In general, the classical inventory problems are designed by considering that the costs of items are constants. But in practical situation, for the owner of the store, some costs are a function of some other variables as quantity (Q) or a length of the cycle (N) …etc., so decision makers is sometimes take into account that some costs are varying.

Many researchers studied probabilistic periodic review model. For example, Sqren and Roger  developed the model with continuous demand and lost sales. Iida  studied the non-stationary model with uncertain production capacity and uncertain demand. Chiang  presented optimal replenishment for the model with two supply modes. Fergany  developed the multi-item inventory model with zero lead-time under constraints and varying ordering cost using Lagrange multiplier technique.

The cost parameters in real inventory systems and other parameters such as price, marketing and service elasticity of demand are imprecise and uncertain in nature. This uncertainty applied the notion of fuzziness. Since the proposed model is in a fuzzy environment, a fuzzy decision should be made to meet the decision criteria, and the results should be fuzzy as well. Many researchers introduced fuzzy sets and its application as a mathematical way of representing impreciseness or vagueness in everyday life. For example, Dubois and Prade  presented a theory and application for fuzzy sets and system. Dey and Chakraborty  introduced fuzzy periodic review system with fuzzy random variable demand.

For the importance of the deterioration in many inventory systems it has been investigated in the inventory science the last twenty years ago. Deterioration is deﬁned as decay, damage, dryness and spoilage. An item loses its utility and becomes useless in the process. Many researchers studied the deterioration in items. For example, Chern et al.  explained an inventory models for deteriorating items with fluctuating and inflation demand with partial backorder shortage. Thangam and Uthayakumar  studied an inventory model for deteriorating items with inflation induced demand and exponential partial backorders. Priya et al.  introduced an inventory model for deteriorating items with exponential demand and time varying holding cost. Maragatham and Lakshmidevi  explained an inventory model for non-instantaneous deteriorating items under conditions of permissible delay in payments for N cycles. Mishra et al.  introduced an inventory model for deteriorating items with time-dependent demand and time varying holding cost under partial backorder.

This study investigates probabilistic periodic review inventory model for deteriorating items under a varying deteriorating cost constraint with the following assumptions: (1) The lead-time is zero. (2) The demand is a random variable that follows Pareto distribution. (3) The Shortages are allowed and its divided into backlogged and lost sales. (4) The deterioration rate follows decreasing distribution as two-parameter exponential distribution. (5) The holding, the deterioration and the salvage costs are varying. (6) The objective function under varying deterioration constraint is imposed in both crisp and fuzzy environments. The main objective is to find the optimal values of four decision variables (the maximum inventory level, the stock-out time, the deteriorating time and the review time) which minimize the expected annual total cost under consideration. Newton's method is used to solve the nonlinear equations. The results are got by Mathematica program.

The following assumptions are made in developing the model

The replenishment rate is instantaneous (without lead-time) The deterioration rate θ(t) follows decreasing distribution as two-parameter exponential distribution.The salvage value γ, 0< γ<1 is associated to deteriorated units during the cycle. Shortages are allowed to occur (divided in backlogged and lost sales). A fraction ρ 0 ≤ ρ ≤ 1 is backorder. Then the remaining fraction 1 – ρ is lost, ρ=11+ϵN-t , 0<ϵ<1.Suppose that the demand for a particular item follows Pareto distribution such as: fx=η δηx+δη+1 , 0≤x<+∞ , η is a continuous shape parameter α>0, δ is a continuous scale parameter δ>0. The model is illustrated in Figure 1

Inventory process for the model with mixture shortage

Figure 1: Inventory process for the model with mixture shortage

Mathematical model

In the interval (0, N), the inventory level gradually decreases to meet demands. During the period 0, nd the inventory depletes due to the demand. In addition, during the period nd, n1 the inventory depletes due to the demand and deteriorating. Then during the interval [n1, N], shortage occurs and it is a mixture of backorder and lost sales. By this process, the inventory level reaches zero level at time n1 and then shortages are allowed to occur in the interval n1,N . The differential equations for the instantaneous inventory qi (t), 0<t<N are given by Equations (1), (2), (5) and (6). The boundary conditions :

q10 =Qm , q2n1= q3n1 =0 , q3N = x̅-Qm

If x≤ Qm, The rate of change of inventory during positive stock period [0,n1] is governed by the following differential equations:

d q1tdt=- D̅=-xnd for 0≤t≤nd 1

d q2tdt+θ(t) q2t=- D̅=-xn1-nd for nd≤t≤n1 2

The probability density function for two-parameter exponential distribution is given by

ft=1σe- t-μσ, μ>t>∞, σ>0. where the parameters α and δ are positive real constants.

