APPMATHISRN Applied Mathematics2090-55722090-5564International Scholarly Research Network65746410.5402/2011/657464657464Research ArticleA Deterministic Inventory Model of Deteriorating Items with Two Rates of Production, Shortages, and Variable Production CycleBhowmickJhuma1SamantaG. P.2ChoS.SpinelloD.1Department of MathematicsMaharaja Manindra Chandra CollegeKolkata 700003Indiammccollege.org2Department of MathematicsBengal Engineering and Science UniversityShibpurHowrah 711103Indiabecs.ac.in201126072011201109032011190420112011Copyright © 2011 Jhuma Bhowmick and G. P. Samanta.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A Continuous production control inventory model is developed for a deteriorating item having shortages and variable production cycle. It is assumed that the production rate is changed to another at a time when the inventory level reaches a prefixed level Q1 and continued until the inventory level reaches the level S(>Q1). The demand rate is assumed to be constant, and the production cycle T is taken as variable. The production is started again at a time when the shortage level reaches a prefixed quantity Q2. For this model, the total cost per unit time as a function of Q1, Q2, S, and T is derived. The optimal decision rules for Q1, Q2, S, and T are computed. The sensitivity of the optimal solution towards changes in the values of different system parameters is also studied. Results are illustrated by numerical examples.

1. Introduction

EOQ inventory models have long been attracting considerable amount of research attention. For the last fifteen years, researchers in this area have extended investigation into various models with considerations of item shortage, item deterioration, demand patterns, item order cycles, and their combinations. The control and maintenance of production inventories of deteriorating items with shortages have received much attention of several researchers in the recent years because most physical goods deteriorate over time. In reality, some of the items are either damaged or decayed or vaporized or affected by some other factors and are not in a perfect condition to satisfy the demand. Food items, drugs, pharmaceuticals, and radioactive substances are examples of items in which sufficient deterioration can take place during the normal storage period of the units, and consequently this loss must be taken into account when analyzing the system. Research in this direction began with the work of Whitin  who considered fashion goods deteriorating at the end of a prescribed storage period. Ghare and Schrader  were the two earliest researchers to consider continuously decaying inventory for a constant demand. An order-level inventory model for items deteriorating at a constant rate was discussed by Shah and Jaiswal . Aggarwal  developed an order-level inventory model by correcting and modifying the error in Shah and Jaiswal’s analysis  in calculating the average inventory holding cost. In all these models, the demand rate and the deterioration rate were constants, the replenishment rate was infinite, and no shortage in inventory was allowed.

Researchers started to develop inventory systems allowing time variability in one or more than one parameters. Dave and Patel  discussed an inventory model for replenishment. This was followed by another model by Dave  with variable instantaneous demand, discrete opportunities for replenishment, and shortages. Bahari-Kashani  discussed a heuristic model with time-proportional demand. An economic order quantity (EOQ) model for deteriorating items with shortages and linear trend in demand was studied by Goswami and Chaudhuri . It is a common belief that a large pile of goods attracts more customers in the supermarket. This phenomenon is termed as stock-dependent demand rate. Baker and Urban  established an economic order quantity model for a power-form inventory-level-dependent demand pattern. Mandal and Phaujdar  then developed an economic production quantity model for deteriorating items with constant production rate and linearly stock-dependent demand. Later on, Datta and Pal  presented an EOQ model in which the demand rate is dependent on the instantaneous stocks displayed until a given level of inventory S0 is reached, after which the demand rate becomes constant. Other papers related to this area are Urban , Giri et al. , Padmanabhan and Vrat , Pal et al. , Ray and Chaudhuri , Urban and Baker , Ray et al. , Giri and Chaudhuri , Datta and Paul , Ouyang et al. , Teng et al. , Roy and Samanta , and others.

