Fuzzy demand three-warehouse deteriorating inventory model with carbon trading and trade credit

Introduction

Classical inventory models from the economic order quantity of Harris (1913) [7] to its many deteriorating-item, multi-warehouse, and financially augmented descendants are formulated under the convenient but restrictive assumption that demand parameters are known with certainty. In practice, a retailer estimating the scale and growth rate of ramp-type demand for a new or seasonal product relies on managerial judgment, limited historical data, and market forecasts that are inherently imprecise rather than genuinely random. Probability theory requires a fully specified distribution that is rarely available in such settings, whereas fuzzy set theory (Zadeh, 1965) [15] offers a natural language for encoding expert judgment of the form "demand will most likely be around a₂ units per week, almost certainly no lower than a₁ and no higher than a₃" precisely the semantics of a Triangular Fuzzy Number (TFN). This paper develops a fuzzy extension of the three-warehouse non-instantaneously deteriorating (NID) inventory framework studied for crisp ramp-type iso-elastic demand, Exponential partial backlogging, carbon cap and trade regulation, and two-part trade credit. Rather than treating fuzziness as a peripheral robustness check, we develop it as the organizing methodological contribution of the paper, we prove that the fuzzy total profit function is itself a triangular fuzzy number (Proposition 1), we prove a certainty equivalence result showing exactly when and why the signed distance-optimal decision coincides with the crisp modal optimum (Proposition 2), we establish existence of an interior optimal price (Proposition 3), and we verify second-order optimality via a signed, alternating Hessian (Proposition 4). We further introduce a risk sensitive extension using Yager's (1981) [13] optimism-pessimism ranking index, which unlike the signed-distance criterion is shown numerically to shift the optimal preservation investment and selling price as the decision maker's risk attitude changes. The remainder of the paper is organised as follows. Section 2 reviews the relevant literature on fuzzy inventory control and situates the contribution. Section 3 develops the fuzzy-set preliminaries used throughout. Section 4 lists notation and assumptions. Section 5 develops the mathematical model. Section 6 states and proves the main propositions. Section 7 gives the solution algorithm. Section 8 reports a numerical illustration, cost breakdown, and multi-dimensional sensitivity analysis. Section 9 discusses managerial insights, and Section 10 concludes. Zadeh (1965) [15] introduced fuzzy set theory as an alternative to probability for representing imprecise, linguistically expressed information, and Kaufmann and Gupta (1991) [8] developed the associated fuzzy arithmetic that underlies most subsequent fuzzy-EOQ work. Yao and Lee (1996) and Lee and Yao (1998) [14] were among the first to formulate economic order quantity and economic production quantity models with fuzzy demand and fuzzy production rate, defuzzifying the resulting fuzzy total cost using the extension principle and the graded mean integration representation. Chang, Yao and Ouyang (2006) [14] extended the fuzzy treatment to mixture inventory models with variable lead time, comparing several defuzzification indices. De and Sana (2013) developed an intuitionistic fuzzy backlogging EOQ model with promotional-effort and price-sensitive demand, illustrating how fuzzy and intuitionistic fuzzy representations can be embedded directly inside optimisation rather than applied only as a posthoc sensitivity device.

Literature Review 2.1 Fuzzy set theory in inventory control

Fuzzy deteriorating and multi-warehouse inventory models

Halim, Giri and Chaudhuri (2008) formulated a fuzzy economic order quantity model for perishable items with stochastic demand, partial backlogging, and a fuzzy deterioration rate, showing that fuzzifying the deterioration rate alone materially widens the profit ambiguity band relative to fuzzifying demand alone. Mahata and Goswami (2013) [10] considered fuzzy EOQ models for deteriorating items with stock-dependent demand and non-linear holding cost, using the signed distance method for defuzzification the same method adopted in the present paper for its analytical tractability and its property (proved here for the first time in the ramp-type or three-warehouse setting) of reducing to the crisp modal solution under symmetric fuzziness. Multiwarehouse extensions incorporating fuzziness remain comparatively rare, most two and three-warehouse deteriorating-item models (Yang 2004, 2006, Rong et al., 2008, Kumar and Mahapatra, 2021) retain crisp demand, leaving the interaction between warehouse-sequencing decisions and demand ambiguity largely unexplored.

