Abstract

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 [1] who considered fashion goods deteriorating at the end of a prescribed storage period. Ghare and Schrader [2] 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 [3]. Aggarwal [4] developed an order-level inventory model by correcting and modifying the error in Shah and Jaiswal’s analysis [3] 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 [5] discussed an inventory model for replenishment. This was followed by another model by Dave [6] with variable instantaneous demand, discrete opportunities for replenishment, and shortages. Bahari-Kashani [7] 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 [8]. 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 [9] established an economic order quantity model for a power-form inventory-level-dependent demand pattern. Mandal and Phaujdar [10] then developed an economic production quantity model for deteriorating items with constant production rate and linearly stock-dependent demand. Later on, Datta and Pal [11] 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 [12], Giri et al. [13], Padmanabhan and Vrat [14], Pal et al. [15], Ray and Chaudhuri [16], Urban and Baker [17], Ray et al. [18], Giri and Chaudhuri [19], Datta and Paul [20], Ouyang et al. [21], Teng et al. [22], Roy and Samanta [23], and others.

Another class of inventory models has been developed with time-dependent deterioration rate. Covert and Philip [24] used a two-parameter Weibull distribution to represent the distribution of the time to deterioration. This model was further developed by Philip [25] by taking a three-parameter Weibull distribution for the time to deterioration. Mishra [26] 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 [27] studied a model with a finite rate of production and a time-proportional deterioration rate, allowing backlogging. Goswami and Chaudhuri [8] 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 [28] and Raafat [29]. An order-level inventory model for deteriorating items without shortages has been developed by Jalan and Chaudhuri [30]. Ouyang et al. [31] 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 [32], Samanta and Roy [33].

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:(i)π‘Ž is the constant demand rate;(ii)𝑝1 (>a) and 𝑝2 (>𝑝1) are the constant production rates started at time 𝑑=0 and at time 𝑑 = 𝑑1 (>o), respectively;(iii)𝐢1 is the holding cost per unit per unit time;(iv)𝐢2 is the shortage cost per unit per unit time;(v)𝐢3 is the cost of a deteriorated unit. (𝐢1, 𝐢2, and 𝐢3 are known constants);(vi)𝐢4 and 𝐢5 are the constant unit production costs when the production rates are 𝑝1 and 𝑝2 respectively (𝐢4>𝐢5);(vii)𝑄(𝑑) is the inventory level at time t(β‰₯0);(viii)A is the setup cost;(ix)replenishment is instantaneous and lead time is zero;(x)𝑇 is the variable duration of production cycle;(xi)shortages are allowed and backlogged;(xii)C is the average cost of the system;(xiii)the distribution of the time to deterioration of an item follows the exponential distribution 𝑔(𝑑), where 𝑔(𝑑)=πœƒπ‘’-πœƒπ‘‘,for𝑑>0,0,otherwise.(2.1)πœƒ is called the deterioration rate; a constant fraction πœƒ (0 < πœƒβ‰ͺ1) 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 𝑑=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, 𝑑1], the duration of production at the rate 𝑝1, by [𝑑1, 𝑑2], the duration of production at the rate 𝑝2, by [𝑑2, 𝑑3], the duration when there is no production but only consumption by demand at a rate a, by [𝑑3, 𝑑4], the duration of shortage, and by [𝑑4, 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.


Figure 1 

3. Model Formulation and Solution

Let 𝑄(𝑑) be the instantaneous state of the inventory level at any time 𝑑 (0 ≀ 𝑑 ≀ 𝑇),then the differential equations describing the instantaneous states of 𝑄(𝑑) in the interval [0, 𝑇] are given by the following:𝑑𝑄(𝑑)𝑑𝑑+πœƒπ‘„(𝑑)=𝑝1-π‘Ž,0≀𝑑≀𝑑1,𝑑𝑄(𝑑)𝑑𝑑+πœƒπ‘„(𝑑)=𝑝2-π‘Ž,𝑑1≀𝑑≀𝑑2,𝑑𝑄(𝑑)𝑑𝑑+πœƒπ‘„(𝑑)=-π‘Ž,𝑑2≀𝑑≀𝑑3,𝑑𝑄(𝑑)𝑑𝑑=-π‘Ž,𝑑3≀𝑑≀𝑑4,𝑑𝑄(𝑑)𝑑𝑑=𝑝2-π‘Ž,𝑑4≀𝑑≀𝑇.(3.1) The boundary conditions are 𝑄𝑑(0)=0,𝑄1ξ€Έ=𝑄1𝑑,𝑄2𝑑=𝑆,𝑄3𝑑=0,𝑄4ξ€Έ=-𝑄2,𝑄(𝑇)=0.(3.2)