Another class of inventory models has been developed with time-dependent deterioration rate. Covert and Philip  used a two-parameter Weibull distribution to represent the distribution of the time to deterioration. This model was further developed by Philip  by taking a three-parameter Weibull distribution for the time to deterioration. Mishra  analyzed an inventory model with a variable rate of deterioration, finite rate of replenishment, and no shortage, but only a special case of the model was solved under very restrictive assumptions. Deb and Chaudhuri  studied a model with a finite rate of production and a time-proportional deterioration rate, allowing backlogging. Goswami and Chaudhuri  assumed that the demand rate, production rate, and deterioration rate were all time dependent. A detailed information regarding inventory modeling for deteriorating items was given in the review papers of Nahmias  and Raafat . An order-level inventory model for deteriorating items without shortages has been developed by Jalan and Chaudhuri . Ouyang et al.  considered the continuous inventory system with partial backorders. A production inventory model with two rates of production and backorders is analyzed by Perumal and Arivarignan , Samanta and Roy .

In the present paper, we have developed a continuous production control inventory model with variable production cycle for deteriorating items with shortages in which two different rates of production are available, and it is possible that production started at one rate and after some time it may be switched over to another rate. Such a situation is desirable in the sense that by starting at a low rate of production, a large quantum stock of manufactured items at the initial stage is avoided, leading to reduction in the holding cost.

2. Notations and Modeling Assumptions

The mathematical model in this paper is developed on the basis of the following notations and assumptions:

a is the constant demand rate;

p1 (>a) and p2 (>p1) are the constant production rates started at time t  =  0 and at time t = t1 (>o), respectively;

C1 is the holding cost per unit per unit time;

C2 is the shortage cost per unit per unit time;

C3 is the cost of a deteriorated unit. (C1, C2, and C3 are known constants);

C4 and C5 are the constant unit production costs when the production rates are p1 and p2 respectively (C4>C5);

Q(t) is the inventory level at time t(0);

A is the setup cost;

replenishment is instantaneous and lead time is zero;

T is the variable duration of production cycle;

shortages are allowed and backlogged;

C is the average cost of the system;

the distribution of the time to deterioration of an item follows the exponential distribution g(t), where

g(t)={θeθt,for  t>0,0,otherwise.  θ is called the deterioration rate; a constant fraction θ (0 < θ1) of the on-hand inventory deteriorates per unit time. It is assumed that no repair or replacement of the deteriorated items takes place during a given cycle.

In this paper, we have considered a single commodity deterministic continuous production inventory model with a constant demand rate a. The production of the item is started initially at t  =  0 at a rate p1 (>a). Once the inventory level reaches Q1, the rate of production is switched over to p2 (>p1), and the production is stopped when the level of inventory reaches S(>Q1) and the inventory is depleted at a constant rate a. It is decided to backlog demands up to Q2 which occur during stock-out time. Thus, the inventory level reaches Q2 (backorder level is Q2), the production is started at a faster rate p2 so as to clear the backlog, and when the inventory level reaches 0 (i.e., the backlog is cleared), the next production cycle starts at the lower rare p1.

We denote by [0, t1], the duration of production at the rate p1, by [t1, t2], the duration of production at the rate p2, by [t2, t3], the duration when there is no production but only consumption by demand at a rate a, by [t3, t4], the duration of shortage, and by [t4, T], the duration of time to backlog at the rate p2. The cycle then repeats itself after time T. The duration of a production cycle T is taken as variable.

This model is represented by Figure 1.

3. Model Formulation and Solution

Let Q(t) be the instantaneous state of the inventory level at any time t (0 ≤ tT),then the differential equations describing the instantaneous states of Q(t) in the interval [0, T] are given by the following:dQ(t)dt+θQ(t)=p1a,0tt1,dQ(t)dt+θQ(t)=p2a,t1tt2,dQ(t)dt+θQ(t)=a,t2tt3,dQ(t)dt=a,t3tt4,dQ(t)dt=p2a,t4tT. The boundary conditions are Q(0)=0,Q(t1)=Q1,Q(t2)=S,Q(t3)=0,Q(t4)=Q2,Q(T)=0.

The solutions of equations (3.1) are given byQ(t)=1θ(p1-a)(1-e-θt),0tt1=1θ(p2    a)+e-θ(t    t1)[  Q1-  1θ(p2  -  a)],  t1tt2=  -aθ  +(S+aθ  )  e-θ  (t    t2),t2tt3=  -a(tt3),  t3tt4    =  -  Q2  +  (p2    a)(t    t4),t4tT.