And the cumulative probability density function is Ft=1- e-t-μσ, μ>t>∞, σ>0.

Then the instantaneous deterioration rate θt as a function of time can be given by θt=ft1-Ft=1σ , t>0

The solutions of the above differential equations after applying the boundary conditions

q1t=Qm- D̅ t=Qm- x tnd for 0≤t≤nd 3

q2t=σ x n1-nd(1-ⅇn1-tσ) for nd≤t≤n1 (4)

If x> Qm, the following relationships are evident:

Shortage period during the interval [n1 , N] the inventory level depends on the demand and a fraction of demand is backlogged and lost sales.

d q3tdt=- D̅1+ϵN-t=-x-QmN-n111+ϵN-t n1≤t≤N 5

d q4tdt=- D̅1- 11+ϵN-t=-x-QmN-n1 1- 11+ϵN-t n1≤t≤N 6

hence,

q3t=-x-Qmϵ N-n1 ln⁡1+ϵN-n1- ln⁡1+ϵN-t n1≤t≤N 7

q4t=-x-Qmϵ N-n1ϵt-n1+ln1+ϵN-t- ln1+ϵN-n1 n1≤t≤N (8)

The expected annual total cost for the cycle is composed of expected order cost, expected purchase cost, expected varying deteriorating cost, expected varying salvage cost, expected backorder cost, expected lost sales cost and expected varying holding cost

E(TC) = EOC+EPC+EDC N+EVCN+EBC+ELC+EHCN

3.1 The model for crisp environment

The expected order cost per cycle is given by

EOC=Co (9)

The expected purchase cost per cycle is given by

EPC=Cp ∫x=0∞∫0Nx fx dt dx=Cp N∫x=0∞x fx dx (10)

The expected varying deterioration cost per order is given by

EDC= Cd Nβ Qm-∫x=0Qm∫ndn1 Dt fx dt dx

=Cd NβQm-1n1-nd∫x=0Qmn1-nd x fx dx=Cd NβQm-∫x=0Qmx fx dx (11)

The expected varying salvage value of deteriorated units per order is given by

EVC=Cv NβQm-∫x=0Qmx fx dx=Cd γ NβQm-∫x=0Qmx fx dx 12

The expected backorder cost per cycle is given by

EBC=Cb ∫Qm∞∫n1N - q3t fx dtdx =Cb ∫Qm∞x-Qmϵ 1-1ϵ N-n1ln1+ϵN-n1 fx dx (13)

The expected lost sales cost per cycle is given by

ELC=CL∫x=Qm∞∫n1N- q4t fx dt dx=CL∫Qm∞x-Qmϵϵ N-n12-1-1ϵ N-n1ln1+ϵN-n1fx dx 14

The expected varying holding cost at any time per cycle is given by

EHC=Ch N-β∫0Qm∫0ndq1t dt+∫ndn1q2t dt fx dx+∫Qm∞∫n1N-q4t dt fx dx

=Ch N-β∫0QmQm nd- x nd2+σ x1+σ-σ ⅇn1-ndσn1-ndfx dx+∫Qm∞x-Qmϵϵ N-n12-1-1ϵ N-n1ln1+ϵN-n1fx dx (15)

From Equations (9), (10), (11), (12), (13), (14) and (15). The expected annual total cost:

ETCQm,nd,n1,N=Co+Cp N∫x=0∞x fx dx+Cd 1+γ NβQm-∫x=0Qmx fx dx +∫Qm∞x-Qmϵ Cb-Ch N-β-CL1-ln1+ϵN-n1ϵN-n1+Ch N-β+CLN-n1 ϵfx dx+Ch N-β∫0QmQm nd- x nd2+σ x1+σ-σ ⅇn1-ndσn1-ndfx dx (16)

Consider a limitation on the available expected varying deteriorating cost, i.e.

