ARTICLEAn ordering policy for deteriorating items with price-dependentiso-elastic demand under permissible delay in payments andprice inflationPuspita Mahataa, Gour Chandra Mahatab and Avik Mukherjee baDepartment of Commerce, Srikrishna College, Nadia, India; bDepartment of Mathematics,Sidho-Kanho-Birsha University, Purulia, IndiaABSTRACTThis paper considers the problem of dynamic decision-making for aninventory model for deteriorating items under price inflation andpermissible delay in payment. In this paper, we adopt an iso-elasticand selling price dependent demand function to model the finitetime horizon inventory for deteriorating items. The stocks deterioratephysically at a constant fraction of the on-hand inventory. The objective of this paper is to determine the optimal retail price, number ofreplenishments, and the cycle time under two different credit periodsso that the net profit is maximized. We discuss the optimizationproperties and develop an algorithm for solving the problem basedon dynamic programming techniques. Numerical examples are presented to illustrate the validity of the optimal control policy, andsensitivity analysis on major parameters is performed to providemore managerial insights into deteriorating items.ARTICLE HISTORYReceived 14 February 2019Accepted 3 October 2019KEYWORDSEOQ model; deterioratingitems; price inflation;iso-elastic and pricedependent demand;permissible delay inpayment1. IntroductionClassical inventory models have been developed under the assumption that all the costparameters remain unchanged through the planning horizon. The effects of inflation andtime value of money are vital in practical environment, especially in the developing countrieswith double-digit Gross National Product rates, such as India and China. Hence, fromfinancial point of view, it is important to investigate how time value of money influencesan inventory policy. Buzacott [1] was the first to consider an EOQ model with inflation,subject to different types of pricing policies. Misra (1979) [2] developed a discounted-costmodel and included internal (company) and external (general economy) inflation rates forvarious costs associated with an inventory system. Sarker and Pan [3] surveyed the effects ofinflation and the time value of money on order quantity with finite replenishment rate. Boseet al. [4] provided inventory model under inflation and time discounting. Yang et al. [5]discussed various inventory models with time-varying demand patterns under inflation. Houand Lin [6] investigated an EOQ model for deteriorating items with price and stockdependent selling rate under inflation and time value of money. Sarkar and Moon [7]developed an EOQ model for an imperfect production process for time-varying demandCONTACT Gour Chandra Mahata [email protected]; [email protected] Department ofMathematics, Sidho-Kanho-Birsha University, Ranchi Road, Purulia 723104, IndiaMATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS2019, VOL. 25, NO. 6, 575–601https://doi.org/10.1080/13873954.2019.1677724© 2019 Informa UK Limited, trading as Taylor & Francis Groupwith inflation and time value of money. Rameswari and Uthayakumar [8] investigatedeconomic order quantity for deteriorating items under time discounting with inflation.Palanivel and Uthayakumar [9] addressed EOQ model on finite planning horizon for priceand advertisement-dependent demand with backlogging under inflation. Other papers whichalso consider the time value of money and inflation are by Uthayakumar and Geetha [10],Vrat and Padmanabhan [11], Dutta and Pal [12], Hou [13], to name a few. Tiwari et al. [14]provided two-warehouse inventory model for non-instantaneous deteriorating items withstock-dependent demand and inflation using particle swarm optimization. All these papers,however, assumed equal replenishment cycles. Hariga and Ben-Daya [15] developed aninventory model with unequal replenishment cycles taking into account the effects ofinflation and time value of money. Sabahno [16] studied an inventory model with timevalue of money and inflation, finite replenishment rate and unequal replenishment cyclelengths.The assumption of constant demand rate has been used in many earlier investigations.However, for a wide variety of items like fashion goods, medicines, etc. this assumption isviolated as demand is likely to be linear in price and time, and may also depend on the stocklevel. Studies along this line has been conducted by many authors, see, for example, Harigaand Benkherouf [17], Hariga [18], Jalan and Choudhuri [19], Mandal et al. [20], You [21],Chakrabarty et al. [22], Alfares and Ghaithan [23], Heydari and Norouzinasab [24]. Foritems like fruit juice, soft drinks, detergents, paper towels, etc. the demand functions areoften found to be iso-elastic, and seldom linear in price. Iso-elastic demand is widely used inproduction economics as seen in the studies by Petruzzi and Dada [25], and Nilsen [26].Since the deterioration of goods is a realistic phenomenon, it is well known that certainproducts such as medicine, volatile liquids, blood banks, food stuffs and many others,deteriorate (vaporization, damage, spoilage, dryness and so on) during their normal storageperiod. As a result, while determining the optimal inventory policy of that type of products,the loss due to deterioration can not be ignored. In the early stage of the study, most of thedeteriorating rates in the models are constant, such as Ghare and Schrader [27], Shah andJaiswal [28], Aggarwal [29], Padmanabhana and Vratb [30], and Bhunia and Maiti [31]. Inrecent research, more and more studies have begun to consider the relationship betweentime and deteriorating rate. In this