# Optimal Replenishment Policy for Weibull-Distributed Deteriorating Items with Trapezoidal Demand Rate and Partial Backlogging

Authors: Lianxia Zhao
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

An inventory model for Weibull-distributed deteriorating items is considered so as to minimize the total cost per unit time in this paper. The model starts with shortage, allowed partial backlogging, and trapezoidal demand rate. By analyzing the model, an efficient solution procedure is proposed to determine the optimal replenishment and the optimal order quantity and the average total costs are also obtained. Finally, numerical examples are provided to illustrate the theoretical results and a sensitivity analysis of the major parameters with respect to the stability of optimal solution is also carried out.


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## Body

## 1. Introduction

Deteriorating is a general phenomenon for many products, in that fruits or vegetables are spoiled directly while alcohol undergoes physical depletion over time, and electronic products deteriorate through a gradual loss of potential utility as time passed. Based on the above situation, the effect of deteriorating for items can not be disregarded in many inventory systems. An inventory model with a constant decaying rate for items is proposed by Ghare and Schrader in 1963 [1], while Covert and Philip [2] extended Ghare and Schrader’s model to an EOQ model with a two-parameter Weibull-distributed deteriorating rate. Since then, there are many researchers who focused on such topic; Wee et al. [3] considered an inventory model for deteriorating items with quantity discount, pricing, and partial backlogging, and more related articles can be referred to like Yang et al. [4], Shah et al. [5], Yang et al. [6], and so forth.However, it is impossible for deteriorating items that the demand rate increases continuously during their full life cycle. Based on such reasons, Hill [7] proposed an inventory model with ramp-type demand rate, which extended to allow shortage proposed by Mandal and Pal [8]. Further, Wu [9] extended Mandal and Pal’s model to have Weibull-distributed deterioration and time-dependent backlogging. Skouri et al. [10] considered a model with general ramp-type demand rate, partial backlogging, and Weibull deterioration rate. Tan and Weng [11] considered a discrete inventory model for deteriorating items with partial backlogging. Pal et al. [12] considered a production inventory model for deteriorating items with ramp-type demand rate under the effect of inflation and shortages under fuzziness. There are many other related literatures about such inventory model, such as Ahmed et al. [13], Agrawal et al. [14], Singh and Pattnayak [15], Wu et al. [16], and Rabbani et al. [17].While, for some short life cycle products, the demand rate may increase up to a certain level in the early stage of marketing, then reach a stabilized period, and finally decrease to zero and reach the end of their life cycle, Cheng and Wang [18] proposed an inventory model for deteriorating items with trapezoidal type demand rate. Further, Cheng et al. [19] extended the model to be of time-dependent deteriorating items with trapezoidal type demand rate and partial backlogging. Uthayakumar and Rameswari [20] studied a model for defective items with trapezoidal type demand rate to determine the optimal product reliability. Lin [21] and Lin et al. [22] considered the inventory model which was proposed by Cheng and Wang [18]. Zhao [23] studied an inventory model for Weibull-distributed deterioration items with trapezoidal type demand rate and partial backlogging in which the model begins with no-shortage. While, in real situation, customers will respond differently in case of shortages, either to wait until replenishment or to look for alternative products, in this paper, we study an inventory model beginning with shortage, Weibull-distributed deterioration items, trapezoidal type demand rate, and time-dependent partial backlogging. By analyzing the studied model, we propose an optimal replenishment strategy for it and also carry out a sensitivity analysis of the main parameters.The remainder of the paper is organized as follows. Notations and assumptions are described in Section2. Section 3 constructs and analyzes the inventory model starting with shortage. Some numerical examples to illustrate the solution procedure are provided; then sensitivity analysis of the major parameters is also carried out in Section 4. Finally, a conclusion is presented.