Ranking and Defuzzification Methods

Yager (1981) [13] proposed a family of ranking indices for ordering fuzzy subsets of the real line, including the optimismpessimism weighted index used in Section 6.4 of this paper, which nests the pessimistic (worst-case, λ=0), the signed distanceequivalent risk-neutral (λ=0.5, under symmetry), and the optimistic (best-case, λ=1) rankings as special cases. The signed-distance method used as the primary defuzzification criterion in this paper was developed for triangular and trapezoidal fuzzy numbers and has been widely adopted in fuzzy inventory optimisation (Yao and Lee, 1996, Mahata and Goswami, 2013) [10,14] because it preserves linearity, the signed distance of an affine combination of fuzzy numbers equals the affine combination of their signed distances, a property exploited directly in Proposition 1 below.

Positioning of the present study

The crisp antecedent of the present model integrates ramp-type iso-elastic demand, exponential backlogging, carbon cap-andtrade, and two-part trade credit within a three-warehouse NID framework. To the best of the authors' knowledge, no prior study has (i) fuzzified the ramp-demand parameters within a three-warehouse NID system carrying all four of these features simultaneously, (ii) proved that the resulting fuzzy profit is itself triangular and derived a certainty equivalence condition under signed distance defuzzification, or (iii) contrasted a risk-neutral defuzzification criterion against a risk-sensitive Yager-index criterion within this class of models. This paper fills that gap.

Fuzzy set preliminaries Definition 1 (Triangular Fuzzy Number).

A triangular fuzzy number 𝑎𝑎 � = (𝑎𝑎 1 , 𝑎𝑎 2 , 𝑎𝑎 3 ), with 𝑎𝑎 1 ≤ 𝑎𝑎 2 ≤ 𝑎𝑎 3 , is a fuzzy subset of ℝ with membership function

The value 𝑎𝑎 2 is the mode (most-likely value), and [𝑎𝑎 1 , 𝑎𝑎 3 ]is the support (the extreme range regarded as possible). ). These rules follow from the extension principle applied to monotonic functions and are used repeatedly below because every term of the total-profit function is, for fixed decision variables, an affine (monotonic, order-preserving) function of the pair �𝑎𝑎 � , 𝑏𝑏 � �.

Definition 3 (Signed Distance)

For a TFN𝑣𝑣 � = (𝑣𝑣 1 , 𝑣𝑣 2 , 𝑣𝑣 3 ), the signed distance from 𝑣𝑣 �to the origin (Yao and Lee, 1996) [14] is

The signed distance is a linear functional on the space of TFNs: 𝑑𝑑 (𝑘𝑘 1 , 𝑢𝑢 � + 𝑘𝑘 2 𝑣𝑣 � 0) = 𝑘𝑘 1 𝑑𝑑(𝑢𝑢 � , 0) + 𝑘𝑘 2 𝑑𝑑(𝑣𝑣 � , 0), for scalars 𝑘𝑘 1 , 𝑘𝑘 2 and TFNs𝑢𝑢 �, 𝑣𝑣 �. This linearity is the analytical device underlying Proposition 2.

Definition 4 (Yager's Optimism-Pessimism Index)

, 𝑣𝑣 2 , 𝑣𝑣 3 )and optimism parameter𝜆𝜆 ∈ [0 , 1], Yager's (1981) [13] weighted ranking index used here is 𝑅𝑅 𝜆𝜆 (𝑣𝑣 �) = 𝜆𝜆𝑣𝑣 3 + (1 -𝜆𝜆)𝑣𝑣 1 , which ranks 𝑣𝑣 � using only it's optimistic (𝑣𝑣 3 ) and pessimistic (𝑣𝑣 1 ) extremes, weighted by the decision-maker's risk attitude𝜆𝜆. Here, 𝜆𝜆 = 0is the max-min (pessimistic/worst-case) criterion, 𝜆𝜆 = 1is the max-max (optimistic/best-case) criterion, and 𝜆𝜆 = 0.5places equal weight on both extremes while ignoring the mode. (ii) Fuzzy demand, 𝐷𝐷 � (𝑡𝑡 , 𝑝𝑝) = �𝑎𝑎 � + 𝑏𝑏 � 𝑡𝑡�𝑝𝑝 -𝜂𝜂 , for 𝐼𝐼(𝑡𝑡) ≥ 0; during shortage the effective withdrawal rate is, 𝐵𝐵(𝑡𝑡) ⋅ 𝐷𝐷 � (𝑡𝑡 , 𝑝𝑝) with exponential backlogging fraction 𝐵𝐵(𝑡𝑡) = 𝑒𝑒 -𝜅𝜅(𝑇𝑇-𝑡𝑡) .

