Presentation Courtesy of Tor Schoenmeyr. Used With Permission

“Centralized Ordering Policies in a Multi-Warehouse System with Lead Times and Random Demand” System and Problem Description The Allocation Assumption Policy 1: Order up to y every period Policy 2: Order up to y every m periods System Description and Assumptions Total inventory in system: y Depot (no inventory) Supplier lead time L Supplier Transportation lead time l N warehouses Demand (with inventories) (random) Z1 e1 = N ( µ1 , σ 1 ) Z2 e2 = N ( µ2 , σ 2 ) Z3 e3 = N ( µ3 , σ 3 ) Costs to be minimized: Decisions to be made every period: •Holding cost h per unit in inventory •How much, if anything, should be ordered from the supplier •Penalty cost p per unit of unmet demand (placed in backlog) •Fixed cost K for every order placed •How should we distribute the incoming orders at the Depot Why have a Depot? (with no inventory) Problem • Depot Benefit Separate warehouses have little purchasing power Demand fluctuates for the individual warehouse • • It is expensive/ impractical to build a depot • • (Demand can vary also in the aggregate) • • • Exploit quantity discounts from the supplier Fluctuations in different warehouses even out, and you gain “statistical economies of scale” Depot need not to be a physical entity (the point is that goods are allocated after orders completed) (Maybe a depot with inventory can do even better) Applicability of model Good application: Steel for conglomerate Questionable application: Coca-Cola for 7-Eleven Production lead times: Long Short Inventory surplus: Holding costs (expensive) Cheap, not to say desirable (up to shelf capacity) Inventory shortfall: Order placed on “backlog” at some penalty Customer walks (or buys a substitute) System and Problem Description The Allocation Assumption Policy 1: Order up to y every period Policy 2: Order up to y every m periods Allocation Assumption (for every period ordering, normal demand) “Every period t, we can make an allocation (at the depot) such that the probability of running out at each warehouse is the same at period t+l” Depot Supplier “Every period, we can find a constant v, such that the total inventory at and in transit to the i th warehouse is: (l + 1) µi + vσ i l + 1 Transportation lead time l Warehouses Example when Allocation Assumption holds (identical warehouses) Ware- Demand houses Period t L=1 Supplier 8 10 Period t+1 Warehouses 6 l=0 L=1 Supplier 10 ? 10-6+5=9 l=0 4 10-4+3=9 Example when Allocation Assumption is violated (identical warehouses) Ware- Demand houses Period t L=1 Supplier 8 10 Period t+1 Warehouses 9 l=0 L=1 Supplier 10 0 ? 10-9+8=9 l=0 10-0+0=10 The Allocation Assumption holds for high µ/σ and low N Theoretical Result Eppen and Schrage derive a good theoretical approximation formula for the probability of A.A. being true. The The paper paper does does not not explain explain how how “negative “negative demand” demand” should should be be interpreted. interpreted. This This happens happens frequently frequently in in the the lower left corner where lower left corner where my my experiments experiments gave gave different different results results than than those those of of the the paper paper Probability of A.A. being true according to experiment presented in paper (my experiment in parenthesis) Percent µ/σ N ½ 1 3/2 2 5/2 2 32.6 (35.9) 66.3 (66.3) 85.8 (86.5) 95.2 (95.5) 98.8 (98.8) 3 20.1 (19.8) 54.7 (53.2) 79.8 (79.0) 93.0 (93.5) 98.1 (98.2) 4 11.4 (10.0) 43.1 (41.0) 73.3 (72.0) 90.1 (90.5) 97.3 (97.4) 5 7.6 (4.9) 36.5 (30.6) 68.6 (65.8) 88.3 (88.0) 96.5 (96.9) 6 4.6 (2.5) 29.9 (22.3) 63.2 (60.7) 86.4 (85.2) 96.1 (95.8) 7 2.8 (1.2) 24.5 (15.8) 59.1 (53.5) 84.1 (83.2) 95.5 (95.5) 8 1.6 (0.5) 20.4 (11.0) 54.3 (46.6) 82.0 (80.5) 94.6 (95.3) System and Problem Description The Allocation Assumption Policy 1: Order up to y every period Policy 2: Order up to y every m periods Policy 1: Order Every Period (fixed ordering costs K = 0) Problem: Intuitive answer: But… How should we distribute the goods that come in to the depot every period? We should distribute goods so that total goods at and en route to every factory is “the same” But is this always possible? If we make the A.A., then yes! How much should we order from the factory every period? We should order so that the same total inventory level y is achieved every period. (=order last period’s demand) What should be the value of y? Eppen and Schrage find an analytical expression for the inventory at each warehouse We We know know how how much much is is ordered ordered every every period period (as (as aa function function of of y) y) We We know know how how the the incoming incoming goods goods are are split split up up at at the the Depot Depot We We know know the the (random (random function function for) for) demand demand at at each each warehouse warehouse Eppen and Schrage derive this expression for the inventory S at each warehouse (simplified form for the case of identical warehouses): y S j = (l + 1) µ + − (l + 1) µ − N Fixed component ∑ L t =1 ∑ N N e i =1 it − ∑ t = L +1 e jt L +l Random component The problem is now equivalent to the newsboy problem, and can be solved analytically Newsboy problem “The newsboy buys i newspapers, at a cost c each. He sells what is demanded d (random variable), or all he has got i, whichever is less, at a price r. Any surplus is lost.” Deterministic inventory Newsboy Random inventory i -d e y L +l ∑ t =1 ∑i =1 it (l + 1)µ + − (l + 1)µ − − ∑t =L+1 e jt N N L Our system (at warehouse) Cost of surplus (per unit) Cost of shortage (per unit) c r-c h p N System and Problem Description The Allocation Assumption Policy 1: Order up to y every period Policy 2: Order up to y every m periods Policy 2: Order Up to level y every m periods + We can select m and y to minimize total costs, including fixed ordering costs K + Periodic ordering policy easy to implement in practice + Authors claim that theoretical results on this policy has wider applicability – Certain approximations have to be made to find the best m and y – Even if the best m and y were to be found, the periodical ordering policy isn’t necessarily optimal Several new assumptions lead to an analytical solution for the periodic ordering policy 2. 2. The The allocation allocation rule rule (l + m) µi + vσ i is is used, used, but but not not proven proven to to be be optimal optimal 1. “Stock can only run out in the last period before order arrivals” Analytical expression for optimal y and cost, given m In In typical typical cases, cases, we we can can then then find find the the best best solution solution by by optimizing optimizing yy for for m=1,2,3… m=1,2,3… 3. As before, we make the a.a. – it is always possible to make this allocation