EDC=Cd NβQm-∫x=0Qmx fx dx≤kd (17)

It may written as: Min E TCQm,nd,n1,N

Subject to inequality constraint EDCQm,nd,n1,N≤kd

To find the optimal values Qm*, nd*,n1* and N* which minimize E(TC) under the constraint (17), the Lagrange multipliers technique with the Kuhn-Tacker conditions is used as follows: -

ELQm,nd,n1,N=Co+Cp N∫x=0∞x fx dx+∫Qm∞x-Qmϵ Cb-Ch N-β-CL1-ln1+ϵN-n1ϵN-n1+Ch N-β+CLϵ N-n12fx dx+Ch N-β∫0QmQm nd- x nd2+σ x n1-ndn1-nd+σ-σ ⅇn1-ndσfx dx+Cd 1+γ NβQm-∫x=0Qmx fx dx+λd Cd NβQm-∫x=0Qmx fx dx-Kd 18

The optimal values Qm*,n1*, nd*,N* can be calculated by setting the corresponding first partial derivatives of Equation (18) equal to zero, and then the following equations are obtained.

∂ELQm,nd,n1,N∂QmQm=Qm*=0

CL+ChN-β δη δ+Qm-η n1-N2+Cb-CL-ChN-βδη δ+Qm-ηn1-Nϵ+ln⁡1+ϵN-n1n1-Nϵ2

+NβCd+γ1+λdδ+Qm-1-ηδδ+Qmη+Qm-ηδη+δ+Qmη

+Ch N-β2n1-ndδ+Qm-1-η-2n1-ndndδδη-δ+Qmη+Qmn1-nd2ηδησ+ndη-2δη+2δ+Qmη+2ηδησ21-ⅇn1-ndσ=0 (19)

∂ELQm,nd,n1,N∂ndnd=nd*=0

-12-1+ηn1-nd2Chⅇ-ndσN-βQm+δ-ηⅇndσδ-δη+Qm+δη+Qm-2+ηδη-2-1+ηQm+δηnd2-2ⅇn1/σn1δQm+δη-ⅇndσQmηδη+δδη-Qm+δησσ1-ⅇn1-ndσ-2ndⅇndσn1δ-δη+Qm+δη+Qm-2+ηδη-2-1+ηQm+δη-ⅇn1/σδQm+δησ1-ⅇn1-ndσ+ⅇn1/σδηQmη+δσ'1-ⅇn1-ndσ+n1ⅇndσn1δ-δη+Qm+δη+Qm-2+ηδη-2-1+ηQm+δη+2ⅇn1/σδηQmη+δσ'1-ⅇn1-ndσ=0 20

∂ELQm,nd,n1,N∂n1n1=n1*=0

Cb-CL-ChN-βδηQm+δ1-η+ϵN-n11+ϵN-n1-Ln1+ϵN-n1-1+ηϵ2N-n12-CL+ChN-βδηQm+δ1-η2-1+η

+Chⅇ-ndσN-βQm+δ-ηQmηδη+δδη-Qm+δη-ⅇn1σnd+ⅇnd/σσ+ⅇn1σn1σ1-ⅇ-nd+n1σ-1+ηnd-n12=0 (21)

∂ELQm,nd,n1,N∂NN=N*=012η-12Cp δ+Ch βδηQm+δ1-ηn1-NN-1-β+δηQm+δ1-ηCL+ChN-β

+δηQm+δ1-ηCb-CL-ChN-βLn1+ϵN-n1+ϵN-n1ϵn1-N-1ϵ2η-1n1-N2

+ChβδηQm+δ-ηN-1+βη-1 Qm+δ1+Ln1+ϵN-n1ϵn1-Nϵ-12n1-ndn1-ndQm+δη2ndQmη-1-ndδ+2δσ+δηnd-Qmη-2+δ-2Qmη+δσ-2-δQm+δη+δηQmη+δσ21-ⅇn1-ndσ

+βQm+δ-ηη-1δδη-Qm+δη+Qm-Qm+δη+ηδη+Qm+δηCd+γ+λdN-1+β=0 (22)

∂ELQm,nd,n1,N∂λdλd=λd*

Nβ(Qm+δ)-η(δ(δη-(Qm+δ)η)+Qm(-(Qm+δ)η+η(δη+(Qm+δ)η)))η-1-Kd=0 (23)

