10/24/2012
I. Introduction:
The Specialty Toys Company faces a challenge of deciding how many units of a new toy should be purchased to meet anticipated sales demand. If too few are purchased, sales will be lost; if too many are purchased, profits will be reduced because of low prices realized in clearance sales. Here, I will help to analyze an appropriate order quantity for the company. II. Data Analysis:

1.

20,0
00
.025
10,0
00
30,0
00

.025
.95

20,0
00
.025
10,0
00
30,0
00

.025
.95

Since the expected demand is 2000, thus, the mean µ is 2000. Through Excel, we get the z value given a 95% probability is 1.96. Thus, we have: z= (x-µ)/ σ=(30000-20000)/ σ=1.96, so we get the standard deviation σ=(30000-20000)/1.96=5102. The sketch of distribution is above. 95.4% of the values of a normal random variable are within plus or minus two standard deviations of its mean.

2. At order quantity of 15,000, z= (15000-20000)/5102=-0.98, P(stockout) = 0.3365 + 0.5 = 0.8365
At order quantity of 18,000, z= (18000-20000)/5102=-0.39,
P(stockout) = 0.1517 + 0.5= 0.6517
At order quantity of 24,000, z= (24000-20000)/5102=0.78,
P (stockout) = 0.5 - 0.2823 = 0.2177
At order quantity of 28,000, z= (28000-20000)/5102=1.57,
P (stockout) = 0.5 - 0.4418 = 0.0582

3.
Order Quantity = 15,000|
Unit Sales| Total Cost| Sales at $24| Sales at $5| Profit| 10,000| 240,000| 240,000| 25,000| 25,000|
20,000| 240,000| 360,000| 0| 120,000|
30,000| 240,000| 360,000| 0| 120,000|

Order Quantity = 18,000|
Unit Sales| Total Cost| Sales at $24| Sales at $5| Profit| 10,000| 288,000| 240,000| 40,000| -8000|
20,000| 288,000| 432,000| 0| 144,000|
30,000| 288,000| 432,000| 0| 144,000|

Order Quantity = 24,000|
Unit Sales| Total Cost| Sales at $24| Sales at $5| Profit| 10,000| 384,000| 240,000| 70,000| -74,000|
20,000| 384,000|...

...000) / 1.96 = 5,102 units.
So, we have a distribution with a mean of 20,000 and a standard deviation of 5,102.
2. Compute the probability of a stock-out for the order quantities suggested by members of the management team.
Using the normal distribution theory, we discover that as the ordered quantity increases the probability of stockout decreases.
At 15,000 the probability of stockout will be 0.8365
At 18,000 the probability of stockout will be 0.6517
At 24,000 the probability of stockout will be 0.2177
At 28,000 the probability of stockout will be 0.0582
3. Compute the projected profit for the order quantities suggested by the management team under three scenarios: worst case in which sales = 10,000 units, most likely case in which sales = 20,000 units and best case in which sales = 30,000 units:
Order Quantity: 15,000 were cost price is $16, selling price $24 & after holiday selling price $5
|Unit Sales |Profit |
|10,000 |25,000 |
|20,000 |120,000 |
|30,000 |120,000 |
Order Quantity: 18,000 were cost price is $16, selling price $24 & after holiday selling price $5
|Unit Sales |Profit |
|10,000 |-8,000 |
|20,000 |144,000 |
|30,000 |144,000...

