,EVf(I) = E\tilde{f}_{0} (I)( + )^{\prime}\left( {0,V(I)} \right)$$
(19)

where \(E\tilde{f}_{0} \left( I \right) = \left( {Ef_{1} \left( I \right),Ef_{2} \left( I \right),Ef_{3} \left( I \right)} \right)\) with

$$\begin{aligned} Ef_{k} (I) & = \sum\limits_{i = 1}^{n} {\left\{ {V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} } \right.} \left( {\left( {1 - 0.5I_{e} I_{p} M^{2} } \right)S_{i} FT^{ - 1} + \left( {0.5 - \gamma_{i} } \right)I_{e} S_{i} P^{2} T - \left( {\gamma_{i} - \gamma_{i} I_{e} I_{p} M^{2} } \right)S_{i} } \right. \\ & \quad - \;\left( {1 - \gamma_{i} } \right)PS_{i} + \gamma_{i} \left( {G_{i} + E_{i} } \right) + \left( {1 - \gamma_{i} } \right)P\left( {G_{i} + E_{i} } \right) + 0.5\left( {\left( {h_{ik} + \mu_{{h_{i} }} } \right) + \left( {\pi_{ik} + \mu_{{\pi_{i} }} } \right)\gamma_{i} + sH} \right)P^{2} T \\ & \quad + \;\left( {C_{i} - \left( {g_{ik} + \mu_{{g_{i} }} } \right) - C_{i} I_{p} M} \right)\left( {1 - \gamma_{i} } \right)P\, - \left( {\pi_{ik} + \mu_{{\pi_{i} }} } \right)\gamma_{i} PT \\ & \quad + \;\left( {C_{i} \gamma_{i} + \left( {1 - \gamma_{i} } \right)\left( {g_{ik} + \mu_{{g_{i} }} } \right) - C_{i} \gamma_{i} I_{p} M} \right) + 0.5I_{p} M^{2} \left( {1 + 0.5I_{e} M} \right)S_{i} T^{ - 1} \\ & \quad - \;0.5\left( {I_{e} + I_{p} } \right)S_{i} F^{2} T^{ - 1} + \left( {1 - \gamma_{i} } \right)I_{e} S_{i} FP + C_{i} \gamma_{i} I_{p} F + C_{i} I_{p} \left( {1 - \gamma_{i} } \right)PF \\ & \quad \left. {\left. { - \;\gamma I_{e} S_{i} PT + \gamma_{i} I_{e} \left( {1 - I_{p} M} \right)SF + 0.5\gamma_{i} \left( {\pi_{ik} + \mu_{{\pi_{i} }} } \right)T} \right)} \right\} + \left( {A_{k} + \mu_{A} + sa} \right)T^{ - 1} \quad {\text{k = 1,2,3}} \\ \end{aligned}$$

and

$$\begin{aligned} V(I) & = \sum\limits_{i = 1}^{n} {0.25V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} \left( {\left( {\sigma_{{h_{i} }}^{2} + \sigma_{{\pi_{i} }}^{2} \gamma_{i}^{2} } \right)P^{4} T^{2} + \left( {P^{2} + 1} \right)\sigma_{{\pi_{i} }}^{2} \gamma_{i}^{2} T^{2} + 4\sigma_{{g_{i} }}^{2} \left( {1 - \gamma_{i} } \right)^{2} \left( {1 + P^{2} } \right)} \right)} \\ & \quad + \;\sigma_{A}^{2} T^{ - 2} \\ \end{aligned}$$

Referring to Panda et al. (2008), the approximated value of triangular fuzzy number \(\tilde{b} \equiv \left( {b_{1} ,b_{2} ,b_{3} } \right)\) is calculated as \(\hat{b} = \frac{{b_{1} + 2b_{2} + b_{3} }}{4}\). Therefore, an approximated value of \(E\tilde{f}_{0} (I)\) is as follows:

