The EOQ Model with Planned Backorders

1
The EOQ Model with Planned Backorders
2

• Demand does not have to be satisfied immediately
  (from on-hand inventory).
• Customers are willing to wait.
• A penalty cost b is incurred per unit backordered
  per unit time.
• Orders are received L units of time after they
  are ordered (typo in your handout)

3
Objective

• Minimize purchasing ordering holding
  backordering cost

4
Notation

• I(t) inventory level at time t
• B(t) number of backorders at time t
• B Average backorder level
• IN(t) Net inventory at time IN(t) I(t) -
  B(t)
• PB(t) Stock-out indicator at time t
  PB(t)1IN(t) lt 0
• PB Average fraction of time there is a stock-out
  (stock-out probability)
• B(t) Number of units that are backordered in
  time interval 0, t

5
(No Transcript)
6
Qss
L
I(t)
r
0
ss
2Q/D
3Q/D
4Q/D
Q/D
t
7
Let ss r - DL, then 1. If ss gt 0, I(t) gt 0
and B(t) 0, 2. If ss lt - Q, I(t) 0 and B(t)
gt 0 3. If -Q ? ss ? 0, then both I(t) and B(t)
can be positive Only case 3
makes sense!
8

• T time interval between orders
• T1 time interval within T during which we have
  positive inventory
• T2 Time interval within T during which
  backorders are positive

9
Qss
I(t)
0
t
ss
2Q/D
3Q/D
4Q/D
Q/D
10

• T Q/D
• T1 (Q ss)/D
• T2 -ss/D
• PB T2/T -ss/Q
• I (1-PB)(Qss)/2 (Qss)2/2Q
• B PB(-ss/2) ss2/2Q

11
Total Cost
12
Total Cost
Let a b/(bh), then
13
The Optimal Order Quantity
14
The Optimal Cost and the Optimal Stockout
Probability
15
Systems with Service Level Constraints

• Minimize purchasing ordering holding
  cost, subject to a constraint on the probability
  of a stock-out.

16
Formulation

• Minimize AD/Q h(Qss)2/2Q cD
• Subject to
• PB -ss/Q ? 1- ao

17

• Since cost is increasing in ss while PB is
  decreasing in ss, the constraint is binding.
• ss -Q(1- ao)
• Y(Q, ss) AD/Q hao2Q/2 cD

18
Systems with Backorder Penalties per Occurrence

• Instead of a cost b per backorder per unit time,
  we incur a one time cost k per backorder.

19
Total Cost

• B DPB -Dss/Q
• ?
• Y(Q,ss) cD AD/Q h(Qss)2/2Q -kDss/Q.
• ?
• ss kD/h Q
• ?
• Y(Q,ss) (ck)D 2ADh (kD)2/2hQ

20
Two Cases

• 1. (kD)2 ? 2ADh ? ss 0 and
• 2. (kD)2 lt 2ADh ? Q ? and ss -?

21
Systems with Lost Sales

• No backorders are allowed
• Demand that arrives when no on-hand inventory is
  available is considered lost
• There is a penalty cost k (opportunity cost)
  for each unit of lost demand

22

• ss amount of demand lost per order cycle
• Q amount of total demand per order cycle
• Q amount of demand satisfied per order cycle
  Q-ss
• Average number of orders per unit time D/Q
• Average inventory (Q-ss)2/2Q
• PB ss/Q
• Average amount of demand lost per unit time
  DPB Dss/Q

23
Total cost

• Y(Q,ss) cD(1-ss/Q) AD/Q h(Q-ss)2/2Q
  kDss/Q.
• cD-cDss/Q AD/Q h(Q-ss)
  2/2Q kDss/Q
• cD AD/Q h(Q-ss) 2/2Q
  (k-c)Dss/Q
• The total cost has the same form as in the case
  with costs per backorder occurrence ? a similar
  solution approach applies.