Uncertain demand

This section deals with production lines for more than one product. In the process industries it is often a problem to assign the capacity of one production line to several products, all or some of which have uncertain demand. We want to optimize the overall service level for such a production line. 

The core algorithm assigns the production capacity to the competing products of the same production line, such that the overall expected sales are maximal. If the capacity is critical this basically means that production capacity is designed to the more likely parts of the uncertain demand. If there is plenty of production capacity, safety stock is allocated reflecting the product-specific uncertainty of demand. When looking at all products, usually the situation is between these extremes and the algorithm provides an optimal compromise. 

The software tool performs an optimal calculation of lot sizes incorporating uncertain demand from forecasts or history as well as up-to-date inventory and open order data. The effort for the regular user is negligible because of the interface to SAP R/3. Various technical constraints can be included. Specific training to use the software is not necessary because it looks like the familiar Excel format to the... 

Currently Bayer Technology Services considers extending the software BayAPS PP to compute the optimal split of a product between several production lines or factories that can produce it. This split is influenced by uncertain demand with different characteristics in different regions, different cost oftransport and different production cost in the factories. This means that different marginal incomes for the same product occur depending on the place of production and/or the customer group which receives it. The mathematical formulation ofthe optimization criterion again is to maximize the expected service. This has already been solved for several types of constraints. 

The scheduling problem is subject to uncertainties in the demands. The demands di in period i are only known precisely after the period i. Thus, the production decision %s has to be made under uncertainty without knowing the demand exactly for the current and for later periods. Table 9.1 provides a model of the uncertain demands. The model consists of two possible outcomes of the demands for each period i d- and df. We assume a probability distribution with equal probabilities pj and pj for all outcomes. 

Table 9.1 Uncertainties model of the uncertain demands for four periods.

The sequence of decisions obtained from the scheduler (Figure 9.4) has a tree structure. This structure results from the scenario tree of the uncertain demand parameters (Figure 9.3). Due to the moving horizon scheme, the decisions and the observations alternate at each period and the decisions are functions of the observations. Each point in time where a decision is made is called a stage. The result is a multi-stage tree where each stage corresponds to a period. 

When these uncertainties are not considered in the computation of a schedule, the uncertainties in the capacity may lead to infeasible schedules, e.g., a schedule requires more capacity than available, whereas the uncertain demands have an effect on the value of the profit, e.g., when a schedule results in more or less product than demanded. 

Gupta/Maranas (2003) as one example for a demand uncertainty model present a demand and supply network planning model to minimize costs. Production decisions are made here and now and demand uncertainty is balanced with inventories independently incorporating penalties for safety stock and demand violations. Uncertain demand quantity is modeled as normally distributed random variables with mean and standard deviation. The philosophy to have one production plan separated from demand uncertainty can be transferred to the considered problem. Penalty costs for unsatisfied demand and normally distributed demand based on historical data... 

Zenios (1995) to the problem of capacity expansion of power systems. The problem was formulated as a large-scale nonlinear program with variance of the scenario-dependent costs included in the objective function. Another application using variance is employed by Bok, Lee, and Park (1998), also within a robust optimization framework of Mulvey, Vanderbei, and Zenios (1995), for investment in the long-range capacity expansion of chemical process networks under uncertain demands. 

Robust investment model for long-range capacity expansion of chemical processing networks under uncertain demand forecast scenarios. Computers el Chemical Engineering, 22, 1037. 

Cost structure equivalent to the structure in figure 5 and uncertain demand - see Bjorkvoll (1994)... 

Jordan and Graves (1995) analyze volume flexibility that can be achieved via product-plant assignment choices in a multi-plant, multiproduct production network when faced with uncertain demand. Based on a 10 plants/10 products example they demonstrate that, if correctly designed, a network with partial flexibility can yield almost the same volume flexibility benefits as a totally flexible network where all plants are able to produce all products. Their recommendation is that products should be allocated to plants in a "chain pattern" with the complete network ideally creating a single chain instead of several shorter chains (cf. Fig. 5). For more complex networks their recommendation is to equalize the number of plants a product is directly connected to and the number of products to which each plant is directly connected and create a circuit that encompasses as many plants and products as possible. 