$$\begin{aligned} AEf_{0} (I) & = \frac{1}{4}\left( {Ef_{1} (I) + 2Ef_{2} (I) + Ef_{3} (I)} \right) \\ & = \sum\limits_{i = 1}^{n} {\left\{ {V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} } \right.} \left( {\left( {1 - 0.5I_{e} I_{p} M^{2} } \right)S_{i} FT^{ - 1} + \left( {0.5 - \gamma_{i} } \right)I_{e} S_{i} P^{2} T - \left( {\gamma_{i} - \gamma_{i} I_{e} I_{p} M^{2} } \right)S_{i} } \right. \\ & \quad - \;\left( {1 - \gamma_{i} } \right)PS_{i} + \gamma_{i} \left( {G_{i} + E_{i} } \right) + \left( {1 - \gamma_{i} } \right)P\left( {G_{i} + E_{i} } \right) + 0.5\left( {\left( {\hat{h}_{i} + \mu_{{h_{i} }} } \right) + \left( {\hat{\pi }_{i} + \mu_{{\pi_{i} }} } \right)\gamma_{i} + sH} \right)P^{2} T \\ & \quad + \;\left( {C_{i} - \left( {\hat{g}_{i} + \mu_{{g_{i} }} } \right) - C_{i} I_{p} M} \right)\left( {1 - \gamma_{i} } \right)P\, - \left( {\hat{\pi }_{i} + \mu_{{\pi_{i} }} } \right)\gamma_{i} PT \\ & \quad + \;\left( {C_{i} \gamma_{i} + \left( {1 - \gamma_{i} } \right)\left( {\hat{g}_{i} + \mu_{{g_{i} }} } \right) - C_{i} \gamma_{i} I_{p} M} \right) + 0.5I_{p} M^{2} \left( {1 + 0.5I_{e} M} \right)S_{i} T^{ - 1} \\ & \quad - \;0.5\left( {I_{e} + I_{p} } \right)S_{i} F^{2} T^{ - 1} + \left( {1 - \gamma_{i} } \right)I_{e} S_{i} FP + C_{i} \gamma_{i} I_{p} F + C_{i} I_{p} \left( {1 - \gamma_{i} } \right)PF \\ & \quad \left. {\left. { - \;\gamma_{i} I_{e} S_{i} PT + \gamma_{i} I_{e} \left( {1 - I_{p} M} \right)SF + 0.5\gamma_{i} \left( {\hat{\pi }_{i} + \mu_{{\pi_{i} }} } \right)T} \right)} \right\} + \left( {A_{k} + \mu_{A} + sa} \right)T^{ - 1} \\ \end{aligned}$$
(20)

The problem (19) transforms into a multi-objective fuzzy nonlinear programming problem using the Eq. (21) as follows:

$$\begin{aligned} & \hbox{min} \,\,\,\,\,\,\left[ {AEf_{0} (I),\,\,V(I)} \right] \\ & {\text{s}} . {\text{t}} .\quad I = \left( {S_{i} ,G_{i} ,E_{i} ,F,P,T} \right)\quad {\text{and}}\;{\text{Constrains}}\left( {14} \right){-}\left( {15} \right){\text{ and (18)}} \\ \end{aligned}$$
(21)

After defining surprise function approach (see Sect. 3.6 of Panda et al. (2008), the above problem converts to a multi-objective crisp unconstrained nonlinear programming problem as:

$$\hbox{min} \,\,\,\,\,\left[ {W_{1} (I),V_{1} (I)} \right]$$
(22)

where

$$W_{1} (I) = AEf_{0} (I) + \xi (I)$$
(23)
$$V_{1} (I) = V(I) + \xi (I)$$
(24)
$$\begin{aligned} \xi (I) & = \left( {\frac{{\sum\nolimits_{i = 1}^{n} {\left( {C_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} - B_{2} }}{{B_{3} - \sum\nolimits_{i = 1}^{n} {\left( {C_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} }}} \right)^{2} \\ & \quad + \,\left( {\frac{{\sum\nolimits_{i = 1}^{n} {\left( {f_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} - X_{2} }}{{X_{3} - \sum\nolimits_{i = 1}^{n} {\left( {f_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} }}} \right)^{2} \\ \end{aligned}$$

The two Eqs. (23) and (24) can be rewritten as the following equations, by introducing some additional variables:

$$W_{2} (I^{\prime}) = AEf_{0} (I^{\prime}) + \vartheta_{1}^{2} \vartheta_{2}^{ - 2} + \vartheta_{3}^{2} \vartheta_{4}^{ - 2}$$
(25)
$$V_{2} (I^{\prime}) = V(I^{\prime}) + \vartheta_{1}^{2} \vartheta_{2}^{ - 2} + \vartheta_{3}^{2} \vartheta_{4}^{ - 2}$$
(26)
$$\sum\limits_{i = 1}^{n} {\left( {C_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} - B_{2} = \vartheta_{1}$$
(27)
$$B_{3} - \sum\limits_{i = 1}^{n} {\left( {C_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} = \vartheta_{2}$$
(28)
$$\sum\limits_{i = 1}^{n} {\left( {f_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} - X_{2} = \vartheta_{3}$$
(29)
$$X_{3} - \sum\limits_{i = 1}^{n} {\left( {f_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} = \vartheta_{4}$$
(30)

\(I^{\prime} = \left( {S_{i} ,G_{i} ,E_{i} ,F,P,T,y_{m} } \right)\) and \(I^{\prime} \ge 0\)

