Single-Echelon Systems: Deterministic Lot Sizing

Abstract

When using an (R, Q) policy we need to determine the two parameters R and Q. We shall first consider the determination of the batch quantity Q. (When using an (s, S) policy, Q essentially corresponds to S—s.) We shall in this chapter assume that the future demand is deterministic and given. If the lead-time is constant, it does then not affect the problem and we can therefore just as well assume that the lead-time L = 0. The only difference in case of a positive lead-time is that we need to order L time units earlier. See also the discussion in Sect. 4.1.3. (If the demand is stochastic, we cannot disregard the lead-time.)

Problems

(*Answer and/or hint in Appendix 1)

• 4.1* Consider the following data for two products:

Product	A	B
Demand per year, units	48000	72000
Unit cost, $	50	85
Ordering cost, $	125	125

The carrying charge is 24 %.

• a) Use the classical economic lot size model to determine batch quantities.

• b) Assume that the real ordering cost is $ 200 instead of $ 125. How large are the additional costs incurred by the bad estimate of the ordering cost?

• 4.2* We consider a company using the classical economic lot size model. How high is the relative real cost increase if the holding cost in the model is only 60 % of the real holding cost?

• 4.3 Consider the classical economic order quantity model in Sect. 4.1. Let I be the inventory on hand. Assume that the linear holding cost hI is only valid for I ≤ I´. For I > I´ the holding costs are obtained as hI´ + h´(I–I´), where h´ > h. An interpretation is that we need to rent additional and more expensive storage space when the inventory on hand is higher than I´. Determine the costs per unit of time as a function of the batch quantity Q.

• 4.4 Assume that the classical economic order quantity model in Sect. 4.1 suggests Q ^* = 25 for a certain item. However, for some practical reasons the order quantity has to be a multiple of 10. What is then the best order quantity? Explain why.

• 4.5 Consider the model with finite production rate in Sect. 4.2. Let p → d from above. What will happen with Q ^* and C ^*? Explain.

• 4.6* Consider again the model with finite production rate in Sect. 4.2. Change the assumptions so that the units of a batch can meet demand first when the whole batch is complete. Determine expressions for Q ^* and C ^*.

• 4.7 Use the data in Example 4.2 in Sect. 4.3, except for v´ < v, which is considered as a problem parameter. Determine for which values of this parameter the optimal solution is obtained from Q´, Q´´, and Q ₀, respectively.

• 4.8 Consider again Example 4.2 in Sect. 4.3. Assume instead incremental discounts, i.e., the lower price v´ is only for those units that exceed Q ₀. Note that this also affects the holding costs. Determine the optimal solution.

• 4.9 In the model in Sect. 4.4, replace the backorder cost per unit and time unit, b ₁, by a backorder cost per unit, b ₂. This means that there is no difference between long and short delays.

• a) Show that for a given 0 ≤ x ≤ 1 the optimal costs can be expressed as

$$ C=\sqrt{2Adh}(1-x)+{{b}_{2}}dx $$

.

• What is then the decision rule for x? Explain.

• b) Consider the data in Example 4.3, i.e., the constant demand is d = 300 units per year. The ordering cost is A = $ 200 and the holding cost h = $ 20 per unit and year. What is the optimal solution for different values of b ₂?

• 4.10 Consider again the model in Sect. 4.4 but assume that demands that can not be met from stock are lost. For each unit that is lost there is a lost sales cost b ₂ per unit. Determine the optimal solution.

• 4.11* We consider the model in Sect. 4.4 with the data A = 6, d = 1, h = 1, b ₁= 3.

• a) Determine the optimal solution.

• b) Show that the average waiting time for a customer is 1/8.

• c) Consider now the problem without the backorder cost b ₁ per unit and time unit but with 1/8 as an upper bound for the average waiting time for a customer. Show that the policy in a) is optimal.

• 4.12 Consider again the model in Sect. 4.4 without the backorder cost b ₁ per unit and time unit. Assume now that there is an upper limit W´ for the maximum waiting time for a customer. Determine the optimal policy. Compare with the optimal policy for the original problem.

• 4.13 Consider again the model in Sect. 4.4 without the backorder cost b ₁ per unit and time unit. Assume now that there is a lower limit for the fill rate, 1–x, i.e., the fraction of demand that can be satisfied immediately from stock on hand. Let the constraint be x ≤ x´. Determine the optimal policy. Compare with the optimal policy for the original problem.

• 4.14 Consider the classical dynamic lot size problem. Assume that the period demand takes place in the middle of the period, while the deliveries are still in the beginning of a period. Show how the costs are affected, and that the optimal solution is still the same.

• 4.15 Consider a more general version of the classical dynamic lot size problem where the ordering and holding costs A _i , h _i  > 0 may vary with i. Provide a proof that Property 1, the zero-inventory property, is still valid.

• 4.16* The demand for a product over five periods is 25, 40, 40, 40, and 90 units. The ordering cost is $ 100 and the holding cost $ 1 per unit and period. Determine batch quantities by using

• a) The Wagner-Whitin algorithm,

• b) The Silver-Meal algorithm,

• c) The heuristic in Sect. 4.8.

• 4.17* The demand for a product over six periods is 10, 40, 95, 70, 120, and 50 units. The ordering cost is $ 100 and the holding cost $ 0.50 per unit and period. Determine batch quantities by using

• a) The Wagner-Whitin algorithm,

• b) The Silver-Meal algorithm.

• 4.18* The demand for a product over seven periods is 10, 24, 12, 7, 5, 4, and 3 units. The ordering cost is $ 100 and the holding cost $ 4 per unit and period. Determine batch quantities by using

• a) The Wagner-Whitin algorithm,

• b) The Silver-Meal algorithm.

• 4.19 In Sect. 4.6 we use Dynamic Programming with Forward Recursion, i.e., we start with the solution for period 1. Given this solution we determine the solution for periods 1 and 2, etc. It is also possible to use Backward Recursion, i.e., start with the solution for period T. Given this solution, determine the solution for periods T–1 and T, etc. Formulate such an alternative Dynamic Program and solve Example 4.5 in this way.

• 4.20 The demand for an item over 7 periods is given.

Period	1	2	3	4	5	6	7
Demand	20	30	20	10	30	50	20

The holding cost is 2 per unit and period and the ordering cost is 100.

• a. Determine all optimal solutions by the Wagner-Whitin algorithm.

• b. Determine the optimal solutions if deliveries in period 5 are not allowed.