## 2. Notations and Assumptions

The fundamental notations and assumptions used in this paper are given as follows.(i)
I(t) is the inventory level at time t, 0≤t≤T, where T is the length of inventory cycle.(ii)
t1 denotes the replenishment time, and t1∗ denotes the optimal t1.(iii)
S is the maximum inventory level for each ordering cycle.(iv)
Q∗ denotes the optimal ordering quantity.(v)
A0 denotes the fixed cost per order.(vi)
c1 is the cost of each deteriorated item, c2 is the unit inventory holding cost per unit time, c3 is the unit shortage cost per unit time, and c4 is the unit lost sales cost.(vii)
Ci(t1), i=1,2,3, denotes the average total cost per unit time under different conditions, respectively.(viii)
TC(t1) denotes the average total cost per unit time.(ix)
The demand rate,D(t), which is positive and consecutive, is assumed to be a trapezoidal type function of time; that is,(1)Dt=ft,t≤μ1,D0,μ1<t<μ2,gt,μ2≤t<T,whereμ1 is time point changing from the increasing demand function f(t) to the constant demand D0, and μ2 is time point changing from the constant demand D0 to the decreasing demand function g(t).(x)
The replenishment rate is infinite; that is, replenishment is instantaneous.(xi)
The time to deterioration of the item is distributed as Weibull(α,β); that is, the deterioration rate is θ(t)=αβtβ-1, where α is the scale parameter, 0<α<1, and β is the shape parameter, 0<β.(xii)
Shortage is allowed and lete-δt be fraction of shortages backlogged, where t is the waiting time up to the next replenishment and δ is a tiny positive constant; that is, δ≪1/T.


## 3. Inventory Model Starting with Shortage

The system of model begins with shortage and untilt1 the shortage achieves its maximum; then replenishment occurs at time t=t1. All of the shortage demand during (0,t1) is partial backlogged (Figure 1). At t1, the total number of backlogged items will be satisfied by the replenishment and the inventory level up to S, while the deteriorating occurs during the time of [t1,T]. According to the notations and assumptions mentioned above, the model is described as shown in Figure 1:(2)dItdt=-e-δt1-tDt,0<t<t1,-θtIt-Dt,t1<t<T,with boundary condition I(0)=0,  I(t1)=S,  I(T)=0.Figure 1
Graphical representation of inventory level over the cycle starting with shortage.Based on the values oft1, μ1, and μ2, three possible cases are presented as follows.Case 1 (0<t1≤μ1).
If the replenishment timet1∈(0,μ1], from (2), we have(3)dItdt=-e-δt1-tft,0<t<t1,-αβtβ-1It-ft,t1<t<μ1,-αβtβ-1It-D0,μ1<t<μ2,-αβtβ-1It-gt,μ2<t<T.Solving the differential equations (3) with I(0)=0, we have(4)It=-∫0teδx-t1fxdx,0<t<t1,∫tμ1eαxβ-tβfxdx+D0∫μ1μ2eαxβ-tβdx+∫μ2Teαxβ-tβgxdx,t1<t<μ1,D0∫tμ2eαxβ-tβdx+∫μ2Teαxβ-tβgxdx,μ1<t<μ2,∫tTeαxβ-tβgxdx,μ2<t<T.The total replenishment quantity can be computed as(5)S=It1=∫t1μ1eαxβ-t1βfxdx+D0∫μ1μ2eαxβ-t1βdx+∫μ2Teαxβ-t1βgxdx.The total shortage quantity BT during the interval [0,t1) is(6)BT=∫0t1t1-teδt-t1ftdt.The total number