Since \(\vartheta_{m} > 0,\,\,m = 1,2,3,4,\) we can rewrite the multi-objective problem (22) as:

$$\begin{aligned} & \hbox{min} \,\,\,\,\,\left[ {W_{2} (I^{\prime}),V_{2} (I^{\prime})} \right] \\ & {\text{s}} . {\text{t}} .\quad \sum\limits_{i = 1}^{n} {\left( {C_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} - B_{2} \le \vartheta_{1} \\ & \quad \quad B_{3} - \sum\limits_{i = 1}^{n} {\left( {C_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} \ge \vartheta_{2} \\ & \quad \quad \sum\limits_{i = 1}^{n} {\left( {f_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} - X_{2} \le \vartheta_{3} \\ & \quad \quad X_{3} - \sum\limits_{i = 1}^{n} {\left( {f_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} \ge \vartheta_{4} \\ & \quad \quad I^{\prime} = \left( {S_{i} ,G_{i} ,E_{i} ,F,P,T,y_{m} } \right)\quad {\text{and}}\quad I^{\prime} \ge 0 \\ \end{aligned}$$
(31)

Finally, the problem (31) converts to the following multi-objective problem that each objective is as a SGP problem:

$$\hbox{min} \,\,\,\,\,\left[ {W_{2} (I^{\prime}),V_{2} (I^{\prime})} \right]$$
(32)
$${\text{s}} . {\text{t}} .\quad \sum\limits_{i = 1}^{n} {\left( {C_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} B_{2}^{ - 1} - \vartheta_{1} B_{2}^{ - 1} \le 1$$
(33)
$$B_{3}^{ - 1} \vartheta_{2} + \sum\limits_{i = 1}^{n} {\left( {C_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} B_{3}^{ - 1} \le 1$$
(34)
$$\sum\limits_{i = 1}^{n} {\left( {f_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} X_{2}^{ - 1} - X_{2}^{ - 1} \vartheta_{3} \le 1$$
(35)
$$X_{3}^{ - 1} \vartheta_{4} + \sum\limits_{i = 1}^{n} {\left( {f_{i} V_{i} E_{i}^{{\delta_{i} }} G_{i}^{{\chi_{i} }} S_{i}^{{ - \alpha_{i} }} T\left( {\gamma_{i} + P\left( {1 - \gamma_{i} } \right)} \right)} \right)} X_{3}^{ - 1} \le 1$$
(36)
$$M \le F \le PT$$
(37)
$$I^{\prime} = \left( {S,G,M,P,T,y} \right)\quad {\text{and}}\quad I^{\prime} \ge 0$$
(38)

Now we solve the multi-objective problems (32)–(38) by the following steps, respectively.

Step 1 Solve the problem (32)–(38) with considering only one objective function, say \(W_{2} \left( {I^{\prime } } \right)\), by using of the proposed approach of Xu (2014). Let \(I^{\prime (2)} = (S^{(2)} ,G^{(2)} ,M^{(2)} ,P^{(2)} ,T^{(2)} ,y^{(2)} )\) be the optimal solutions for decision variables and so the optimal amount of objective function is \(W_{2} (I^{\prime (1)} )\). Next calculate the amount of the second objective function \(V_{2} \left( {I^{\prime } } \right)\) in \(I^{\prime (1)}\), say \(V_{2} \left( {I^{\prime (1)} } \right)\).

Step 2 Consider just the second objective function \(V_{2} \left( {I^{\prime } } \right)\) and solve it using SGP approach said in Step 1 and obtain the optimal solutions for decision variables and objective function as \(I^{\prime (2)} = (S^{(2)} ,G^{(2)} ,M^{(2)} ,P^{(2)} ,T^{(2)} ,y^{(2)} )\) and \(V_{2} (I^{\prime (2)} )\), respectively. Next compute the amount of the first objective function \(W_{2} \left( {I^{\prime } } \right)\) in \(I^{\prime (2)}\), say \(W_{2} \left( {I^{\prime (2)} } \right)\).

Step 3 The following relations hold for the objective functions: \(W_{2} (I^{\prime (1)} )\,\, < \,\,W_{2} (I^{\prime } ) < \,\,W_{2} (I^{\prime (2)} )\) and \(V_{2} (I^{\prime (2)} ) < \,\,V_{2} (I^{\prime } ) < \,\,V_{2} (I^{\prime (1)} )\).

