Gallego 1 Production Planning with Time - Varying Demand

In this lecture we present a few key results in production planning with known time-varying requirements under convex or concave production costs. Convex production costs arise when we need to utilize increasingly more expensive resources to achieve higher levels of production. As an example, consider a factory with a regular workforce. If demand is beyond the capacity of the regular workforce, management may employ overtime at a higher cost, and, if needed, it may subcontract production at an even higher cost. Convex production costs therefore reflect diseconomies of scale. The main insight, when production costs are convex, is that production smoothing is optimal. This means that the production rate (production per period) should be kept as even as possible. Thus, production plans under convex production costs mitigate or smooth out variability in demand. On the other hand, concave production costs reflect economies of scale in production. For example, if the production costs are fixed (setup cost) plus linear (variable cost), as in the EOQ, then the unit cost decreases with production volume and this acts as an incentive to order/produce in large batches. As we shall see, under concave production costs, optimal plans tends to amplify the demand variability as production is concentrated in a few periods over the planning horizon. Under concave production costs, optimal production plans tend to propagate demand variability through the supply chain. Thus, a retailer may experience fairly leveled customer demand, but because of economies of scale he or she may order in batches from the warehouse, which in turn may aggregate orders from different retailers and order in even larger batches from the factory. This phenomena is known as the bullwhip effect. Later in the course we will talk about supply chain management and how increased efficiencies may be obtained by coordinating orders and communicating demand information across the supply chain to mitigate the bullwhip effect.