If g1Qm,nd,n1,N=∂ELQm,nd,n1,N∂QmQm=Qm*=0,

g2Qm,nd,n1,N=∂ELQm,nd,n1,N∂ndnd=nd*=0,

g3Qm,nd,n1,N=∂ELQm,nd,n1,N∂n1n1=n1*=0,

g4Qm,nd,n1,N=∂ELQm,nd,n1,N∂NN=N*=0,

and g5Qm,nd,n1,N=∂ELQm,nd,n1,N∂λdλd=λd*=0

The goal is to solve the previous multivariable nonlinear system by using Newton’s method using Mathematica program. The following Algorithm is applied.

g1Qm,nd,n1,N=0, g2Qm,nd,n1,N=0,

g3Qm,nd,n1,N=0, g4Qm,nd,n1,N=0 and g5Qm,nd,n1,N=0

Step 1: Deﬁne G(y) and J(y): Let F be a function which maps Rn to Rn and The Jacobian matrix is a matrix of ﬁrst order partial derivatives without the equation of the constraint g5 to find the value of Kd as:

Gx=g1Qm,nd,n1,Ng2Qm,nd,n1,Ng3Qm,nd,n1,Ng4Qm,nd,n1,N, Jx=∂g1∂Qm∂g1∂nd∂g2∂Qm∂g2∂nd ∂g1∂n1∂g1∂N∂g2∂n1∂g2∂N∂g3∂Qm∂g3∂nd∂g4∂Qm∂g4∂nd ∂g3∂n1∂g3∂N∂g4∂n1∂g4∂N

Step 2: Let y∈Rn. Then y represents the vector Qmndn1Nλd

Step3: Assume any initial value y0 for Qm,nd,n1,N when λd=0.

Step4: Calculate G(y0 ), J(y0) and then find the inverse matrix J-1(y0), for y0

Step 5: Solve the system y1=y0-J-1y0 G(y0).

Step 6: Use the results of y1 to ﬁnd the next iterationy2 by using the same procedure.

Step 7: Keep repeating the process until finding the same results for two consecutive values of Qm,N, n1, nd. Then from Equation (17) it is possible to calculate Kd.

Step 8: Repeat step 1, 2, 3 and 4 with changing λd and adding g5Qm,nd,n1,N to the system until obtaining the same results for two consecutive values of Qm, nd, n1,N. Then these values are the optimal values of Qm*, nd*, n1*,N*.

Step 9: Thus, the optimal value of the annual expected total cost TCQm,N, n1, nd can be easily calculated.

The Model when all parameters are triangular fuzzy numbers

The inventory cost coefficients, elasticity parameters and other coefficients in the model are fuzzy in nature. Therefore, the decision variables and the objective function should be fuzzy as well. To solve this inventory model using Lagrange multiplier technique, this paper should ﬁnd the right and the left shape functions of the objective function and decision variables, by finding the upper bound and the lower bound of the objective function, i.e. L̃L(∝) and L̃R(∝). Recall that L̃L(∝) and L̃R(∝) represent the largest and the smallest values (The left and right ∝cuts) of the optimal objective function L̃∝. For example using approximated value of TFN of C̃owhich observe in Figure 2.

Order cost as triangular fuzzy number

Consider the model when all parameters are triangular fuzzy numbers (TFN) as given

Cp=Cp-ω1, Cp , Cp+ω2, Co=Co-ω3, Co, Co+ω4, Ch=Ch-ω5, Ch, Chr+ω6,

Cb=Cb-ω7, Cb ,Cb+ω8, CL=CL-ω9, CL ,CL+ω10 and Cd=Cd-ω11, Cd, Cd+ω12.

where ωi , i=1,2,……,10 are arbitrary positive numbers under the following restrictions:

0≤ω1≤Cp,ω2≥0, 0≤ω3≤Co,ω4≥0, 0≤ω5≤Ch,ω6≥0 ,

0≤ω7≤Cb,ω8≥0, 0≤ω9≤CL,ω10≥0 and 0≤ω11≤Cd,ω12≥0.

