The concept of marketing literature, especially innovation diffusion concept, plays a pivotal role in developing EOQ models in the field of inventory management. The integration of marketing parameters, especially the idea of diffusion of new products with the inventory models, makes the models more realistic which is most essential while building the economic ordering policies of the products. Also, because of rapid technological development, the diffusion of technology can also be viewed as an evolutionary process of replacement of an old technology by a new one. Therefore, the effect of technological substitution along with the diffusion of new products must be taken into account while formulating economic ordering policies in an inventory model. In this paper, a mathematical model has been developed for obtaining the Economic Order Quantity (EOQ) in which the demand of the product is assumed to follow an innovation diffusion process as proposed by Fourt and WoodLock (1960). The idea of effect of technological substitution of products has been incorporated in the demand model to make the economic ordering policies more realistic. A numerical example with sensitivity analysis of the optimal solution with respect to different parameters of the system is performed to illustrate the effectiveness of the model.
Technological breakthroughs are continuously being experienced in every field of business management and because of this new, products are constantly entering into the market system. Also, the penetration of new products into the market system generally come in successive generations. This happens due to speedy technological progress because of rapid development of information and communication technology market. The impact of globalization plays one of the pivotal roles to spread awareness about the new products among societies. Also, the changing needs of the society increases the demand of new products which encourage innovations of the products. Therefore, the theory of innovation diffusion is highly desired for attentive management of the new products in order to minimize the total cost and maximize the benefits. To make the business fascinating and demanding, the importance of innovations in business and industry is highly significant. The innovations, especially technological innovations, have made the firms to upgrade their products skillfully for surviving in the market. Diffusion is defined as the process by which an innovation is communicated through certain channels over time among members of a social system (Rogers [
Fourt and WoodLock Curve (Source: Lilien et al. [
Also, diffusion of new products cannot be justified without taking into account the effects of successive generations of products because an important feature of most modern new technologies is that they come in successive generations. New products are substituted with more advanced products and technological generations which creates heterogeneity in the adopting population. Moreover, it is observed that any given generation of technology will end up being replaced by a new generation. The old products are not straight away replaced by the new products because of constant innovation, but new products intend to substitute and start competing with the old products which creates parallel diffusion of both the old and the new products in the market. This makes the decreasing pattern of diffusion of old products. The series of technological generations have been well explained through the Figure
Series of technological generations (Source: Norton and Bass [
The replenishment rate is infinite, and thus, replenishments are instantaneous.
Lead time is zero.
The planning horizon is finite.
Shortages are not allowed.
Demand rate is time dependent governed by innovation diffusion process and is affected by the introduction of substitute products because of technological innovations.
The size of the potential market of total number of adopters remains constant. Here, the potential market size includes the number of initial purchases for the time interval for which replacement purchases are excluded.
The coefficient of innovation remains constant, that is, the likelihood that somebody who is not yet using the product will start using it because of mass media coverage or other external factors will act as constant throughout the cycle length.
There is only one product bought per new adopter.
The innovation’s sales are confined to a single geographic area.
The impact of marketing strategies by the innovator is adequately captured by the model’s parameters.
The basic assumption considered by different researchers in marketing literature for a fundamental diffusion model is that the rate of diffusion or the number of adopters at any given point in time is directly proportional to the number of remaining potential adopters at that moment. Mathematically, this can be represented as follows:
It has also been assumed that
Depending on the importance of each source of influence, different versions can be derived from the fundamental diffusion model (Mahajan and Peterson [
When
The basic assumptions used in the Bass Model are that the adoption of a new product spreads through a population primarily due to contact with prior adopters. Hence, the probability that an initial purchase occurs at time
If we define
The basic demand model that has been used in the EOQ model is based on the following assumptions.
Adoptions take place due to innovationdiffusion effect and it is influenced by the innovationeffect (mass media), that is, external influence only.
Adoptions of the first generation product is diminishing by the introduction of the second generation product.
Here, the diffusion of new products which is spread through the external influence has been considered for making economic ordering policies of the products. The demand of first generation products, that is, products which are supposed to be introduced at the beginning of the planning period also known as old products, is affected (diminished) due to introduction of the second generation products, that is, products which are supposed to be new in comparison to the first generation products and affecting the demand of old products because of technological innovations after a certain interval of time. Therefore, using the above assumptions, explanations, and (
Now by model assumption, replenishment is instantaneous and shortages are not allowed. Thus, the inventory level at the initial point of the planning horizon can be assumed to be the cumulative adoption of the product during the cycle time
The solution of the differential equation (
Now,
The different cost elements involved in the inventory system per unit time can be defined as
Thus, the total cost of the inventory system per unit time
The necessary criterion for
Now, for
The solution of the equation
When
For optimum total cost, the necessary criterion is
Now, for
The solution of the equation
The solution procedure has been summarized in the following algorithm.
The above steps are used for all replenishment cycles using appropriate parameter values. In order to obtain the appropriate values of “
The effectiveness of the proposed model has been shown by the following numerical examples. A hypothetical example has the following parameter values in appropriate units:
The results have been well presented in the following different tables. Also, to prove the validity of the model numerically and to get the appropriate parameter values, the references have been considered as Chanda and Kumar [
The mean value of the coefficient of innovation for a new product usually lies between 0.0007 and 0.03 (Sultan et al. [
The mean value of the coefficient of innovation for a new product is usually 0.01for developed countries and 0.0003 for developing countries (Talukdar et al. [
A hypothetical example has the following parameter values in appropriate units:
The results obtained from different numerical tables in Section
As the coefficient of innovation increases keeping other parameters constant then the optimal cycle length
As the cycle length
As the value of
Sensitivity analysis on coefficient of innovation “





