Download 132 Amarillo Fort Worth 84 minutes 90 138 348 66 90 156 48 120

Course: SCI2910 Logistics - Planning, Management, and Analysis
Examples
Example 1: (Decision making) The bicycle problem involves n people who have to travel a distance of
ten miles, and have one single-seat bicycle at their disposal. The data specificed by the walking speed wj
and the bicycling speed bj of each person j, j = 1, · · · , n; the task is to minimize the arrival time of the
last person. Formulate this problem as a solvable mathematical problem.
Example 2: (Decision making) The caterer problem. A caterer knows that, in connection with the meals
to be served during the next n days, rt ≥ 0 fresh napkins will be needed on the tth day, wiht t = 1, 2, · · · , n.
Laundering normally takes p days, i.e., a soiled napkin sent for laundering immediately after use on the tth
day is returned in time to be used again on the (t + p)th day. However, the laundry also has a higher-cost
service which returns the napkins in q < p days (p and q integers). Having no usable napkins on hand or
in the laundry, the caterer will meet early demands by purchasing napkins at a cents each. Laundering
expense is b and c cents per napkin for the normal slow and high-cost (fast) service, respectively. We
assume a > b and a > c > b. How does the caterer arrange matters to meet the needs and minimize the
outlays for the next n days? Formulate the problem as a mathematical problem.
Example 3: (Vehicle routing) Suppose that we have the problem shown in the following figure. We seek a
minimum time route between Amarillo, Texas, and Fort Worth, Texas. Each link has an associated driving
time between nodes, and the nodes are road junctions.
Amarillo
90
138
84 minutes
66
120
60
90
348
156
84
132
126
126
132
48
48
150
Fort Worth
A highway network between Amarillo and Fort Worth
Example 4: (Tranportation) The Energetic company needs to make plans for the energy systems for a
new building. The energy needs in the building fall into three categories: (1) electricity, (2) heating water,
and (3) heating space in the building. The daily requirement for these three categories (all measured in
the same units) are
Electricity
Water heating
Space heating
20 units
10 units
30 units
The three possible sources of energy to meet these needs are electricity, natural gas, and a solar heating
unit that can be installed on the roof. The size of the roof limits the largest possible solar heater to 30
units, but there is no limit to the electricity and natural gas available. Electricity needs can be met only
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by purchasing electricity (at a cost of $50 per unit). Both other energy needs can be met by any source or
combination of sources. The unit costs are
Water heating
Space heating
Electricity
$90
$80
Natural Gas
$60
$50
Solar Heater
$30
$40
The objective is to minimize the toal cost of meeting the energy needs.
a) Formulate this problem as a mathematical problem.
b) Find the optimal solution to the above problem.
Example 5: (Inventory) A television manufacturing company produces its own speakers, which are used
in the production of its television sets. The television sets are assembled on a continuous production line
at a rate of 8,000 per month, with one speaker needed per set. The speakers are produced in batches
because they do not warrant setting up a continuous production line, and relatively large quantities can
be produced in a short time. Therefore, the speakers are placed into inventory until they are needed for
assembly into television sets on the production line. The company is interested in determining when to
produce a batch of speakers and how many speakers to produce in each batch. Several costs must be
considered:
1) Each time a batch is produced, a setup cost of $12,000 is incurred. This cost includes the cost of
“tooling up” administrative costs, record keeping, and so forth. Note that the existence of this cost
argues for producing speakers in large batches.
2) The unit production cost of a single speaker (excluding the setup cost) is $10, independent of the
batch size produced. (In general, however, the unit production cost need not be constant and may
decrease with batch size.)
3) The production of speakers in large batches leads to a large inventory. The estimated holding cost
of keeping a speaker in stock is $0.30 per month. This cost includes the cost of captial tied up in
inventory. Since the money invested in inventory cannot be used in other productive ways, this cost
of captial consists of the lost return (referred to as the opportunity cost) because alternative uses of
the money must be forgone. Other components of the holding cost include the cost of leasing the
storage space, the cost of insurance against loss of inventory by fire, theft, or vandalism, taxes based
on the value of the inventory, and the cost of personnel who oversee and protect the inventory.
4) Company policy prohibits deliberately planning for shortage of any of its components. However, a
shortage of speakers occasionally crops up, and it has been estimated that cost includes the extra cost
of installing speakers after the television set is fully assembled otherwise, the interest lost because of
the delay in receiving sales revenue, the cost of extra record keeping, and so forth.
Example 6: (Inventory) The Li family drinks a case of Watson’s distilled water every day, 365 days a year.
Fortunately, a local distributor offers quantity discounts for large orders as in the table below, where the
price for each category applies to every case purchased. Considering the cost of gasoline, Mr. Li estimates
it costs him about $50 to go to pick up an order of Watson’s distilled water. Mr. Li also is an investor in
the stock market, where he has been earning a 20 percent average annual return. He considers the return
lost by the buying the Watson’s distilled water instead of stock to be the only holding cost for the Watson’s
distilled water.
a) Determine the optimal order quantity. What is the resulting total cost per year?
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Discount
Category
1
2
3
Quantity
Purchased
1 to 49
50 to 99
100 or more
Price
(per case)
$40
$39
$38
b) With this order quantity, how many orders need to be placed per year? What is the time interval
between orders?
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