The left and right limits of ∝cuts of Cp,Co,Ch,Cb,CL and Cd are given by

C̃pL∝=Cp-1-∝ω1,

C̃pR∝=Cp+(1-∝)ω2,

C̃oL∝=Co-(1-∝)ω3,

C̃oR∝=Co+(1-∝)ω4,

C̃hL∝=Ch-(1-∝)ω5,

C̃hR∝=Ch+(1-∝)ω6,

C̃bL∝=Cb-(1-∝)ω7,

C̃bR∝=Cb+(1-∝)ω8,

C̃LL∝=CL-1-∝ω9,

C̃LR∝=CL+(1-∝)ω10,

and C̃dL∝=Cd-(1-∝)ω11,

C̃dR∝=Cd+(1-∝)ω12

where C̃p=Cp+14(ω2-ω1), C̃o=Co+14(ω4-ω3),

C̃h=Ch+14(ω6-ω5), C̃b=Cb+14(ω8-ω7),

C̃L=CL+14(ω10-ω9) and C̃d=Cd+14(ω12-ω11).

Likewise, the same steps as in the crisp case will be applied here, except that the crisp costs Cp,Co,Ch,Cb,CL and Cd will be replaced by the fuzzy costs C̃p,C̃o,C̃h,C̃b,C̃L and C̃d. Then the optimal values Qm*, nd* , n1* and N* which minimize expected annual total cost for fuzzy case can be calculated by using the same previous Algorithm.

Application

Petro-chemical store, sells Alcohol that follows a periodic review "order up to Qm" with zero lead-time. The parameters with 60 samples indicate to demand (See the Appendix) are estimated when α=0.05 in Table 1. The optimal solutions of the crisp environment and triangular fuzzy number TFN using Mathematica program are given in Table 2, Figure 3, when ft follows exponential distribution σ=0.6, ϵ=0.5, γ=0.4, η=2, δ=0.8351.

Crisp and fuzzy values for Exponential deterioration
 ? ?d Qm nd n1 N E(TC) E(TC)/ Qm 0.1 174.223 2.965 0.2523 0.253 3.5322 312.3331 105.36 0.2 179.93 3.2725 0.382 0.382 3.6122 313.7664 95.881 0.3 191.09 3.4863 0.54 0.542 3.692 315.314 90.45 0.4 210.2 3.595 0.73 0.73233 3.772 316.819 88.14 0.5 242.11 3.588 0.953 0.953 3.85 318.113 88.67 0.6 297.05 3.449 1.206 1.21 3.93 318.963 92.49 0.7 400.96 3.186 1.492 1.492 4.004 319.243 100.213 0.8 633.85 2.8 1.8 1.812 4.1 318.774 113.858 0.9 1381.45 2.3438 2.154 2.1546 4.152 317.704 135.552 ?? ??d Q?m n?d n?1 N? E(TC?) E(TC?)/Q?m 0.1 113.51 2.9333 0.29 0.293 3.513 312.99 106.7 0.2 112.36 3.262 0.43 0.43125 3.594s 314.48 96.42 0.3 113.74 3.496 0.595 0.599 3.675 316.111 90.431 0.4 117.4 3.574 0.7814 0.8113 3.758 317.454 88.83 0.5 124.48 3.5043 0.988 1.064 3.842 318.38 90.853 0.6 138.81 3.441 1.279 1.303 3.915 319.61 92.88 0.7 161.56 3.1932 1.5823 1.591 3.991 319.934 100.194 0.8 200.85 2.8 1.901 1.922 4.0675 319.367 114.054 0.9 275.63 2.2613 2.2 2.313 4.1468 317.677 140.487
Crisp and fuzzy value for the results

Conclusion

For the importance of deteriorating in units, this paper investigates probabilistic periodic review inventory model for deteriorating items with varying costs under a constraint. The demand is a random variable follows Pareto distribution for crisp and fuzzy environment without lead-time. Here, this paper calculated optimal values of the four decision variables (maximum inventory level, stock-out time, deteriorating time and review time) which minimize the expected annual total cost. A numerical analysis method (Newton's method) was used to solve the system consisting of some nonlinear equations for different values of β. Then the model is illustrated with an application. It can be concluded that fuzzy environment is closer to the practical situation than crisp case. In addition, when β equal 0.4 it obtains the best value for minimum expected annual total cost.

Competing interests

The author declare that no competing interests exist.

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