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)ğ‘Ž is the constant demand rate;(ii)𝑝1 (>a) and 𝑝2 (>𝑝1) are the constant production rates started at time 𝑡=0 and at time 𝑡 = 𝑡1 (>o), respectively;(iii)𝐶1 is the holding cost per unit per unit time;(iv)𝐶2 is the shortage cost per unit per unit time;(v)𝐶3 is the cost of a deteriorated unit. (𝐶1, 𝐶2, and 𝐶3 are known constants);(vi)𝐶4 and 𝐶5 are the constant unit production costs when the production rates are 𝑝1 and 𝑝2 respectively (𝐶4>𝐶5);(vii)𝑄(𝑡) is the inventory level at time t(≥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(𝑡), where 𝑔(𝑡)=𝜃𝑒-𝜃𝑡,for𝑡>0,0,otherwise.(2.1)𝜃 is called the deterioration rate; a constant fraction 𝜃 (0 < 𝜃≪1) 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 ≤ 𝑡 ≤ 𝑇),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)=𝑝1-ğ‘Ž,0≤𝑡≤𝑡1,𝑑𝑄(𝑡)𝑑𝑡+𝜃𝑄(𝑡)=𝑝2-ğ‘Ž,𝑡1≤𝑡≤𝑡2,𝑑𝑄(𝑡)𝑑𝑡+𝜃𝑄(𝑡)=-ğ‘Ž,𝑡2≤𝑡≤𝑡3,𝑑𝑄(𝑡)𝑑𝑡=-ğ‘Ž,𝑡3≤𝑡≤𝑡4,𝑑𝑄(𝑡)𝑑𝑡=𝑝2-ğ‘Ž,𝑡4≤𝑡≤𝑇.(3.1) The boundary conditions are 𝑄𝑡(0)=0,𝑄1=𝑄1𝑡,𝑄2𝑡=𝑆,𝑄3𝑡=0,𝑄4=-𝑄2,𝑄(𝑇)=0.(3.2)

The solutions of equations (3.1) are given by1𝑄(𝑡)=𝜃𝑝1âˆ’ğ‘Žî€¸î€·1−𝑒−𝜃𝑡,0≤𝑡≤𝑡1=1(3.3)𝜃𝑝2-ğ‘Ž+𝑒−𝜃(𝑡-𝑡1)𝑄1−1𝜃𝑝2î€¸î‚„âˆ’ğ‘Ž,𝑡1≤𝑡≤𝑡2ğ‘Ž(3.4)=−𝜃+î‚€ğ‘Žğ‘†+𝜃𝑒−𝜃(𝑡-𝑡2),𝑡2≤𝑡≤𝑡3(3.5)=âˆ’ğ‘Žğ‘¡âˆ’ğ‘¡3,𝑡3≤𝑡≤𝑡4(3.6)=−𝑄2+𝑝2-ğ‘Žî€¸î€·ğ‘¡-𝑡4,𝑡4≤𝑡≤𝑇.(3.7)

From (3.2) and (3.3), we have1𝜃𝑝1âˆ’ğ‘Žî€¸î€·1−𝑒−𝜃𝑡1=𝑄1(3.8a)∴𝑡1=1𝜃log1+𝜃𝑄1𝑝1+ğœƒâˆ’ğ‘Ž2𝑄21𝑝1î€¸âˆ’ğ‘Ž2=𝑄(neglectinghigherpowersof𝜃,0<𝜃≪1)1𝑝1+âˆ’ğ‘Žğœƒğ‘„212𝑝1î€¸âˆ’ğ‘Ž2(neglectinghigherpowersof𝜃,0<𝜃≪1).(3.8b)