Step 4 Formulate the membership functions for the objective functions of problems (32)–(38) as follows:

$$\begin{aligned} \mu_{{W_{2} }} (I^{\prime } ) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad W_{2} (I^{\prime } ) \le W_{2} (I^{\prime (1)} )} \hfill \\ {\frac{{W_{2} (I^{\prime (2)} ) - W_{2} (I^{\prime } )}}{{W_{2} (I^{\prime (2)} ) - W_{2} (I^{\prime (1)} )}}} \hfill & {\quad W_{2} (I^{\prime (1)} ) \le W_{2} (I^{\prime } ) \le W_{2} (I^{\prime (1)} )} \hfill \\ 0 \hfill & {\quad W_{2} (I^{\prime } )\,\, \ge \,\,W_{2} (I^{\prime (2)} } \hfill \\ \end{array} } \right. \hfill \\ \hfill \\ \end{aligned}$$
(39)
$$\begin{aligned} \mu_{{V_{2} }} (I^{\prime } ) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad V_{2} (I^{\prime } ) \le V_{2} (I^{\prime (2)} )} \hfill \\ {\frac{{V_{2} (I^{\prime (1)} ) - V_{2} (I^{\prime } )}}{{V_{2} (I^{\prime (1)} ) - V_{2} (I^{\prime (2)} )}}} \hfill & {\quad V_{2} (I^{\prime (2)} ) \le V_{2} (I^{\prime } ) \le V_{2} (I^{\prime (1)} )} \hfill \\ 0 \hfill & {\quad V_{2} (I^{\prime } )\,\, \ge \,V_{2} (I^{\prime (1)} )} \hfill \\ \end{array} } \right. \hfill \\ \hfill \\ \end{aligned}$$
(40)

Step 5 The membership functions are maximizing by max-convex combination operator through the following equations:

$$\begin{aligned} & \hbox{max} \,\,\,\,MF(I^{\prime}) = \omega_{1} \mu_{{W_{2} }} (I^{\prime}) + \omega_{2} \mu_{{V_{2} }} (I^{\prime}) \\ & {\text{s}} . {\text{t}}\quad {\text{Constrains }}\left( {33} \right){-}\left( {38} \right) \\ \end{aligned}$$
(41)

where the weights \(\omega_{1}\) and \(\omega_{2}\) \(\left( {\omega_{1} + \omega_{2} = 1} \right)\) are corresponding to the member functions \(\mu_{{W_{2} }} (I^{\prime})\) and \(\mu_{{V_{2} }} (I^{\prime})\), respectively. The problem (41) can be rewritten as the following constrained SGP problems with \(DD > 2\):

$$\begin{aligned} & \hbox{min} \,\,\,F(I^{\prime } ) = \frac{{\omega_{1} }}{{W_{2} (I^{\prime (2)} ) - W_{2} (I^{\prime (1)} )}}W_{2} (I^{\prime } ) + \frac{{\omega_{2} }}{{V_{2} (I^{\prime (1)} ) - V_{2} (I^{\prime (2)} )}}V_{2} (I^{\prime } ) \\ & {\text{s}} . {\text{t}}\quad {\text{Constrains }}\left( {33} \right) \, - \, \left( {38} \right). \\ \end{aligned}$$
(42)

Now problem (41) can be solved using global optimization of SGP problem as discussed in “Appendix 2”.

Numerical examples

In this Section, three numerical illustrations are designed to demonstrate the application of the model and solution procedures proposed above for a particular retailer that orders two types of products from the supplier. This retailer has two fuzzy constraints on the total warehouse space and production costs. The total production cost is taken as (311, 352, 410) $ and total warehouse area is (285, 305, 335) sq.m. According to the past reorders, the annual demand rate of two items are calculated as \(10^{6} E_{1}^{0.009} G_{1}^{0.008} S_{1}^{ - 3.5}\) and \(10^{6} E_{2}^{0.01} G_{2}^{0.006} S_{2}^{ - 3.6}\). The specific and general data of these examples are provided in Tables 2 and 3. To solve the problem and show the effectiveness of the proposed algorithm, multi-objective genetic algorithm (MOGA), multi-objective genetic algorithm with varying population (MOGAVP), and hybrid heuristic algorithm (HA) of Jana and Das (2017) are employed. All algorithms are coded in MATLAB R2014b software and implemented on an Intel Core i5 PC with CPU of 1.4 GHz and 4.00 GB RAM.

Table 2 General input data for precise parameters
Table 3 General input data for hybrid parameters

Example 1

In first example, cost parameters are considered as hybrid parameters. The payoff matrix of problem (17), which is needed to transform problem (17) into problem (42), is as following:

$$\left( {\begin{array}{*{20}c} {W_{2} (I^{\prime (1)} )} &\quad {V_{2} (I^{\prime (1)} )} \\ {W_{2} (I^{\prime (2)} )} &\quad {V_{2} (I^{\prime (2)} )} \\ \end{array} } \right) = \left( {\begin{array}{*{20}l} { - 18.9869} \hfill & {\quad 8.0945} \hfill \\ { \, 220.2000} \hfill & {\quad 5} \hfill \\ \end{array} } \right)$$

Calculating this payoff matrix and considering the weights 0.9 and 0.1 plus the provided data, it is possible to solve the problem (42) using global optimization of SGP approach by GGPLAB solver (Mutapcic et al. 2006). The optimal values of decision variables for each product along with the optimal values of mean profit function \(\left( {AEf^{\prime}_{0} } \right)\), variance profit function \(\left( V \right)\), and the amount carbon emissions from all methods are reported in Tables 4 and 5.

