Question

The Fashion Store sells fashion items. The store has to order these items many months in advance of the fashion season in order to get a good price on the items. Each unit costs Fashion $100. These units are sold to customers at a price of $250 per unit. Items not sold during the season can be sold to the outlet store at $80 per unit. If the store runs out of an item during the season it has to obtain the item from alternative sources and the cost including air freight to Fashion is $190 per unit. What is the initial order quantity that maximizes the profit for the Fashion Store? x f(x) 61 0.01 62 0.01 63 0.01 64 0.01 65 0.01 66 0.02 67 0.02 68 0.02 69 0.02 70 0.03 71 0.03 72 0.03 73 0.04 74 0.04 75 0.04 76 0.04 77 0.04 78 0.04 79 0.04 80 0.04 81 0.04 82 0.04 83 0.04 84 0.04 85 0.03 86 0.02 87 0.02 88 0.03 89 0.02 90 0.02 91 0.02 92 0.02 93 0.01 94 0.01 95 0.01 96 0.01 97 0.01 98 0.01 99 0.00 100 0.00 101 0.00 102 0.00 103 0.00 104 0.00

Answer 1

Step 1: First of all we arrange the given data into tabular format,and we can determine mean and StandardDeviation .

SalesDemand (X)	FrequencyF(x)	P( SalesDemand) P(x)	Cum.prob
61	1	0.01	0.01
62	1	0.01	0.02
63	1	0.01	0.03
64	1	0.01	0.04
65	1	0.01	0.05
66	2	0.02	0.07
67	2	0.02	0.09
68	2	0.02	0.11
69	2	0.02	0.13
70	3	0.03	0.16
71	3	0.03	0.19
72	3	0.03	0.22
73	4	0.04	0.26
74	4	0.04	0.3
75	4	0.04	0.34
76	4	0.04	0.38
77	4	0.04	0.42
78	4	0.04	0.46
79	4	0.04	0.5
80	4	0.04	0.54
81	4	0.04	0.58
82	4	0.04	0.62
83	4	0.04	0.66
84	4	0.04	0.7
85	3	0.03	0.73
86	2	0.02	0.75
87	2	0.02	0.77
88	3	0.03	0.8
89	2	0.02	0.82
90	2	0.02	0.84
91	2	0.02	0.86
92	2	0.02	0.88
93	1	0.01	0.89
94	1	0.01	0.9
95	1	0.01	0.91
96	1	0.01	0.92
97	1	0.01	0.93
98	1	0.01	0.94
99	0	0	0.94
100	0	0	0.94
101	0	0	0.94
102	0	0	0.94
103	0	0	0.94
104	0	0	0.94
	100		0.94

Here we can see ,the ovbsevation total frequency F(X)=100.Formule discrption

Here we can see the given data sets containing zero frequency's,so we are neglecting that and creating the frequency distribution table.

Probabulity of SalesDemand=P(X)/total no. of freq

Cumulative probabulity of SalesDemand is Given in last Column,we can get by adding probablity of sales Demand column.

Step2:

Here the mean of the given observitation =

mean=82.5

Step3:

Suppose we are trying to decide whether to order an inventory of 82 units for the start of the season or whether we should increase the inventory ordered to 83 units. If the extra item is not sold we incur a marginal cost of 100− 80 =20 . On the other hand, if the extra item is sold we earn the difference between the contribution we make by having the item on hand and the contribution we would have made if we would not have had the item on hand. Thus we gain a marginal contribution of (250 − 100) − (250 − 190) = 190 − 100 = 90.

  In order to compute the marginal net contribution, we need the probabilities

P(The 83th item is not sold) = P(SalesDemand ≤ 82)

  P(The 83th item is sold) = P(SalesDemand > 83)

which we can look up in Table . With these probabilities, we can compute

Expected Marginal Cost = P(SalesDemand ≤ 82) × 20

Expected Marginal Profit = P(SalesDemand > 82) × 90

Thus we would increase the order from 82 to 83 if

Expected Marginal Cost < Expected Marginal Profit

P(SalesDemand ≤ 82) × 20 < P(SalesDemand > 82) × 90

0.62(20) < 0.32(90)

12.4<28.8

Letting Ce be the cost of excess and Cs be the cost of shortage, in general we increase our inventory B while

P(SalesDemand ≤ B)Ce < P(SalesDemand > B)Cs.

  We would stop increasing B at B* , the optimal stocking level, when P(SalesDemand ≤ B * )

Ce = P(SalesDemand > B*)Cs.

Rewriting this last equation we get

P(SalesDemand ≤ B * )Ce = (1 − P(SalesDemand ≤ B * ))Cs

0 = Cs − P(SalesDemand ≤ B * )(Cs + Ce),

which becomes

P(SalesDemand ≤ B * ) = Cs /(Cs + Ce) ..(a)

Since the left-hand side is by definition the service level, (a) is equivalent to (??).

From our Fashion Store example we have Ce = 20 and Cs = 90. So,

SL* = Cs /(Cs + Ce) = 90 /(90 + 20) = .8182

  From the frequency table, for a service level of .8182, we get B∗ = 89

CONCLUSION:

  89  is the initial order quantity that maximizes the profit for the Fashion Store .