Again, from (3.2) and (3.4), we have 1𝜃𝑝2-ğ‘Ž+𝑒𝜃(𝑡1−𝑡2)𝑄1−1𝜃𝑝2-ğ‘Ž=𝑆(3.9)   ∴𝑒𝜃(𝑡2−𝑡1)=1−𝑆𝜃𝑝2î€¸îƒ­âˆ’ğ‘Žâˆ’1𝑄1−1𝜃𝑝2î€¸îƒ­î€·âˆ’ğ‘Ž=1+𝑆−𝑄1𝜃𝑝2+ğœƒâˆ’ğ‘Ž2𝑆−𝑄1𝑆𝑝2î€¸âˆ’ğ‘Ž2(neglectinghigherpowersof𝜃),(3.10a)∴𝑡2−𝑡1=1𝜃log1+𝑆−𝑄1𝜃𝑝2+ğœƒâˆ’ğ‘Ž2𝑆−𝑄1𝑆𝑝2î€¸âˆ’ğ‘Ž2=𝑆−𝑄1𝑝2+ğœƒî€·ğ‘†âˆ’ğ‘Ž2−𝑄122𝑝2î€¸âˆ’ğ‘Ž2(neglectinghigherpowersof𝜃),(3.10b)𝑡2=𝑄1𝑝1+âˆ’ğ‘Žğœƒğ‘„212𝑝1î€¸âˆ’ğ‘Ž2+𝑆−𝑄1𝑝2+ğœƒî€·ğ‘†âˆ’ğ‘Ž2−𝑄122𝑝2î€¸âˆ’ğ‘Ž2by(3.8b).(3.11)

From (3.5), we have−1𝜃1ğ‘Ž+𝑆+ğœƒğ‘Žî‚ğ‘’ğœƒ(𝑡2−𝑡3)𝑡=0,since𝑄3=0(3.12a)⟹𝑡3−𝑡2=1𝜃log1+ğœƒğ‘†ğ‘Žî‚=ğ‘†ğ‘Žâˆ’ğ‘†2𝜃2ğ‘Ž2(neglectinghigherpowersof𝜃)(3.12b)∴𝑡3=𝑄1𝑝1+âˆ’ğ‘Žğœƒğ‘„212𝑝1î€¸âˆ’ğ‘Ž2+𝑆−𝑄1𝑝2+ğœƒî€·ğ‘†âˆ’ğ‘Ž2−𝑄122𝑝2î€¸âˆ’ğ‘Ž2+ğ‘†ğ‘Žâˆ’ğ‘†2𝜃2ğ‘Ž2by(3.11).(3.13)

Again, from (3.6), we haveğ‘Žî€·ğ‘¡3-𝑡4=−𝑄2𝑡,since𝑄4=−𝑄2,(3.14)∴𝑡4=𝑡3+𝑄2ğ‘Ž.(3.15)