situation there are several scenarios: deteriorating rate isa linear increasing function of time [32], deteriorating rate is three-parameter Weibulldistributed [33], and deteriorating rate is other function of time. Under fuzzy environment,the readers are referred to Taleizadeh et al. (2015) [34] and their references. Researchersincluding [35], Tiwari et al. [36], Teng et al. [37], and Tiwari et al. [38], developed economicorder quantity models that focused on deteriorating items with varying demand. Recently,Goyal and Giri [39], wrote an excellent survey on the recent trends in modelling ofdeteriorating inventory since early 1990s.A notable feature in today’s business transactions is the permissible delay period allowedby the supplier to the inventory manager to pay his dues. If the manager pays up within thegrace period allowed to him, he does not have to pay any interest. However, he is chargedan interest if he settles his dues after the permissible period. In such a situation, it makeseconomical sense for the manager to delay his payment to the end of the grace period sinceduring the period he can sell his stock and accumulate revenue on it. The first study alongthis line was carried out by Goyal [40]. Thereafter many authors investigated inventorymodels. See, for example, Shinn et al. [41], Hwang and Shinn [42], Jamal et al. [43], Shah576 P. MAHATA ET AL.and Shah [44], Mahata and Goswami [45], Mahata and Mahata [46], Mahata [47,48],Huang et al. [49], Liao et al. [50,51], Mahata et al. [52,53], Mukherjee and Mahata [54],Shaikh et al. [55], Mishra et al. [56]. Inventory models allowing permissible delay inpayment and also taking into account inflation and time value of money have beenconsidered by Shah [57], Mishra et al. [58]. Under the given retailer’s upper limit ofaccount payable, Jia et al. [59] proposed an EOQ model with inventory level-dependentdemand, which aims to explore its influences on the retailer’s optimal order policy. Tiwariet al. [60] developed an inventory model for non-instantaneous deteriorating items ina two-warehouse environment under trade credit and also taking into account of inflation.They, however, considered equal replenishment cycles. Recently, Lin et al. [61],establisheda generalized model on the basis of the integrated inventory system for a deteriorating item,which incorporates both operational elements (e.g. in-transit and retail deterioration,production capacity) and financial factors (i.e. delay in payments).Pricing is a major strategy for a seller to achieve its maximum profit. Consequently, Tsaoand Sheen [62] established a joint pricing and lot sizing model allowing partial backloggingand trade credit. Tsao [63] developed the model with a multivariate demand function ofprice and time, but restricted the replenishment periods to be of equal length and useda fixed fraction of demand rate to model partial backlogging. In addition, he considered thedifferent prices for in-stock and stock-out periods and obtained the decision with low priceduring stock-out periods and high price during in-stock periods. The result is reasonableunder constant unit purchasing cost because the retailer may offer a lower price duringstock-out periods to compensate for a customer’s waiting time and recapture some lostdemand to increase his/her sales and profit. To characterize the more practical situation,Chen et al. [64,65] dealt with the inventory model under the demand function following theproduct-life-cycle shape. They employed the Nelder–Mead algorithm to solve the nonlinearprogramming problem and determined the optimal replenishment number and the optimal replenishment strategy. Meanwhile, Hsieh and Dye [66] extended the models of Chenet al. [64,65] with deteriorating items and partial backlogging by considering a multivariatedemand function of price and time. Dye and Ouyang [67] then presented a retailer’soptimal pricing and lot-sizing problem for deteriorating items with fluctuating demandunder trade credit financing. They relaxed the restrictive assumption of replenishmentcycles of equal length, but shortages are not allowed. Recently, Recently, Tiwari et al. [36]established an optimal pricing and lot-sizing policy for supply chain system with deteriorating items under limited storage capacity.However, all of the above-mentioned studies assumed that the unit purchasing cost isconstant. In practice, the unit cost of high-tech products might drop due to new productintroduction. When the cost of purchases as a percentage of sales is often substantial, it isnecessary to include fluctuating purchasing cost for the inventory system. Khouja andPark [68] analysed the problem of optimizing the lot size with constant demand andequal length for the entire horizon. Khouja and Goyal [69] then relaxed the restriction ofequal length to allow varying cycle times. In contrast to the EOQ/EPQ model withconstant demand, the models presented by Yang et al. [70] extended the result ofKhouja and Park [68] to the case of infinite horizon and time-varying demand andcost. Recently, Tsao et al. [71] developed a finite time horizon inventory model for price,warranty and time-dependent market demand under trade credit.MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 577This research uses the dynamic programming to solve the pricing and inventoryproblems. Dynamic programming is a method for solving complex problems by breakingthem down into simpler sub-problems, which has also been applied to solve problems ineconomics and finance. Grüne and Semmler [72] used dynamic programming withadaptive grid scheme for optimal control problems in economics. Their study examplesinclude economic growth, investment