of perished items during the interval [t1,T], say DT, is(7)DT=∫t1μ1eαtβ-t1β-1ftdt+D0∫μ1μ2eαtβ-t1β-1dt+∫μ2Teαtβ-t1β-1gtdt.The total number of inventories carried during the interval [t1,T], say HT, is(8)HT=∫t1μ1∫tμ1eαxβ-tβfxdxdt+D0∫t1μ1∫μ1μ2eαxβ-tβdxdt+∫t1μ2∫μ2Teαxβ-tβgxdxdt+D0∫μ1μ2∫tμ2eαxβ-tβdxdt+∫μ2T∫tTeαxβ-tβgxdxdt.The total of lost sales during the interval [0,t1], say LT, is(9)LT=∫0t11-eδt-t1ftdt.Therefore, the average total cost per unit time under the condition t1≤μ1 can be given by(10)
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.Taking the first and the second order derivative of C1(t1) with respect to t1, respectively, we have(11)dC1t1dt1=1Tc3∫0t11-δt1+δteδt-t1ftdt+c4δ∫0t1eδt-t1ftdt-c2+c1αβt1β-1∫t1Teαtβ-t1βDtdt,(12)d2C1t1dt12=1Tc3ft1+δ∫0t1δt1-δt-2eδt-t1ftdt+c4δft1-δ∫0t1eδt-t1ftdt+c1αβt1β-1+c2ft1+αβt1β-2c1αβt1β-β+1+c2t1∫t1Teαtβ-t1βDtdt.Then, we have (13)dC1t1dt1t1=0=-c2T∫0TeαtβDtdt<0,and let(14)Δ1=c3∫0μ11-δμ1+δteδt-μ1ftdt+c4δ∫0μ1eδt-μ1ftdt-c2+c1αβμ1β-1D0∫μ1μ2eαtβ-μ1βdt+∫μ2Teαtβ-μ1βgtdt.Therefore, (i) if Δ1>0 and d2C1(t1)/dt12>0, there exists a unique solution t1∗∈(0,μ1) satisfying dC1(t1)/dt1=0, and C1(t1) obtains its minimum at t1=t1∗; (ii) if Δ1<0 and d2C1(t1)/dt12>0, it means that C1(t1) is a nonincreasing function and obtains its minimum value at point t1∗=μ1; (iii) if the sign of d2C1(t1)/dt12 is indefinite, then the curve of C1(t1) is W-shape, and C1(t1) may obtain its minimum at more than one point; without loss of generality, set t1=t1∗, where t1∗ is one of the solutions of dC1(t1)/dt1=0.
Thus, the optimal value of the order level,S=I(t1∗), is(15)S∗=∫t1∗μ1eαxβ-t1∗βfxdx+D0∫μ1μ2eαxβ-t1∗βdx+∫μ2Teαxβ-t1∗βgxdx,and the optimal order quantity Q∗ is(16)Q∗=S∗+∫0t1∗eδt-t1∗ftdt.Case 2 (μ1≤t1≤μ2).
If the replenishment timet1∈[μ1,μ2], then from (2), we have(17)dItdt=-e-δt1-tft,0<t<μ1,-e-δt1-tD0,μ1<t<t1,-αβtβ-1It-D0,t1<t<μ2,-αβtβ-1It-gt,μ2<t<T.Solving the differential equation (17) with I(0)=I(T)=0, we have(18)It=-∫0teδx-t1fxdx,0<t<μ1,-D0δeδt-t1-eδμ1-t1-∫0μ1eδx-t1fxdx,μ1<t<t1,D0∫tμ2eαxβ-tβdx+∫μ2Teαxβ-tβgxdx,t1<t<μ2,∫tTeαxβ-tβgxdx,μ2<t<T.From (18), the total replenishment quantity can be computed:(19)S=It1=D0∫t1μ2eαxβ-t1βdx+∫μ2Teαxβ-t1βgxdx.The total shortage quantity during the interval (0,t1] is(20)BT=∫0μ1t1-teδt-t1ftdt+D0δ∫μ1t1eδt-t1-eδμ1-t1dt.The total number of perished items in the interval [t1,T] is(21)DT=D0∫t1μ2eαtβ-t1β-1dt+∫μ2Teαtβ-t1β-1gtdt.The total number of inventories carried during the interval [t1,T] is(22)HT=D0∫t1μ2∫tμ2eαxβ-tβdxdt+∫t1μ2∫μ2Teαxβ-tβgxdxdt+∫μ2T∫tTeαxβ-tβgxdxdt.The total of lost sales during the interval [0,t1] is(23)LT=∫0μ11-eδt-t1ftdt+D0∫μ1t11-eδt-t1dt.Therefore, the average total cost per unit time under the condition μ1<t1≤μ2 can be given by(24)C2t1=1TA0+c1DT+c2HT+c3BT+c4LT=1TA0+c1D0∫t1μ2eαtβ-t1β-1dt+∫μ2Teαtβ-t1β-1gtdt+c2D0∫t1μ2∫tμ2eαxβ-tβdxdt+∫t1μ2∫μ2Teαxβ-tβgxdxdt+∫μ2T∫tTeαxβ-tβgxdxdt+c4∫0μ11-eδt-t1ftdt+D0∫μ1t11-eδt-t1dt+c3∫0μ1t1-teδt-t1ftdt+D0δ∫μ1t1eδt-t1-eδμ1-t1dt.