From (3.7) and 𝑄(𝑇) = 0, we have−𝑄2+𝑝2-ğ‘Žî€¸î€·ğ‘‡-𝑡4=0∴𝑄2=𝑝2-ğ‘Žğ‘‡-𝑡3-𝑄2ğ‘Žî‚¶î€ºî€»,𝑄by(3.15)(3.16)2=ğ‘Žî€·ğ‘2î€¸âˆ’ğ‘Žğ‘2𝑄𝑇−1𝑝1î€¸âˆ’âˆ’ğ‘Žğœƒğ‘„212𝑝1î€¸âˆ’ğ‘Ž2−𝑆−𝑄1𝑝2î€¸âˆ’ğœƒî€·ğ‘†âˆ’ğ‘Ž2−𝑄122𝑝2î€¸âˆ’ğ‘Ž2âˆ’ğ‘†ğ‘Ž+𝑆2𝜃2ğ‘Ž2,[]=𝑝by(3.13)(3.17)𝐷=Totalnumberofdeteriorateditemsin0,𝑇1î€¸ğ‘¡âˆ’ğ‘Ž1+𝑝2ğ‘¡âˆ’ğ‘Žî€¸î€·2-𝑡1+𝑡−𝑆𝑆-ğ‘Ž3-𝑡2=𝑝1î€¸îƒ¬ğ‘„âˆ’ğ‘Ž1𝑝1+âˆ’ğ‘Žğœƒğ‘„212𝑝1î€¸âˆ’ğ‘Ž2+𝑝2î€¸îƒ¬î€·âˆ’ğ‘Žğ‘†âˆ’ğ‘„1𝑝2+ğœƒî€·ğ‘†âˆ’ğ‘Ž2−𝑄122𝑝2î€¸âˆ’ğ‘Ž2îƒ­î‚µğ‘†âˆ’ğ‘Žğ‘Žâˆ’ğ‘†2𝜃2ğ‘Ž2=𝜃using(3.8b),(3.10b),and(3.12b)2𝑄12𝑝1+î€·ğ‘†âˆ’ğ‘Ž2−𝑄12𝑝2+ğ‘†âˆ’ğ‘Ž2ğ‘Žîƒ­,𝑆(3.18)1[]=TotalShortageovertheperiod0,𝑇=−𝑡4𝑡3ğ‘Žî€·ğ‘¡âˆ’ğ‘¡3𝑑𝑡+𝑇𝑡4−𝑄2+𝑝2âˆ’ğ‘Žî€¸î€·ğ‘¡âˆ’ğ‘¡4𝑝𝑑𝑡by(3.6)and(3.7)=−2𝑄22𝑝2ğ‘Ž2,ğ¼âˆ’ğ‘Žbyusing(3.6)and(3.7)(3.19)𝑇[]==Totalinventorycarriedovertheperiod0,𝑇𝑡10𝑄(𝑡)𝑑𝑡+𝑡2𝑡1𝑄(𝑡)𝑑𝑡+𝑡3𝑡2𝑄(𝑡)𝑑𝑡.(3.20)Now,𝑡101𝑄(𝑡)𝑑𝑡=𝜃𝑝1î€¸î€œâˆ’ğ‘Žğ‘¡101−𝑒−𝜃𝑡=1𝑑𝑡by(3.3)𝜃𝑝1î€¸î‚ƒğ‘¡âˆ’ğ‘Ž1+1𝜃𝑒−𝜃𝑡1=𝑝−11î€¸îƒ¬ğ‘¡âˆ’ğ‘Ž1𝜃−𝑄1𝜃𝑝1=î€·ğ‘âˆ’ğ‘Žby(3.8a)11âˆ’ğ‘Žğœƒ2log1+𝜃𝑄1𝑝1+ğœƒâˆ’ğ‘Ž2𝑄21𝑝1î€¸âˆ’ğ‘Ž2