Table 4 The optimal solutions for the first example
Table 5 The optimal solutions for the first example (continued)

Example 2

In second example, we omit fuzziness in hybrid numbers and obtain the results for the model when the cost parameters only expressed by random numbers as shown in Tables 6 and 7. The payoff matrix of this example is as following:

$$\left( {\begin{array}{*{20}c} {W_{2} (I^{\prime (1)} )} &\quad {V_{2} (I^{\prime (1)} )} \\ {W_{2} (I^{\prime (2)} )} &\quad {V_{2} (I^{\prime (2)} )} \\ \end{array} } \right) = \left( {\begin{array}{*{20}l} { - 30.5472} \hfill & {\quad 7.4509} \hfill \\ { \, 107.7000} \hfill & {\quad 5} \hfill \\ \end{array} } \right)$$
Table 6 The optimal solutions for the second example
Table 7 The optimal solutions for the second example (continued)

Example 3

In third example, we omit randomness in cost parameters and obtain the results for the model when the cost parameters only expressed by fuzzy numbers as shown in Tables 8 and 9. The payoff matrix of this example is as following:

$$\left( {\begin{array}{*{20}c} {W_{2} (I^{\prime (1)} )} &\quad {V_{2} (I^{\prime (1)} )} \\ {W_{2} (I^{\prime (2)} )} & \quad{V_{2} (I^{\prime (2)} )} \\ \end{array} } \right) = \left( {\begin{array}{*{20}l} { - 29.9529} \hfill & {\quad 0} \hfill \\ { \, 194.7071} \hfill & {\quad 0} \hfill \\ \end{array} } \right)$$
Table 8 The optimal solutions for the third example
Table 9 The optimal solutions for the third example (continued)

Although the results are not comparable as the nature varies from problem to problem, it is observed that profit is maximum when costs parameters are hybrid numbers. A comparison of the results in Tables 4, 5, 6, 7, 8 and 9, displays that geometric programming method provides better results than the MOGA, MOGAVP, and HA. For this reason, sensitivity analyses are done using only geometric programming approach. In case of HA, it provides better results than MOGA and MOGAVP.

Sensitivity analysis

Sensitivity analyses for the proposed problem are done to analyze the impacts of changes in the key parameter values on the optimal solutions by geometric programming approach only. For simplicity, we choose product 1 as case. We first examine the effects of the supplier’s different permissible delay times on the optimal solutions; the calculated results are shown in Fig. 3. We observe from Fig. 3e that when the supplier offers a longer credit period, the retailer will decrease selling price to share profit with his/her customers, and therefore the market demand and order quantity increase (see Fig. 3f, g). Moreover, higher demand and order quantity leads to release more carbon into the atmosphere. Therefore, the value of carbon emissions per year increases and retailer’s total profit decreases as the credit period increases. From Fig. 3a and b, we can find that when the credit period \(\left( M \right)\) increases from 0 to 0.12 year, the optimal cycle time \(\left( T \right)\) and the optimal delay in payment time \(\left( F \right)\) increase; however, when the credit period increases from 0.12 to 0.2, the optimal cycle time is equal to 1 and remains constant, while the optimal value of \(F\) depends on the amount supplier’s permissible delay time. These occur because when the supplier offers the retailer a credit period that is not sufficiently large, i.e. \(M \le 0.12\), the amount of retailer’s interest charged will be less than his/he interest earned. In this situation, to obtain more interest, it would be better for the retailer to pay his/her payment at the point \(P^{*} \left( {M < P^{*} < T^{*} } \right)\); however, if the permissible delay period \(M\) is sufficiently large, i.e. \(M \ge 0.12\), the marginal cost increase will be more than the marginal revenue increase, so the retailer will pay his/her payment at the time \(P^{*} = M\).