Taking the first and the second order derivative ofC2(t1) with respect to t1, respectively, we have(25)dC2t1dt1=1Tc4δ∫0μ1eδt-t1ftdt+D0∫μ1t1eδt-t1dt-c1αβt1β-1+c2D0∫t1μ2eαtβ-t1βdt+∫μ2Teαtβ-t1βgtdt+c3∫0μ11-δt1+δteδt-t1ftdt-D0μ1-t1eδμ1-t1,(26)d2C2t1dt12=1Tαβt1β-2c1αβt1β-β+1+c2t1D0∫t1μ2eαtβ-t1βdt+∫μ2Teαtβ-t1βgtdt+c1αβt1β-1+c2D0+c4δD0eδμ1-t1-δ∫0μ1eδt-t1ftdt+c3∫0μ1δ2t1-δ2t-2δeδt-t1ftdt+D01+δμ1-δt1eδμ1-t1.Let(27)Δ2=c3∫0μ11-δμ2+δteδt-μ2ftdt+D0μ2-μ1eδμ1-μ2+c4δ∫0μ1eδt-μ2ftdt+D0∫μ1μ2eδt-μ2dt-c2+c1αβμ2β-1∫μ2Teαtβ-μ2βgtdt.Therefore, (i) if Δ1<0, Δ2>0, and d2C2(t1)/dt12>0, there exists a unique solution t1∗∈(μ1,μ2) satisfying dC2(t1)/dt1=0, and C2(t1) obtains its minimum at t1=t1∗; (ii) if Δ1<0,  Δ2≤0, and d2C2(t1)/dt12>0, it means that C2(t1) is a nonincreasing function and obtains its minimum at point t1∗=μ2; (iii) if Δ1≥0, Δ2>0, and d2C2(t1)/dt12>0, it means that C2(t1) is a nondecreasing function and obtains its minimum at point t1∗=μ1; (iv) if the sign of d2C2(t1)/dt12 is indefinite, then the curve of C2(t1) is W-shape, and C2(t1) may obtain its minimum value at more than one point; without loss of generality, set t1=t1∗, where t1∗ is one of the solutions of dC2(t1)/dt1=0.
Thus, the optimal value of the order level,S=I(t1∗), is(28)S∗=D0∫t1∗μ2eαtβ-t1∗βdt+∫μ2Teαtβ-t1∗βgtdt,and the optimal order quantity Q∗ is(29)Q∗=S∗+∫0μ1eδt-t1∗ftdt+D0∫μ1t1∗eδt-t1∗dt.Case 3 (μ2≤t1<T).
