# Supplier Replacement Model in a One-Level Assembly System under Lead-Time Uncertainty
## 3 Problem Description  
The problem of replenishment in assembly systems under component lead time uncertainties is complex. This is due to the fact that it is necessary to have all the components to start the assembly process. If delivery of a component is delayed, the entire process is stopped and it is necessary to pay additional holding costs for the components which are already in stock, and backlogging costs for the finished product. Thus, the randomness of component delivery times should be reduced and, if this is impossible, it should be taken into account in production planning at all decision levels. A supplier is considered as more reliable if there is less randomness in its lead times. In this paper, an assembly system is considered to assemble a given product. The demand in finished product is known and fixed, i.e., we know how many finished products it is necessary to assemble by the end of each period (fixed demand). To assemble one product, \(n\) different components are necessary. The components are ordered from external independent suppliers. The lead times (the time between order release and component delivery to assembly system) of suppliers are independent random discrete variables with known distributions of probabilities. There is no assumption about the form of distributions of probabilities. Any discrete distributions can be used, but these distributions are known before optimization. They are obtained from statistics on the past deliveries of similar components from the same suppliers. The assembly capacity is considered as infinite. We know the unit holding cost per component and per period and the unit backlogging cost per period for the finished product. The orders for components are given at the beginning of each period. The ordered components are delivered at the end of a period. We also know the unit purchasing cost of each component from the selected supplier. The purchasing cost can be increased if the supplier agrees to reduce the randomness of its lead time. The objective is to find the planned lead time and unit purchasing cost, minimizing the total cost. This is a problem of both replenishment and pricing in assembly systems with stochastic component lead times. It appears at the contracting stage when the planned lead times and prices are negotiated with suppliers. The model developed can also be used for supplier selection and replacement. In our previous work, we have already studied the replenishment part of the problem. The developed approach was based on modeling the possible states of the system (orders in progress) with Markov chains, analysis of the convergence and steady state probabilities and finally obtaining steady state expressions of expected cost composed of expected holding cost for components and backlogging cost for finished products [37]. The obtained cost functions were nonlinear with discrete decision variables (planned lead times). To optimize them, a B&B and genetic algorithms were developed and tested [3]. In addition, their efficiency is demonstrated. In this paper, we will use our previous results and extend the models to the case of optimization of both replenishment and pricing. First of all, we will present this extended problem, the cost function for the steady state (mathematical expectation of the cost composed of holding cost for components, purchasing cost for components and backlogging cost for finished products), an optimization approach to determine the best decision variables (prices and planned lead times) and finally numerical tests and managerial insights. In an OLAS with \(n\) different components replenished from \(n\) independent suppliers, the uncertainty of component replenishment lead times causes a high component inventory level and a backlog for the finished product. In this context, the producer may be ready to pay an additional purchasing cost (APC) if the supplier can decrease the uncertainty of their lead times. In this paper, we consider a purchasing and replenishment optimization model for such systems.