Fig. 3
figure3

Effect of credit period on optimal solutions for product 1

To investigate the effect of carbon tax rate \(\left( s \right)\) on the optimal solutions, it is assumed that carbon tax rate changes from 0 to 0.2. Figure 4 show the optimal solutions in changed conditions under different carbon tax rate. From Fig. 4c–e, it can be observed that when the carbon tax rate increases, the retailer will decrease the marketing expenditure and service expenditure and increase the selling price to make up the profit. So, the order quantity and demand rate reduce as the carbon tax rate increases. In addition, because the market demand and order quantity per year decrease, we can observe from Fig. 4h andi, that the amount of carbon emissions per year and total profit decrease as the carbon tax rate increases. From Fig. 4b, we also infer that increasing the amount of carbon tax results in a decrease in the optimal delay in payment time \(\left( F \right)\). This occurs because a higher carbon tax rate leads to increase the unit inventory holding cost and purchasing cost; thus, it is not to the advantage of the retailer to delay his/her payment. Therefore, the optimal delay in payment decreases as the carbon tax rate increases. We also investigate the sensitivity analyses on the optimal solutions due to the parameters \(I_{e}\), \(I_{p}\), \(\gamma_{1}\), and \(\tilde{X}\). The impact of the changes is reported in Table 10 and the following results can be viewed:

Fig. 4
figure4

Effect of carbon tax rate on optimal solutions for product 1

Table 10 Sensitivity analysis on the parameters \(I_{e}\), \(I_{p}\), \(\gamma_{1}\), and \(\tilde{X}\)
  • When the parameter \(I_{p}\) increases, the amount of \(S^{*}\) will increase, whereas the amounts of \(T^{*}\), \(\lambda^{*}\), \(Q^{*}\), \(CO_{2}\), \(AEf^{\prime}_{0}\), \(P^{*}\), and \(F^{*}\) will decrease.

  • When the parameter \(I_{e}\) increases, the amounts of \(F^{*}\), \(E^{*}\), \(G^{*}\), and \(\lambda^{*}\) will increase, whereas the amounts of \(T^{*}\), \(S^{*}\), \(Q^{*}\), \(CO_{2}\), \(AEf^{\prime}_{0}\), and \(P^{*}\) will decrease.

  • When the parameter \(\gamma_{1}\) increases, the amounts of \(F^{*}\), \(S^{*}\), \(G^{*}\), \(CO_{2}\), and \(AEf^{\prime}_{0}\) will increase, whereas the amounts of \(T^{*}\), \(\lambda^{*}\), \(Q^{*}\), and \(P^{*}\) will decrease.

  • When the parameter \(\tilde{X}\) increases, the amounts of \(\lambda^{*}\), \(CO_{2}\), and \(F^{*}\) will increase, whereas the amounts of \(E^{*}\), \(G^{*}\), \(S^{*}\), \(P^{*}\), \(Q^{*}\), \(AEf^{\prime}_{0}\), and \(T^{*}\) will decrease.

Finally, the changes in mean and variance profit function with respect to weight parameter \(\omega_{1} \left( { = 1 - \omega_{2} } \right)\) are illustrated in Fig. 5. From this figure, when \(\omega_{1} \left( { = 1 - \omega_{2} } \right)\) increases from 0.55 to 0.85, both objective functions will increase; however, when \(\omega_{1} \left( { = 1 - \omega_{2} } \right)\) increases from 0.85 to 1, both objective functions will decrease. The optimal values of objective functions as shown in Tables 4, 5, and 6 are with \(\omega_{1} = 0.9\) and \(\omega_{2} = 0.1\). Decision makers may decide weight parameters to be given for mean and variance profit functions with respect to these sensitivity analyses.

Fig. 5
figure5

Effect of weight parameter \(\omega_{1}\) on mean and variance profit function

Conclusions

In this study, for first time a sustainable EOQ model for multi-items with hybrid cost parameters and fuzzy resources has been developed under permissible delay in payments incorporating marketing expenditure—service expenditure—selling price dependent demand. Under carbon tax regulation, the aim of this study is to determine the optimal payment time, service and marketing expenditure, selling price, and inventory replenishment strategies in order to maximize retailer’s profit function. The proposed problem has been formulated as fuzzy signomial geometric programming model and solved using a global optimization approach. Efficiency of this algorithm is tested and compared with multi-objective genetic algorithm, multi-objective genetic algorithm with varying population, and hybrid heuristic algorithm. Finally, several numerical examples and a sensitivity analysis of the main parameters have been provided to demonstrate the formulated model. Results of sensitivity analysis have provided the valuable references to decision makers in planning service and marketing activities, controlling inventory level, and also reducing carbon emissions simultaneously. For example, when credit period increases, the optimal marketing and service expenditure, selling price, and mean total profit will decrease and the amount of carbon emissions, order quantity, and demand will increase, but when credit period increase from 0 to 0.12, the optimal cycle time and payment time will increase; however, when credit period increases from 0.12 to 0.2, the optimal payment time is equal credit period and the optimal cycle time is equal to 1 and remains constant. Moreover, an increase in carbon tax and interest payable reduces the amount of carbon emissions and payment time.