From (3.2) and (3.3), we have1θ(p1  -  a)(1  -  e-θt1)  =  Q1  t1=1θlog  [1+θQ1(p1-a)+θ2Q12(p1-a)2](neglecting  higher  powers  of  θ,  0<θ1)=Q1(p1-a)+θQ122(p1-a)2(neglecting  higher  powers  of  θ,  0<θ1).

Again, from (3.2) and (3.4), we have 1θ(p2    a  )  +  e  θ(t1-t2)[Q1  -1θ(p2    a)]  =  Seθ(t2-t1)=[1-Sθ(p2-a)]-1  [1-Q1θ(p2-a)]=1  +(S-Q1)θ(p2-a)  +  θ2(S-Q1)S(p2-a)2(neglecting  higher  powers  of  θ),    t2-t1=  1θlog[1+(S-Q1)θ(p2-a)+θ2(S-Q1)S(p2-a)2]=(S-Q1)(p2-a)+  θ(S2-Q12)2(p2-a)2(neglecting  higher  powers  of  θ),  t2  =  Q1(p1-a)+θQ122(p1-a)2+(S-Q1)(p2-a)+θ(S2-Q12)2(p2-a)2[by  (3.8b)].

From (3.5), we have-1θa  +  (S  +1θa)eθ(t2-  t3)  =  0,          since  Q(t3)  =  0t3    -  t2=  1θlog(1  +θSa)=  Sa-S2θ2a2(neglecting  higher  powers  of  θ)t3=  Q1(p1-a)+θQ122(p1-a)2+  (S-Q1)(p2-a)  +θ(S2-Q12)2(p2-a)2+Sa-S2θ2a2[by  (3.11)].

Again, from (3.6), we havea(t3    t4)  =  -Q2,  since  Q(t4)=  -Q2,  t4  =  t3  +    Q2a.

From (3.7) and Q(T) = 0, we have-  Q2  +  (p2    a)(T    t4)=0Q2=  (p2    a)(T    t3    Q2a)[by  (3.15)],Q2  =a(p2-a)p2  [θ(S2-Q12)2(p2-a)2T-Q1(p1-a)-θQ122(p1-a)2-  (S-Q1)(p2-a)-  θ(S2-Q12)2(p2-a)2-Sa+S2θ2a2][by  (3.13)],D=  Total  number  of  deteriorated  items  in  [0,T]=  [(p1  -  a)t1+(p2-  a)(t2    t1)-  S]+  [S    a(t3    t2)]=(p1-a)[Q1(p1-a)+θQ122(p1-a)2]+(p2-a)[(S-Q1)(p2-a)+θ(S2-Q12)2(p2-a)2]-a  (Sa-S2θ2a2)[using  (3.8b),  (3.10b),  and  (3.12b)]=  θ2[  Q12(p1-a)+(S2-Q12)(p2-a)+S2a  ],  S1=  Total  Shortage  over  the  period  [0,T]=  -t3t4a(t-t3)dt+t4T[-Q2+(p2-a)(t-t4)]dt  [by  (3.6)  and  (3.7)]=-p2Q222a(p2-a)[by  using  (3.6)  and  (3.7)],  IT=  Total  inventory  carried  over  the  period  [0,  T]=  0t1Q(t)dt+t1t2Q(t)dt+t2t3Q(t)dt.  Now,0t1Q(t)dt=  1θ(p1-a)0t1(1-e-θt)dt[by  (3.3)]=  1θ(p1-a)[t1+1θ(e-θt1-1)]=(p1-a)[t1θ-Q1θ(p1-a)][by  (3.8a)]=(p1-a)[1θ2log{1+θQ1(p1-a)+θ2Q12(p1-a)2+θ3Q13(p1-a)3}-Q1θ(p1-a)][by  (3.8a)  and  neglecting  higher  powers  of  θ]=(p1-a)[Q12(p1-a)2+θQ13(p1-a)3-Q122(p1-a)2-θQ13(p1-a)3+θQ133(p1-a)3](neglecting  higher  powers  of  θ)=  Q122(p1-a)+θQ133(p1-a)2,t1t2Q(t)dt=t1t2[1θ(p2-a)+{Q1-1θ(p2-a)}e-θ(t-t1)]dt[by  (3.4)]=1θ(p2-a)(t2-t1)-1θ{Q1-1θ(p2-a)}{e-θ(t2-t1)-1}=1θ2(p2-a)log[1+(S-Q1)θ(p2-a)+θ2(S-Q1)S(p2-a)2+θ3S2(S-Q1)(p2-a)3]-(S-Q1)θ[by  (3.9)    and    (3.10a)]=(p2-a)[(S-Q1)θ(p2-a)+(S-Q1)S(p2-a)2+S2(S-Q1)θ(p2-a)3-(S-Q1)22(p2-a)2-(S-Q1)2Sθ(p2-a)3+(S-Q1)3θ3(p2-a)3]-(S-Q1)θ(neglecting  higher  powers  of  θ)=(S2-Q12)2(p2-a)+θ(S3-Q13)3(p2-a)2,t2t3Q(t)dt=t2t3[-aθ+(S+aθ)e-θ(t-t2)]dt[by  (3.5)]=-aθ(t3-t2)-1θ(S+aθ){e-θ(t3-t2)-1}=-aθ2log(1+Sθa)+Sθ[by  (3.12a)]=-aθ2[Sθa-S2θ22a2+S3θ33a3]+Sθ(neglecting  higher  powers  of  θ)=S22a-S3θ3a2.

