Question

Question 4

A shop is planning an order for a popular Christmas festive season product. Demand for the product usually starts from first week of December till first week of January and reduces sharply thereafter. For this reason, and to stimulate sales for leftovers, the product is sold at a significantly reduced price from the second week of January to the fourth week of January. Any leftover after the fourth week of January goes waste. The table below gives past data on total demand for the period from first week of December to first week of January, and from second week of January to fourth week of January, together with their respective chances of occurrence. The product can be purchased at a wholesale price of GHS60 per unit for a pack containing 600 products, GHS57 per unit for a pack containing 800 products, and GHS52 per unit for a pack containing 1000 products. The shop plans to sell the product for GHS80 per unit from first week of December to first week of January, and at a reduced price of 30% thereafter.

  

1st week of December to 1st week of January

2nd week of January to 4th week of January

  

  

  

  

Demand

Probability

Demand

Probability

  

  

  

  

500

0.1

320

0.5

  

  

  

  

600

0.3

180

0.3

  

  

  

  

750

0.4

130

0.2

  

  

  

  

850

0.2

i. What is the expected profit when a pack containing 600 products is ordered?

ii. What the expected profit when a pack containing 800 products is ordered?

iii. What is the expected profit when a pack containing 1000 products is ordered?

iv. What is the standard deviation of the profit when a pack of 800 is ordered? State any assumption you used in calculating this value and with justification.

v. What is the standard deviation of the profit when a pack of 1000 is ordered? Apply the same assumption you used in (iv) above.

vi. Which of the two, a pack of 800 or a pack of 1000 products should be ordered in order to minimize risk associated with profit? Give reasons for your answer.

Answer 1

Sol:

If we see the expected units to be sold:

E(X= 1st December to 1st week of January) = 500*0.1+600 *0.3+ 750*0.4 +850*0.2

= 50 + 180 +300 + 170

= 600

We expect 600 units to be sold in first week of december to first week of January

As we are suppose to sale the GHS 80 per unit so total revenue would be GHS 48000 (600*80) in the above period

E(X= 2nd January to 4th week of January) = 320*0.5+180 *0.3+ 130*0.2

= 160 + 54 +26

= 240

We expect 240 units to be sold in 2nd January to 4th week of January.

As we are suppose to sale the GHS 48 per unit (30% discounte price of GHS 80 is 48)  so total revenue would be GHS 11520 (240*48) in the above period

Total revenue would be form 1st wekk of december to 4th week of January = 11520 + 48000

Total revenue would be form 1st wekk of december to 4th week of January = GHS 59520

Answer a)

If we order the pack containg 600 units, we need to order two such pack to sale 840 units.

So total cost would be GHS 72000 (60*1200) (one unit cost GHS 60 )

so total lost would be = 72000- 59520

=12480

So total loss would be GHS 12480

Answer B)

If we order the pack containg 800 units, we need to order two such pack to sale 840 units.

So total cost would be GHS 91200 (57*1600) (one unit cost GHS 60 )

so total lost would be = 91200- 59520

=31680

So total loss would be GHS 31680

Answer C)

If we order the pack containg 1000 units, we need to order one such pack to sale 840 units.

So total cost would be GHS 52000 (52*1000) (one unit cost GHS 60 )

so total profit would be = 59520 - 52000

=7520

So total Profit would be GHS 7520