The solutions of equations (3.1) are given by1𝑄(𝑑)=πœƒξ€·π‘1βˆ’π‘Žξ€Έξ€·1βˆ’π‘’βˆ’πœƒπ‘‘ξ€Έ,0≀𝑑≀𝑑1=1(3.3)πœƒξ€·π‘2ξ€Έ-π‘Ž+π‘’βˆ’πœƒ(𝑑-𝑑1)𝑄1βˆ’1πœƒξ€·π‘2ξ€Έξ‚„βˆ’π‘Ž,𝑑1≀𝑑≀𝑑2π‘Ž(3.4)=βˆ’πœƒ+ξ‚€π‘Žπ‘†+πœƒξ‚π‘’βˆ’πœƒ(𝑑-𝑑2),𝑑2≀𝑑≀𝑑3ξ€·(3.5)=βˆ’π‘Žπ‘‘βˆ’π‘‘3ξ€Έ,𝑑3≀𝑑≀𝑑4(3.6)=βˆ’π‘„2+𝑝2-π‘Žξ€Έξ€·π‘‘-𝑑4ξ€Έ,𝑑4≀𝑑≀𝑇.(3.7)

From (3.2) and (3.3), we have1πœƒξ€·π‘1βˆ’π‘Žξ€Έξ€·1βˆ’π‘’βˆ’πœƒπ‘‘1ξ€Έ=𝑄1(3.8a)βˆ΄π‘‘1=1πœƒξƒ¬log1+πœƒπ‘„1𝑝1ξ€Έ+πœƒβˆ’π‘Ž2𝑄21𝑝1ξ€Έβˆ’π‘Ž2ξƒ­=𝑄(neglectinghigherpowersofπœƒ,0<πœƒβ‰ͺ1)1𝑝1ξ€Έ+βˆ’π‘Žπœƒπ‘„212𝑝1ξ€Έβˆ’π‘Ž2(neglectinghigherpowersofπœƒ,0<πœƒβ‰ͺ1).(3.8b)

Again, from (3.2) and (3.4), we have 1πœƒξ€·π‘2ξ€Έ-π‘Ž+π‘’πœƒ(𝑑1βˆ’π‘‘2)𝑄1βˆ’1πœƒξ€·π‘2ξ€Έξ‚„-π‘Ž=𝑆(3.9)   βˆ΄π‘’πœƒ(𝑑2βˆ’π‘‘1)=1βˆ’π‘†πœƒξ€·π‘2ξ€Έξƒ­βˆ’π‘Žβˆ’1𝑄1βˆ’1πœƒξ€·π‘2ξ€Έξƒ­ξ€·βˆ’π‘Ž=1+π‘†βˆ’π‘„1ξ€Έπœƒξ€·π‘2ξ€Έ+πœƒβˆ’π‘Ž2ξ€·π‘†βˆ’π‘„1𝑆𝑝2ξ€Έβˆ’π‘Ž2(neglectinghigherpowersofπœƒ),(3.10a)βˆ΄π‘‘2βˆ’π‘‘1=1πœƒξƒ¬ξ€·log1+π‘†βˆ’π‘„1ξ€Έπœƒξ€·π‘2ξ€Έ+πœƒβˆ’π‘Ž2ξ€·π‘†βˆ’π‘„1𝑆𝑝2ξ€Έβˆ’π‘Ž2ξƒ­=ξ€·π‘†βˆ’π‘„1𝑝2ξ€Έ+πœƒξ€·π‘†βˆ’π‘Ž2βˆ’π‘„12ξ€Έ2𝑝2ξ€Έβˆ’π‘Ž2(neglectinghigherpowersofπœƒ),(3.10b)𝑑2=𝑄1𝑝1ξ€Έ+βˆ’π‘Žπœƒπ‘„212𝑝1ξ€Έβˆ’π‘Ž2+ξ€·π‘†βˆ’π‘„1𝑝2ξ€Έ+πœƒξ€·π‘†βˆ’π‘Ž2βˆ’π‘„12ξ€Έ2𝑝2ξ€Έβˆ’π‘Ž2ξ€Ίξ€»by(3.8b).(3.11)