Notation and Assumptions

(iii) NID structure: items placed in a warehouse begin deteriorating only after the freshness period 𝑡𝑡 𝑑𝑑 has elapsed since that warehouse began serving demand; preservation investment 𝜁𝜁 scales the deterioration rate down via the multiplier𝑚𝑚(𝜁𝜁).

(iv) Depletion priority RW1 → RW2 → OW (the highest holding-cost warehouse is depleted first). (v) Each warehouse's order-up-to quantity 𝑄𝑄 1 , 𝑄𝑄 2 , 𝑄𝑄 3 is chosen so that it exactly reaches zero at its designated depletion time �𝑡𝑡 1 , 𝑡𝑡 2 , 𝑡𝑡 3 respectively�, subject to the capacity upper bounds𝑁𝑁 1 , 𝑁𝑁 2 , 𝑁𝑁 3 .

(vi) Total emissions,𝐸𝐸 = 𝑒𝑒 0 𝐿𝐿 are compared with the cap 𝐶𝐶 ̅ under a market-based cap-and-trade instrument; the net cost, 𝑝𝑝 𝑐𝑐 (𝐸𝐸 -𝐶𝐶 ̅ ) may be negative (a credit) if 𝐸𝐸 < 𝐶𝐶 ̅ .

(Vii) Two-part trade credit, Case𝑇𝑇 > 𝑀𝑀.

(viii) All costs are discounted at continuous inflation rate𝑟𝑟; deteriorated units are written off.

(ix) 𝑎𝑎 �And 𝑏𝑏 � are triangular fuzzy numbers with 𝑎𝑎 1 ≤ 𝑎𝑎 2 ≤ 𝑎𝑎 3 and 𝑏𝑏 1 ≤ 𝑏𝑏 2 ≤ 𝑏𝑏 3 , independent of 𝑡𝑡and𝑝𝑝.

Fuzzy model formulation 5.1 Fuzzy inventory dynamics

For a fixed representative pair (𝑎𝑎 , 𝑏𝑏) later taken to be each of(𝑎𝑎 1 , 𝑏𝑏 1 ), (𝑎𝑎 2 , 𝑏𝑏 2 ), (𝑎𝑎 3 , 𝑏𝑏 3 )in turn the inventory dynamics replicate the crisp NID structure with demand 𝐷𝐷 𝑡𝑡 = (𝑎𝑎 + 𝑏𝑏𝑡𝑡)𝑝𝑝 -𝜂𝜂 .

With each warehouse's order-up-to quantity fixed by the terminal condition that it depletes to exactly zero at its own designated time. Solving these linear first-order ODEs (identical technique to the crisp case) gives, with the auxiliary functions

the required order-up-to quantities

Each 𝑄𝑄 𝑖𝑖 and 𝑅𝑅is manifestly an affine function of (𝑎𝑎 , 𝑏𝑏)for fixed (𝑡𝑡 1 , 𝑡𝑡 2 , 𝑡𝑡 3 , 𝑝𝑝 , 𝜁𝜁) a fact used directly in Proposition 1.

Cost and Revenue Components

With all costs discounted at continuous inflation rate 𝑟𝑟, the ordering, purchasing, holding, deterioration, shortage, transportation, carbon cap and trade, preservation, trade-credit, and revenue components are computed exactly as in the crisp model, using the fuzzy-parametrised 𝑄𝑄 𝑖𝑖 , 𝑅𝑅, 𝐿𝐿 above:

Since these holds for every 𝑥𝑥in the feasible region, the maximiser of 𝐷𝐷(⋅) over 𝑥𝑥 coincides exactly with the maximiser of 𝑇𝑇𝑃𝑃(⋅ , 𝑎𝑎 (𝑎𝑎 2 + 𝑏𝑏 2 𝑡𝑡) 𝑑𝑑𝑡𝑡 is strictly positive.