If the replenishment timet1∈[μ2,T), from (2), we have(30)dItdt=-e-δt1-tft,0<t<μ1,-e-δt1-tD0,μ1<t<μ2,-e-δt1-tgt,μ2<t<t1,-αβtβ-1It-gt,t1<t<T.Solving (30) with I(0)=I(T)=0, we have(31)It=-∫0teδx-t1fxdx,0<t<μ1,D0δeδμ1-t1-eδt-t1-∫0μ1eδx-t1fxdx,μ1<t<μ2,-∫μ2teδx-t1gxdx+D0δeδμ1-t1-eδμ2-t1-∫0μ1eδx-t1fxdx,μ2<t<t1,∫tTeαxβ-tβgxdx,t1<t<T.From the last equation of (31), the total replenishment quantity can be computed:(32)S=It1=∫t1Teαxβ-t1βgxdx.The total shortage quantity during the interval (0,t1] is(33)BT=∫0μ1t1-teδt-t1ftdt+∫μ2t1t1-teδt-t1gtdt+D0δ2eδμ1-t1-eδμ2-t1+D0δt1-μ1eδμ1-t1-t1-μ2eδμ2-t1.The total number of perished items in the interval [t1,T] is(34)DT=∫t1Teαxβ-t1β-1gxdx.The total number of inventories carried during the interval [t1,T] is(35)HT=∫t1T∫tTeαxβ-tβgxdxdt.The total of lost sales during the interval [0,t1] is(36)LT=∫0μ11-eδt-t1ftdt+D0∫μ1μ21-eδt-t1dt+∫μ2t11-eδt-t1gtdt.Therefore, the average total cost per unit time under the condition μ2<t1≤T can be given by(37)C3t1=1TA0+c1DT+c2HT+c3BT+c4LT=1TA0+c1∫t1Teαxβ-t1β-1gxdx+c2∫t1T∫tTeαxβ-tβgxdxdt+c4∫0μ11-eδt-t1ftdt+D0∫μ1μ21-eδt-t1dt+∫μ2t11-eδt-t1gtdt+c3∫0μ1t1-teδt-t1ftdt+∫μ2t1t1-teδt-t1gtdt+D0δ2eδμ1-t1-eδμ2-t1+D0δt1-μ1eδμ1-t1-t1-μ2eδμ2-t1.
Taking the first and the second order derivative ofC3(t1) with respect to t1, respectively, we have(38)dC3t1dt1=1T-c2+c1αβt1β-1∫t1Teαtβ-t1βgtdt+c4δ∫0μ1eδt-t1ftdt+D0∫μ1μ2eδt-t1dt+∫μ2t1eδt-t1gtdt+c3∫0μ11+δt-δt1eδt-t1ftdt+∫μ2t11+δt-δt1eδt-t1gtdt+D0t1-μ1eδμ1-t1-t1-μ2eδμ2-t1,(39)d2C3t1dt12=1Tαβt1β-2c1αβt1β-β+1+c2t1∫t1Teαtβ-t1βgtdt+c2+c1αβt1β-1gt1+c4δδ∫0μ1eδt-t1ftdt+D0δ∫μ1μ2eδt-t1dt+δ∫μ2t1eδt-t1gtdt+gt1+c2αβt1β-1∫t1Teαtβ-t1βgtdt+gt1+c3∫0μ1δ2t1-δ2t-2δeδt-t1ftdt+∫μ2t1δ2t1-δ2t-2δeδt-t1gtdt+gt1+D01+δμ1-δt1eδμ1-t1-1+δμ2-δt1eδμ2-t1.
We know that(40)dC3t1dt1t1=T=1Tc4δ∫0μ1eδt-Tftdt+D0∫μ1μ2eδt-Tdt+∫μ2Teδt-Tgtdt+c3∫0μ11+δt-δTeδt-Tftdt+∫μ2T1+δt-δTeδt-Tgtdt+D0T-μ1eδμ1-T-T-μ2eδμ2-T>0.Therefore, (i) if Δ2<0 and d2C3(t1)/dt12>0, there exists a unique solution t1∗∈(μ2,T) satisfying dC3(t1)/dt1=0, and C3(t1) obtains its minimum at t1=t1∗; (ii) if Δ2≥0 and d2C3(t1)/dt12>0, it means that C3(t1) is a nondecreasing function and obtains its minimum value at point t1∗=μ2; (iii) if the sign of d2C3(t1)/dt12 is indefinite, then the curve of C3(t1) is W-shape, and C3(t1) may obtain its minimum value at more than one point; without loss of generality, set t1=t1∗, where t1∗ is one of the solutions of dC3(t1)/dt1=0.
