Q.1a)The following graph is EOQ model with planned shortages.

Let the parameters from the basic EOQ model.

d= constant demand rate
K= setup cost for placing one order
Q= order quantity
h=inventory holding cost per unit of product per unit of time p= shortage cost per unit of product per unit of time
S= inventory level just after an order of size Q arrives

So,
Q– S = Shortage in inventory just before an order of Q units is added

Production or ordering cost per cycle = K+cQ, where cQ is the unit cost

During each cycle, the inventory level is positive for a time: = Sd

The average inventory level during this time is: = ( S + 0)2= S2 units

The corresponding cost is = hS2per unit time

Therefore, the Holding cost per cycle= hS2 × Sd=hS22d

During each cycle, the shortage occurs for a time: = (Q – S)d

The average amount of Shortages during this time is: = ( 0 + Q - S)2= (Q - S)2 units

The corresponding cost is = p (Q-S)2per unit time

Therefore, the Shortage cost per cycle = p (Q-S)2 × (Q-S)d=p(Q-S)22d Q.1a)Hence, Total cost per cycle = K+cQ+ hS22d+ p(Q-S)22d

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