Proof (sketch)

Collecting all 𝑝𝑝-dependent terms of 𝑇𝑇𝑃𝑃(𝑝𝑝), every cost term other than the 𝜁𝜁-and 𝑡𝑡-dependent constants (𝑂𝑂𝐶𝐶, 𝑃𝑃𝑇𝑇𝐼𝐼, and the constant part of 𝑇𝑇𝐶𝐶) scales as either 𝑝𝑝 1-𝜂𝜂 (revenue 𝑆𝑆𝑅𝑅and interest-earned 𝐼𝐼𝐸𝐸) or 𝑝𝑝 -𝜂𝜂 (purchasing 𝑃𝑃𝐶𝐶, holding, deterioration, shortage, interest-charged 𝐼𝐼𝐶𝐶, and the 𝐿𝐿-proportional part of 𝑇𝑇𝐶𝐶, since 𝑄𝑄 𝑖𝑖 , 𝑅𝑅 ∝ 𝑝𝑝 -𝜂𝜂 ). Writing 𝑇𝑇𝑃𝑃(𝑝𝑝) = 𝑝𝑝 1-𝜂𝜂 𝐾𝐾 𝑅𝑅 𝑇𝑇 -𝑝𝑝 -𝜂𝜂 𝐾𝐾 𝐶𝐶 -𝐹𝐹, with 𝐾𝐾 𝐶𝐶 > 0 collecting all 𝑝𝑝 -𝜂𝜂 -scaled cost coefficients and 𝐹𝐹the 𝑝𝑝-independent constant terms, we obtain

Setting this to zero gives the unique interior critical point

, Which is strictly positive and finite whenever 𝜂𝜂 > 1 and 𝐾𝐾 𝑅𝑅 , 𝐾𝐾 𝐶𝐶 > 0, both guaranteed under Assumptions (ii) and positivity of𝑎𝑎 2 ,𝑏𝑏 2 ,𝑐𝑐.

The second derivative at 𝑝𝑝 * is negative (direct differentiation of the bracketed term, which is strictly decreasing in 𝑝𝑝since𝜂𝜂 > 1), so 𝑝𝑝 * is a local maximiser. Joined with 𝐶𝐶 2 continuity and the sign pattern of negative definite equivalently, its five leading principal minors alternate in sign starting negative,

Proof. This is the standard second-order sufficient condition for unconstrained optimisation, applied to the 𝐶𝐶 2 function 𝐷𝐷(⋅)(𝐷𝐷 inherits smoothness from the underlying exponential and polynomial building blocks of Section 5, away from the boundary 𝑡𝑡 𝑑𝑑 = 𝑡𝑡 1 or 𝑡𝑡 1 = 𝑡𝑡 2 , etc.). A symmetric matrix is negative definite if and only if its eigenvalues are all negative, which holds if and only if its leading principal minors alternate in sign beginning with a negative first minor a standard linear-algebra equivalence (Sylvester's criterion applied to -𝐻𝐻). Section 8.2 verifies this condition numerically at the reported optimum: all five eigenvalues of 𝐻𝐻(𝑥𝑥 * ) are strictly negative, and the leading principal minors are (Δ 1 , … , Δ 5 ) = (-, 45.95 , , 1247.21 , -204042.24 259825.73 -434.34), exhibiting exactly the required alternating pattern, confirming that 𝑥𝑥 * is a strict local (and, given the unimodal numerical profiles of Section 8.3, the global) maximiser.

Solution Algorithm

Step 1: Input crisp parameters and the fuzzy triples�𝑎𝑎 � , 𝑏𝑏 � �.

Step2: For any candidate 𝑥𝑥 = (𝑡𝑡 (𝑇𝑇𝑃𝑃 𝐿𝐿 + 2𝑇𝑇𝑃𝑃 𝑀𝑀 + 𝑇𝑇𝑃𝑃 𝑈𝑈 ).

Step 3: Solve the five first-order conditions

Simultaneously (Newton-Raphson or a derivative-free simplex method) to obtain a candidate 𝑥𝑥 * .

Step 4: Verify the second-order sufficiency condition of Proposition 4 (signed, alternating leading principal minors of the Hessian of 𝐷𝐷at𝑥𝑥 * ).