Therefore, the optimal value of the order level,S=I(t1∗), is(41)S∗=∫t1∗Teαxβ-t1∗βgxdx,and the optimal order quantity Q∗ is(42)Q∗=S∗+∫0μ1eδt-t1∗ftdt+D0∫μ1μ2eδt-t1∗dt+∫μ2t1∗eδt-t1∗gtdt.
From the above analysis, we obtain that the total average cost of the system over the interval[0,T] is(43)TCt1=C1t1,0<t1≤μ1,C2t1,μ1<t1≤μ2,C3t1,μ2<t1<T,where C1(t1),  C2(t1), and C3(t1) are obtained from (10), (24), and (37), respectively.Basing on the analyzed results above, we provide the results which ensure the existence of a uniquet1, say t1∗, to minimize the total average cost for the model system starting with shortages. The procedure is as follows:Step 0. Input the parameters, μ1,  μ2,  T,  a1,  a2,  b1,  α,  β,  δ,  D0, and ci,  i=1,2,3,4.Step 1. Calculate Δ1 and d2C1(t1)/dt12.Step 1.1. If Δ1>0 and d2C1(t1)/dt12>0, solve a unique t1∗ from dC1(t1)/dt1=0, and the total average cost and the order quantity can be obtained by (10) and (16), respectively.Step 1.2. If Δ1≤0 and d2C1(t1)/dt12>0, then the optimal average cost and the optimal order quantity can be obtained by (10) and (16) at t1∗=μ1, respectively.Step 1.3. If the sign of d2C1(t1)/dt12 is indefinite, then solve equation dC1(t1)/dt1=0. If the solution is not unique, by comparing their average costs C1(t1) to obtain the optimal replenishment point, compute the order quantity by (16).Step 2. Calculate Δ2 and d2C2(t1)/dt12.Step 2.1. If Δ1<0,  Δ2>0, and d2C2(t1)/dt12>0, then seek a unique t1∗ from equation dC2(t1)/dt1=0, and the optimal total average cost and the optimal order quantity can be obtained by (24) and (29) at t1=t1∗, respectively.Step 2.2. If Δ1>0,  Δ2>0, and d2C2(t1)/dt12>0, then C2(t1) obtains its minimum at t1∗=μ1, and compute the order quantity by (29).Step 2.3. If Δ1<0,  Δ2≤0, and d2C2(t1)/dt12>0, then C2(t1) obtains its minimum at t1∗=μ2, and compute the order quantity by (29).Step 2.4. If the sign of d2C2(t1)/dt12 is indefinite, then solve equation dC2(t1)/dt1=0. If the solution is not unique, by comparing their average costs C2(t1) to obtain the optimal replenishment point, compute the optimal order quantity by (29).Step 3. Calculate d2C3(t1)/dt12.Step 3.1. If Δ2<0 and d2C3(t1)/dt12>0, seek the unique t1∗ from equation dC3(t1)/dt1=0, and the total average cost and the order quantity can be obtained by (37) and (42), respectively.Step 3.2. If Δ2≥0 and d2C3(t1)/dt12>0, then C3(t1) obtains its minimum at t1∗=μ2, and compute the order quantity by (42).Step 3.3. If the sign of d2C3(t1)/dt12 is indefinite, then solve equation dC3(t1)/dt1=0. If the solution is not unique, by comparing their average costs C3(t1) to obtain the optimal replenishment point, compute the order quantity by (42).Step 4. Find TC(t1∗)=min{C1(t1∗),C2(t1∗),C3(t1∗)} and accordingly select the optimum t1∗, and then obtain corresponding optimal order quantity Q∗.