theory, environmental and resource economics.Grüne et al. [73] examined a company’s default risk in the context of a dynamic decisionproblem where companies can borrow from the credit market for investment. Then,Semmler and Bernard [74] employed an algorithm [36] created by Grüne and Semmler[72] to solve some dynamic optimizations. They studied the role of complex securities inthe financial market meltdown and how they exacerbate leverage cycles. The majorassumptions used in the above research articles are summarized in Table 1.In this paper, we follow the discounted cash flow approach to develop a lot-sizingmodel with generalized demand for instantaneous deteriorating items with varyinglengths of replenishment cycles, taking into account the effects of inflation and timevalue of money. Demand is assumed to be price dependent iso-elastic. In addition, wealso integrate trade credit policy to fit more complex situations.To our best knowledge, the impacts of price dependent iso-elastic demand andfluctuating unit purchasing cost, dynamic pricing and trade credit on the configurationof the inventory system with finite planning horizon have not been addressed. The mainemphasis of this paper is to determine the optimal replenishment scheme and sellingprices over the planning horizon which maximize the total profit over the finite-planninghorizon. The optimization properties and solution procedures based on dynamic programming techniques are provided for solving the problem. The proposed model can beused as an add-in optimizer like advanced planner and optimizer in an enterpriseresource planning (ERP) system. The paper is organized as follows. Section 2 gives theTable 1. Contribution of authors.Author(s)InventoryControlSystemiso-elastic pricedependentdemand rateOtherdemand InflationDelaypayment DeteriorationInfinite/finiteplanninghorizonKhouja andGoyal [69]EOQ p FiniteTsao & Sheen[62]EOQ p p p FiniteTsao et al. [71] EOQ pppp FiniteHsieh and Dye[66]EOQ FiniteDye andOuyang[67]EOQ p p p FiniteTiwari et al.[60]EOQ p p p p InfiniteTiwari et al.[36]EOQ p p InfiniteShaikh et al.[55]EOQ p p p InfiniteHsieh & Dye[66]EOQ p p p FiniteYang et al.[70]EOQ p p FiniteThis paper EOQ p p p p Finite578 P. MAHATA ET AL.notations used and the assumptions made in the study. Section 3 analyses the model. InSection 4, numerical examples are presented to illustrate the validity of the optimalcontrol policy, and sensitivity analysis on major parameters is performed to providemore managerial insights. Finally, a discussion on the model is given in Section 5.2. Assumptions and notationsThe notations used in the paper are as follows:ð0; HÞ ¼ planning horizon in yearsn ¼ number of replenishment periods during the planning horizon½tj1; tj ¼ j th reorder interval, 1 j n, t0 ¼ 0, tn ¼ HTj¼ tj tj1, 1 j npt ¼ nominal selling price per item in dollars in inventory at time tf ¼ discount rater ¼ inflation rateR ¼ f r ¼ net discount rate of inflationpt ¼ ert ¼ general price level at time t (p0 ¼ 1)zt ¼pt pt ¼ real selling price in dollars for every item in inventory at time tD(t) = aztb ¼ aðptertÞb ¼ ap t bebrt ¼ iso-elastic price dependent demand rate attime t, b > 1A0 ¼ ordering cost per order in dollars at time t ¼ 0h0 ¼ unit inventory holding cost per year in dollars at time t ¼ 0C0 ¼ purchasing cost per unit of item in dollars at time t ¼ 0M ¼ permissible delay period in years for settling account.θ ¼ constant fraction of the on-hand inventory deteriorating per unit timeIe¼ interest that can be earned per dollar during the planning horizonIp¼ interest paid per dollar investment in stocks during the planning horizonThe policy is to place orders at time points 0 ¼ t0 < t1 < t2 < ::::::::: < tn1 < tnð¼ HÞ, andthe nominal, unadjusted selling price remains fixed at pj in the jth reorder interval½tj1; tj. It is assumed that pertj1 pj pertj, where pert is the inflation adjusted initialprice at t. The real price per item at time point t in the jth interval is zj ¼ pjert, and thedemand rate is DjðtÞ ¼ ap j bebrt, which should reflect a real situation: i.e. the demandmay increase when the price decreases, or it may vary through time. The consideration ofthe time-and-price dependent demand of deteriorating items is useful for the fashiongoods, high-tech product and fast food industries or items with a short shelf life. It isassumed that the demand during a reorder cycle is completely met, and there is no stockon hand or shortage at the end of the cycle.3. Mathematical model formulationLet IjðtÞ denote the inventory level at time t 2 ½tj1; tj. Since depletion from stock occurssimultaneously due to demand and deterioration, the differential equation that describesthe instantaneous state of IjðtÞ is given byMATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 579dIjðtÞdt þ θIjðtÞ ¼ DjðtÞ; tj1 t tj; j ¼ 1; 2; ::::::::; n (1)with boundary condition IjðtjÞ ¼ 0.The solution to the above differential equation is given byIjðtÞ ¼ap j bθ þ br ½eðθþbrÞtjθt ebrt; tj1 < t < tj; j ¼ 1; 2; ::::::::; n: (2)The decision variables are n and tj, pj, j ¼ 1; 2; :::::::; n 1, which are determined so as tomaximize the present value of the total profit over the planning horizon.Let Zðn; t; pÞ denote the present value of the total profit over ð0; HÞ, where t ¼ðt1; t2; ::::::; tn1Þ and p ¼ ðp1; p2; ::::::; pn1; pnÞ. The different components of Zðn; t; pÞare as follows:(i) Ordering cost: There are n replenishment cycles in ð0; HÞ. Hence the presentvalue of the total ordering cost isCR ¼ A0 Xnj¼1eRtj1: (3)(ii) Holding cost: The present value of the inventory holding cost in the n cycles isCh ¼ h0 Xnj¼1ðt tj j1IjðtÞeRtdt¼ h0 aθ þ br Xnj¼1p j b eθðθþþbrRÞtj n o eðθþRÞtj1 eðθþRÞtj R 1 br n o eðθþRÞtj1 eðθþRÞtj : (4)(iii) Deterioration cost: Since a fraction θ of the stock on hand deteriorates per unittime, the present value of inventory deteriorating cost is Cθ ¼ θc0Ch:(iv) Purchase cost: The present value of total purchasing cost is given by(5) Cp¼ c0 Xnj¼1Ijðtj1ÞeRtj1¼ c0abr þ θ Xnj¼1p j bh i eðθþbrÞtjθtj1 eRtj1(6)(v) Sales revenue: The present value of total sales revenue over ð0; HÞ isSR ¼ Xnj¼1ðt tjj1pjDjðtÞeRtdt¼aR br Xnj¼1pj1b eðbrRÞtj1 eðbrRÞtj :(7)580 P. MAHATA ET AL.