From (3.5), we haveβˆ’1πœƒξ‚€1π‘Ž+𝑆+πœƒπ‘Žξ‚π‘’πœƒ(𝑑2βˆ’π‘‘3)𝑑=0,since𝑄3ξ€Έ=0(3.12a)βŸΉπ‘‘3βˆ’π‘‘2=1πœƒξ‚€log1+πœƒπ‘†π‘Žξ‚=π‘†π‘Žβˆ’π‘†2πœƒ2π‘Ž2(neglectinghigherpowersofπœƒ)(3.12b)βˆ΄π‘‘3=𝑄1𝑝1ξ€Έ+βˆ’π‘Žπœƒπ‘„212𝑝1ξ€Έβˆ’π‘Ž2+ξ€·π‘†βˆ’π‘„1𝑝2ξ€Έ+πœƒξ€·π‘†βˆ’π‘Ž2βˆ’π‘„12ξ€Έ2𝑝2ξ€Έβˆ’π‘Ž2+π‘†π‘Žβˆ’π‘†2πœƒ2π‘Ž2ξ€Ίξ€»by(3.11).(3.13)

Again, from (3.6), we haveπ‘Žξ€·π‘‘3-𝑑4ξ€Έ=βˆ’π‘„2𝑑,since𝑄4ξ€Έ=βˆ’π‘„2,(3.14)βˆ΄π‘‘4=𝑑3+𝑄2π‘Ž.(3.15)

