the eoq model with planned backorders
DESCRIPTION
The EOQ Model with Planned Backorders. 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. Objective. - PowerPoint PPT PresentationTRANSCRIPT
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The EOQ Model with Planned Backorders
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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
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Minimize purchasing + ordering + holding + backordering cost
Objective
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Notation
I: Average inventory level at timeB: Average number of backordersPB: Average fraction of time there is a stock-out (stock-out probability)B’: Average umber of units that are backordered
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0
})(1{lim dttBT
B T
TtBB T)('lim'
0
})(1{lim dttPT
P BTB
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Q/D 2Q/D 3Q/D 4Q/D
Q+ss
I(t)
t
ss
0
Lr
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Let ss = r - DL, then
1. If ss > 0, I(t) > 0 and B(t) = 0,
2. If ss < - Q, I(t) = 0 and B(t) > 0
3. If -Q ss 0, then both I(t) and B(t) can be positive
Only case 3 makes sense!
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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
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Q/D 2Q/D 3Q/D 4Q/D
Q+ss
I(t)
tss
0
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T= Q/D T1= (Q + ss)/D T2= -ss/D
PB = T2/T = -ss/Q
I = (1-PB)(Q+ss)/2 = (Q+ss)2/2Q
B = PB(-ss/2) = ss2/2Q
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Total Cost
Qssb
QssQhQADcDssQY
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22
0)(),(
Qbss
QssQh
ssssQY
02
)(),(2
2
2
22
2
Qbss
QssQh
QAD
QssQY
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Total Cost
2 2 2
2 2 2
2 2 2 2 2
2 2 2
( , ) ( , ) ( )2
( (1 ) ) (1 ) 02
Y Q ss Y Q ss AD h Q ss bssQ Q Q Q QAD h Q Q b QQ Q Q
( ) 0h Q ss bss hQssQ Q h b
Let = b/(b+h), then
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*
2*
* (1 )
* *
ADQh
ss Q
r DL ss
The Optimal Order Quantity
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( *, *) 2
*( *, *) 1*B
Y Q ss ADh cD
ss hP Q sQ b h
The Optimal Cost and the Optimal Stockout Probability
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Systems with Service Level Constraints
Minimize purchasing + ordering + holding cost, subject to a constraint on the probability of a stock-out.
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Formulation
Minimize AD/Q + h(Q+ss)2/2Q + cD
Subject to: PB = -ss/Q 1- o
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Since cost is increasing in ss while PB is decreasing in ss, the constraint is binding.
ss* = -Q(1- o
Y(Q, ss*) = AD/Q + ho2Q/2 + cD
22*ohADQ
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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.
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B’ = DPB = -Dss/Q
Y(Q,ss) = cD + AD/Q + h(Q+ss)2/2Q -kDss/Q.
ss* = kD/h – Q
Y(Q,ss*) = (c+k)D + [2ADh – (kD)2]/2hQ
Total Cost
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1. + (kD)2 2ADh ss* = 0 and 2. + (kD)2 < 2ADh Q* = and ss* = -
Two Cases
* 2 /Q AD h
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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
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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’
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Total costY(Q’,ss’) = cD(1-ss’/Q’) + AD/Q’ + h(Q’-ss’)2/2Q’ +k’Dss’/Q’.
= cD-cDss’/Q’ + AD/Q’ + h(Q’-ss’) 2/2Q’ +k’Dss’/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.