## 4. Numerical Examples and Sensitivity Analysis

In order to demonstrate the procedure to obtain the optimal solution of the model, we present four examples for the model. Examples are based on piece-wise demand rate, such asf(t)=a1+b1t and g(t)=a2e-b2t.Example 1.
The parameter values of an inventory system are given as follows:T=12 weeks, μ1=4 weeks, μ2=8 weeks, α=0.005, β=2, δ=0.04, a1=30 units, b1=5 units, a2=100 units, A0 = $500, c1 = $2, c2 = $3, c3 = $12, c4 = $8.
SinceΔ1=18.809>0, Δ2=222.869>0, d2C1(t1)/dt12>0 for all t1∈(0,μ1), d2C2(t1)/dt12>0 for all t1∈[μ1,μ2), and d2C3(t1)/dt12>0 for all t1∈[μ2,T), then we get local minimum value C2(μ1)=763.891 in [μ1,μ2) and C3(u2)=1263.58 in [μ2,T). By solving equation dC1(t1)/dt1=0, we have t1∗=3.664. From (10), we obtain C1(t1∗)=762.759. Therefore, the optimal ordering quantity is Q∗=637.839 and the total average cost is TC(t1∗)=762.759.Example 2.
The parameter values of an inventory system are given as follows:T=12 weeks, μ1=4 weeks, μ2=8 weeks, α=0.005, β=2, δ=0.02, a1=30 units, b1=5 units, a2=100 units, A0 = $500, c1 = $5, c2 = $10, c3 = $12, c4 = $8.
SinceΔ1=-257.889<0, Δ2=139.269>0, d2C1(t1)/dt12>0 for all t1∈(0,μ1], d2C2(t1)/dt12>0 for all t1∈[μ1,μ2), and d2C3(t1)/dt12>0 for all t1∈(μ2,T), then we get local minimum value C1(μ1)=1805.883 in (0,μ1] and C3(u2)=1582.32 in [μ2,T). By solving equation dC2(t1)/dt1=0, we have t1∗=6.559. From (24), we obtain C2(t1∗)=1480.324. Therefore, the optimal ordering quantity is Q∗=571.827 and the total average cost is TC(t1∗)=1480.324.Example 3.
The parameter values of an inventory system are given as follows:T=12 weeks, μ1=4 weeks, μ2=6 weeks, α=0.005, β=2, δ=0.06, a1=30 units, b1=5 units, a2=100 units, A0 = $500, c1 = $5, c2 = $10, c3 = $12, c4 = $8.
SinceΔ1=-249.329<0, Δ2=-121.869<0, d2C1(t1)/dt12>0 for all t1∈(0,μ1], d2C2(t1)/dt12>0 for all t1∈(μ1,μ2], and d2C3(t1)/dt12>0 for all t1∈[μ2,T), then we get local minimum value C1(μ1)=1417.134 in (0,μ1] and C2(u2)=1052.016 in (μ1,μ2]. By solving equation dC3(t1)/dt1=0, we have t1∗=9.120. From (37), we obtain C3(t1∗)=889.767. Therefore, the optimal ordering quantity is Q∗=278.960 and the total average cost is TC(t1∗)=889.767.Example 4.
The parameter values of an inventory system are given as follows:T=12 weeks, μ1=4 weeks, μ2=6 weeks, α=0.005, β=1.6, δ=0.02, a1=30 units, b1=5 units, a2=100 units, A0 = $500, c1 = $5, c2 = $10, c3 = $12, c4 = $8.
