# Operations Research

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Operations Research (MTH601)
63
Rs.5×747×0.5
=
Shortage cost
2×12
= Rs. 77.75 per order
Total cost per order
= 3857 + 400 + 77.75 + 322.14
= Rs. 4656.89
Annual cost = Number of orders/year × cost
= 4.66 × 4656.89
= Rs. 21701.
Model 3: Manufacturing model with no shortages
In this model the following assumptions are made:
(1)
Demand is at a constant rate (D).
All cost coefficients (C1, C2, C3) are constants.
(2)
(3)
There is no shortage cost, or C4 = 0.
(4)
The replacement rate is finite and greater than the demand rate. This is also called replenishment rate
or manufacturing rate, denoted by R.
Schematically, this model is illustrated in fig 4
Slope= (R-D)
Slope = D
Q
Im
t2
Time
t1
t
Fig. 4
The total cost of inventory per period is the sum of three components: item cost, order cost and items holding cost.
Let Im be the maximum inventory, t1 be the time of manufacture and t2 be the time during which there is no supply.
In this model, all items required for a cycle are not stored at the beginning as in Wilson's Model. The items are
manufactured at a higher rate than the demand so that the difference (R­D) is the existing inventory till the items are
exhausted.
Item cost/period = C1Q
(32)
Order cost/period = C2
(33)
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Operations Research (MTH601)
64
= C × I × (t1 + t2 ) 2
Item holding cost/period
(34)
m
3
= C3 I  m × t 2
(35)
= t ( R - D)
(36)
I
1
m
t =Q R
(37)
But
1
= (Q R)( R - D)
Therefore
(38)
I
m
Substituting the value of Im, we get the total cost of inventory per period.
C′ = C Q + C2 + C3 (Q R)( R - D) × t 2
(39)
1
Total cost of inventory per unit time
C = Ct
(40)
t + C3 (Q R)( R - D) × t
= C Q t +C
(41)
2
1
2
But t = Q/D
Substituting the value of t we get
C = C D + C D Q + C3 (Q R) ( R-D) 2
(42)
1
2
Differentiating C with respect to Q and setting equal to zero for minimum C, we get,
C D  C ( R-D)
dC
= 0-  2  +  3
=0
(43)
dQ
2R
Q2
Solving equation (43), we get
2C2D
R
Q* =
(44)
R-D
C3
This gives the economic order quantity and is a balance between holding and set up costs.
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Operations Research (MTH601)
65
Example: The demand for an item in a company is 18000 units/year and the company can produce at the rate of
3000 per month. The cost of one set up is Rs. 500 and the holding cost of 1 unit per month is 15 paisas. Determine:
(a)
The optimum manufacturing quantity.
(b)
The maximum inventory.
(c)
The time between orders.
(d)
The number of orders/year.
(e)
The time of manufacture.
(f)
The optimum annual cost if the cost of the item per unit is Rs. 2.
Assume no shortages.
Solution
C1 = Rs. 2 per item.
C2 = Rs. 500 per order.
C3 = Rs. 0.15 per item per month
D = 18000/year = 1500/month
R = 3000/month
a)
Optimum manufacture quantity
2C2D
R
Q* =
R-D
C3
2×500×1500×3000
=
= 4470 units
C 0.15(3000-1500)
b)
The maximum inventory
= Q ( R-D) R = 4470 × 1500 3000 = 2235 units
I
m
c)
The time between orders
t = Q D = 4470 1500 = 2.98 months
3 months
d)
The number of orders/year
N = 12 3 = 4
e)
The time of manufacture
t = Q R = 4470 3000 = 1.490 months
1
f)
The optimum annual cost
= Item cost + Ordering cost + Holding cost
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