There are some opportunities for further research. First, we consider only carbon tax regulation. One may develop the problem by considering other carbon emission policies. Secondly, we assume that lead time is equal to zero. One may extend the model with non -zero lead time. Third, this study has been formulated based on the EOQ model. One may generalize the proposed EOQ model to economic production quantity model with scrap and rework.

Notes

  1. 1.

    DD = The number of decision variables + the numbers of terms in objective functions and constraints − 1.

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Appendices

Appendix 1: Hybrid numbers

Let \(\underline{{\tilde{R}}} = \left( {\tilde{R},\underline{r} } \right)\) be a hybrid number, which the couple \(\left( {\tilde{R},\underline{r} } \right)\) displays the addition to a fuzzy number with a random variable without changing characteristic of each one and without reducing the amount of variable information, where \(\tilde{R}\) is a fuzzy number and \(\underline{r} = \left( {\mu ,\sigma^{2} } \right)\) is the random variable with density function \(f_{{\underline{r} }} \left( r \right)\) whose mean and variance are \(\mu\) and \(\sigma^{2}\) respectively. In this paper, we denote the couple \(\left( {\tilde{R},\underline{r} } \right)\) by the symbol \(\tilde{R}( + )^{\prime } \underline{r}\).

Appendix 2: Transforming SGP problems into a series of standard GP problems

As mentioned earlier, a global optimization method is applied for solving SGP problem proposed in Steps 1, 2, and 5. We first present a SGP problem, and then explain this approach in detail for transforming the SGP problem to a series of standard GP problems according to the type of the proposed problems.

A SGP program is equal to an optimization problem as follows

$$\begin{aligned} & \hbox{min} \,\,\xi_{0} (y) = \sum\limits_{k = 1}^{{n_{0} }} {\theta_{0k} c_{0k} } \prod\limits_{i = 1}^{m} {y_{i}^{{\alpha_{oik} }} } \quad \,c_{0k} > 0, \, \theta_{0k} = \pm 1 \\ & {\text{s}} . {\text{t}} .\quad \xi_{j} (y) = \sum\limits_{k = 1}^{{n_{j} }} {\theta_{jk} c_{jk} } \prod\limits_{i = 1}^{m} {y_{i}^{{\alpha_{jik} }} \le 1} \quad c_{jk} > 0, \, \theta_{jk} = \pm 1, \, \alpha_{jik} \in R,j = 1,2, \ldots ,t \\ & \quad \quad y_{i} > 0,i = 1,2, \ldots ,m, \\ \end{aligned}$$
(43)

where \(n_{j} \left( {j = 0,1,2, \ldots ,t} \right)\) shows the number of elements of the objective function and restrictions, and \(\xi_{j} (y)\) is Signomial function.

Global optimization approach

This method defines all functions \(\xi_{j}\) as:

$$\xi_{j} (y) = \xi_{j}^{ + } (y) - \xi_{j}^{ - } (y)\quad j = 0,1,2, \ldots ,t$$
(44)

where \(\xi_{j}^{ - } (y)\) and \(\xi_{j}^{ + } (y)\) are formulated as:

$$\xi_{j}^{ + } (y) = \sum\limits_{k = 1}^{{n_{j} }} {\theta_{jk} c_{jk} } \prod\limits_{i = 1}^{m} {y_{i}^{{\alpha_{jik} }} } \quad \theta_{jk} = + 1,\,j = 0,1,2, \ldots ,t$$
(45)
$$\xi_{j}^{ - } (y) = \sum\limits_{k = 1}^{n_{j} } {\theta_{jk} c_{jk} } \prod\limits_{i = 1}^{m} {y_{i}^{{\alpha_{jik} }} } \quad \theta_{jk} = - 1, \, j = 0,1,2, \ldots ,t$$
(46)

Next it defines a large number, \(L > 0\), so that \(\xi_{0}^{ + } (y) - \xi_{0}^{ - } (y) + L > 0\) and rewrites the model (43) as the following problem:

$$\hbox{min} \,\,\,\xi_{0} (y) = \xi_{0}^{ + } (y) - \xi_{0}^{ - } (y) + L$$
(47)
$${\text{s}} . {\text{t}} .\quad \xi_{j}^{ + } (y) - \xi_{j}^{ - } (y) \le 1$$
(48)
$$y_{i} > 0,\quad j = 1,2, \ldots ,t,\,\,i = 1,2, \ldots ,m$$
(49)

The model (43) converts to the following optimization problem, by introducing an extra variable \(y_{0}\) in order to express restrictions and objective function as quotient and linear form, respectively.

