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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 1 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? 2 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? 3