A retailer needs to find out the best order size for the coming season. The reta

Question

A retailer needs to find out the best order size for the coming season. The retail price is $12 per unit of sold product. The ordering cost for each unit is $4. The savage value of each unsold product is $2. The retailer estimates the future demand will be random variable which follows a normal distribution with mean=1000 and standard deviation=100.
What is the retailer’s best order size decision (you don’t need to find the exact value but need to show all steps to find this value, including excel functions needed)
How can you measure the risk in the retailer’s net profit? (You need to show the steps of analysis conceptually)

Explanation / Answer

Retailer’s best order size decision

Given

Cost =4

Price =12

Salvage=2

Cost of Overage (Co) = cost - salvage = $`4 - $2= $ 2

Cost of Underage (Cu) = Price - cost = $ 12- $4 = $ 8

Mean, u= 1000

Standard Deviation, s =    100

STEPS

1.             Find critical ratio.

2.             Then Find z value using critical ratio

3.             Convert Z into order quantity, Q= u + z*standard deviation

1.             Critical ratio, F (Q) =Cu/(Cu + Co) =8/(8+2) = 0.8 . Critical ratio offers the service level required. Here it is 85%.

2.             Now we find z-value corresponding to critical ratio=0.8000.

If the critical ratio falls between two values in the standard normal distribution table, choose the greater z-statistic. Z-value corresponding to 0.8000 is critical ratio is   0.85

3.             Profit-maximizing order quantity

= u + z*standard deviation

=1000+0.85(100)

=1085 units

Measuring the risk in the retailer’s net profit

The real risk is the number of units left over in the inventory. The risk in lost sales is an opportunity cost. So I have not included it here.

We order Q= 1085 units

Expected demand, (mean) µ= 1000(given)

STEP 1

1.             Z= (Q-µ)/s =( 1085-1000) / 100=0.85

2.             Use z to find L (z) use lost table. L (Z)=L(0.85) = 0.1100

3.             Expected lost sales = Standard deviation * L (z) =100*0.1100=11 units

STEP 2

1.             Expected sale= µ (expected demand) - expected lost sales = 1000-11 units =989 units

STEP 3

Expected Leftovers = Q – Expected Sale = 1085- 989 = 96 units

Loss due to excess inventory= [(Cost - Salvage value) X Expected leftover inventory]

= = [(4 - 2) X 96] = $192

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