Step 5: Compute 𝐿𝐿 * , 𝑄𝑄 1

Base Parameter Setting

All computations below were carried out by direct numerical implementation of the closed-form expressions of Section 5 (quadrature for the residual definite integrals, Nelder-Mead simplex search from multiple starting points for the optimisation, and central finite differences for the Hessian), so that every reported figure is internally consistent and independently reproducible. As λ increases from 0 to 1 i.e., as the planner's stance shifts from pessimistic to optimistic the optimal preservation investment ζ* rises monotonically (from $184.47 to $197.25 per cycle, +6.9%), while the optimal price p* falls slightly (from $56.48 to $56.44). The economic logic is that an optimistic planner, anticipating the higher end of the demand band, finds it worthwhile to invest more in preservation technology to protect a larger expected throughput against deterioration, and can sustain a marginally lower price because higher anticipated volume compensates the margin. The depletion times (t₁*,t₂*,t₃*) are essentially insensitive to λ, indicating that warehouse sequencing decisions are governed primarily by relative holding/deterioration costs rather than by risk attitude, whereas the continuous investment variable ζ absorbs most of the risk-attitude adjustment.

Symbol

Parameter Sensitivity Analysis

Each parameter is varied by -20%, -10%, 0, +10%, +20% about its base value (Table 1) while holding all others fixed, with reoptimisation of (t₁, t₂, t₃, p, ζ) at each setting and a fixed fuzzy spread of 10%. Table 4C. Because the firm operates below its emission cap (E*<C ̄) at the optimum, it is a net seller of carbon allowances, a higher carbon-trading price p c therefore increases carbon-credit revenue and is profit-enhancing (+0.5% for p c +20%) rather than cost-inducing the opposite qualitative direction from a firm operating above its cap, and a direct managerial signal that carbon policy exposure should be assessed relative to the firm's own emissions to cap ratio, not treated as a uniformly negative cost shock. 2), consequently profit is essentially insensitive to β₁ at the optimum, with the model responding mainly through a small compensating increase in ζ* rather than through a change in t₁*. ) is optimal regardless of how wide the demand ambiguity band is. This is reassuring for practitioners: elaborate fuzzy elicitation exercises need not be repeated every planning cycle purely to re-derive the operating policy, though they remain valuable for the risk diagnostics of point (ii).

Change

Change in M

(ii) Profit-at-risk quantification. The width of the profit band grows linearly with the fuzzy spread and provides a directly interpretable profit-at-risk metric: at a 20% symmetric demand spread, weekly profit could plausibly range from about $2 254 to $3 069 around a central estimate of $2 662 information a crisp model cannot supply, and which is useful for cash-flow planning, covenant compliance, or communicating forecast uncertainty to stakeholders. (iii) Risk attitude enters through preservation investment, not warehouse timing. The Yager-index analysis (Section 8.5) shows that as planners become more optimistic about demand, the single lever that responds materially is preservation technology investment ζ*, which should be increased to protect the larger anticipated throughput from deterioration, selling price and warehouse depletion timing are comparatively insensitive to risk attitude and can be set from the risk-neutral (signed-distance) solution with little loss. (iv) Carbon policy exposure is directionally firm-specific. Because the optimal solution keeps emissions below the regulatory cap, the firm is a net seller of carbon allowances, and rising carbon prices are profit-enhancing rather than cost-inducing (Table 4C) a reminder that the sign of a firm's exposure to carbon price risk depends on its emissions to cap ratio at the optimum, not on the mere presence of a cap and trade regime. (v) Supplier credit negotiation remains a high-value, low-risk lever. Consistent with the crisp literature, extending the supplier credit period M is unambiguously profit-improving and is nearly costless to implement relative to operational changes, making it a priority item for supply-chain finance negotiations.

(vi) RW1 should be run down quickly. The optimal RW1 depletion time (t₁*≈2.05 weeks) sits close to the freshness period t d =1.5 weeks, meaning the model recommends depleting the costliest and most deterioration exposed warehouse rapidly once its freshness window closes, consistent with a "deplete the-most expensive-tier first" heuristic that generalises the two-warehouse RW before OW rule to a three-tier hierarchy.