Therefore, from (3.20), total inventory carried over the cycle [0,T]=Q122(p1-a)+θQ133(p1-a)2+(S2-Q12)2(p2-a)+θ(S3-Q13)3(p2-a)2+S22a-S3θ3a2,P=Production  cost  over  the  period  [0,T]=C4p1t1+C5{p2(t2t1)+p2(Tt4)}=C4p1Q1(p1-a)[1+θQ12(p1-a)]+C5p2(p2-a)[S+Q2-Q1+θ(S2-Q12)2(p2-a)][by  (3.8b),(3.10b),  and(3.16)].

Average cost of the system=C(Q1,S,T)=1T[C1IT-C2S1+C3D+P+A]=1T[A+C5p2(p2-a)S+{(C1+C3θ)p22a(p2-a)+C5p2θ2(p2-a)2}S2+C1θ(2a-p2)p23a2(p2-a)2S3+C5p2(p2-a)Q2+C2p22a(p2-a)Q22+{C4p1(p1-a)-C5p2(p2-a)}Q1+{f2(C1+C3θ)+θ2(C4p1(p1-a)2-C5p2(p2-a)2)}Q12+C1dfθ3kQ13][using  (3.18),(3.19),(3.24),  and(3.25)], where d=  p1  +  p2    2a,k=  (p1    a)(p2    a),f=    (p2-p1)k, and from (3.17),Q2=a(p2-a)p2T-S-a(p2-p1)p2(p1-a)Q1-θ(2a-p2)2a(p2-a)S2-adfθ2p2(p1-a)Q12.

The necessary conditions for C(Q1, S, T) to be minimum are CQ1=0,  CS=0,  CT  =0, that is, C1dfθkQ12+{(C1+C3θ)f+θ(C4p1(p1-a)2-C5p2(p2-a)2)}Q1-1k(C5a+C2Q2)(p2-p1+θdfQ1)+C4p1(p1-a)-C5p2(p2-a)=0,C1θp2(2ap2)S2+ap2{(C1+C3θ)(p2a)+C5aθ}S+C5a2p2(p2a)p2(aC5+C2Q2){a(p2a)+θ(2ap2)S}=0,T(C5a+C2Q2)-A-{C4p1(p1-a)-C5p2(p2-a)}Q1-p2(p2-a)[C5(Q2+S)+C22aQ22+{(C1+C3θ)a+C5θ(p2-a)}×S22+C1θ(2a-p2)3a2(p2-a)S3]-{(C1+C3θ)f+θ(C4p1(p1-a)2-C5p2(p2-a)2)}Q122-C1dfθ3kQ13=0. Solving these and using (3.28), we get the optimal values Q1*, Q2*, S*, and T* of Q1, Q2, S, and T, respectively, which minimize C(Q1, S, T) provided they satisfy the following sufficient condition: H=The  Hessian  Matrix  of  C=(2CQ122CQ1S2CQ1T2CSQ12CS22CST2CTQ12CTS2CT2) is positive definite.

