In this research work, we have developed an imperfect production inventory model under the screening process in the manufacturing system to meet the demand of quality items for the customer. Here, the novelty lies in the screening rate depending on the number of cycles and the learning effect of the workers. The motivation behind such consideration is to increase the screening rate in each subsequent cycle on account of the learning of the workers. The virtue of government policy is to keep the environment free from pollution, nowadays the production of green products is very essential although this can lead to higher production costs for the manufacturer. As a result, customers also have to pay higher purchasing costs. So, to motivate the customers extensively to buy a green product, awareness about such product is very essential. In this regard, the manufacturer adopts an innovative advertising policy, which is another novelty of this research work. Ultimately, the demand function depends on the green degree, advertisement frequency and the retailer's selling price. It is also noted that in the present competitive business market the manufacturer offers an incentive to their retailers to carry on their business smoothly. In this regard, a credit period policy of logistic nature has been taken. Finally, an integrated profit function has been developed and its feasibility has been justified taking a numerical example. The result indicates that not only an appropriate advertisement frequency but also a suitable level of the green degree of the product is more important for maintaining the integrated profit of the system and individual profit of the retailer and the manufacturer also. We have analyzed the impacts of the retailer's advertisement frequency on the optimal credit period and the retailer's profit. We have also analyzed how the learning rate of the workers affects the screening cost and the manufacturer's profit.

Plankton occupy a vital place in the marine ecosystem due to their essential role. However small or microscopic, their absence can bring the entire life process to a standstill. In this work, we have proposed a prey–predator ecological model consisting of phytoplankton, zooplankton, and fish, incorporating the cannibalistic nature of zooplankton harvesting the fish population. Due to differences in their feeding habits, zooplankton are divided into two sub-classes: herbivorous and carnivorous. The dynamic behavior of the model is examined for each of the possible steady states. The stability criteria of the model have been analyzed from both local and global perspectives. Hopf bifurcation analysis has been accomplished with the growth rate of carnivorous zooplankton using cannibalism as a bifurcation parameter. To characterize the optimal control, we have used Pontryagin’s maximum principle. Subsequently, the optimal system has been derived and solved numerically using an iterative method with Runge–Kutta fourth-order scheme. Finally, to facilitate the interpretation of our mathematical results, we have proceeded to investigate it using numerical simulations.

In today’s business world, advertising is one of the most important policies to attract more customers. This policy increases the retailer’s sales and makes the retailer’s business position strong. In this paper, we have considered an integrated inventory model for non-instantaneous deteriorating items with a single supplier and single retailer, where a supplier sells his/her products in the market through a retailer who faces a stochastic demand depending on both retail price and advertisement. Here, to increase the retailer’s demand, the supplier wholesales his/her products to the retailer with a credit period. Since the long credit period increases the demand rate but at the same time, it increases the supplier’s opportunity loss. In this paper, we have discussed about the credit policy and find out, how this policy effective on profit of the supplier, the retailer as well as the whole system. After purchasing, the retailer sells his/her products to his customers with a markup. This markup is based on the supplier’s wholesale price and also on advertising costs. Finally, an integrated profit function has been developed and we have illustrated numerical examples to justify the feasibility of the proposed model. The result indicates that not only the credit period but also an appropriate number of advertisements is more important for maintaining the profit of the supplier, the retailer as well as the integrated profit of the system. We have analyzed the effect of markup on the profit function. We have also analyzed how the profit structure of the suppliers and retailers changes along with the changes in the length of the credit period. An effective algorithm has been presented in the solution procedure to find the optimal solutions of the proposed model. Also, the numerical example with uniform distribution has been carried out. Finally, sensitivity analysis of major parameters has been illustrated to provide managerial insights