Policy

You are a salesperson of a durable product. You rely on system recommendations to manage
procurement and selling. Yet your manager is unhappy with your performance. To save your job,
you must figure out why your performance is poor and how to improve.
The current system uses the (s, S) policy. Here s is the reordering point and S is the base-stock
level, with s ≤ S. In each period t = 1, 2, . . . , T, the system works as follows:
• PROCUREMENT: Upon observing the initial inventory level xt
, you have two situations: (i)
if xt < s, you will order Qt = S − xt , pay fixed ordering cost K, and bring the post-order inventory level yt to the order-up-to level S (i.e., yt = S); (ii) if xt ≥ s, you do not order and inventory remains yt = xt . Hence, your ordering cost is K · 1(xt < s). • SELLING: with yt inventory on hand, you satisfy customer demand Dt ∼ N (µ, σ2 ) as much as possible. There are two situations: (i) if Dt ≤ yt , the leftover inventory (yt − Dt) carries over to the next period at unit holding cost h; (ii) if Dt > yt
, the unsatisfied demand (Dt −yt)
is booked at unit backorder cost b. Hence, your inventory holding and backorder costs are1
h(yt − Dt)

• + b(Dt − yt)
  +.
  • The remaining inventory carries over to the next period t + 1, with initial inventory level
  xt+1 = yt − Dt
  .
  The demand has µ = 3, σ = 1, while the cost structure has h = 1, b = 10, K = 20. The current
  policy is set to s = 3, S = 15. You suspect that the current (s, S) policy is responsible for your
  poor performance. To convince the manager that the policy is the culprit, you must address the
  following questions, with your favorite Excel and simulation.

1. Simulation the system for a year, with T = 365 and x1 = 0. Find the following performance
   measures
   total.cost =
   ∑T
   t=1 (
   K · 1 ∗ (xt < s) + h(yt − Dt)

• + b(Dt − yt)
  +
  )
  P(in.stock) =
  ∑
  t
  1 ∗ (xt < s)
  T
  fill.rate =
  ∑
  t min(yt
  , Dt)
  ∑
  t Dt
  P(ordering) =
  ∑T
  t=1 1 ∗ (xt < s)
  T
  .

1. SAW-TOOTHED inventory dynamics: plot the initial inventory and post-order inventory levels for January, i.e., two time-series (xt)
   31
   t=1,(yt)
   31
   t=1.