If the solutions obtained from equations (3.30), (3.31), and (3.32) do not satisfy the sufficient condition (3.33), we may conclude that no feasible solution will be optimal for the set of parameter values taken to solve equations (3.30), (3.31), and (3.32). Such a situation will imply that the parameter values are inconsistent and there is some error in their estimation.

4. Numerical ExamplesExample 4.1.

Let A=200, C1=1.5, C2=2, C3=18, C4=15, C5=13, a=3, p1=4, p2=6, and θ=0.002 in appropriate units.

Based on these input data, the computer outputs are

Q1*=6.2681,  S*=13.9149,Q2*=10.8676,T*  =20.7353,  C*=60.7351.

These results satisfy the sufficient condition.

Example 4.2.

The parameters are similar to those in Example 4.1, except that p2 is changed to 8 units.

Based on these input data, the computer outputs are

Q1*=8.2123,  S*=14.5851,Q2*=11.3927,  T*=20.4743,  C*=61.7855.

These results satisfy the sufficient condition.

Example 4.3.

The parameters are similar to those in Example 4.1, except that p2 is changed to 10 units.

Based on these input data, the computer outputs are

Q1*=8.9252,  S*=14.875,  Q2*=11.62,T*  =20.3248,C*=62.2401.

These results satisfy the sufficient condition.

These examples reveal that a higher value of p2 causes higher values of Q1*, S*, Q2*, and C*, but lower value of T*.

5. Sensitivity Analysis

Sensitivity analysis depicts the extent to which the optimal solution of the model is affected by changes or errors in its input parameter values. In this section, we study the sensitivity of the optimal inventory levels Q1*, S*, backorder lever Q2*, cycle length T*, and average system cost   C* with respect to the changes in the values of the parameters C1, C2, C3, C4, C5,θ, A, a, p1, and p2. The results are shown in Tables 13.

Sensitivity Analysis of Example 4.1.

 Changing parameter Change (%) Change (%) in Q1* S* Q2* T* C* C1 +10 −3.786 −6.510 +2.467 −2.826 +0.883 +5 −1.916 −3.376 +1.262 −1.462 +0.452 − 5 +1.961 +3.649 −1.328 +1.571 −0.475 −10 +3.958 +7.608 −2.726 +3.263 −0.975 C2 +10 +3.628 +1.624 −7.616 −1.191 +0.581 +5 +1.881 +0.842 −3.960 −0.622 +0.301 − 5 −2.028 −0.908 +4.306 +0.683 −0.325 −10 −4.223 −1.891 +9.009 +1.437 −0.677 C3 +10 −0.089 −0.168 +0.062 −0.072 +0.022 + 5 −0.045 −0.084 +0.031 −0.036 +0.011 −5 +0.045 + 0.085 −0.031 +0.036 −0.011 −10 +0.091 +0.169 −0.063 +0.072 −0.022 C4 +10 −80.468 +5.337 + 5.337 −12.126 +1.910 +5 −38.354 +3.478 +3.478 −5.062 +1.245 −5 +35.143 −4.861 −4.861 +3.348 −1.740 −10 +67.384 −10.973 −10.975 +5.140 −3.927 C5 +10 +58.906 −9.329 −9.179 +4.747 +3.137 + 5 +30.574 −4.196 −4.115 +2.965 +1.738 −5 −33.032 +3.165 +3.079 −4.258 −2.109 −10 −68.877 +5.148 +4.973 −9.992 −4.642 θ +10 −0.108 −0.294 +0.102 −0.101 +0.036 + 5 −0.054 −0.147 +0.052 −0.051 +0.018 −5 +0.056 +0.148 −0.052 +0.051 −0.018 −10 +0.110 +0.296 −0.102 +0.103 −0.036 A +10 +9.654 +4.321 +4.320 +5.422 +1.546 + 5 +4.890 +2.189 +2.189 +2.746 +0.783 −5 −5.027 −2.250 −2.252 −2.823 −0.805 −10 −10.204 −4.568 −4.570 −5.729 −1.635 a +10 +2.527 −4.392 −4.400 +10.160 +4.847 + 5 +2.127 −1.809 −1.814 +4.189 +2.562 −5 −3.529 +1.179 +1.812 −2.983 −2.787 −10 −8.259 +1.820 +1.182 −5.084 −5.767 p1 +10 −36.526 +4.298 +4.296 -9.557 +1.538 + 5 −14.757 +2.565 +2.564 -5.313 +0.918 − 5 + 8.642 −3.632 −3.634 +7.214 −1.300 −10 +11.062 −8.804 −8.804 +18.052 −3.150 p2 +10 +13.856 +2.009 +2.016 −0.440 +0.721 + 5 +7.752 +1.098 +1.100 −0.223 +0.394 − 5 −10.207 −1.355 −1.360 +0.210 −0.486 − 10 −24.350 −3.081 −3.090 +0.358 −1.106