...1. Let X be the demand for the toy. Then X follows normal distribution with mean μ = 20000 and standard deviation σ. Then
P(10000 < X < 30000) = 0.95
P( X < 20000)=0.5
P(10000 < X < 20000) = 0.475
P( X < 10000)=0.025
NORM.S.INV(0.025)=-1.96
NORM.S.INV(0.975)=1.96
Z-score of 10000 =-1.96
Z-score of 30000=1.96
σ = (30000-20000)/1.96 =10000/1.96 = 5102
Standard Deviation of 5102
The graph above shows the distribution for the demand for the Weather Teddy Bear usingSpecialtyToys’ forecasts based off of sales histories for similar products. This forecast predicts that this toy will have a demand of 20,000 units. However, the forecasts also predict that the probability of selling between 10,000 to 30,000 units is equal to 0.95. Using this information, the forecast suggests a mean sale of 20,000 units with a range of 10,000 to 30,000 units. Using the normal standard inverse function in excel, the standard deviation for this forecast is calculated to be 5,102 units. Using 20,000 units as the given mean, this additional information can be used to generate the graph showing 14,898 and 25,102 units within one standard deviation of the mean, and 9,796 and 30,204 units within 2 standard deviations of the mean. 9,796 and 30,204 are outside of the 0.95 probabilty range.
2.
Order (X)
z-score=(X-20000)/5102
P(X)
Stock out=P(1-X)
15000
-0.980
0.164
0.836
18000
-0.392
0.348
0.652
24000
0.784
0.783
0.217
28000
1.568
0.942...

...Southeast
2000
North
2300
Northwest
600
West
1100
Southwest
500
Total
6500
Total profit as per the production plan:
The objective function to be maximized is:
Constraints
On Land:
On Water Irrigation Limit:
Parcel
Water Irrigation
Required
Maximum capacity
Southeast
3200
North
3400
Northwest
800
West
500
Southwest
600
In addition maximum water available is 7400
The constraints are:
On Sales:
Per acre production
Total Production
Maximum Sales
Wheat
50 bushels
110,000
Alfalfa
1.5 tons
1800
Barley
2.2 tons
2200
Constraints:
Thus mathematical model is:
Maximize:
Subject to:
Solving the problem using solver of MS Excel we get the solution as follows:
Variable
Solution
547.21
544.90
422.71
311.34
373.84
65.52
65.52
0.00
Variable
Max. Profit
Solution
0.00
0.00
586.66
376.95
35.33
0.53
0.53
315862.07
Crop Plan
Parcel
Cultivation Area (Acre)
Wheat
Alfalfa
Barley
Total Area
Southeast
547.21
65.52
586.66
1199.39
North
544.90
65.52
376.95
987.36
Northwest
422.71
0.00
35.33
458.04
West
311.34
0.00
0.53
311.87
Southwest
373.84
0.00
0.53
374.37
Profit
$220,000.00
$7,862.07
$88,000.00
$315,862.07
...

...Case Study Analysis:
Abstract
This analysis examines freight cost and cleaning fluid supplies at two locations; Cincinnati and Oakland, to determine the optimal distribution network to supply the cleaning fluid to Great North American at minimal cost to Solutions Plus. Based on projected cost a bid recommendation is made and decision factors related to the analysis are discussed.
Keywords: Solutions Plus, Cost minimization, Breakeven, Bid, Shipping Cost
Background
Solutions Plus is an industrial chemicals company that produces cleaning fluids and solvents for many applications. Great North American railroad is taking bids for delivery of a cleaning fluid for its locomotives at eleven different locations. Solutions Plus can produce the cleaning fluid for $1.20 per gallon at their Cincinnati plant. The management team has elected to limit capacity at the Cincinnati plant to 500,000 gallons. Solutions Plus also negotiated with an industrial chemical company in Oakland, California to produce and ship up to 50,000 gallons of the locomotive cleaning fluid at $1.65 per gallon. Higher cost at the Oakland plant can be offset by lower shipping costs to some of the customer’s locations. Total supply of cleaning fluid available to Solutions Plus equates to 550,000 gallons. Great North American railroad demand for the cleaning fluid for the eleven locations is...

