Global Economy

OPRE 6366. SCM : 4. Inventory Management

1. Inventory Management Certainty: 2. Long Term Quantity Discounts

2.1 All-unit quantity discount

The price of each unit in an order of Q is the same.     Price per unit =   

αδ the textbook

That price depends on the magnitude of Q as follows.  p1 if q0 ≤ Q < q1    p2 if q1 ≤ Q < q2 ... ...    p N if q N −1 ≤ Q < q N

where we set q0 = 0 and q N = ∞. Prices are monotonically ordered by pn−1 ≥ pn for 2 ≤ n ≤ N.     p1 Q if 0 ≤ Q < q1     p2 Q if q1 ≤ Q < q2 Purchasing cost of Q units = ...   ...     p N Q if q N −1 ≤ Q < q N We coin the term region n for order sizes in qn−1 , qn ]. If the order size is in region n, i.e., qn−1 ≤ Q < qn , then define the total costs in region n as TCn ( Q) := h R R ( p n Q ) + S + ( p n Q ). 2 Q Q (1)

Then the total cost over all regions is obtained by patching all TCn together, i.e.,     TC1 ( Q) if 0 ≤ Q < q1     TC2 ( Q) if q1 ≤ Q < q2 TC ( Q) = ...   ...     TCN ( Q) if q N −1 ≤ Q < q N In the marginal-units quantity discount problem, we want to solve min TC ( 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 1 if 0 ≤ Q < q1 if q1 ≤ Q < q2 ... if q N −1 ≤ Q < q N       

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