Sensitivity Analysis of Example 4.2.

 Parameter Change % Change % in Q1* S* Q2* T* C* C1 10 −4.5943 −6.4038 2.5824 −3.1601 0.9522 5 −2.3550 −3.3212 1.3201 −1.6357 0.4865 −5 2.4780 3.5920 −1.3798 1.7607 −0.5090 −10 5.0899 7.4926 −2.8281 3.6622 −1.0426 C2 10 2.4512 1.3747 −7.8410 −1.0047 0.5069 5 1.2725 0.7137 −4.0824 −0.5250 0.2632 −5 −1.3784 −0.7727 4.4502 0.5773 −0.2850 −10 −2.8762 −1.6119 9.3209 1.2152 −0.5946 C3 10 −0.1120 −0.1659 0.0641 −0.0806 0.0238 5 −0.0560 −0.0830 0.0325 −0.0401 0.0120 −5 0.0560 0.0836 −0.0316 0.0405 −0.0118 −10 0.1108 0.1673 −0.0641 0.0806 −0.0238 C4 10 −43.0659 8.6979 8.6994 −8.1600 3.2080 5 −20.6130 4.8550 4.8575 −3.4526 1.7907 −5 18.9557 −5.7662 −5.7669 2.3200 −2.1267 −10 36.3674 −12.3825 −12.3851 3.5786 −4.5673 C5 10 31.7463 −10.6369 −10.4892 3.2802 2.4439 5 16.4668 −5.0024 −4.9233 2.0421 1.3403 −5 −17.7368 4.3277 4.2413 −2.8973 −1.5920 −10 −36.8618 7.9142 7.7348 −6.7284 −3.4595 θ 10 −0.1705 −0.2935 0.1053 −0.1114 0.0387 5 −0.0852 −0.1467 0.0527 −0.0557 0.0194 −5 0.0852 0.1481 −0.0527 0.0562 −0.0194 −10 0.1717 0.2955 −0.1053 0.1123 −0.0388 A 10 7.4498 4.1769 4.1781 5.2598 1.5405 5 3.7724 2.1152 2.1163 2.6633 0.7801 −5 −3.8759 −2.1728 −2.1733 −2.7351 −0.8015 −10 −7.8638 −4.4086 −4.3686 −5.5479 −1.6261 a 10 −3.0734 −4.3647 −4.3695 10.4272 4.7005 5 −0.7781 −1.7559 −1.7573 4.2653 2.5075 −5 −0.4420 1.0710 1.0735 −2.9881 −2.7602 −10 −1.9093 1.5646 1.5721 −5.0468 −5.7329 p1 10 −7.0102 5.7998 5.8002 −9.1725 2.1392 5 −2.4147 3.2567 3.2591 −5.2099 1.2011 −5 −0.1413 −4.2530 −4.2536 7.2799 −1.5687 −10 −3.6117 −9.9986 −9.9994 18.3850 −3.6878 p2 10 4.2741 0.9565 0.9611 −0.3409 0.3540 5 2.3160 0.5129 0.5161 −0.1802 0.1899 −5 −2.7861 −0.5999 −0.6021 0.2032 −0.2222 −10 −6.2029 −1.3130 −1.3184 0.4322 −0.4860