From (3.7) and 𝑄(𝑇) = 0, we haveβˆ’π‘„2+𝑝2-π‘Žξ€Έξ€·π‘‡-𝑑4ξ€Έ=0βˆ΄π‘„2=𝑝2ξ€Έξ‚΅-π‘Žπ‘‡-𝑑3-𝑄2π‘Žξ‚Άξ€Ίξ€»,𝑄by(3.15)(3.16)2=π‘Žξ€·π‘2ξ€Έβˆ’π‘Žπ‘2ξƒ¬π‘„π‘‡βˆ’1𝑝1ξ€Έβˆ’βˆ’π‘Žπœƒπ‘„212𝑝1ξ€Έβˆ’π‘Ž2βˆ’ξ€·π‘†βˆ’π‘„1𝑝2ξ€Έβˆ’πœƒξ€·π‘†βˆ’π‘Ž2βˆ’π‘„12ξ€Έ2𝑝2ξ€Έβˆ’π‘Ž2βˆ’π‘†π‘Ž+𝑆2πœƒ2π‘Ž2ξƒ­ξ€Ίξ€»,[]=𝑝by(3.13)(3.17)𝐷=Totalnumberofdeteriorateditemsin0,𝑇1ξ€Έπ‘‘βˆ’π‘Ž1+𝑝2π‘‘βˆ’π‘Žξ€Έξ€·2-𝑑1ξ€Έξ€»+ξ€Ίξ€·π‘‘βˆ’π‘†π‘†-π‘Ž3-𝑑2=𝑝1ξ€Έξƒ¬π‘„βˆ’π‘Ž1𝑝1ξ€Έ+βˆ’π‘Žπœƒπ‘„212𝑝1ξ€Έβˆ’π‘Ž2ξƒ­+𝑝2ξ€Έξƒ¬ξ€·βˆ’π‘Žπ‘†βˆ’π‘„1𝑝2ξ€Έ+πœƒξ€·π‘†βˆ’π‘Ž2βˆ’π‘„12ξ€Έ2𝑝2ξ€Έβˆ’π‘Ž2ξƒ­ξ‚΅π‘†βˆ’π‘Žπ‘Žβˆ’π‘†2πœƒ2π‘Ž2ξ‚Άξ€Ίξ€»=πœƒusing(3.8b),(3.10b),and(3.12b)2𝑄12𝑝1ξ€Έ+ξ€·π‘†βˆ’π‘Ž2βˆ’π‘„12𝑝2ξ€Έ+π‘†βˆ’π‘Ž2π‘Žξƒ­,𝑆(3.18)1[]ξ€œ=TotalShortageovertheperiod0,𝑇=βˆ’π‘‘4𝑑3π‘Žξ€·π‘‘βˆ’π‘‘3ξ€Έξ€œπ‘‘π‘‘+𝑇𝑑4ξ€Ίβˆ’π‘„2+𝑝2βˆ’π‘Žξ€Έξ€·π‘‘βˆ’π‘‘4𝑝𝑑𝑑by(3.6)and(3.7)=βˆ’2𝑄22𝑝2π‘Ž2ξ€Έξ€Ίξ€»,πΌβˆ’π‘Žbyusing(3.6)and(3.7)(3.19)𝑇[]=ξ€œ=Totalinventorycarriedovertheperiod0,𝑇𝑑10ξ€œπ‘„(𝑑)𝑑𝑑+𝑑2𝑑1ξ€œπ‘„(𝑑)𝑑𝑑+𝑑3𝑑2ξ€œπ‘„(𝑑)𝑑𝑑.(3.20)Now,𝑑101𝑄(𝑑)𝑑𝑑=πœƒξ€·π‘1ξ€Έξ€œβˆ’π‘Žπ‘‘10ξ€·1βˆ’π‘’βˆ’πœƒπ‘‘ξ€Έξ€Ίξ€»=1𝑑𝑑by(3.3)πœƒξ€·π‘1ξ€Έξ‚ƒπ‘‘βˆ’π‘Ž1+1πœƒξ€·π‘’βˆ’πœƒπ‘‘1ξ€Έξ‚„=ξ€·π‘βˆ’11ξ€Έξƒ¬π‘‘βˆ’π‘Ž1πœƒβˆ’π‘„1πœƒξ€·π‘1ξ€Έξƒ­ξ€Ίξ€»=ξ€·π‘βˆ’π‘Žby(3.8a)11βˆ’π‘Žπœƒ2ξƒ―log1+πœƒπ‘„1𝑝1ξ€Έ+πœƒβˆ’π‘Ž2𝑄21𝑝1ξ€Έβˆ’π‘Ž2+πœƒ3𝑄31𝑝1ξ€Έβˆ’π‘Ž3ξƒ°βˆ’π‘„1πœƒξ€·π‘1ξ€Έξƒ­ξ€Ίξ€»=ξ€·π‘βˆ’π‘Žby(3.8a)andneglectinghigherpowersofπœƒ1ξ€Έξƒ¬π‘„βˆ’π‘Ž12𝑝1ξ€Έβˆ’π‘Ž2+πœƒπ‘„31𝑝1ξ€Έβˆ’π‘Ž3βˆ’π‘„212𝑝1ξ€Έβˆ’π‘Ž2βˆ’πœƒπ‘„13𝑝1ξ€Έβˆ’π‘Ž3+πœƒπ‘„313𝑝1ξ€Έβˆ’π‘Ž3ξƒ­=𝑄(neglectinghigherpowersofπœƒ)212𝑝1ξ€Έ+βˆ’π‘Žπœƒπ‘„313𝑝1ξ€Έβˆ’π‘Ž2,ξ€œ(3.21)𝑑2𝑑1ξ€œπ‘„(𝑑)𝑑𝑑=𝑑2𝑑11πœƒξ€·π‘2ξ€Έ+ξ‚†π‘„βˆ’π‘Ž1βˆ’1πœƒξ€·π‘2ξ€Έξ‚‡π‘’βˆ’π‘Žβˆ’πœƒ(π‘‘βˆ’π‘‘1)ξ‚„ξ€Ίξ€»=1𝑑𝑑by(3.4)πœƒξ€·π‘2π‘‘βˆ’π‘Žξ€Έξ€·2βˆ’π‘‘1ξ€Έβˆ’1πœƒξ‚†π‘„1βˆ’1πœƒξ€·π‘2ξ€Έξ‚‡ξ€½π‘’βˆ’π‘Žβˆ’πœƒ(𝑑2βˆ’π‘‘1)ξ€Ύ=1βˆ’1πœƒ2𝑝2ξ€Έξƒ¬ξ€·βˆ’π‘Žlog1+π‘†βˆ’π‘„1ξ€Έπœƒξ€·π‘2ξ€Έ+πœƒβˆ’π‘Ž2ξ€·π‘†βˆ’π‘„1𝑆𝑝2ξ€Έβˆ’π‘Ž2+πœƒ3𝑆2ξ€·π‘†βˆ’π‘„1𝑝2ξ€Έβˆ’π‘Ž3ξƒ­βˆ’ξ€·π‘†βˆ’π‘„1ξ€Έπœƒξ€Ίξ€»=𝑝by(3.9)and(3.10a)2ξ€Έξƒ¬ξ€·βˆ’π‘Žπ‘†βˆ’π‘„1ξ€Έπœƒξ€·π‘2ξ€Έ+ξ€·βˆ’π‘Žπ‘†βˆ’π‘„1𝑆𝑝2ξ€Έβˆ’π‘Ž2+𝑆2ξ€·π‘†βˆ’π‘„1ξ€Έπœƒξ€·π‘2ξ€Έβˆ’π‘Ž3βˆ’ξ€·π‘†βˆ’π‘„1ξ€Έ22𝑝2ξ€Έβˆ’π‘Ž2βˆ’ξ€·π‘†βˆ’π‘„1ξ€Έ2π‘†πœƒξ€·π‘2ξ€Έβˆ’π‘Ž3+ξ€·π‘†βˆ’π‘„1ξ€Έ3πœƒ3𝑝2ξ€Έβˆ’π‘Ž3ξƒ­βˆ’ξ€·π‘†βˆ’π‘„1ξ€Έπœƒ(=𝑆neglectinghigherpowersofπœƒ)2βˆ’π‘„21ξ€Έ2𝑝2ξ€Έ+πœƒξ€·π‘†βˆ’π‘Ž3βˆ’π‘„31ξ€Έ3𝑝2ξ€Έβˆ’π‘Ž2,ξ€œ(3.22)𝑑3𝑑2ξ€œπ‘„(𝑑)𝑑𝑑=𝑑3𝑑2ξ‚ƒβˆ’π‘Žπœƒ+ξ‚€π‘Žπ‘†+πœƒξ‚π‘’βˆ’πœƒ(π‘‘βˆ’π‘‘2)ξ‚„ξ€Ίξ€»π‘Žπ‘‘π‘‘by(3.5)=βˆ’πœƒξ€·π‘‘3βˆ’π‘‘2ξ€Έβˆ’1πœƒξ‚€π‘Žπ‘†+πœƒξ‚ξ€½π‘’βˆ’πœƒ(𝑑3βˆ’π‘‘2)ξ€Ύπ‘Žβˆ’1=βˆ’πœƒ2ξ‚€log1+π‘†πœƒπ‘Žξ‚+π‘†πœƒξ€Ίξ€»π‘Žby(3.12a)=βˆ’πœƒ2ξ‚Έπ‘†πœƒπ‘Žβˆ’π‘†2πœƒ22π‘Ž2+𝑆3πœƒ33π‘Ž3ξ‚Ή+π‘†πœƒ=𝑆(neglectinghigherpowersofπœƒ)2βˆ’π‘†2π‘Ž3πœƒ3π‘Ž2.(3.23)

