Chopra

OPRE 6366. SCM : 4. Inventory Planning

1 2

2.1

Inventories with Certainty:

αδ Chapter 10 of Chopra

Long Term Quantity Discounts

All-unit quantity discount

  p1   p2 Price per unit =  ...   pN if q0 ≤ Q < q1 if q1 ≤ Q < q2 ... if qN −1 ≤ Q < qN       

where we set q0 = 0 and qN = ∞. Prices are monotonically ordered by pn−1 ≥ pn for 2 ≤ n ≤ N .   if 0 ≤ Q < q1  p1 Q      p2 Q if q1 ≤ Q < q2 Purchasing cost of Q units = ...  ...      pN Q if qN −1 ≤ Q < qN If qn−1 ≤ Q < qn , define the total costs in region n as T Cn (Q) := h R R (pn Q) + S + (pn Q). 2 Q Q (1)

Then the total cost over all regions is obtained by patching all T Cn together, i.e.,   if 0 ≤ Q < q1  T C1 (Q)      T C2 (Q) if q1 ≤ Q < q2 T C(Q) = ...  ...      T CN (Q) if qN −1 ≤ Q < qN In the marginal-units quantity discount problem, we want to solve min T C(Q).

Q≥0

We propose the ordering algorithm in Table 1 to find the optimal order quantity. Note that this algorithm does not generate any candidate solution after step A2 and step C because the flag CandidatesComplete becomes true and stops the while loop. We validate this algorithm in the exercises.

2.2

Marginal-units quantity discount

  p1   p2 Price for an additional unit = given that Q is already purchased  ...   pN

if 0 ≤ Q < q1 if q1 ≤ Q < q2 ... if qN −1 ≤ Q < qN

      

Let Vn be the cost of buying exactly qn units. Then V0 = 0, V1 = p1 (q1 − q0 ) and V2 = p1 (q1 − q0 ) + p2 (q2 − q1 ). In general, Vn = p1 (q1 − q0 ) + p2 (q2 − q1 ) + · · · + pn (qn − qn−1 ).

1

Initialize n := N and CandidatesComplete = F alse. While n ≥ 1 and CandidatesComplete = F alse do Find EOQn . If qn−1 ≤ EOQn ≤ qn A1: Add EOQn to the set of candidate solutions; A2: CandidatesComplete = T rue; else if EOQn < qn−1 B: Add qn−1 to the set of candidate solutions; else if C: CandidatesComplete = T rue; If CandidatesComplete = F alse n:=n-1; Evaluate the total cost at each...