Sensitivity Analysis of Example 4.3.

 Parameter Change % Change % in Q1* S* Q2* T* C* C1 10 −4.7428 −6.3395 2.6515 −3.2960 0.9900 5 −2.4358 −3.2881 1.3537 −1.7063 0.5053 −5 2.5770 3.5570 −1.4139 1.8377 −0.5278 −10 5.3086 7.4192 −2.8924 3.8229 −1.0800 C2 10 2.1053 1.2585 −7.9466 −0.9186 0.4700 5 1.0935 0.6541 −4.1394 −0.4802 0.2442 −5 −1.1865 −0.7092 4.5172 0.5284 −0.2649 −10 −2.4784 −1.4810 9.4647 1.1119 −0.2521 C3 10 −0.1165 −0.1654 0.0680 −0.0836 0.0246 5 −0.0583 −0.0827 0.0336 −0.0418 0.0124 −5 0.0583 0.0834 −0.0327 0.0423 −0.0124 −10 0.1154 0.1661 −0.0654 0.0841 −0.0247 C4 10 −33.843 9.8266 9.8287 −6.8744 3.6698 5 −16.218 5.3284 5.3305 −2.9216 1.9900 −5 14.9308 −6.0887 −6.0912 1.9744 −2.2741 −10 28.6470 −12.895 −12.897 3.0524 −4.8160 C5 10 24.9866 −11.109 −10.962 2.7853 2.1724 5 12.9599 −5.2894 −5.2117 1.7319 1.1870 −5 −14.171 4.7287 4.6429 −2.4468 −1.3999 −10 −28.957 8.8491 8.6695 −5.6635 −3.0291 θ 10 −0.1860 −0.2931 0.1084 −0.1151 0.0400 5 −0.0930 −0.1466 0.0542 −0.0576 0.0199 −5 0.0930 0.1472 −0.0525 0.0581 −0.0201 −10 0.1871 0.2951 −0.1084 0.1161 −0.0402 A 10 6.9018 4.1264 4.1274 5.2060 1.5408 5 3.4946 2.0894 2.0904 2.6357 0.7802 −5 −3.5898 −2.1452 −2.1446 −2.7056 −0.8014 −10 −7.2805 −4.3523 −4.3537 −5.4879 −1.6256 a 10 −4.5355 −4.4437 −4.4484 10.6200 4.6049 5 −1.5260 −1.7768 −1.7780 4.3356 2.4685 −5 0.3339 1.0629 1.0654 −3.0219 −2.7352 −10 −0.3328 1.5267 1.5318 −5.0893 −5.6941 p1 10 −1.4812 6.2783 6.2806 −9.2306 2.3446 5 0.0560 3.4931 3.4923 −5.2601 1.3045 −5 −2.0862 −4.4867 −4.4880 7.3826 −1.6758 −10 −6.9959 −10.463 −10.463 18.6688 −3.9079 p2 10 2.3775 0.6158 0.6170 −0.2480 0.2306 5 1.2739 0.3281 0.3305 −0.1309 0.1228 −5 −1.4902 −0.3765 −0.3778 0.1486 −0.1414 −10 −3.2537 −0.8148 3.1885 0.3178 −0.1334

Careful study of Table 1 reveals the following facts.

It is seen that Q1* is insensitive to changes in the values of parameters C3 and θ, moderately sensitive to changes in the values of parameters C1, C2, A, and a, and highly sensitive to changes in the values of parameters C4, C5, p1, and p2.