(vi) Interest paid and interest earned: In any replenishment cycle, say the jth cycle, theinterest earned or paid by the inventory manager depends on whether Tj Mor Tj < M.Case 1: Tj M, i.e. tj1 þ M tj.In this case, the inventory manager uses his sales revenue to earn interest throughout thecycle. However, the unsold stock remaining after the end of the grace period has to befinanced at the specified rate of interest.Thus, the present value of the total interest earned in the jth cycle is given byIE1ðjÞ ¼ Ieðt tjj1pjDjðtÞteRtdt¼ IeaðR brÞ2 pj1bh i eðbrRÞtj1 ðR brÞtj1 þ 1 eðbrRÞtj ðR brÞtj þ 1(8)And, the total interest payable isIP1ðjÞ ¼ c0Ipðt Mj þtj1IjðtÞeRtdt¼ c0Ipaθ þ br p j beθðθþþbrRÞtj n o eðθþRÞðMþtj1Þ eðθþRÞtj 1R br n o eðbrRÞðMþtj1Þ eðbrRÞtj (9)Case 2: Tj M, i.e. tj1 þ M tj.In this case, the inventory manager earns interest on his sales revenue till the end of thepermissible delay period and does not have to pay any interest.Thus, the interest earned is given byIE2ðjÞ ¼ Iepj ð t jt jtj1tj1¼ aIepj1bðR brÞ2“n o eðbrRÞtj1 ðR brÞtj1 þ 1 eðbrRÞtj ðR brÞtj þ 1 DjðtÞteRtdt þ ðM TjÞðD” # jðtÞeRtdt1þM TjR br n o eðbrRÞtj1 eðbrRÞtj (10)Therefore, the present value of the total profit over the planning horizon isZðn; t; pÞ ¼ SR þ Xnj¼1δjIE 1ðjÞ þ ð1 δjÞIE2ðjÞ CR þ Ch þ Cθ þ Cp þ Xnj¼1δjIP! 1ðjÞ ;(11)whereMATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 581δj¼1; if tj1 þ M tj0; if tj1 þ M tj:Then, the optimal values of n, t and p satisfy the following:Zðn þ 1; t; pÞ Zðn; t; pÞ 0 Zðn; t; pÞ Zðn 1; t; pÞ (12)@Zðn; t; pÞ@tj ¼ 0 for j ¼ 1; 2; ::::::; n 1; (13)@Zðn; t; pÞ@pj ¼ 0 for j ¼ 1; 2; ::::::; n: (14)For simplicity we consider the following two situations:Situation 1: δj ¼ 1 for all j ¼ 1; 2; ::::::::; n.In this case, @Zð@np;jt;pÞ ¼ 0 givespj ¼bb 1F2ðtj1; tjÞF1ðtj1; tjÞ ; (15)whereF1ðtj1; tjÞ ¼ eðbrRÞtj1 eðbrRÞtjR br þ IeeðbrRÞtj1 ðR brÞtj1 þ 1 eðbrRÞtj ðR brÞtj þ 1” ðR brÞ2 #;F2ðtj1; tjÞ ¼ ðθC0 þ h0Þθ þ br eðθþbrÞtj eðθþRÞtj1 eðθþRÞtj R 1 br eðbrRÞtj1 eðbrRÞtjθ þ R C0θ þ br eðθþbrÞtjðθþRÞtj1 eðbrRÞtj1þ Ip eðθþbrÞtj1 eðθþRÞðMþtj1Þ eðθþRÞtj eðbrRÞðMþtj1Þ eðbrRÞtjθ þ RR br :Also, we [email protected]ðn; tÞ@pj2 ¼ ðb 1ÞF1ðtj1; tjÞ < 0:As a result, Zðn; t; pÞ is concave in pj. Expressing pj as (15) in Zðn; t; pÞ, we haveZðn; tÞ ¼ a b( ) b 1 1b b b 1 b Xj¼n1 F2ðtj1; tjÞ 1b½F1ðtj1; tjÞb A0 Xj¼n1 eRtj1: (16)Thus, for fixed n, the problem becomes:582 P. MAHATA ET AL.maximize Zðn; tÞsubject to tj > M þ tj1; 1 tj n; t0 ¼ 0; tn ¼ H;and bb 1F2ðtj1; tjÞF1ðtj1; tjÞ pertj1:subject to tj > M þ tj1; 1 tj n; t0 ¼ 0; tn ¼ H;and bb 1F2ðtj1; tjÞF1ðtj1; tjÞ pertj1:Situation 2: δj ¼ 0 for all j ¼ 1; 2; :::::::::::::::; n.In this case, @Zð@np;jt;pÞ ¼ 0 givespj ¼bb 1G2ðtj1; tjÞG1ðtj1; tjÞ ; (17)whereG1ðtj1; tjÞ ¼ eðbrRÞtj1 eðbrRÞtjR br þ IeeðbrRÞtj1 ðR brÞtj1 þ 1 eðbrRÞtj ðR brÞtj þ 1” ðR brÞ2þM TjR br eðbrRÞtj1 eðbrRÞtj G2ðtj1; tjÞ ¼ ðθc0 þ h0Þθ þ br eðθþbrÞtj eðθþRÞtj1 eðθþRÞtj R 1 br eðbrRÞtj1 eðbrRÞtjθ þ R c0θ þ br eðθþbrÞtjðθþRÞtj1 eðbrRÞtj1 :Also, we [email protected]ðn; t; pÞ@pj2 ¼ ðb 1ÞG1ðtj1; tjÞ < 0:As a result, Zðn; t; pÞ is concave in pj. Hence expressing pj as (17) in Zðn; t; pÞ, we canwrite the profit function asZðn; tÞ ¼ a b( ) b 1 1b b b 1 b Xj¼n1 G2ðtj1; tjÞ 1b G1ðtj1; tjÞ b A0 Xj¼n1 eRtj1:(18)So, for fixed n, our problem becomes:MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 583maximize Zðn; tÞsubject to M þ tj1 > tj; 1 tj n; t0 ¼ 0; tn ¼ H;and bb 1G2ðtj1; tjÞG1ðtj1; tjÞ pertj1;where p is the selling price in the first cycle.Let ZðnÞ ¼ maxt Zðn; tÞ. Then, we have the following theorems:Theorem 1: ZðnÞ is concave in n.Proof: See Appendix A.Theorem 2: For br R 0, and eðbrRÞM θ þ br, optimal reorder intervalssatisfy T1 > T2 > ::::::::: > Tn.Proof: See Appendix B.The above theorem gives sufficient conditions to have T1 > T2 > ::::::::: > Tn. The conditions are, however, not necessary as is evident from the example in Section 4.3.1. Algorithm to find optimal solutionThe above theorem proves that the optimal solution exists uniquely. It can be easilydeduced for situation 1 thatTi > M ) H ¼ tn t0 ¼ Xni¼1Ti > nM:Thus, situation 1 is admissible for n < MH and situation 2 elsewhere. For a fixed n,max Zðn; tÞ can be obtained by any standard search algorithm. Hence, our problemreduced to find the optimal value of n which will be unique by Theorem 1. Now to findthe optimal solution, the stepwise procedure of the Algorithm is shown as below:AlgorithmStep 1: Check if Zð1Þ > Zð2Þ. If so then n ¼ 1, find t by any standard tool. If not proceedfurther.Step 2: Use any standard search algorithm and find Zð2Þ; Zð3Þ; :::; ZðkÞ until ZðkÞ > Zðk þ 1Þ using (16) if k MH 1 and (18) otherwise. In that case n ¼ k. 4. Numerical examplesIn this section, in order to show the applicability of the presented model and also thesolution procedure, two numerical examples are presented. In addition, these examplesprovide the materials for sensitivity analysis as well as extracting some managerialinsights, which will be discussed in the next section.Recently we have visited some Milk-Chocolate Product and Dairy companies toinspect the business scenario and after a deep analytical synthesis some facts have beengenerated as follows.584 P. MAHATA ET AL.