The main issue is planning the company’s production capacity for the coming year. At issue is the overall level of capacity and the type of capacity?for example, the degree of flexibility in the manufacturing system. The main tool used to aid the company’s planning process is a mixed integer linear programming (MILL) model. A mixed integer program has both integer and continuous variables. Problem Statement The Giant Motor Company (GUM) produces three lines of cars for the domestic (U. S. ) market: Lyres, Libra, and Hydras.
The Lyre is a relatively inexpensive subcompact car that appeals mainly to first-time car owners and to households using it as a second car for commuting. The Libra is a sporty compact car that is sleeker, faster, and roomier than the Lyre. Without any options, the Libra costs slightly more than the Lyre; additional options increase the price. The Hydra is the luxury car of the GUM line. It is significantly more expensive than the Lyre and Libra, and it has the highest profit margin of the three cars.
Retooling Options for Capacity Expansion Currently GUM has three manufacturing plants in the United States. Each plant is dedicated to producing a single line of cars. In its planning for the coming year, GUM is considering the retooling of its Lyre and/or Libra plants. Retooling either plant would represent a major expense for the company. The retooled plants would have significantly increased production capacities. Although having greater fixed costs, the retooled plants would be more efficient and have lower marginal production costs?that is, higher marginal profit contributions.
In addition, the retooled plants would be flexible?they would have the capability of producing more than one line of cars. A dash in the profit margin section indicates that the plant cannot manufacture that line of car. For example, the new Lyre plant would be capable of producing both Lyres and Libra but not Hydras. The new Libra plant would be capable of producing any of the three lines of cars.
Note, however, that the new Libra plant has a slightly lower profit margin for producing Hydras than the Hydra plant. The flexible new Libra plant is capable of producing the luxury Hydra model but is not as efficient as the current Hydra plant that is dedicated to Hydra production. The fixed costs are annual costs incurred by GUM, independent of the number of cars produced by the plant. For the current plant configurations, the fixed costs include property taxes, insurance, payments on the loan that was taken out to construct the plant, and so on.
Partially offsetting the lack of capacity is the phenomenon of demand diversion. If a potential car buyer walks into a GUM dealer showroom wanting to buy a Lyre but the dealer is out of stock, frequently the salesperson can convince the customer to purchase the better Libra car, which is in stock. Unsatisfied demand for the Lyre is said to be diverted to the Libra.
Only rarely in this situation can the salesperson convince the customer to switch to the luxury Hydra model. Table 2. 2 Demand for GUM Cars Demand (in sass) 1400 1100 From past experience, GUM estimates that 30% of unsatisfied demand for Lyres is diverted to demand for Libra and 5% to demand for Hydras. Similarly, 10% of unsatisfied demand for Libra is diverted to demand for Hydras. For example, if the demand for Lyres is cars, then the unsatisfied demand will be 400,000 if no capacity is added.
Out of this unsatisfied demand, 120,000 (= 400,000 * 0. ) will materialize as demand for Libra, and 20,000 (= 400,000 * 0. 05) will materialize as demand for Hydras. Similarly, if the demand for Libra is 1 , 220,000 cars (1, 100,000 original demand plus 120,000 demand diverted from Lyres), then the unsatisfied demand for Lyres would be 420,000 if no capacity is added. Out of this unsatisfied demand, 42,000 (= 420,000 * 0. 1) will materialize as demand for Hydras. All other unsatisfied demand is lost to competitors.
GUM wants to decide whether to retool the Lyre and Libra plants. In addition, GUM wants to determine its production plan at each plant in the coming year. Based on the previous data, formulate a MILL model for solving Gem’s reduction planning-capacity expansion problem for the coming year. CASE TWO Ski Jacket Production Egress, Inc. , is a small company that designs, produces, and sells ski jackets and other coats. The creative design team has labored for weeks over its new design for the coming winter season.
It is now time to decide how many ski jackets to produce in this production run. Because of the lead times involved, no other production runs will be possible during the season. Predicting ski jacket sales months in advance of the selling season can be tricky. Egress has been in operation for only 3 years, and its ski jacket designs were quite successful in 2 of those years. Based on realized sales from the last 3 years, current economic conditions, and professional judgment, 12 Egress employees have independently estimated demand for their new design for the upcoming season.
Arrivals at the Credit Union The Indiana University Credit Union Eastland Plaza branch was having trouble getting the correct staffing levels to match customer arrival patterns. On some days, the number of tellers was too high relative to the customer traffic, so that tellers were often idle. On other days, the opposite occurred; long customer waiting lines formed because the relatively few tellers could not keep up with the number of customers. The credit union manager, James Clinton, knew that there as a problem, but he had little of the quantitative training he believed would be necessary to find a better staffing solution.
James figured that the problem could be broken down into three parts. First, he needed a reliable forecast of each days number of customer arrivals. Second, he needed to translate these forecasts into staffing levels that would make an adequate trade-off between teller idleness and customer waiting. Third, he needed to translate these staffing levels into individual teller work assignments?who should come to work when. The last two parts of the problem require analysis tools (queuing and scheduling) that we will not pursue here.
However, you can help James with the first part? forecasting. The file Credit Union Arrivals. XSL (the file is uploaded on DEL) lists the number of customers entering this credit union branch each day of the past year. It also lists other information: the day of the week, whether the day was a staff or faculty payday, and whether the day was the day before or after a holiday. Use this data set to develop one or more forecasting models that James could use to help solve his problem. Based on your model(s), make any recommendations about staffing that appear reasonable.