$$\hbox{min} \,\,\,y_{0}$$
(50)
$${\text{s}} . {\text{t}} .\quad \frac{{\xi_{0}^{ + } (y) + L}}{{\xi_{0}^{ - } (y) + y_{0} }} \le 1$$
(51)
$$\frac{{\xi_{j}^{ + } (y)}}{{\xi_{j}^{ - } (y) + 1}} \le 1\quad j \in j_{1}$$
(52)
$$\frac{{\xi_{j}^{ + } (y)}}{{\xi_{j}^{ - } (y) + 1}} \le 1\,\quad \,j \in j_{2}$$
(53)
$$y_{i} > 0,\quad j = 1,2, \ldots ,t,i = 1,2, \ldots ,m$$
(54)

where \(j_{1} = \{ \left. j \right|\xi_{j}^{ - } (y) + 1\,\) are monomials \(\}\) and \(j_{2} = \left\{ {\left. j \right|j \notin j_{1} } \right\}\). In the above model the objective function (50) is a posynomial function, restriction (54) is a posynomial inequality that both equations are allowable in standard GP problem, but restrictions (51)–(53) are not permitted in a standard GP problem. So this method used from arithmetic–geometric mean approximation to approximate every denominator of restrictions (51)–(53) with monomial functions as follows:

$$f(y) \ge \hat{f}(y) = \prod\limits_{u} {\left( {\frac{{v_{u} (y)}}{{\omega_{u} (x)}}} \right)}^{{\omega_{u} (x)}}$$
(55)

where the parameters \(\omega_{u} (x)\) can be computed as:

$$\omega_{u} (x) = \frac{{v_{u} (x)}}{f(x)}\quad \forall u$$
(56)

where \(f(y) = \sum\nolimits_{u} {v_{u} (y)}\) is a posynomial function, \(v_{u} (y)\) are monomial terms, and \(x > 0\) is a fixed point. Using the proposed monomial approximation approach to every denominator of restrictions (51)–(53), finally we have:

$$\hbox{min} \,\,\,y_{0}$$
(57)
$${\text{s}} . {\text{t}} .\quad \frac{{\xi_{0}^{ + } (y) + L}}{{\xi_{0}^{ - } (y,y_{0} )}} \le 1$$
(58)
$$\frac{{\xi_{j}^{ + } (y)}}{{\xi_{1j}^{ - } (y)}} \le 1\quad j \in j_{1}$$
(59)
$$\frac{{\xi_{j}^{ + } (y)}}{{\xi_{2j}^{ - } (y)}} \le 1\quad \,j \in j_{2}$$
(60)
$$y_{i} > 0,\quad j = 1,2, \ldots ,t,\;\;i = 1,2, \ldots ,m$$
(61)

where \(\xi_{0}^{ - } (y,y_{0} )\), \(\xi_{1j}^{ - } (y)\), and \(\xi_{2j}^{ - } (y)\) are the corresponding monomial functions approximated using Eq. (55). Now, the problem (57)–(61) is a standard geometric programming model than can be optimized efficiently using GGPLAB solver in MATLAB (Mutapcic et al. 2006). So, the proposed algorithm can be summarized as an iterative algorithm as follows:

Step 0 Select an initial solution for decision variables \(y_{0}\) and \(y\),\(y_{0}^{(0)}\) and \(y^{(0)}\) respectively. Consider a solution accuracy \(\varepsilon > 0\) and put iteration counter \(r = 0\).

Step 1 In iteration \(r\), calculate the monomial components in the denominator posynomials of Eqs. (51)–(53) by the determined \(y_{0}^{(r - 1)}\) and \(y^{(r - 1)}\). Calculate their corresponding parameters \(\omega_{u} (y_{0}^{(r - 1)} ,y^{(r - 1)} )\) using Eq. (56).

Step 2 Do the condensation on the denominator posynomials of Eqs. (51)–(53) using Eq. (55) by parameters \(\omega_{u} (y_{0}^{(r - 1)} ,y^{(r - 1)} )\).

Step 3 Solve the standard GP (57)–(61) to obtain \(\left( {y_{0}^{(r)} ,y^{(r)} } \right)\).

Step 4 If \(\left\| {y^{(r)} - y^{(r - 1)} } \right\| \le \varepsilon\), then stop. Else \(r = r + 1\) and return to Step 1.

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Jabbarzadeh, A., Aliabadi, L. & Yazdanparast, R. Optimal payment time and replenishment decisions for retailer’s inventory system under trade credit and carbon emission constraints. Oper Res Int J 21, 589–620 (2021). https://doi.org/10.1007/s12351-019-00457-5

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Keywords

  • Inventory model
  • Signomial geometric programming
  • Optimal payment time
  • Delayed payments
  • Carbon emissions