Fixed lead times

At the end of period t we can decide to produce a subset of the known orders, to produce all the known orders, or we can decide to delay production. If we produce in period t+1, then we have set-up costs s and if we produce orders that have to be delivered during one of the periods t+2.t+/V, we have holding costs h per order per period for the orders that are manufactured too early. If there are required deliveries for period t+l or even earlier, and we decide to delay the production, then we have penalty costs at the end of period f+1. The penalty costs at the end of this period will be p units per order for the required deliveries for period t+l or earlier. We have chosen for this linear form because this allows a tremendous simplification of the state space, due to an aggregation of all late orders. The late orders are not lost, but backlogged. 

The objective is to minimise the long term average costs per period. First we will model the problem as a Maikov Decision Problem. Using the method of successive approximations, the optimal production rule can be determined as well as the average costs of this rule. Then we will describe some heuristic methods and consider their performance in a few examples, as well as the performance of a cyclic service rule. In these examples all heuristics will turn out to perform reasonably well. The main differences between the heuristics are in the complexity of the decision rules and the possibility to compute the average costs. 

In order to give a good description of the problem, we shall model it as a Markov Decision Problem (MDP). Markov Decision Processes have been studied initially by Bellmann (1957) and Howard (1960). We will first give a short description of an MDP in general. Suppose a system is observed at discrete points of time. At each time point the system may be in one of a finite number of states, labeled by 1,2. M. If, at time t, the system is in state i, one may choose an action a, fix m a finite space A. This action results in a probability PJ- of finding the system in state j at time r+1. Furthermore costs qf have to be paid when in state i action a is taken. 

Returning to our problem, we find that we can only give an adequate description of the state if we use a state vector instead of a single integer value. In this state vector we want to express the required deliveries for the various periods. However, at the end of period t, there is no difference in future costs between required deliveries for period H-1 and required deliveries for earlier periods, such as r, r-1, etc.. Therefore we can limit the order state vector to N components. At the end of an arbitrary period t, the first component denotes the required deliveries for period r+l and earlier periods, the second component denotes the required deliveries for period r+2, and so on, until the IV-th and last component, which denotes the required deliveries for period r+iV. We will denote the state we observe, r=(ri,r2.o ), the order state vector. The set of all the possible states is denoted by R. 

The second element of the MDP is the action space A(r). Each state r R is associated with a finite non-empty set of actions A (r). Since we have no capacity constraints, we will always produce the demand for an integral number of periods. Therefore, the meaning of action a is that we produce the orders with a residual lead time of a 

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In this section we return to making decisions over continuous time. Demands each period ace modeled as following a distribution. We first consider the case where we can only mtike decisions once each period. The costs consist of the costs of holding inventory each period and the costs of shortage each period. We include the case where there may be a fixed lead time L for the supplier to deliver an order. The basic idea is that in the presence of uncertainty and lead time for delivery, orders have to be placed well in advance of inventory nmning out in order to guarantee a high level of availability. 

A simple model of this form to illustrate the basic types of issues that crm be modeled is given below. Let the decision variables denote the amount of product i to be produced in period t. We will let I l denote the amount of inventory on hand at the end of period t and S, the amount of backlog at the end of period t. The product j produced in period t is assumed to become available ij periods later, that is, we assume a fixed lead time of Tj for product j. The model will rilso determine the amounts of regular and overtime labor to be used, which will be denoted by LR, and LO, respectively. Costs are also incurred when we increase and decrease our labor force, rmd are proportional to the amount of the increase or decrease. Denoting the amount of labor force increase or decrease in period t by A, and A, respectively, we can then state a basic model as follows. 

A common approach to lead time quotation is to promise a constant lead time to all customers, regardless of the characteristics of the order and the current status of the system [66] [112]. Despite its popularity, there are serious shortcomings of fixed lead times [63]. When the demand is high, these lead times will be understated leading to missed due dates and disappointed customers, or to higher costs due to expediting. When the demand is low, they will be overstated and some customers may choose to go elsewhere. 

R. Kaufman. End fixed lead times. Manufacturing Systems, pages 68-72, January 1996. 

MRP schedules have two logical flaws that, according to critics, limit their effectiveness. The first is that MRP requirements are based on forecasts. With cloudy crystal balls, MRP causes overstocking of items that don t sell and xmderstocking of those that do. The second flaw is the assumption of infinite capacity. Associated with this is the use of fixed lead times. As a result, forecasts may make impossible demands on the production process. Actual lead times vary with the workload. When backlogs go up, lead times are likely to go up as well. (Refer to the "3C" alternative to MRP in Chapter 29 for an approach that addresses tirese two shortcomings.)... 

We can achieve a lead time for an order in three different ways. In the first place, the firm and the client may have a long-term agreement according u> which the lead time of the order is fixed, the so-called fixed lead times. This inq)lies that every time a client orders a certain item, the lead time for the order will be the same. In the second place, the firm can offer the client a lead time for the order, to which the client can agree or disagree, the so-called firm-initiated lead time. And in the third place, clients can ask the firm for a certain lead time fm an order, the client-initiated lead time. The firm can refuse the order with this lead time or accept it... 