Therefore, from (3.20), total inventory carried over the cycle [0,𝑇]=𝑄212𝑝1ξ€Έ+βˆ’π‘Žπœƒπ‘„313𝑝1ξ€Έβˆ’π‘Ž2+𝑆2βˆ’π‘„21ξ€Έ2𝑝2ξ€Έ+πœƒξ€·π‘†βˆ’π‘Ž3βˆ’π‘„31ξ€Έ3𝑝2ξ€Έβˆ’π‘Ž2+𝑆2βˆ’π‘†2π‘Ž3πœƒ3π‘Ž2[],(3.24)𝑃=Productioncostovertheperiod0,𝑇=𝐢4𝑝1𝑑1+𝐢5𝑝2𝑑2-𝑑1ξ€Έ+𝑝2𝑇-𝑑4=𝐢4𝑝1𝑄1𝑝1ξ€Έξƒ¬βˆ’π‘Ž1+πœƒπ‘„12𝑝1ξ€Έξƒ­+πΆβˆ’π‘Ž5𝑝2𝑝2ξ€Έξƒ¬βˆ’π‘Žπ‘†+𝑄2βˆ’π‘„1+πœƒξ€·π‘†2βˆ’π‘„21ξ€Έ2𝑝2ξ€Έξƒ­ξ€Ίξ€».βˆ’π‘Žby(3.8b),(3.10b),and(3.16)(3.25)