...Transportation Problem and Solution in Case of Bangladesh
An adequate and efficient transport system is a pre-requisite for both initiating and sustaining economic development. Investment in improving transport efficiency is the key to expansion and integration of markets - sub-national, national and international. It also helps the generation of economies of scale, increased competition, reduced cost, systematic urbanization, export-led faster growth and a larger share of international trade.
The transport system of Bangladesh consists of roads, railways, inland waterways, two sea ports, maritime shipping and civil aviation catering for both domestic and international traffic. Presently there are about 21,000 km of paved roads; 2,706 route-kilometers of railways (BG-884 km and MG -1,822 km); 3,800 km of perennial waterways which increase to 6,000 km during the monsoon, 2 seaports and 2 international (Dhaka and Chittagong) and 8 domestic airports.
In Bangladesh, development and maintenance of transport infrastructure is essentially the responsibilities of the public sector as are the provision of railways transportation services and air transport. The public sector is involved in transport operations in road, inland water transport (IWT) and ocean shipping alongside the private sector. In the road transport and IWT sub-sectors, the private sector is dominant. In ocean shipping, however, public sector still...

...Operational Research (OR)
DR. SUPRIYA KUMAR DE
ASSIGNMENT
XLRI-PGCBM-18
NAME: GH. RASOOL WANI
SMS ID: 2217429
Table of Contents
1. Problem: 1
2. Solution: 2
2.1 Manual Approach 2
2.2 Linear programming approach 2
2.2.1 Decision Variables: 3
2.2.2 Objective Function: 3
2.2.3 Constraints: 3
2.3 Excel Solution. 6
2.3.1 Excel Solution Embedded: 6
3. Analysis: 6
3.1 Sensitivity Analysis: Objective Function 6
3.2 Sensitivity Analysis: Right Hand Side of the constraints 7
Problem:
Mr. Ramesh Chandra is a Software Development Project Manager in one of the renowned Indian Software Services Company, namely “ABC Technologies”. The company’s business division has recently won a project from a major European telecom company. This Project is very important for the “ABC Technologies” as it is first time the company is entering in the telecom sector for the software services.
The “ABC Technologies” has been given whole end to end responsibility for this software package from Requirement Analysis till final Deployment.
Customer has communicated recently to the business director of “ABC Technologies” that they would like to see the final Project Plan with a Breakdown of the tasks, Total Cost of the Project and Total duration of the Project by end of next week.
“ABC Technologies” management has decided to assign Ramesh...

... Case 13: Southeastern Specialty, Inc.
Financial Risk (1, 2, 3, 4, & 6)
1. Is the return on the one-year T-bill risk free?
No, the return on the one-year T-bill is not risk free. Financial risk is related to the probability of earning a return less than expected and the larger the chance of earning a return far below that expected, the greater the amount of financial risk. Risk free assumes 100% probability that the investment will earn the total percent of return that is expected.
2. Calculate the expected rate of return on each of the five investment alternatives listed in Exhibit 13.1. Based solely on expected returns, which of the potential investments appear best?
Based on the expected returns, the potential investment that appears the best is 15% with S & P 500 Fund.
(Probability of Return 1 x Return 1) + (Probability of Return 2 x Rate 2) = Expected Rate of Return
1-Year T-Bill
(0.10 x .07) + (0.20 x .07) + (0.40 x .07) + (0.20 x .07) + (0.10 x .07) = .07 = 7%
Project A
(0.10 x [-.08]) + (0.20 x .02) + (0.40 x .14) + (0.20 x .25) + (0.10 x .33) = .135 = 13.5%
Project B
(0.10 x .18) + (0.20 x .23) + (0.40 x .07) + (0.20 x [-.03]) + (0.10 x .02) = .088 = 8.8%
S & P 500 Fund
(0.10 x [-.15]) + (0.20 x 0) + (0.40 x .15) + (0.20 x .30) + (0.10 x .45) = .15 = 15%
Equity in SSI
(0.10 x 0) + (0.20 x .05) + (0.40 x .10) + (0.20 x .15) + (0.10 x .20) = .10 = 10%
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