Conclusion

This paper has developed a fuzzy three-warehouse inventory model for non-instantaneously deteriorating items with ramp-type iso-elastic demand under triangular fuzzy scale and growth parameters, integrated with exponential time-dependent backlogging, carbon cap and trade, two-part trade credit, and continuous inflation. Its principal methodological contributions are: (a) a proof that the fuzzy total profit is itself a triangular fuzzy number, obtained simply by evaluating the crisp profit at the three defining demand triples (Proposition 1), (b) a certainty equivalence theorem showing that, under symmetric fuzzification, signed-distanceoptimal decisions coincide exactly with the crisp modal optimum, so that fuzziness manifests purely as a profit-ambiguity band rather than as a shift in policy (Proposition 2), (c) existence of an interior optimal price via the classical iso-elastic markup argument (Proposition 3), and (d) numerically verified second-order optimality via a negative-definite, sign alternating Hessian (Proposition 4). A Yager-index extension shows that risk-sensitive planners can and should deviate from the risk-neutral policy, primarily through the preservation-investment decision. The numerical study confirms an economically sensible, capacity feasible optimum with well-behaved comparative statics, and a five-parameter sensitivity analysis together with a tornado chart identifies ramp-demand growth and supplier credit period as the most actionable levers, while showing that carbon-price and deteriorationrate exposure are comparatively benign at the optimum given the firms below cap emissions position. Future research directions include: (i) extending the fuzzy treatment to trapezoidal or intuitionistic fuzzy numbers to separate ambiguity from hesitancy, (ii) fuzzifying the deterioration rates and preservation-effectiveness function alongside demand, as in Halim et al. (2008) [6] , to compare the relative contribution of each source of imprecision to the profit-ambiguity band, (iii) developing a fully Pareto-style bi-criterion formulation that simultaneously reports the signed-distance profit and the Yager-index profit rather than requiring the planner to pre-commit to λ, (iv) relaxing the fixed depletion-priority assumption (RW1→RW2→OW) to allow the sequencing itself to be optimised jointly with (t₁, t₂, t₃, p, ζ), and (v) validating the ramp-demand and fuzzy-parameter specification against real point of sale data using fuzzy regression or fuzzy time-series estimation.

Declaration of Competing Interest

The author(s) declare no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

1. Chang HJ, Yao JS, Ouyang LY. Fuzzy mixture inventory model with variable lead-time based on probabilistic fuzzy set and triangular fuzzy number. Math Comput Model. 2006; 43(1-2):172-188.
2. Chern MS, Yang HL, Teng JT, Papachristos S. Partial backlogging inventory lot-size models for deteriorating items with fluctuating demand under inflation. Eur J Oper Res. 2008; 191(1):127-141.
3. De SK, Sana SS. Backlogging EOQ model for promotional effort and selling price sensitive demand an intuitionistic fuzzy approach. Ann Oper Res. 2013;233:57-76.
4. Ghare PM, Schrader GF. A model for an exponentially decaying inventory. J Ind Eng. 1963;14:238-243.
5. Goyal SK. Economic order quantity under conditions of permissible delay in payments. J Oper Res Soc. 1985;36(4):335-338.
6. Halim KA, Giri BC, Chaudhuri KS. Fuzzy economic order quantity model for perishable items with stochastic demand, partial backlogging and fuzzy deterioration rate. Int J Oper Res. 2008;3(1-2):77-96.
7. Harris FW. How many parts to make at once. Factory Mag Manag. 1913;10(2):135-136.
8. Kaufmann A, Gupta MM. Introduction to fuzzy arithmetic: theory and applications. New York: Van Nostrand Reinhold; 1991.
9. Lee HM, Yao JS. Economic production quantity for fuzzy demand quantity and fuzzy production quantity. Eur J Oper Res. 1998;109(1):203-211.
10. Mahata GC, Goswami A. Fuzzy EOQ models for deteriorating items with stock dependent demand and non-linear holding costs. Int J Appl Comput Math. 2013;1(2):187-201.
11. Nahmias S. Perishable inventory theory: a review. Oper Res. 1982;30(4):680-708.
12. Routray AK, Dehury S, Sahu SK. Three-warehouse inventory model for non-instantaneously deteriorating items with ramptype demand, carbon cap and trade, exponential back and two-part trade credit under inflation. Asian J Probab Stat. 2026;28(6):43-64. doi:10.9734/ajpas/2026/v28i6905.
13. Yager RR. A procedure for ordering fuzzy subsets of the unit interval. Inf Sci. 1981;24(2):143-161.
14. Yao JS, Lee HM. Fuzzy inventory with backorder for fuzzy order quantity. Inf Sci. 1996;93(3-4):283-319.
15. Zadeh LA. Fuzzy sets. Inf Control. 1965;8(3):338-353.
16. Zhou YW, Chen C, Li C, Zhong Y. A synergic economic order quantity model with trade credit, shortages, imperfect quality and inspection errors. Appl Math Model. 2016;40(2):1012-1028.