It is observed that S* is insensitive to changes in the values of parameters C2, C3,  and  θ and moderately sensitive to changes in the values of parameters C1, C4, C5, A, a, p1, and p2.

It is seen that Q2* is insensitive to changes in the values of parameters C3, θ and moderately sensitive to changes in the values of parameters C1, C2, C4, C5, A, a, p1 and p2.

Table 1 reveals that T* is insensitive to changes in the values of parameters C2, C3θ, and p2, moderately sensitive to changes in the values of parameters C1, C5, A, and a, and highly sensitive to changes in the values of parameters C4 and p1.

It can be seen that the optimum total cost C* is insensitive to changes in the values of parameters C1,  C2, C3, θ, A, and p2 and moderately sensitive to changes in the values of parameters C4,  C5, a, and p1.

Careful study of Table 2 reveals the following facts.

It is seen that Q1* is insensitive to changes in the values of parameters C3 and θ, moderately sensitive to changes in the values of parameters C1, C2, A, a, p1, and p2, and highly sensitive to changes in the values of parameters C4 and   C5.

It is observed that S* is insensitive to changes in the values of parameters C2, C3, θ,and p2, moderately sensitive to changes in the values of parameters C1, C5, A, a, and p1, and highly sensitive to changes in the values of parameters C4.

It is seen that Q2* is insensitive to changes in the values of parameters C3, θ, and p2 moderately sensitive to changes in the values of parameters C1, C2, C5, A, a, and p1, and highly sensitive to changes in the value of parameter C4.

Table 2 reveals that T* is insensitive to changes in the values of parameters C2, C3, θ, and p2, moderately sensitive to changes in the values of parameters C1, C4, C5, A, and a, and highly sensitive to changes in the value of parameter p1.

It can be seen that the optimum total cost C* is insensitive to changes in the values of parameters C1, C2, C3, θ, A, and p2 and moderately sensitive to changes in the values of parameters C4, C5, a, and p1.

Careful study of Table 3 reveals the following facts.

It is seen that Q1* is insensitive to changes in the values of parameters   C3 and θ, moderately sensitive to changes in the values of parameters C1, C2, A, a, p1, and p2, and highly sensitive to changes in the values of parameters C4 and C5.

It is observed that S* is insensitive to changes in the values of parameters C2, C3, and θ, p2, moderately sensitive to changes in the values of parameters C1, A, a, and p1, and highly sensitive to changes in the values of parameters C4 and C5.

It is seen that Q2* is insensitive to changes in the values of parameters C3 and θ, moderately sensitive to changes in the values of parameters C1, C2, C5, A, a, p1, and p2, and highly sensitive to changes in the value of parameter C4.

Table 3 reveals that T* is insensitive to changes in the values of parameters C2, C3, and θ, p2, moderately sensitive to changes in the values of parameters C1, C4, C5, A, and a, and highly sensitive to changes in the value of parameter p1.

It can be seen that the optimum total cost C* is insensitive to changes in the values of parameters C1, C2, C3, θ, A, and p2 and moderately sensitive to changes in the values of parameters C4, C5, a, and p1.

6. Concluding Remarks

In the present paper, we have dealt with a continuous production inventory model for deteriorating items with shortages in which two different rates of production are available, and it is possible that production started at one rate and after some time it may be switched over to another rate. It is assumed that the demand and production rates are constant and the distribution of the time to deterioration of an item follows the exponential distribution. Such a situation is desirable in the sense that by starting at a low rate of production, a large quantum stock of manufactured item, at the initial stage is avoided, leading to reduction in the holding cost. The variation in production rate provides a way resulting consumer satisfaction and earning potential profit. For this model, we have derived the average system cost and the optimal decision rules for Q1, Q2, S, and T when the deterioration rate θ is very small. Results are illustrated by numerical examples.

However, success depends on the correctness of the estimation of the input parameters. In reality, however, management is most likely to be uncertain of the true values of these parameters. Moreover, their values may be changed over time due to their complex structures. Therefore, it is more reasonable to assume that these parameters are known only within some given ranges.

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