+𝜃3𝑄31𝑝1î€¸âˆ’ğ‘Ž3−𝑄1𝜃𝑝1=î€·ğ‘âˆ’ğ‘Žby(3.8a)andneglectinghigherpowersof𝜃1î€¸îƒ¬ğ‘„âˆ’ğ‘Ž12𝑝1î€¸âˆ’ğ‘Ž2+𝜃𝑄31𝑝1î€¸âˆ’ğ‘Ž3−𝑄212𝑝1î€¸âˆ’ğ‘Ž2−𝜃𝑄13𝑝1î€¸âˆ’ğ‘Ž3+𝜃𝑄313𝑝1î€¸âˆ’ğ‘Ž3=𝑄(neglectinghigherpowersof𝜃)212𝑝1+âˆ’ğ‘Žğœƒğ‘„313𝑝1î€¸âˆ’ğ‘Ž2,(3.21)𝑡2𝑡1𝑄(𝑡)𝑑𝑡=𝑡2𝑡11𝜃𝑝2+î‚†ğ‘„âˆ’ğ‘Ž1−1𝜃𝑝2î€¸î‚‡ğ‘’âˆ’ğ‘Žâˆ’ğœƒ(𝑡−𝑡1)=1𝑑𝑡by(3.4)𝜃𝑝2ğ‘¡âˆ’ğ‘Žî€¸î€·2−𝑡1−1𝜃𝑄1−1𝜃𝑝2î€¸î‚‡î€½ğ‘’âˆ’ğ‘Žâˆ’ğœƒ(𝑡2−𝑡1)=1−1𝜃2𝑝2î€¸îƒ¬î€·âˆ’ğ‘Žlog1+𝑆−𝑄1𝜃𝑝2+ğœƒâˆ’ğ‘Ž2𝑆−𝑄1𝑆𝑝2î€¸âˆ’ğ‘Ž2+𝜃3𝑆2𝑆−𝑄1𝑝2î€¸âˆ’ğ‘Ž3−𝑆−𝑄1𝜃=𝑝by(3.9)and(3.10a)2î€¸îƒ¬î€·âˆ’ğ‘Žğ‘†âˆ’ğ‘„1𝜃𝑝2+î€·âˆ’ğ‘Žğ‘†âˆ’ğ‘„1𝑆𝑝2î€¸âˆ’ğ‘Ž2+𝑆2𝑆−𝑄1𝜃𝑝2î€¸âˆ’ğ‘Ž3−𝑆−𝑄122𝑝2î€¸âˆ’ğ‘Ž2−𝑆−𝑄12𝑆𝜃𝑝2î€¸âˆ’ğ‘Ž3+𝑆−𝑄13𝜃3𝑝2î€¸âˆ’ğ‘Ž3−𝑆−𝑄1𝜃(=𝑆neglectinghigherpowersof𝜃)2−𝑄212𝑝2+ğœƒî€·ğ‘†âˆ’ğ‘Ž3−𝑄313𝑝2î€¸âˆ’ğ‘Ž2,(3.22)𝑡3𝑡2𝑄(𝑡)𝑑𝑡=𝑡3𝑡2î‚ƒâˆ’ğ‘Žğœƒ+î‚€ğ‘Žğ‘†+𝜃𝑒−𝜃(𝑡−𝑡2)î‚„î€ºî€»ğ‘Žğ‘‘ğ‘¡by(3.5)=−𝜃𝑡3−𝑡2−1ğœƒî‚€ğ‘Žğ‘†+𝜃𝑒−𝜃(𝑡3−𝑡2)î€¾ğ‘Žâˆ’1=−𝜃2log1+ğ‘†ğœƒğ‘Žî‚+ğ‘†ğœƒî€ºî€»ğ‘Žby(3.12a)=−𝜃2î‚¸ğ‘†ğœƒğ‘Žâˆ’ğ‘†2𝜃22ğ‘Ž2+𝑆3𝜃33ğ‘Ž3+𝑆𝜃=𝑆(neglectinghigherpowersof𝜃)2−𝑆2ğ‘Ž3𝜃3ğ‘Ž2.(3.23)