(1) The products consists a low but static deterioration rate. It depends on thematerial and thus only static deterioration factor over stock.(2) This consist specific discount rates and involves factors of inflation. Hencediscounting rate including inflation rate should have to be considered.(3) Prices to be adjusted regarding stock to promote a better sell and acceptancetowards customer in this competitive market. Considering the initial prize itshould readjusted according to the factor of stock and demand.(4) The demand is price dependent and iso-elastic in nature over discount adjustedprice.Based on these factors we consider the following example consistent with this realsituation inculcated through the survey.Example 1: Let us assume that H ¼ 5years, θ ¼ 0.01, A0 ¼ $10, C0 ¼ $5, h0 ¼ $2,p ¼ $1.4, h0 ¼ $2, r ¼ 0.045, R ¼ 0.1, M ¼ 1 years, Ie ¼ 0.05, Ip ¼ 0.02. Here we usethe Demand rate as DðtÞ ¼ aztb ¼ aðptertÞb ¼ ap t bebrt where a ¼ 1000 and b ¼ 1.5Solution: Here we use the software Lingo 16.0 × 64 in order to solve the problem.In Table 2, it can be seen that the optimal replenishment frequency holds for n ¼ 14.Based on the replenishment periods we obtain the optimal price list from Equations(15,17) as shown in Table 3.The Figure 1 shows that the optimal profit is concave in nature with replenishmentfrequencies and thus by the algorithm to selective search always find the optimalreplenishment and price of each reorder cycle of a certain frequency. This is moredeterministic to obtain the replenishment frequency replenishment intervals and intervalprices of the product in a collective manner. Based on the optimal results, a realisticexample of the stock-holding time proposal can be elaborated in accordance withtheorem 2.Based on Table 3, we obtain the stock holding period (Tj) as described in the followingTable 4.A pie chart as shown in Figure 2 is also given to describe the slightly descendingmonotonic nature of the optimal holding time along with retaining the expected uniformity in nature.It describes that the product in a reorder cycle should be retained in stock nearlya certain period of time but in a quite more frequent reordering manner as going with theplanning horizon.Example 2: In the previous problem we set the model parameters f ¼ 0:15, r ¼ 0:08,Ie¼ 0:12, Ip ¼ 0:15, θ ¼ 0:02, a ¼ 200000, b ¼ 1:5, a0 ¼ 200, c ¼ 20, M ¼ 0:2, h ¼ 1,H ¼ 5, p ¼ 25:00.Solution: By using the software Lingo 16.0 × 64 we get the optimal reordering frequencyn ¼ 19 with the optimal profit is $147594:90.Hence, optimal value of n is 19, and the optimal values of tj, Tj and selling price pj aregiven in Table 5. Table 5 shows that the length of the replenishment cycle decreases whilethe unadjusted price increases with increase in the cycle number.MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 585Table 2. Optimal replenishment frequency.n Z t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t142 1880.046 0 6.479356 18.545373 2490.239 0 6.479356 13.49273 204 2788.977 0 5.695359 10.74232 15.45953 205 2999.764 0 4.611422 8.760048 12.64543 16.53111 206 3159.623 0 3.877586 7.406375 10.7211 13.89627 16.97857 207 3285.178 0 3.349051 6.423766 9.320494 12.09534 14.78453 17.41323 208 3386.385 0 2.951101 5.678954 8.255662 10.72504 13.11612 15.44941 17.74011 209 3469.621 0 2.641275 5.095774 7.419487 9.647661 11.80419 13.90622 15.96655 17.99516 2010 3539.188 0 2.39367 4.627465 6.746213 8.77898 10.74587 12.66151 14.53694 16.38076 18.19989 2011 3598.099 0 2.191594 4.243708 6.193158 8.064421 9.874737 11.6368 13.36027 15.05274 16.72031 18.368 2012 3648.536 0 2.02382 3.923978 5.731371 7.46699 9.14585 10.77913 12.37542 13.9415 15.48284 17.00395 18.5086 2013 3692.118 0 1.882515 3.653883 5.340514 6.960692 8.527673 10.05143 11.53969 12.99859 14.43308 15.84728 17.24462 18.62802 2014 3730.175 0 1.761981 3.422908 5.005712 6.526527 7.997198 9.426721 10.82211 12.18897 13.53185 14.85453 16.16018 17.45147 18.73074 2015 3685.233 0 1.692947 3.290355 4.81331 6.276785 7.691854 9.066973 10.40877 11.72254 13.01263 14.28262 15.53553 16.77391 18 19Note: optimal replenishment frequency holds for n*=14586 P. MAHATA ET AL.4.1. Managerial insightSensitivity analysis has been performed in order to determine the robustness of themodel presented above. Using the same data as those in Example 1, we study thesensitivity analysis on the optimal solutions with respect to the parameters in appropriate units. The computational results are shown in Table 6. The sensitivity analysisreveals that(1) The Profit increases with increasing initial demand of the planning horizondescribed by the parameter “a”. This is evident from the general aspect that asdemand of the customer increases profit increases gradually. The increasingTable 3. Optimal price list.jReplenishmentPeriods (tj) Prize (p j )1 1.761981 12.803292 3.422908 11.334613 5.005712 10.211314 6.526527 9.3205435 7.997198 8.5941996 9.426721 7.9886687 10.82211 7.4746168 12.18897 7.0316689 13.53185 6.64504910 14.85453 6.30401911 16.16018 6.00034312 17.45147 5.72755713 18.73074 5.48097814 20.00000 5.256515Figure 1. Optimal profits over different replenishment frequencies.Table 4. Optimal reorder intervals.T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T141.762 1.661 1.583 1.521 1.471 1.430 1.395 1.367 1.343 1.323 1.306 1.291 1.279 1.269MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 587characteristic is quite linear with 10:165314% increasing in profit as 10% parameter increase.