Average cost of the system𝑄=𝐢1ξ€Έ=1,𝑆,𝑇𝑇𝐢1πΌπ‘‡βˆ’πΆ2𝑆1+𝐢3ξ€»=1𝐷+𝑃+𝐴𝑇𝐢𝐴+5𝑝2𝑝2ξ€Έξƒ―ξ€·πΆβˆ’π‘Žπ‘†+1+𝐢3πœƒξ€Έπ‘2𝑝2π‘Ž2ξ€Έ+πΆβˆ’π‘Ž5𝑝2πœƒ2𝑝2ξ€Έβˆ’π‘Ž2𝑆2+𝐢1πœƒξ€·2π‘Žβˆ’π‘2𝑝23π‘Ž2𝑝2ξ€Έβˆ’π‘Ž2𝑆3+𝐢5𝑝2𝑝2ξ€Έπ‘„βˆ’π‘Ž2+𝐢2𝑝2𝑝2π‘Ž2ξ€Έπ‘„βˆ’π‘Ž22+𝐢4𝑝1𝑝1ξ€Έβˆ’πΆβˆ’π‘Ž5𝑝2𝑝2ξ€Έξƒ°π‘„βˆ’π‘Ž1+𝑓2𝐢1+𝐢3πœƒξ€Έ+πœƒ2𝐢4𝑝1𝑝1ξ€Έβˆ’π‘Ž2βˆ’πΆ5𝑝2𝑝2ξ€Έβˆ’π‘Ž2𝑄ξƒͺξƒ°21+𝐢1π‘‘π‘“πœƒπ‘„3π‘˜31ξƒ­ξ€Ίξ€»,using(3.18),(3.19),(3.24),and(3.25)(3.26) where 𝑑=𝑝1+𝑝2𝑝-2π‘Ž,π‘˜=1𝑝-π‘Žξ€Έξ€·2ξ€Έ,𝑝-π‘Žπ‘“=2βˆ’π‘1ξ€Έπ‘˜,(3.27) and from (3.17),𝑄2=π‘Žξ€·π‘2ξ€Έβˆ’π‘Žπ‘2π‘Žξ€·π‘π‘‡βˆ’π‘†βˆ’2βˆ’π‘1𝑝2𝑝1ξ€Έπ‘„βˆ’π‘Ž1βˆ’πœƒξ€·2π‘Žβˆ’π‘2𝑝2π‘Ž2ξ€Έπ‘†βˆ’π‘Ž2βˆ’π‘Žπ‘‘π‘“πœƒ2𝑝2𝑝1ξ€Έπ‘„βˆ’π‘Ž21.(3.28)

The necessary conditions for 𝐢(𝑄1, 𝑆, 𝑇) to be minimum are πœ•πΆπœ•π‘„1=0,πœ•πΆπœ•π‘†=0,πœ•πΆπœ•π‘‡=0,(3.29) that is, 𝐢1π‘‘π‘“πœƒπ‘˜π‘„21+𝐢1+𝐢3πœƒξ€Έξƒ©πΆπ‘“+πœƒ4𝑝1𝑝1ξ€Έβˆ’π‘Ž2βˆ’πΆ5𝑝2𝑝2ξ€Έβˆ’π‘Ž2𝑄ξƒͺξƒ°1βˆ’1π‘˜ξ€·πΆ5π‘Ž+𝐢2𝑄2𝑝2βˆ’π‘1+πœƒπ‘‘π‘“π‘„1ξ€Έ+𝐢4𝑝1𝑝1ξ€Έβˆ’πΆβˆ’π‘Ž5𝑝2𝑝2ξ€ΈπΆβˆ’π‘Ž=0,(3.30)1πœƒπ‘2ξ€·2π‘Ž-𝑝2𝑆2+π‘Žπ‘2𝐢1+𝐢3πœƒπ‘ξ€Έξ€·2ξ€Έ-π‘Ž+𝐢5ξ€Ύπ‘†π‘Žπœƒ+𝐢5π‘Ž2𝑝2𝑝2ξ€Έ-π‘Ž-𝑝2ξ€·π‘ŽπΆ5+𝐢2𝑄2π‘Žξ€·π‘ξ€Έξ€½2ξ€Έξ€·-π‘Ž+πœƒ2π‘Ž-𝑝2𝑆𝑇𝐢=0,(3.31)5π‘Ž+𝐢2𝑄2ξ€Έξƒ―πΆβˆ’π΄βˆ’4𝑝1𝑝1ξ€Έβˆ’πΆβˆ’π‘Ž5𝑝2𝑝2ξ€Έξƒ°π‘„βˆ’π‘Ž1βˆ’π‘2𝑝2ξ€Έξƒ¬πΆβˆ’π‘Ž5𝑄2ξ€Έ+𝐢+𝑆2𝑄2π‘Ž22+𝐢1+𝐢3πœƒξ€Έπ‘Ž+𝐢5πœƒξ€·π‘2ξ€Έξƒ°Γ—π‘†βˆ’π‘Ž22+𝐢1πœƒξ€·2π‘Žβˆ’π‘2ξ€Έ3π‘Ž2𝑝2ξ€Έπ‘†βˆ’π‘Ž3ξƒ­βˆ’ξƒ―ξ€·πΆ1+𝐢3πœƒξ€Έξƒ©πΆπ‘“+πœƒ4𝑝1𝑝1ξ€Έβˆ’π‘Ž2βˆ’πΆ5𝑝2𝑝2ξ€Έβˆ’π‘Ž2𝑄ξƒͺξƒ°212βˆ’πΆ1π‘‘π‘“πœƒπ‘„3π‘˜31=0.(3.32) Solving these and using (3.28), we get the optimal values 𝑄1βˆ—, 𝑄2βˆ—, π‘†βˆ—, and π‘‡βˆ— of 𝑄1, 𝑄2, 𝑆, and 𝑇, respectively, which minimize 𝐢(𝑄1, 𝑆, 𝑇) provided they satisfy the following sufficient condition: =βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπœ•π»=TheHessianMatrixof𝐢2πΆπœ•π‘„21πœ•2πΆπœ•π‘„1πœ•πœ•π‘†2πΆπœ•π‘„1πœ•πœ•π‘‡2πΆπœ•π‘†πœ•π‘„1πœ•2πΆπœ•π‘†2πœ•2πΆπœ•πœ•π‘†πœ•π‘‡2πΆπœ•π‘‡πœ•π‘„1πœ•2πΆπœ•πœ•π‘‡πœ•π‘†2πΆπœ•π‘‡2⎞⎟⎟⎟⎟⎟⎟⎟⎠(3.33) 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 Examples