Therefore, from (3.20), total inventory carried over the cycle [0,𝑇]=𝑄212𝑝1+âˆ’ğ‘Žğœƒğ‘„313𝑝1î€¸âˆ’ğ‘Ž2+𝑆2−𝑄212𝑝2+ğœƒî€·ğ‘†âˆ’ğ‘Ž3−𝑄313𝑝2î€¸âˆ’ğ‘Ž2+𝑆2−𝑆2ğ‘Ž3𝜃3ğ‘Ž2[],(3.24)𝑃=Productioncostovertheperiod0,𝑇=𝐶4𝑝1𝑡1+𝐶5𝑝2𝑡2-𝑡1+𝑝2𝑇-𝑡4=𝐶4𝑝1𝑄1𝑝1î€¸îƒ¬âˆ’ğ‘Ž1+𝜃𝑄12𝑝1+ğ¶âˆ’ğ‘Ž5𝑝2𝑝2î€¸îƒ¬âˆ’ğ‘Žğ‘†+𝑄2−𝑄1+𝜃𝑆2−𝑄212𝑝2.âˆ’ğ‘Žby(3.8b),(3.10b),and(3.16)(3.25)

Average cost of the system𝑄=𝐶1=1,𝑆,𝑇𝑇𝐶1𝐼𝑇−𝐶2𝑆1+𝐶3=1𝐷+𝑃+𝐴𝑇𝐶𝐴+5𝑝2𝑝2î€¸îƒ¯î€·ğ¶âˆ’ğ‘Žğ‘†+1+𝐶3𝜃𝑝2𝑝2ğ‘Ž2+ğ¶âˆ’ğ‘Ž5𝑝2𝜃2𝑝2î€¸âˆ’ğ‘Ž2𝑆2+𝐶1𝜃2ğ‘Žâˆ’ğ‘2𝑝23ğ‘Ž2𝑝2î€¸âˆ’ğ‘Ž2𝑆3+𝐶5𝑝2𝑝2î€¸ğ‘„âˆ’ğ‘Ž2+𝐶2𝑝2𝑝2ğ‘Ž2î€¸ğ‘„âˆ’ğ‘Ž22+𝐶4𝑝1𝑝1î€¸âˆ’ğ¶âˆ’ğ‘Ž5𝑝2𝑝2î€¸îƒ°ğ‘„âˆ’ğ‘Ž1+𝑓2𝐶1+𝐶3𝜃+𝜃2𝐶4𝑝1𝑝1î€¸âˆ’ğ‘Ž2−𝐶5𝑝2𝑝2î€¸âˆ’ğ‘Ž2𝑄21+𝐶1𝑑𝑓𝜃𝑄3𝑘31,using(3.18),(3.19),(3.24),and(3.25)(3.26) where 𝑑=𝑝1+𝑝2𝑝-2ğ‘Ž,𝑘=1𝑝-ğ‘Žî€¸î€·2,𝑝-ğ‘Žğ‘“=2−𝑝1𝑘,(3.27) and from (3.17),𝑄2=ğ‘Žî€·ğ‘2î€¸âˆ’ğ‘Žğ‘2ğ‘Žî€·ğ‘ğ‘‡âˆ’ğ‘†âˆ’2−𝑝1𝑝2𝑝1î€¸ğ‘„âˆ’ğ‘Ž1−𝜃2ğ‘Žâˆ’ğ‘2𝑝2ğ‘Ž2î€¸ğ‘†âˆ’ğ‘Ž2âˆ’ğ‘Žğ‘‘ğ‘“ğœƒ2𝑝2𝑝1î€¸ğ‘„âˆ’ğ‘Ž21.(3.28)

