Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 8162865, 10 pages

https://doi.org/10.1155/2017/8162865

## Neimark-Sacker Bifurcation in Demand-Inventory Model with Stock-Level-Dependent Demand

^{1}Institute of Logistics and Warehousing, Poznań, Poland^{2}Lodz University of Technology, Łódź, Poland^{3}University of Białystok, Białystok, Poland

Correspondence should be addressed to Ewa Schmeidel

Received 4 September 2016; Accepted 20 November 2016; Published 1 January 2017

Academic Editor: Zhan Zhou

Copyright © 2017 Piotr Hachuła et al. 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.

#### Abstract

An analysis of dynamics of demand-inventory model with stock-level-dependent demand formulated as a three-dimensional system of difference equations with four parameters is considered. By reducing the model to the planar system with five parameters, an analysis of one-parameter bifurcation of equilibrium points is presented. By the analytical method, we prove that nondegeneracy conditions for the existence of Neimark-Sacker bifurcation for the planar system are fulfilled. To check the sign of the first Lyapunov coefficient of Neimark-Sacker bifurcation, we use numerical simulations. We give phase portraits of the planar system to confirm the previous analytical results and show new interesting complex dynamical behaviours emerging in it. Finally, the economical interpretation of the system is given.

#### 1. Introduction

Great economic development after the second world war has released a need of development of mathematical methods to support optimization of economic and business processes. Material flow, production, and inventory are aspects of a business, which, to make it profitable, need to be optimized. Therefore, many models of supply chain were created in the mid-20th century. To mention the most noticeable, we ought to list works of Wagner and Whitin [1], Brown [2], and Holt et al. (HMMS model) [3]. They have laid the foundation for supply chain modelling. Those models, despite being relatively simple, have become an inspiration for contemporary researchers: to redefine models in order to fit to the current challenges and to analyse them using available computation power, for example, [4–6].

Prediction of future demand and inventory is an important aspect of running and managing manufacturing or trade company. Methods supporting those tasks have been developed by economists already in the mid-20th century; nonetheless, they are still being improved, as economy still changes and creates new challenges. One of those methods is modelling of economic phenomena using mathematical formulations. The topic of demand and inventory needs a contextual approach, since many factors can influence it and different views may be needed. Therefore, the models are created with the usage of different mathematical tools. We can mention here recent works related to demand and inventory that investigate and describe specific economic cases: Chen and Hu in [7] consider an inventory and pricing model and develop polynomial time algorithms to maximize the total profit; Mondal et al. in [8] consider generic algorithm for the case of inventory of a deteriorating item; Qi et al. in [4] consider supply chain that experiences a disruption in demand during the planning horizon.

Ma and Feng in [6] proposed the dynamical model of demand and inventory with mechanism of demand stimulation and inventory limitation. Nowakowska in [9] showed application of such a model to electrical energy market. Hachuła et al. analysed the stability of equilibrium points in [10, 11].

The model describes demand and inventory of a product at one echelon of supply chain at the retailer. The considered supply chain consists of three echelons: manufacturer, retailer, and customers. The following rules are applied to the model: customers buy a good from a retailer; a retailer orders a product in the forecasted amount and forecast is prepared using single exponential smoothing Brown model [12]; a manufacturer produces and delivers exactly the ordered amount and the production capacity is unlimited; customers’ demand depends on a retail price, which can be changed by a discount; price cannot be arbitrarily changed but the retailer can offer a discount depending on stock volume.

The model takes a form of the following system of difference equations:where indicates instance of time; is a stock volume at ; is a real number; interpretation of nonnegative values is obvious and the negative value of stock informs what amount is missing to fulfill demand and needs to be overcompensated in the next periods; is a demand volume at , which assumes constant demand elasticity (see [13, p. 280]), ; is a forecast of demand at and order placed by a retailer at a manufacturer; moreover, by assumption of unlimited capacity, it is also the delivered quantity at , ; is a parameter for discount steering, ; is a parameter for defining the target stock, ; is a price elasticity coefficient that regulates dependence between price, discount, and demand, ; is a forecast smoothing coefficient of Brown model, .

The first equation describes the influence of relation between current stock and target stock on future demand , in a way that when stock is high, the retailer offers a discount to encourage customers to buy a product. The second equation lets us calculate future stock as a sum of current stock and delivery of amount of forecasted demand diminished by actual sales . The third equation describes a method of demand forecasting. Future demand is forecasted using single exponential smoothing by Brown [2]. Smoothing coefficient split importance between actual demand (sales) and forecasted demand .

We analyse properties of a given model (1) on the ground of theory of difference equations. Three-dimensional difference systems were studied by many authors, for example, in [14–17]. Under some assumptions, such systems can be rewritten as third-order difference equations. Asymptotic properties of these types of equations were investigated in [18–23]. For the background of difference equations theory, see monographs [24, 25]. Firstly, we construct a positive invariant set of the three-dimensional system. We notice that the given system has an invariant plane and can be reduced to the family of the planar systems, which we construct positive invariant sets and provide the equilibrium points for. We recall the properties of equilibrium points obtained in [11]. The aim of this paper is to analyse the dynamics of planar systems on the ground of bifurcation theory (see [26]) and numerical simulations. It was shown that nondegeneracy conditions for Neimark-Sacker bifurcation are fulfilled and existence of transcritical bifurcation is suggested. The analytical results are confirmed by numerical simulations. Finally, the economical interpretation of the system is given.

#### 2. Main Results

For the sake of convenience, let us write the model as a system of three first-order difference equations with , , and replacing , , and , respectively, in system (1),with , , or as with a mapping:It is easy to see that for and for are equilibrium points of the system .

Let us shortly elaborate on the meaning of those equilibrium points. The goal of the retailer is to reach target stock . If additionally actual demand and forecasted demand are equal and greater than zero, the retailer’s stock level does not change in the next period, which means that system (2) is in the equilibrium point . On the other hand, when , the retailer’s stock level cannot be changed. This means that the retailer remains in equilibrium point .

We recall that is the positive invariant set of the system if and .

Proposition 1 (see [11]). *Here, is the positive invariant set of the system , where is given by (3).*

One can observe that the second and the third equation of system (2) are linear ones. Therefore, we treat them with some linear transformations in order to obtain a more convenient form for analysis: both sides of the second equation multiplied by and the second and the third equation added. As an output, we getwhich enables us to state the following.

Corollary 2. *System (2) has the invariant plane for any . Hence, system (2) can be reduced to a family of planar systems:with , where , .*

*Proof. *From (5), we get the first-order recursion of the form which implies that . Constraint of the expression is a direct consequence of the positive invariant set construction.

Let us later write our systems (6) in the form of a family of mappings:for . Taking into consideration economic interpretation, , where .

Proposition 3. *Let , , and . The set is the positive invariant set of the system , where is given by (7).*

*Proof. *Let . For any , , we have and

Mapping (7) may have at most two equilibrium points:(i)(ii) iff

Stability of the equilibrium points of (2) has been examined in [11]. Based on the findings of [11], one can formulate the conclusions about the existence and stability of equilibrium points of planar systems (6), which are presented in Table 1. Now, using these results, we analyse the dynamics of the planar systems using bifurcation theory. We are especially interested in proving the existence of Neimark-Sacker bifurcation.