0.001  0.86  16400  39 
0.002  0.59  32705  57 
0.003  0.48  48684  70 
0.004  0.41  64499  81 
0.005  0.37  80211  91 
0.006  0.34  95848  101 
0.007  0.31  111430  109 
0.008  0.29  126967  117 
0.009  0.27  142468  124 
0.01  0.26  157938  131 
0.02  0.19  311593  190 
For
Sensitivity analysis on “





0.10  0.59  68778  124 
0.11  0.61  69032  128 
0.12  0.63  69279  131 
0.13  0.64  69520  135 
0.14  0.66  69756  139 
0.15  0.67  69987  143 
0.16  0.69  70213  146 
0.17  0.7  70435  150 
0.18  0.72  70653  153 
0.19  0.73  70866  157 
0.20  0.75  71076  160 
For
Sensitivity analysis on “





0.019  0.38  1802327  2228 
0.020  0.41  1649195  2180 
0.021  0.44  1495228  2121 
0.022  0.48  1340344  2050 
0.023  0.52  1184432  1965 
0.024  0.58  1027337  1865 
0.025  0.65  868835  1747 
0.026  0.73  708581  1605 
0.027  0.86  545998  1433 
0.028  1.07  379973  1217 
0.029  1.54  207669  918 
For
Sensitivity analysis on coefficient of innovation “





0.001  0.76  17865  38 
0.002  0.54  34044  54 
0.003  0.44  49942  67 
0.004  0.38  65695  77 
0.005  0.34  81354  86 
0.006  0.31  96946  95 
0.007  0.29  112487  102 
0.008  0.27  127987  110 
0.009  0.26  143453  117 
0.01  0.24  158891  123 
0.02  0.17  312305  178 
Special case.
Costtime graphs showing convexity of the cost functions.
The goodwill of any organization is entirely dependent on its effective management and this is possible only when the problem of the organization is analyzed properly from all perspectives. Here, our problem is concerned with the effective management of products which are newly introduced in the market and along with this, the effect of second generation products on the demand of first generation products is also realized; the idea included here is that the effect of substitute products decreases the demand of old products after a certain interval of time because of technological innovations and other related factors. Therefore, to overcome this problem, a mathematical model has been developed to know the actual situation of the economic ordering policies which will help the inventory manager to take effective action to maintain the optimal cost. The results obtained in the model create a pool of knowledge on the diffusion processes of innovations and its effective management which are most valuable to both researchers and managers. For managers, this pool of knowledge is important because it provides useful tools and useful information for managing and scheduling the inventories of new products when its demand is affected by the generation of other new products which substitute products after a certain time period. The uniqueness of this model is that how the inventory manager should keep the optimal cycle time of any fixed lot size of one product entering into the inventory system when its substitute product enters into the system after a fixed interval so that optimal cost is maintained and the cost of inventory obsolescence is minimized.
The technological breakthrough is constantly being experienced in various products across the globe which is necessary for the products to gain competitive advantage. There are various marketing strategies to promote and establish products which are newly introduced into the market, but keeping its inventories efficiently and effectively play a greater role. Here, the role of inventory models becomes significant. The primary objective of developing inventory models is to take efficient action concerned with economic ordering policies and to understand well the behaviour of parameters associated with it. In this paper, a mathematical model has been developed for obtaining the Economic Order Quantity (EOQ) in which the demand of the product is assumed to follow an innovation diffusion process as proposed by Fourt and WoodLock [
The research presented in this paper was carried out at the Department of Operational Reseach, Faculty of Mathematical Sciences, University of Delhi. The commercial identities mentioned in the paper such as LINGO and ExcelSolver software have only been used for the numerical analysis of the model discussed in the paper which is only for the academic purpose. The authors of this paper do not have any direct financial relation with the commercial identities mentioned in the paper and there is no conflict of interests regarding financial gains for all the authors.