Single-Echelon Systems: Integration–Optimality

Abstract

In practice it is most common to determine the batch quantity from a deterministic model. The stochastic demand is then replaced by its mean. In Chap. 4 we have considered different methods for determination of batch quantities under the assumption of deterministic demand. Stochastic variations in the demand, and possibly in the lead-time, are then only taken into account when determining the reorder point. As discussed in Chap. 4 this procedure is, in general, an adequate approximation. In Chap. 5 we have described various techniques for determining the reorder point for a given batch quantity.

Problems

(*Answer and/or hint in Appendix 1)

• 6.1* Consider Example 6.1 and the iterations in Table 6.1. What is the fill rate in the different iteration steps? Why?

• 6.2 Verify the transformation in Sect. 6.1.3.

• 6.3* Consider an item which is controlled by a continuous review (R, Q) policy. The forecast and MAD have just been updated by exponential smoothing as $\hat a = 100$ and MAD = 40. The forecast period is one month. The lead-time is two months. When adjusting the standard deviation to a different time, the constant c is set to 0.7. The lead-time demand is normally distributed.

  • a) Determine the reorder point for S _1 = 90 %,

  • b) For this reorder point determine S ₂ for Q = 25, Q = 100, and Q = 1200.

• 6.4* Simple exponential smoothing is used for updating the forecast each week. The smoothing constant is 0.2. MAD is also updated by exponential smoothing with smoothing constant 0.3. The demand during the past five weeks is given in the table. Before week 1 the forecast was 100 and MAD was 10.

  • a) Update forecast and MAD for weeks 1–5. Determine the expected demand and variance for week 6.

  • b) Determine (after the update in period 5) batch quantity by the classical economic order quantity model and reorder point under the following assumptions:

    • Ordering cost: 100

    • Holding cost: 1 per unit and week

    • Lead-time: 2 weeks

    • S ₁ ≥ 95 %

    • Forecast errors in different periods are assumed to be independent.

• 6.5* The demand during the past five weeks is given.

  Table 3

  Forecasts are determined by both simple exponential smoothing and by exponential smoothing with trend. For simple exponential smoothing, the smoothing constant is 0.2. The same smoothing constant is used when updat- ing the mean with the trend model. The smoothing constant for the trend is 0.4. When updating MAD the smoothing constant is 0.2. The forecast errors are assumed to be independent and normally distributed. Before the first week (16) the forecasted demand was 100.0 and the trend was assumed to be zero. MAD was 7.

  • a) Update the forecasts by both methods. Determine mean and standard dev- iation for week 21 after the update in week 20. Assume stationary stocha- stic demand. Use the forecast from simple exponential smoothing. Deter- mine batch quantity by the classical economic lot size formula. Determine reorder point such that the fill rate is approximately 95 %. Make the follo- wing assumptions:

    • Ordering cost: 2500

    • Holding cost: 10 per unit and week

    • Lead-time: 3 weeks

    • Continuous review

  • b) Determine S ₁ for the chosen reorder point.

• 6.6 Consider a continuous review (R, Q) policy. The batch quantity Q = 500. The lead-time is two weeks. Both S ₁ and S ₂ must be at least 95 %. Before week 1 the forecast was ${\hat x_{0,1}} = 100$and MAD _0 = 8. Use simple exponential smoothing with α = 0.2 (for both $\hat a$ and MAD) to update the forecast for weeks 1–5. The demands are:

  Table 4

  Use the forecast from week 5 when determining the reorder point. The demand is normally distributed and deviations in different periods are independent.

• 6.7* Simple exponential smoothing is used for updating the forecast. The smoothing constant is 0.2. MAD is also updated by exponential smoothing with smoothing constant 0.3. The demands during the past five weeks are given as 112, 96, 84, 106, 110. Before week 1 the forecast was 100 and MAD was 10.

  Update forecast and MAD for weeks 1–5. Determine the expected demand and variance for week 6. Determine (after the update in week 5) batch quant-tity by the classical economic order quantity model and reorder point under the following assumptions: ordering cost 100, holding cost 1 per unit and week, lead-time 2 weeks, S₁ ≥ 95 %.

  Table 5

  Forecast errors in different weeks are assumed to be independent.