SinceΔ1=-197.263<0, Δ2=-86.675<0, d2C4(t1)/dt12>0 for all t1∈(0,μ1], d2C2(t1)/dt12>0 for all t1∈(μ1,μ2], and d2C3(t1)/dt12>0 for all t1∈[μ2,T), then we get local minimum value C1(μ1)=1252.59 in (0,μ1] and C2(u2)=1052.016 in (μ1,μ2]. By solving equation dC3(t1)/dt1=0, we have t1∗=8.627. From (37), we obtain TC(t1∗)=842.935. Therefore, the optimal ordering quantity is Q∗=241.524 obtained from (42), and the minimum cost TC(t1∗)=842.935.In order to clearly indicate the effects of parameters such asδ,  α, and β on S∗, Q∗, and TCi(t1∗)  (i=1,2), respectively, we study the sensitivity of different parameters under the studied inventory model. The sensitivity analysis is performed on the base of Example 1, and the results are shown in Tables 1–3, respectively.Table 1
The sensitivity ofδ for the model in Example 1.
δ
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
t
1
∗
3.4366
3.4905
3.5463
3.6042
3.6642
3.7263
3.7907
3.8574
3.9265
S
∗
515.0829
515.0829
511.4210
507.5965
503.6066
499.4433
495.0985
490.5641
485.8321
Q
∗
648.1034
648.1034
644.8369
641.4184
637.8392
634.0900
630.1614
626.0436
621.7267
T
C
2
t
1
∗
775.2147
771.8437
768.4699
764.9983
761.4252
757.7473
753.9611
750.063
746.0498Table 2
The sensitivity ofα for the models in Example 1.
α
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
t
1
∗
3.0141
3.1325
3.2569
3.3870
3.5229
3.6642
3.8105
3.9614
4.1171
S
∗
415.8882
432.0629
448.9624
466.5543
484.7907
503.6066
522.9206
542.6345
575.5923
Q
∗
522.8982
543.9279
565.9745
589.0106
612.9883
637.8392
663.4729
689.7782
723.0000
T
C
2
t
1
∗
638.5314
661.4087
685.1585
709.7639
735.1984
761.4252
788.3972
816.0570
845.0147Table 3
The sensitivity ofβ for the models in Example 1.
β
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
t
1
∗
3.1114
3.1995
3.2513
3.3176
3.40332
3.5148
3.6613
3.8551
4.1128
S
∗
427.5053
440.1612
447.1691
456.2029
467.9084
483.1709
503.6066
529.5334
564.1669
Q
∗
538.4988
554.7959
563.9494
575.7494
591.0495
611.0248
637.8392
672.0262
717.9964
T
C
2
t
1
∗
644.4596
670.3559
680.1679
692.9359
709.5918
731.4743
761.4252
797.6508
845.0147From Table1, we observe that inventory level S∗, order quantity Q∗, and the total average cost TCi(t1∗)  (i=1,2) gradually decrease as the shortage parameter δ increases for the model, and the replenishment time t1∗ increases as δ increases for the model starting with shortage. Also, we observe that the values of t1∗, S∗, Q∗, and TC(t1∗) are slightly sensitive to the changes of δ.From Table2, we find that when the deterioration parameter α increases, then S∗, Q∗, and TC(t1∗) increase for both models, and the replenishment time t1∗ increases for the model starting with shortage. We see that the values of t1∗, S∗, Q∗, and TC(t1∗) all are temperately sensitive to the changes of α for the model.From Table3, we find that S∗, Q∗, and TC(t1∗) increase as the deterioration parameter β increases for both models. Also, we find that the replenishment time t1∗ increases as β increases for the model starting with shortage. We see that the values of t1∗, S∗, Q∗, and TC(t1∗) all are moderately sensitive to the changes of β for the model.


## 5. Conclusion

This paper studies an inventory model for Weibull-distributed deterioration with trapezoidal type demand rate and partial backlogging. First, an optimal replenishment strategy for deteriorating items with the model starting with shortage is proposed. Then, numerical examples are provided to illustrate the theoretical results. From the analysis of Tables1–3, it can be found that the replenishment time point t1∗, order quantity Q∗, and the total average cost TC(t1∗) are moderately sensitive to the changes of α and β and lowly sensitive to the changes of δ, respectively. However, if the order cycle (T) is a decision variable, such inventory model will be of more practical significance, which provides us an interesting topic in future.



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