Example 4.1. Let 𝐴=200, 𝐢1=1.5, 𝐢2=2, 𝐢3=18, 𝐢4=15, 𝐢5=13, π‘Ž=3, 𝑝1=4, 𝑝2=6, and πœƒ=0.002 in appropriate units.
Based on these input data, the computer outputs are
𝑄1βˆ—=6.2681,π‘†βˆ—=13.9149,𝑄2βˆ—=10.8676,π‘‡βˆ—=20.7353,πΆβˆ—=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
𝑄1βˆ—=8.2123,π‘†βˆ—=14.5851,𝑄2βˆ—=11.3927,π‘‡βˆ—=20.4743,πΆβˆ—=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
𝑄1βˆ—=8.9252,π‘†βˆ—=14.875,𝑄2βˆ—=11.62,π‘‡βˆ—=20.3248,πΆβˆ—=62.2401.
These results satisfy the sufficient condition.

These examples reveal that a higher value of 𝑝2 causes higher values of 𝑄1βˆ—, π‘†βˆ—, 𝑄2βˆ—, and πΆβˆ—, but lower value of π‘‡βˆ—.

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 𝑄1βˆ—, π‘†βˆ—, backorder lever 𝑄2βˆ—, cycle length π‘‡βˆ—, and average system cost πΆβˆ— with respect to the changes in the values of the parameters 𝐢1, 𝐢2, 𝐢3, 𝐢4, 𝐢5,ΞΈ, 𝐴, π‘Ž, 𝑝1, and 𝑝2. The results are shown in Tables 1–3.

Table 1 
Sensitivity Analysis of Example 4.1.
Table 2 
Sensitivity Analysis of Example 4.2.
Table 3 
Sensitivity Analysis of Example 4.3.