Three cases with one or more objectives from maximization of net present value (NPV) and optimizing two other criteria (1) production delay/advance and (2) flexibility criteria. Multi-Objective GA (MOGA) A fuzzy approach was proposed to account for uncertain demand in the optimization of batch plant design for multiple objectives. Dietz et al. (2007)... 

Chemical readers with uncertain demand of mathematics, may decide at this point to skip the analysis of history, physics and mathematics of hybridization, and proceed directly to the summary that restates the mathematical results in words. 

Speculation refers to decisions (regarding inventory or capacity) made in advance of demand realization. Price variation may suggest use of speculation as a strategy, with purchases during low price points in anticipation of price increases. Long lead times for supply may suggest buffer safety stock and thus speculative inventory. Uncertain demands may require capacity buffers or speculative capacity. Product supply disruptions may imply stocks to be purchased whenever product is available. Seasonal demand or supply may demand that products are purchased and inventoried when in season. Inventories may also have to be held to smooth production. 

To develop intuition regarding the optimal inventory levels carried by retailers in the presence of competitor, consider a retailers inventory decision when faced with uncertain demand. Because the general model is... 

The capacity for apparel manufacturing is dispersed globally. In addition, apparel manufacturing takes place in smaller firms with little ability to take business risks. Textile quotas historically require early order placement and delivery to guarantee that the volumes shipped to the US market remain within the quota limits. The net effect of uncertain demands and limited capacity is the presence of long lead times. 

The common objective of all such methods and models is to determine the best capacity planning policies which advise manufacturers on which, when, where, and how to expand/ reduce capacity in response to varying and often uncertain demand. 

Cao, D.M. and Yuan, X.G. (2002) Optimal design of batch plants with uncertain demands considering switch over of operating modes of parallel units. Industrial Engineering Chemistry Research, 41(18), 4616 625. 

When a demand is realized online from a market, an order fulfillment request is generated. If the product is available at the local DC, the item is shipped out from the local DC to the customer. If there are no units available at the local DC, transshipment is requested to other DCs with stock on hand. Unlike the traditional inventory-sharing of offline stores, the transshipped item is shipped out directly to the customer from the selected DC. Our goal is to minimize the total expected delivery cost in this uncertain demand environment. 

The plan calls for a sales price of 100 with an attractive gross margin before distribution costs of 40. Distribution costs add 10 per unit to the cost. It is likely that the supply chain manager is measured on whether the costs of distribution meet the 10 expectation. The first year sales forecast calls for 100,000 units, producing expected revenues of 10 million. However, this is an innovative product, so actual demand is uncertain. Demand, in the case of the widget, may in fact be significantly more or less than the forecast of 100,000 units. 

The important message is that supply chain cost under uncertain demand has two components. The first is the traditional cost associated with physical distribution. But distribution cost does not fully capture the economic impact of supply chain design decisions. One should also consider the effect of price markdowns and lost profit opportunities. Later chapters, particularly Chapter 28, are dedicated to adding flexibility to the supply chain s design. 

Innovative product An innovative product has high margins and uncertain demand. The supply chain for such products should be designed for responsiveness to demand, rather than efficiency. (See functional product.)... 

Wang, Y. and Y. Gerchak. 2003. Capacity games in assembly systems with uncertain demand. Manufacturing Service Operations Management, Vol.5, No.3, 252-267. 

Petruzzi and Dada [117] analyze the problem of determining inventory and pricing decisions in a two-period retail setting when an opportunity to refine information about uncertain demand is available. The model extends the... 

S. Subrahmanyan and R. Shoemaker. Developing optimal pricing and inventory policies for retailers who face uncertain demand. Journal of Retailing, 72(l) 7-30, 1996. 

Lovejoy, W.S. 1990. Myopic policies for some inventory models with uncertain demand distributions. Management Set 36 (6) 724-738. 

Erhun, F., P. Keskinokak and S. Tayur. 2001. Sequential procurement in a capacitated supply chain facing uncertain demand. Working Paper. GSIA, Carnegie Mellon University, Pittsburgh. 

---