Posted: September 13th, 2017

EMIS 8371 Homework Assignment 1

For the first two problems, please provide a) formulation of the basic problem (no extensions), b) formulations for specified extensions (i.e., for first problem, you

will have 4 models). For the last problem, there’s only the one formulation to be developed. Good luck & have fun. Individual work only please.

1.    A 24-hour supermarket has the following minimal requirements for cashiers:
Period     1    2    3    4    5    6
Time of Day (24-h Clock)    3-7    7-11    11-15    15-19    19-23    23-3
Minimum No.    7    20    14    20    10    5

Period 1 follows immediately after period 6. A cashier works eight consecutive hours, starting at the beginning of one of the six periods. The interest is to determine

a daily employee worksheet which satisfies the requirements with the least number of personnel.

EXTENSIONS:
i)    Suppose that the supermarket would like to be sure that there is a balance of experienced and less experienced cashiers on duty during each shift. How would

your model change?
ii)    Extend model in i) to additionally support the supermarket’s interest to broaden the model to include 10 different roles (i.e., stock clerks, bakers, deli,

etc.) How would your model need to be modified to accommodate this?
iii)    Go back to the basic model (without i) or ii) considerations) and suppose the supermarket additionally has daily minimum requirements for each shift (i.e.,

Monday 7-11 is likely lower than Saturday 7-11 shift). How you could incorporate this into your model?

2.    A plastics manufacturer has 1200 boxes of transparent wrap in stock in one factory and another 1000 boxes at its second factory. The manufacturer has orders

for this product from three different retailers, in quantities of 1000, 700, and 500 boxes, respectively. The unit shipping costs (in cents per box) from the factories

to the retailers are as follows:
Retailer 1    Retailer 2    Retailer 3
Factory 1    14    13    11
Factory 2    13    13    12

The interest is determine a minimum cost shipping schedule for satisfying all demands from current inventory.
EXTENSIONS:
i)    Suppose that there are 10 other types of products available from the factories to ship to retailers. How could you incorporate this into your model?
ii)    Provide two additional extensions that could you imagine arise with this type of problem in true applications and provide the formulations for each.

3.    Like the rest of Germany after her unconditional surrender in May of 1945, the city of Berlin was divided into an American, British, French and a Russian

sector. Berlin was geographically located in the “Soviet” zone and access to the western (American, British, and French) sectors of the city was assured by agreement

with the Soviets via air, rail and three highways from Hamburg, Hannover and Hof in Northern Bavaria, respectively, until 1990 when history took a different turn

again. When the three western sectors of the city of Berlin adopted in early 1948 the “new” currency that he just been created in the “tri-zone”, i.e., the American,

British and French zones of Germany, the Soviets – in the person of Joseph Vissarionovich Djugashvili, a.k.a. Joseph Stalin (1879-1953) – reacted with a total blockade

of road and rail access to the city of Berlin for all westerners, including American and British forces. A massive and costly “airlift” to Berlin was organized by the

American and British forces that reportedly transported over 2.3 million tons of food, fuel, clothing, etc. into West Berlin to prevent West-Berliners from starving

and freezing. The Berlin Airlift lasted 463 days or about 15 months –from June 1948 to September 1949 when the Soviets finally backed off. Clearly, like modern

warfare, such a massive effort requires a great deal of logistics planning on the part of the military.

The challenge is to create a very simplified linear programming model to aid military planners in such an effort while keeping the cost of such an operation “down”. In

view of the fact that computing power was completely insufficient to permit “cost-minimal” logistics planning by linear programming at the necessary level of detail in

1949, your Berlin Airlift model would have been labeled “hypothetical”.

Suppose we want to plan the logistics for the Berlin Airlift at a very “aggregate” level and that we plan operations for four consecutive “quarters”, e.g. four three-

month periods. For each quarter, we have forecasted cargo that must be airlifted to Berlin. One unit of cargo corresponds to 100,000 tons, say and in quarters 1, 2, 3

and 4, we need to airlift 2, 3, 3 and 4 units of cargo, respectively. We also have an initial number of 330 pilots – personnel that can operate planes or train new

pilots – and an initial number of 110 planes. In each quarter, we “loose” a certain number of pilots and airplanes, i.e., planes that go down in the Soviet zone before

reaching the tri-zone or that simply have to be repaired and are of no use for the rest of the planning period. Thus, procurement of new crew and aircraft much be

planned as well ahead of time. For simplicity, “lost” crew is equated to the corresponding lost aircraft. Finally, pilots or crew that have operated an aircraft during

one quarter are given leave the next quarter and are available after their rest period for a new round of duty. Assume that each unit of airlifted cargo requires 50

airplanes and that three “pilots” are necessary to operate one plane. Moreover, pilots that do not operate an aircraft are either idle or train new pilots – at a

training ratio of 1 to 20 – each trainer “produces” 20 new pilots including himself (or herself if the WASPS were involved) that are ready to operate aircraft in the

following quarter. We have a high “attrition” rate of 20% for crew and aircraft lost “forever” to service per quarter. Morale among the crew is high nevertheless and

all personnel that were given leave return to service after their rest period.

In terms of cost, tonnage to Berlin must be shipped at any cost; so no costs for moving it to Berlin needs to be considered. Naturally, aircraft and crews cost money.

New aircraft costs money (200 monetary units per plane) as does training crews (10 MUs). The unit costs for idle and resting are 7 and 5 MUs, respectively. Note that

these are approximate “numbers” used to aid the planning and are not for use in actual accounting of the cost of the operation.
The interest is to find a minimum-cost logistics schedule for the four quarters.  Provide a   formulation of the Berlin Airlift problem.

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