In this monograph we will not consider any rules for making agreements about fixed lead times. In Dellaert (1987) this has been done for several examples. The lead time that the firm offers a client can depend on several factms ... 

In the situation of fixed lead times we have to make agreements, but also in oth situations in which the clients have some insight into their future demand it might be recommendable to make some agreements with regular clients about the future demand and thus decrease the uncertainty for the firm and also for these clients. These agreements may contain different elements ... 

A more complex situation will be considered in Chapter 5. We will consider a situation in which several types of products are produced on one machine, with a limited capacity. The capacity is fixed in one situation and can be extended by making overtime in another situation. In both situations the (x,7>-rule will be extended and compared with a more complex production rule, based on well-known methods. There will be different groups of clients with different fixed lead times. In some examples we will compare the two production mles with the cyclic production rules that have been described in Chapter 3. 

In Section 4.2. we have studied a situation with fixed lead times and with no capacity restrictions. The problem for the manufacturer was to produce the orders such that the requested lead times were realised with limited costs and in a way that was satisfactory for the customers. In this chapter we will consider an extension of this situation. The orders will again have fixed lead times, but now there are capacity restrictions. Therefore we can now study the situation in which the customers may order products of several different types. This extension makes the production rules more complex because now several types compete for the same capacity. We will consider two possibilities for the available capacity. The available capacity can be fixed or there may be possibilities for working overtime or other ways of flexible capacity. We will model this situation as a multi-type capacitated problem with periodic review. 

In Chapter 4 we have introduced the (x,r)-rule for an uncapacitated situation. In Chapter 5 we have extended the rule for a multi-type capacitated situation with fixed lead times. In this chapter we want to study the necessary extensions for the use of the (x,7>-nile in more complex situations with firm-initiated lead times. The situations in this chapter are combinations of the more simple situations that we have considered in the previous sections several product types with different demand rates, set-ups between types, orders with different priorities, backlogging, capacity constraints and overtime possibilities. 

We model the supply decisions faced by a catalog retailer for a product with random demand over a sales season of fixed length. The retailer must determine an initial order Qi available at the start of the sales season. At a fixed time t during the season, the retailer updates the demand forecast based on observed sales and places a reorder quantity Q2 that arrives after a fixed lead-time L at time t + L. 

Typically, fixing errors and redesign account for around 30% of product development time, as shown in Figure 5.3, and improving this position provides an opportunity for lead time reduction. This means doing more work early in the process when (Parker, 1997) ... 

Where the need for fire detection is identified, the required performance of the fire detection system is already specified as part of the grading process. Fixed fire detection is typically installed to protect equipment that is high value, long lead time, or likely to be significant fire escalation hazards. The performance specification defines fire size and response time thresholds for alarm and action(s). Fire hazards are defined by radiant heat output (RHO). RHO gives a reasonable indication of the potential damage and the probability that the fire will escalate or cause loss. The RHO should not be used to determine fire thermal loading onto equipment and structures. Table 8-3 compares RHO and flame area for some typical hydrocarbon fires. 

Let us suppose that we are interested in implementing this procedure in our laboratory and we fix the time of agitation at 10 min. So, we want to look for the RC (Xi) and pH (X2) values that provide the largest percentage lead recovery (Y), and we will use the simplex method defined by Spendley etal [16]. 

In general, an objective function in the optimization problem can be chosen, depending on the nature of the problem. Here, two practical optimization problems related to batch operation maximization of product concentration in a fixed batch time and minimization of batch operation time given amount of desired product, are considered to determine an optimal reactor temperature profile. The first problem formulation is applied to a situation where we need to increase the amount of desired product while batch operation time is fixed. This is due to the limitation of complete production line in a sequential processing. However, in some circumstances, we need to reduce the duration of batch run to allow the operation of more runs per day. This requirement leads to the minimum time optimization problem. These problems can be described in details as follows. 

Quirk et al. studied the impact of excess versatic acid [191]. According to this study an excess of versatic acid nexcess acid/ Nd > 0.22 reduces monomer conversion at a fixed polymerization time (1 h) and leads to an increase in Mw as well as PDI and has no effect on cis- 1,4-contents (Table 17). 

The reaction constant k, fixing the time scale, may depend on external conditions such as temperature. Expression (3.3) is commonly called the law of mass action, although this name is also given to an important relationship among product and reactant concentrations at chemical equilibrium. When the molecular mechanism leading to the reaction (3.1) involves intermediate molecular steps, expression... 

Alternative method The member states may use the fixed stagnation time sampling method (see Note 2, below), which takes better account of the local or national situation, provided that in the supply zones, it does not lead to fewer breaches of the parametric values than would be the case using the harmonised method. 

An exception is the special case of deterministic lead times, that is, Lj = E[, L2 = Without loss of generality, assume f) <. For any fixed t, there are three cases. 

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