The necessary conditions for 𝐶(𝑄1, 𝑆, 𝑇) to be minimum are 𝜕𝐶𝜕𝑄1=0,𝜕𝐶𝜕𝑆=0,𝜕𝐶𝜕𝑇=0,(3.29) that is, 𝐶1𝑑𝑓𝜃𝑘𝑄21+𝐶1+𝐶3𝜃𝐶𝑓+𝜃4𝑝1𝑝1î€¸âˆ’ğ‘Ž2−𝐶5𝑝2𝑝2î€¸âˆ’ğ‘Ž2𝑄1−1𝑘𝐶5ğ‘Ž+𝐶2𝑄2𝑝2−𝑝1+𝜃𝑑𝑓𝑄1+𝐶4𝑝1𝑝1î€¸âˆ’ğ¶âˆ’ğ‘Ž5𝑝2𝑝2î€¸ğ¶âˆ’ğ‘Ž=0,(3.30)1𝜃𝑝22ğ‘Ž-𝑝2𝑆2+ğ‘Žğ‘2𝐶1+𝐶3𝜃𝑝2-ğ‘Ž+𝐶5î€¾ğ‘†ğ‘Žğœƒ+𝐶5ğ‘Ž2𝑝2𝑝2-ğ‘Ž-𝑝2î€·ğ‘Žğ¶5+𝐶2𝑄2ğ‘Žî€·ğ‘î€¸î€½2-ğ‘Ž+𝜃2ğ‘Ž-𝑝2𝑆𝑇𝐶=0,(3.31)5ğ‘Ž+𝐶2𝑄2𝐶−𝐴−4𝑝1𝑝1î€¸âˆ’ğ¶âˆ’ğ‘Ž5𝑝2𝑝2î€¸îƒ°ğ‘„âˆ’ğ‘Ž1−𝑝2𝑝2î€¸îƒ¬ğ¶âˆ’ğ‘Ž5𝑄2+𝐶+𝑆2𝑄2ğ‘Ž22+𝐶1+𝐶3ğœƒî€¸ğ‘Ž+𝐶5𝜃𝑝2î€¸îƒ°Ã—ğ‘†âˆ’ğ‘Ž22+𝐶1𝜃2ğ‘Žâˆ’ğ‘23ğ‘Ž2𝑝2î€¸ğ‘†âˆ’ğ‘Ž3−𝐶1+𝐶3𝜃𝐶𝑓+𝜃4𝑝1𝑝1î€¸âˆ’ğ‘Ž2−𝐶5𝑝2𝑝2î€¸âˆ’ğ‘Ž2𝑄212−𝐶1𝑑𝑓𝜃𝑄3𝑘31=0.(3.32) 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: =âŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽğœ•ğ»=TheHessianMatrixof𝐶2𝐶𝜕𝑄21𝜕2𝐶𝜕𝑄1𝜕𝜕𝑆2𝐶𝜕𝑄1𝜕𝜕𝑇2𝐶𝜕𝑆𝜕𝑄1𝜕2𝐶𝜕𝑆2𝜕2𝐶𝜕𝜕𝑆𝜕𝑇2𝐶𝜕𝑇𝜕𝑄1𝜕2𝐶𝜕𝜕𝑇𝜕𝑆2𝐶𝜕𝑇2⎞⎟⎟⎟⎟⎟⎟⎟⎠(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, 𝐶1=1.5, 𝐶2=2, 𝐶3=18, 𝐶4=15, 𝐶5=13, ğ‘Ž=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, ğ‘Ž, 𝑝1, and 𝑝2. The results are shown in Tables 1–3.

Careful study of Table 1 reveals the following facts.(i)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 ğ‘Ž, and highly sensitive to changes in the values of parameters 𝐶4, 𝐶5, 𝑝1, and 𝑝2.(ii)It is observed that 𝑆∗ is insensitive to changes in the values of parameters 𝐶2, 𝐶3,and𝜃 and moderately sensitive to changes in the values of parameters 𝐶1, 𝐶4, 𝐶5, 𝐴, ğ‘Ž, 𝑝1, and 𝑝2.(iii)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, ğ‘Ž, 𝑝1 and 𝑝2.(iv)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 ğ‘Ž, and highly sensitive to changes in the values of parameters 𝐶4 and 𝑝1.(v)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, ğ‘Ž, and 𝑝1.

Careful study of Table 2 reveals the following facts.(i)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, ğ‘Ž, 𝑝1, and p2, and highly sensitive to changes in the values of parameters 𝐶4 and 𝐶5.(ii)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, ğ‘Ž, and 𝑝1, and highly sensitive to changes in the values of parameters 𝐶4.(iii)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, ğ‘Ž, and 𝑝1, and highly sensitive to changes in the value of parameter 𝐶4. (iv)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 ğ‘Ž, and highly sensitive to changes in the value of parameter 𝑝1. (v)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, ğ‘Ž, and 𝑝1.

Careful study of Table 3 reveals the following facts.(i)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, ğ‘Ž, 𝑝1, and 𝑝2, and highly sensitive to changes in the values of parameters 𝐶4 and 𝐶5.(ii)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, ğ‘Ž, and 𝑝1, and highly sensitive to changes in the values of parameters 𝐶4 and 𝐶5.(iii)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, ğ‘Ž, 𝑝1, and 𝑝2, and highly sensitive to changes in the value of parameter 𝐶4.(iv)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 ğ‘Ž, and highly sensitive to changes in the value of parameter 𝑝1.(v)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, ğ‘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 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.