Careful study of Table 1 reveals the following facts.(i)It is seen that 𝑄1βˆ— is insensitive to changes in the values of parameters 𝐢3 and πœƒ, moderately sensitive to changes in the values of parameters 𝐢1, 𝐢2, 𝐴, and π‘Ž, and highly sensitive to changes in the values of parameters 𝐢4, 𝐢5, 𝑝1, and 𝑝2.(ii)It is observed that π‘†βˆ— is insensitive to changes in the values of parameters 𝐢2, 𝐢3,andπœƒ and moderately sensitive to changes in the values of parameters 𝐢1, 𝐢4, 𝐢5, 𝐴, π‘Ž, 𝑝1, and 𝑝2.(iii)It is seen that 𝑄2βˆ— is insensitive to changes in the values of parameters 𝐢3, πœƒ and moderately sensitive to changes in the values of parameters 𝐢1, 𝐢2, 𝐢4, 𝐢5, 𝐴, π‘Ž, 𝑝1 and 𝑝2.(iv)Table 1 reveals that π‘‡βˆ— is insensitive to changes in the values of parameters 𝐢2, 𝐢3πœƒ, and 𝑝2, moderately sensitive to changes in the values of parameters 𝐢1, 𝐢5, 𝐴, and π‘Ž, and highly sensitive to changes in the values of parameters 𝐢4 and 𝑝1.(v)It can be seen that the optimum total cost πΆβˆ— is insensitive to changes in the values of parameters 𝐢1,𝐢2, 𝐢3, πœƒ, 𝐴, and 𝑝2 and moderately sensitive to changes in the values of parameters 𝐢4,𝐢5, π‘Ž, and 𝑝1.

Careful study of Table 2 reveals the following facts.(i)It is seen that 𝑄1βˆ— is insensitive to changes in the values of parameters 𝐢3 and πœƒ, moderately sensitive to changes in the values of parameters 𝐢1, 𝐢2, 𝐴, π‘Ž, 𝑝1, and p2, and highly sensitive to changes in the values of parameters 𝐢4 and 𝐢5.(ii)It is observed that π‘†βˆ— is insensitive to changes in the values of parameters 𝐢2, 𝐢3, πœƒ,and p2, moderately sensitive to changes in the values of parameters 𝐢1, 𝐢5, 𝐴, π‘Ž, and 𝑝1, and highly sensitive to changes in the values of parameters 𝐢4.(iii)It is seen that 𝑄2βˆ— is insensitive to changes in the values of parameters 𝐢3, πœƒ, and p2 moderately sensitive to changes in the values of parameters 𝐢1, 𝐢2, 𝐢5, 𝐴, π‘Ž, and 𝑝1, and highly sensitive to changes in the value of parameter 𝐢4. (iv)Table 2 reveals that π‘‡βˆ— is insensitive to changes in the values of parameters C2, C3, πœƒ, and 𝑝2, moderately sensitive to changes in the values of parameters 𝐢1, 𝐢4, 𝐢5, 𝐴, and π‘Ž, and highly sensitive to changes in the value of parameter 𝑝1. (v)It can be seen that the optimum total cost πΆβˆ— is insensitive to changes in the values of parameters 𝐢1, 𝐢2, 𝐢3, πœƒ, 𝐴, and 𝑝2 and moderately sensitive to changes in the values of parameters 𝐢4, 𝐢5, π‘Ž, and 𝑝1.

Careful study of Table 3 reveals the following facts.(i)It is seen that 𝑄1βˆ— is insensitive to changes in the values of parameters 𝐢3 and πœƒ, moderately sensitive to changes in the values of parameters 𝐢1, 𝐢2, 𝐴, π‘Ž, 𝑝1, and 𝑝2, and highly sensitive to changes in the values of parameters 𝐢4 and 𝐢5.(ii)It is observed that π‘†βˆ— is insensitive to changes in the values of parameters 𝐢2, 𝐢3, and πœƒ, 𝑝2, moderately sensitive to changes in the values of parameters 𝐢1, 𝐴, π‘Ž, and 𝑝1, and highly sensitive to changes in the values of parameters 𝐢4 and 𝐢5.(iii)It is seen that 𝑄2βˆ— is insensitive to changes in the values of parameters 𝐢3 and πœƒ, moderately sensitive to changes in the values of parameters 𝐢1, 𝐢2, 𝐢5, 𝐴, π‘Ž, 𝑝1, and 𝑝2, and highly sensitive to changes in the value of parameter 𝐢4.(iv)Table 3 reveals that π‘‡βˆ— is insensitive to changes in the values of parameters 𝐢2, 𝐢3, and πœƒ, 𝑝2, moderately sensitive to changes in the values of parameters 𝐢1, 𝐢4, 𝐢5, 𝐴, and π‘Ž, and highly sensitive to changes in the value of parameter 𝑝1.(v)It can be seen that the optimum total cost πΆβˆ— is insensitive to changes in the values of parameters 𝐢1, 𝐢2, 𝐢3, πœƒ, 𝐴, and 𝑝2 and moderately sensitive to changes in the values of parameters 𝐢4, 𝐢5, π‘Ž, 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 πœƒ 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.