(2) The Profit also decreases with the increasing value of parameter “b” of thedemand. It affects quite more on the profit value as it is a highly sensitiveparameter over profit. A 20% decrease in parameter value makes aFigure 2. Pie chart – Holding time of stock between two consecutive replenishments.Table 5. Optimal values of tj, Tj and pj for n ¼ 19.j tj Tj tjþ1 tj pj0 0 251 0.1754 0.175376 25.352 0.3485 0.173111 25.713 0.5194 0.170915 26.064 0.6882 0.168783 26.415 0.8549 0.166713 26.776 1.0196 0.164702 27.127 1.1823 0.162746 27.488 1.3432 0.160844 27.849 1.5022 0.158993 28.1910 1.6594 0.157190 28.5511 1.8148 0.155433 28.9112 1.9685 0.153721 29.2613 2.1206 0.152051 29.6214 2.2710 0.150423 29.9815 2.4198 0.148833 30.3416 2.5671 0.147281 30.7017 2.7129 0.145765 31.0618 2.8572 0.144284 31.4219 3.0000 0.142836 31.78588 P. MAHATA ET AL.Table 6. Sensitivity analysis.Parameter Assumed Value Change of Parameter (in %) Values of Parameter Optimal Profit (Z*)Sensitivity(in %)a 1000 -20.00% 800.0000 2971.85 -20.329556%-10.00% 900.0000 3351.00 -10.164992%0.00% 1000.0000 3730.18 –10.00% 1100.0000 4109.36 10.165314%20.00% 1200.0000 4488.55 20.330816%b 1.5 -20.00% 1.2000 7874.79 111.110471%-10.00% 1.3500 5276.09 41.443578%0.00% 1.5000 3730.18 –10.00% 1.6500 2731.05 -26.785017%20.00% 1.8000 2051.78 -44.995020%A 10 -20.00% 8.0000 3742.52 0.330923%-10.00% 9.0000 3736.34 0.165327%0.00% 10.0000 3730.18 –10.00% 11.0000 3724.02 -0.165006%20.00% 12.0000 3717.88 -0.329716%r 0.045 -20.00% 0.0360 3232.64 -13.338248%-10.00% 0.0405 3469.87 -6.978278%0.00% 0.0450 3730.18 –10.00% 0.0495 4016.08 7.664573%20.00% 0.0540 4330.41 16.091256%R 0.1 -20.00% 0.0800 4685.25 25.603946%-10.00% 0.0900 4173.53 11.885555%0.00% 0.1000 3730.18 –10.00% 0.1100 3345.28 -10.318497%20.00% 0.1200 3010.42 -19.295449%Ie 0.05 -20.00% 0.0400 3369.24 -9.676168%-10.00% 0.0450 3547.87 -4.887358%0.00% 0.0500 3730.18 –10.00% 0.0550 3916.08 4.983841%20.00% 0.0600 4105.51 10.062182%Ip0.02 -20.00% 0.0160 3730.18 0.000161%-10.00% 0.0180 3730.18 0.000080%0.00% 0.0200 3730.18 –10.00% 0.0220 3730.17 -0.000054%20.00% 0.0240 3730.17 -0.000134%C0 5 -20.00% 4.0000 4068.48 9.069360%-10.00% 4.5000 3888.50 4.244359%0.00% 5.0000 3730.18 –10.00% 5.5000 3589.49 -3.771512%20.00% 6.0000 3463.40 -7.151916%h0 2 -20.00% 1.6000 3814.23 2.253353%-10.00% 1.8000 3771.51 1.108045%0.00% 2.0000 3730.18 –10.00% 2.2000 3690.16 -1.072738%20.00% 2.4000 3651.39 -2.112046%θ 0.01 -20.00% 0.0080 3734.76 0.123024%-10.00% 0.0090 3732.47 0.061498%0.00% 0.0100 3730.18 –10.00% 0.0110 3727.88 -0.061418%20.00% 0.0120 3725.60 -0.122782%M 1 -20.00% 0.8000 3730.14 -0.000831%-10.00% 0.9000 3730.16 -0.000375%0.00% 1.0000 3730.18 –10.00% 1.1000 3730.19 0.000295%20.00% 1.2000 3730.19 0.000509%H 20 -20.00% 16.0000 2957.49 -20.714363%-10.00% 18.0000 3346.60 -10.283137%0.00% 20.0000 3730.18 –10.00% 22.0000 4106.96 10.101108%20.00% 24.0000 4475.86 19.990590%MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 589111:110471% increase in profit. This high sensitivity reveals an exponentialnature in characteristics.(3) As per the general characteristic, the increasing ordering cost decreases the levelof profit as it increases the cost burden. This should a objective for the retailer toreduce the ordering cost to achieve more profit level though the reduction factoris quite low. A 20% reduction in Ordering Cost only increases only 0:330923% ofprofit.(4) The rate of discount is also effect the profit level as its general way. If rate of thediscount consistent with inflation rate i.e. the real discount rate parameterized by“R” characteristically decrease the profit amount as it reduce the gap betweenselling price and purchasing costs. But this kind of discounts and offers attractsthe customers more and makes a greater chance to make profit. The factor ofeffect of this parameter towards profit is gradually linear in nature and makes a10:318497% decrease in profit as Rate of discount increases by 10%. Despitehigher discount rate attracts more customer the retailer should be very carefulbefore announcement of discounts to the customer. It can be seen that despitehigher discount rate increase the demand rate but it also reduce the profit at analarming rate. Thus the retailers have to balance the discount rate in accordancewith the inflation rate very carefully.(5) In accordance with the rate of discount, the inflation rate increases the profit levelas it increases the purchasing power of the customer. As it depends on theMarket and different financial or fiscal policies of the government, the retailshould adjust this by the discount rate at an optimal level. Increase in inflationrate though increases the purchasing power of the customer but it reduces the noof customer eligible for buying the product. Target customers should be kept inmind at the time of adjusting the discount rate. It can be deduced from thissensitivity analysis that inflation and discount rate play the protagonist and showreciprocal characteristic. Different fiscal and financial policies, which increase theinflation rate, always be fruitful for the retailers.(6) The earning interest rate from the customer after trade credit period effects in itsgeneral manner. With the increasing rate of imposed interest rate gives a moreprofit value. But for a retailer it is important to maintain an optimal interest rateto make product more attractive towards the customer. In analytical point ofview, the earning interest rate Ie increases in a linear manner. Ten percentincrease in parameter makes a 4:983841% increase in profit. Since the interestearned increases, the optimal retail price should decrease so as to stimulate thedemand, then the total profit will increase.(7) The interest rate imposed by the supplier to the retailer decreases with increasingprofit value. But the slope of decreasing is too low. Hence, the suppliers’ interestrate Ip effects hardly to the profit. It shows 0:000054% decrease in profit for 10%increase in parameter. The company tends to replenish fewer goods in order toavoid keeping too much inventory that will result in more interest charges.