Question: 1. Catering Logistics: Jessica's Catering Service bakes and delivers lasagnas to parties and group meetings. In a typical week, Jessica has around 40 orders. Each order involves a specific amount of the required lasagna in pounds. A small group would need about 20 pounds, whereas a large group would need almost triple that amount. The following table shows the different customer orders that have come in this week, grouped into eight categories by weight.
|
Job type
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
|
Weight (pounds)
|
20
|
25
|
30
|
35
|
40
|
45
|
50
|
55
|
|
Number
|
10
|
5
|
4
|
7
|
3
|
9
|
2
|
1
|
(a) Jessica has an inventory of 100 six-pound trays and 80 ten-pound trays. Although she appears to have enough capacity in her trays, she would like to plan her orders so that the amount of excess lasagna is kept to a minimum.
(b) How many six-pound trays and ten-pound trays should Jessica use for each of the orders?
2. Routing Flights: Coastal Air offers discount fares to travelers in the East, including service to its main base in New York City (LGA). The airline serves six cities and wants its aircraft to land at each city once each day, returning to its starting point. The task is to find a route with the minimal total flying time. The flying time between city pairs in minutes is given in the table below. (Times are asymmetric due to different take-off and landing patterns, as well as different traffic conditions at the various airports.)
|
|
|
|
|
To
|
City
|
|
|
|
|
|
GSO
|
CLT
|
JAX
|
RDU
|
BWI
|
LGA
|
|
|
GSO
|
0
|
45
|
114
|
42
|
81
|
114
|
|
|
CLT
|
38
|
0
|
88
|
53
|
93
|
145
|
|
From
|
JAX
|
109
|
102
|
0
|
107
|
176
|
187
|
|
City
|
RDU
|
44
|
51
|
111
|
0
|
82
|
109
|
|
|
BWI
|
86
|
100
|
160
|
80
|
0
|
57
|
|
|
LGA
|
130
|
128
|
204
|
118
|
67
|
0
|
(a) What is the minimum flying time of a route that starts and ends in New York (LGA) and visits all of the other cities once?
(b) What is the optimal route corresponding to the flying time in (a)?
3. Taking Discounts:Universal Technologies, Inc. has identified two qualified vendors with the capability to supply some of its electronic components. For the coming year, Universal has estimated its volume requirements for these components and obtained price-break schedules from each vendor. (These are summarized as "all-units" discounts in the table below.) Universal's engineers have also estimated each vendor's maximum capacity for producing these components, based on available information about equipment in use and labor policies in effect. Finally, because of its limited history with Vendor A, Universal has adopted a policy that permits no more than 60% of its total unit purchases on these components to come from Vendor A.
|
|
|
Vendor A
|
Vendor B
|
|
Product
|
Requirement
|
Unit price
|
Volume required
|
Unit price
|
Volume required
|
|
1
|
500
|
$225
|
0-250
|
$224
|
0-300
|
|
|
|
|
$220
|
250-500
|
$214
|
300-500
|
|
|
2
|
1000
|
$124
|
0-600
|
$120
|
0-1000
|
|
|
|
|
$115
|
600-1000
|
|
(no discount)
|
|
|
3
|
2500
|
$60
|
0-1000
|
$54
|
0-1500
|
|
|
|
|
$56*
|
1000-2000
|
$52
|
1500-2500
|
|
|
|
|
$51
|
2000-2500
|
|
|
|
|
Total capacity (units)
|
|
2500
|
|
2000
|
*For example, if 1400 units are purchased from Vendor A, they cost $56 each, for a total of $78,400.
a) What is the minimum-cost purchase plan for Universal?
b) Suppose that Vendor A provides a new price-discount schedule for component 3. This one is an "incremental" discount, as opposed to an "all-units" discount, as follows.
Unit price = $60 on all units up to 1000
Unit price = $56 on the next 1000 units
Unit price = $51 on the next 500 units
With the change in pricing at Vendor A, what is the minimum purchasing cost for Universal?
4. Coffee Blending and Sales: Hill-O-Beans Coffee Company blends four component beans into three final blends of coffee: One is sold to luxury hotels, another to restaurants, and the third to supermarkets for store-label brands. The company has four reliable bean supplies: Argentine Abundo, Peruvian Colmado, Brazilian Maximo, and Chilean Saboro. The table below summarizes the very precise recipes for the final coffee blends, the cost and availability information for the four components, and the wholesale price per pound of the final blends. The percentages indicate the fraction of each component to be used in each blend.
|
Component
|
Hotel
|
Restaurant
|
Market
|
Cost per pound
|
Pounds
available
|
|
Abundo
|
20%
|
35%
|
10%
|
$0.60
|
40,000
|
|
Colmado
|
40%
|
15%
|
35%
|
$0.80
|
25,000
|
|
Maximo
|
15%
|
20%
|
40%
|
$0.55
|
20,000
|
|
Saboro
|
25%
|
30%
|
15%
|
$0.70
|
45,000
|
|
Wholesale price
|
|
|
|
|
|
|
per pound
|
$1.25
|
$1.50
|
$1.40
|
|
|
The processor's plant can handle no more than 100,000 pounds per week, and Hill-O-Beans would like to operate at capacity. There is no problem in selling the final blends, although the Marketing Department requires minimum production levels of 10,000, 25,000, and 30,000 pounds, respectively, for the hotel, restaurant, and market blends.
a) In order to maximize weekly profit, how many pounds of each component should be purchased?
b) What is the shadow price (from the Sensitivity Report) on the availability of Maximo beans?
c) How much (per pound) should Hill-O-Beans be willing to pay for additional pounds of Maximo beans in order to raise total profit?