(8) The holding and purchasing cost should be decreased in order to make a greaterprofit value. Making a 10% increase in parameter value makes a 1:072738% and3:771512% decrease in profit value. The retailer should take greater attention onthese parameters in order to the reach the target profit.590 P. MAHATA ET AL.(9) When the supplier provides a longer credit period M, the company will decreasehis retail price so as to stimulate the demand. The total profit also increases withthe longer credit period. To take advantage of the longer credit period thecompany tends to replenish more goods and to sell them.(10) The deterioration factor (θ) of the product is a greater issue regarding sensitivityanalysis. This qualifies the consistency of the model among products. In accordance with the current problem regularity in deterioration effect is considered. A10% increase in deterioration rate effects 0:061418% in the profit. The deterioration factor is also a factor of consideration for the retailer to achieve thepresumptive profit.(11) The planning horizon is also a key factor for the retailer to make a better profit. Itcan be deduced vividly that a long-term planning horizon always gives the scopeto spread the product in market to meet the customer demand without applyinga frequent replenishment.Both the examples comparatively lead to the idea that a comparatively long-planninghorizon lowers the price of the product with successive replenishments. It is helpful toattract marginal customers who have a dilemma whether to purchase it or not and thoseyet who wants to wait for a while to make the price affordable to their budget. It alsoincreases the demand rate and gives more profit.Based on the characteristics of the different parameters in this model we are giving thecharacteristic graph in Figure 3 of the parameter values with profit level to make a vivididea about the changes in profit.5. ConclusionThe paper studies a dynamic inventory model for deteriorating items with pricedependent iso-elastic demand, which is observed in common items like paper towels,detergent, soft drinks, etc. Price inflation and time value of money are taken intoconsideration, and the supplier is assumed to allow the inventory manager a grace periodto pay his dues. The model is investigated in a finite planning horizon and the replenishment intervals are allowed to vary. The objective is to determine the optimal numberof replenishments, cycle time and selling price under two different credit periodsthroughout a multi-cycle planning horizon so that the net profit is maximized. Severalpropositions for optimality are developed. The problem is formulated as a bivariateoptimization model and is solved by dynamic programming techniques. From numericalstudies, we show that the results of the numerical analysis are consistent with managerialimplications and economical common sense. It has been shown that under certainsufficient conditions, the optimal lengths of the replenishment cycles decrease as onemoves towards the end of the planning horizon.The findings in this article are important to the real world. The prices of fashion goods,for example, clothes or high-tech products, will be marked down gradually with thepassage of time. These types of products are usually characterized as having a short shelflife. Joint optimal replenishment and price of each reorder cycle of a certain frequencydecision-making while considering price-dependent iso-elastic demand in the fashiongoods industry is therefore important. Our model provides the decision-maker usefulMATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 591and practical insights. The results are also applicable for other types of goods which aresimilar in nature to the price dependent demand.The proposed model can be extended in several ways. One immediate extension couldbe to derive the optimal solutions for inventory models with partial backordering andquantity discount. And further, because the deterioration rate in this model is viewedwhich is not subject to control, we could extend the model with effective investment inpreservation technology to improve the customer service level and increase businesscompetitiveness. Furthermore, delays in product availability are common in real-worldscenarios, hence the stockout compensation policy may be incorporated into the presented model to improve market efficiency and increase the retailer’s sales and profit.Figure 3. Characteristic graph of the parameters.592 P. MAHATA ET AL.This research makes further the optimal dynamic decisions for a finite time horizoninventory system with price dependent time-varying demand. For an infinite timehorizon inventory system, one may integrate our model and nonlinear model predictivecontrol (NMPC) to solve the problem. Since the time horizon is infinite, the parameterssetting of the time-varying demand can be updated in every fixed time. Then, the policieschosen are close-loop policies computed repeatedly and are determined based on thelatest data. This means that the system updates the real-time data at the beginning of eachperiod and makes the optimal decisions to maximize the system profit over the followingperiod. This opens a new direction for our research where NMPC can be used. Theinteresting research proposal is left for future work. 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Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ (1957).Proof of Appendix AWhen tj1 þ M tj for all j ¼ 1; 2; ::::::; n, i.e. δj ¼ 1 for all j ¼ 1; 2; :::::::; n, we can writeZðn; tÞ ¼ QðnÞ þ Tðn; 0; HÞ; (19)whereQðnÞ ¼ A0 Xnj¼1eRtj1 where; A0 Xnj¼1eRtj1 ¼ total ordering cost in ð0; HÞ;Tðn; 0; HÞ ¼ a b” # b 1 1b b b 1 b Xj¼n1 ½F2ðtj1; tjÞ1b½F1ðtj1; tjÞb¼ total profit in ½0; H; excluding the ordering cost; when there are n cycles in½0; H:Clearly, QðnÞ is an integer concave and decreasing function of n.Now by Bellman’s principal [75] of optimality the maximum value Tðn; 0; HÞ of Tðn; 0; HÞwith respect to t isTðn; 0; HÞ ¼ maxt2½0;H½Tðn 1; 0; tÞ þ Tð1; t; HÞ (20)where Tð1; t; HÞ denotes the total profit in the cycle ½t; H, excluding the ordering cost.MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 597Let t ¼ H. Then, we have Tðn; 0; HÞ > Tðn 1; 0; HÞ, since the maximum in (20) occurs at aninterior point. Hence Tðn; 0; HÞ is strictly increasing in n.Recursive application of (20) yields the following relations: ti ðn; 0; HÞ ¼ tiðn j; 0; tnjðn; 0; HÞÞ; i ¼ 1; 2; :::::; n j 1:In order to prove that Tðn; 0; HÞ is strictly concave in n, let us choose H1; H2 > H such thattnðn þ 1; 0; H1Þ ¼ tnðn þ 2; 0; H2Þ ¼ H:Then using Bellman’s principle of optimality, we haveTðn þ 1; 0; H1Þ ¼ maxt2½0;H1½Tðn; 0; tÞ þ Tð1; t; H1Þ ¼ Tðn; 0; HÞ þ Tð1; H; H1Þ;(21)(22)(23) and Tðn þ 2; 0; H2Þ ¼ maxt2½0;H2½Tðn þ 1; 0; tÞ þ Tð1; t; H2Þ ¼ Tðn þ 1; 0; HÞ þ Tð1; H; H2Þ:(24)Since H is an optimal interior point in Tðn þ 1; 0; H1Þ and Tðn þ 2; 0; H2Þ, we [email protected]ðn; 0; tÞ@t þ @Tð1; t; H1Þ@tjt¼H ¼ 0 [email protected]ðn @þt 1; 0; tÞþ @Tð[email protected]; tt; H2Þ jt¼H ¼ 0: (25)Now, Tð1; x; yÞ ¼ ap1b eðbrRÞx eðbrRÞy“eðbrRÞxððR brÞx þ 1Þ eðbrRÞyððR brÞy þ 1ÞðR brÞ2# R br þ Iepb aðθc0 þ h0Þθ þ breðθþbrÞyθ þ R eðθþRÞx eðθþRÞy R 1 br eðbrRÞx eðbrRÞyþac0θ þ br eðθþbrÞyðθþRÞx eðbrRÞx þ IpeθðθþþbrRÞy eðθþRÞðMþxÞ eðθþRÞy 1R br eðbrRÞðMþxÞ eðbrRÞy (26)So,@Tðn; 0; tÞ@t jt¼H ¼ @Tð[email protected]; tt; H1Þ jt¼H ¼ pb aðθc0 þ h0ÞeðbrRÞH ac 0 n o ðbr RÞeðbrRÞH þ IpeðbrRÞðMþHÞθ þ brθ þbr þ ap1b eðbrRÞHþ Ie þ HeðbrRÞH p eðbrRÞHðbr RÞ bθ ac þ0br nðθ þ RÞeðθþRÞHþ IpeðθþRÞðMþHÞo þ aðθθcþ0 þbrh0 eðθþRÞHeðθþbrÞtnðn;0;HÞ(27)598 P. MAHATA ET AL.Similarly,@Tðn þ 1; 0; tÞ@t jt¼H ¼ @Tð[email protected]; tt; H2Þ jt¼H¼ pb aðθc0 þ h0Þ θ þ br eðbrRÞH θ ac þ0br n o ðbr RÞeðbrRÞH þ IpeðbrRÞðMþHÞþ ap1b eðbrRÞH þ Ie eðbrRÞH ðbr RÞ þ HeðbrRÞHpb ac0θ þ br n o ðθ þ RÞeðθþRÞH þ IpeðθþRÞðMþHÞþaðθc0 þ h0θ þ br eðθþRÞHeðθþbrÞtnðnþ1;0;HÞ(28)Substacting (28) from (27), we [email protected] @tðTnðn; 0; tÞ Tðn þ 1; 0; tÞÞ ¼ Dðtnðn; 0; HÞÞ Dðtnðn þ 1; 0; HÞÞ < 0 (29)where,Dðt; HÞ ¼ pb aðθc0 þ h0Þ θ þ br eðbrRÞH θ ac þ0br n o ðbr RÞeðbrRÞH þ IpeðbrRÞðMþHÞþ ap1b eðbrRÞH þ Ie eðbrRÞH ðbr RÞ þ HeðbrRÞH pbθ ac þ0br nðθ þ RÞeðθþRÞHþ IpeðθþRÞðMþHÞo þ aðθθcþ0 þbrh0 eðθþRÞHeðθþbrÞt(30)CLearly, Dðt; HÞ is a decreasing function of t for t < H. Since,@Dðt; HÞ@t ¼ pbðθ þ brÞθ ac þ0br n o ðθ þ RÞeðθþRÞH þ IpeðθþRÞðMþHÞþaðθc0 þ h0θ þ br eðθþRÞHeðθþbrÞt < 0:Hence, Tðn; 0; HÞ Tðn þ 1; 0; HÞ, is a strictly decreasing function in H so thatTðn; 0; HÞ Tðn þ 1; 0; HÞ > Tðn; 0; H1Þ Tðn þ 1; 0; H1Þ: (31)Again, from (20)Tðn; 0; H1Þ Tðn þ 1; 0; H1Þ ¼ maxt2½0;HfTðn 1; 0; tÞ þ Tð1; H; H1Þg Tðn; 0; HÞTð1; H; H1ÞLet t ¼ H. Then, Tðn; 0; H1Þ Tðn þ 1; 0; H1Þ > Tðn 1; 0; HÞ Tðn; 0; HÞFrom (31) and (32)Tðn; 0; H1Þ Tðn þ 1; 0; H1Þ > Tðn 1; 0; HÞ Tðn; 0; HÞ(32)(33) which implies that Tðn; 0; HÞ is integer concave in n, and hence ZðnÞ is also integer concave in n.MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 599Proof of Appendix BWe haveZ1ðn; ftjg; fpjgÞ ¼ A0 Xnj¼1eRtj1 þ a Xnj¼1pð1bÞjðeðbrRÞtðj1Þ eðbrRÞtjÞR brþ IeeðbrRÞtj1ððR brÞtj1 þ 1Þ eðbrRÞtjððR brÞtj þ 1ÞðR brÞ2 #Xnj¼1p j baðθθcþ0 þbrh0 eθðθþþbrRÞtj eðθþRÞtj1eðθþRÞtj 1R br eðbrRÞtj1 eðbrRÞtj þ θ ac þ0br n eðθþbrÞtj 1 eðbrRÞtj1þ IpeðθþbrÞtj θ þ R eðθþRÞðMþtj1Þ eðθþRÞtj 1R br eðbrRÞðMþtj1Þ eðbrRÞtj Let, m ¼ θ þ br; w ¼ R þ θ; l ¼ br RThen,Z1ðn; ftjg; fpjgÞ ¼ A0 Xnj¼1eRtj1 þ a Xnj¼1pð1bÞjðeltðj1Þ eltjÞl Ieeltj1ðltj1 1Þ eltjðltj 1Þ l2a Xnj¼1p j b ðθc0 þ h0memtjwðewtj1ewtj Þ þ 1l ðeltj1 eltjÞ þ mc0 emtjwtj1 eltj1þ Ip emtj1wl ewðMþtj1Þ ewtj elðMþtj1Þ eltj þ @Z1ðn; ftjg; fpjgÞ@tj ¼ 0) pð j1bÞ eltj þ Ietjeltj p j b θc0 þ h0memtjc0mwðewtj1 ewtjÞ þ emtjewtj eltjþmfmemtjwtj1 þ Ipmemtjw ewðMþtj1Þ ewtj þ emtjewtj eltj þ p1 jþ1b eltj Ietjeltj p jþb1θc0mþ h0n o ewtjemtjþ1þeltj þ mc0 n oi wemtjþ1wtj leltj þ Iph i ewðMþtjÞemtjþ1 þ elðMþtjÞ ¼ 0(34) ) ewTj p b ðθc0 þ h0Þþ c0 1 þjw ewM Ip wþ emTjþ1 p jþb1 ðθc0 þ h0Þmþwc0m 1 þ IpewMþ ð1 þ IetjÞðp1 j b p1 jþ1bÞ þ p j b ðθc0 þ h0Þw þ c0Ip p jþb1 ðθc0mþ h0Þ þ mc0 l þ IpelM ¼ 0(35)600 P. MAHATA ET AL.) A1emTjþ1 þ B1tj þ C1 ¼ A2ewTj þ B2tj þ C2;whereTjþ1 ¼ tjþ1 tj B1 ¼ Iep1 j b, B2 ¼ Iep1 jþ1b A1 ¼ p jþb1 m 1 þ IpewM , A2 ¼ p b m 1 þ IpewMj ð Þ θc0mþh0 þ wc 0 ð Þ θc0mþh0 þ wc 0 C1 ¼ p1 j b þ p j b ð Þ θc0wþh0 þ c0Ip , C2 ¼ p1 jþ1b þ p jþb1 ð Þ θc0mþh0 þ cm0 l þ IpelMNow, since pjþ1 pj and b > 1, we have B1 B2 ¼ Ie p1 j b p1 jþ1b 0 Again, since m w, we get ð Þ θc0wþh0ðθc0mþh0Þ And since pjþ1 pj and elM m, we get p j b emlM p jþb1 0, so thatc0Ipp j b þ p jþb1 c0ml c0mIpelMp jþb1 ¼ p jþb1 c0ml þ c0Ip p j b elMm pjþ1 b 0:HenceC1 C2 ¼ p1 j b p1 jþ1b þ p j b ð Þ θc0 þ h0w þ c0Ip p jþb1 ð Þ θc0mþ h0 þ mc0 l þ IpelM 0:Similarly A1 A2 ¼ p jþb1 mm 1 þ IpewM ð Þ θc0 þ h0þwc0p j b ð Þ θc0 þ h0mþwc0m 1 þ IpewM 0:Hence A1emTjþ1 þ ðB1 B2Þtj þ ðC1 C2Þ ¼ A2ewTj) A1emTjþ1 A2ewTj ) A1emTjþ1 A2ewTj A1ewTj ) mTjþ1 wTj mTj ) Tjþ1 Tj.MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 601Copyright of Mathematical & Computer Modelling of Dynamical Systems is the property ofTaylor & Francis Ltd and its content may not be copied or emailed to multiple sites or postedto a listserv without the copyright holder’s express written permission. 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