1. You are working as an operations planner in a company which has two factories, one in Atlanta, Georgia and one in St. Jose, California. In addition to the factories your company has four depots with storage facilities in Miami, Baltimore, Denver and Seattle. The company sells its products to six companies C1, C2..., C6. Customers can be supplied either from a depot or from the factory directly as seen in the following figure:
The distribution costs are known and given in the Table (in $ per ton delivered), certain customers have expressed preferences for being supplied from factories or depots which they are used to. The preferred supplier(s) for each customer are given below
C1 Atlanta (factory)
C2 Miami (depot)
C3 No preferences
C4 No preferences.
C5 Baltimore (depot)
C6 Seattle or Denver
Each factory has a monthly capacity. Namely the capacities in Atlanta and St. Jose are 150,000 and 200,000 tons respectively. The maximum throughput for depots are 70000, 50000, 100000 and 40000 for Miami, Baltimore, Denver and Seattle respectively.
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Supplied to
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Atlanta
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St. Jose
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Miami
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Baltimore
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Denver
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Seattle
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Depots
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|
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|
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|
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Miami
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0.5
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-
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|
|
|
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Baltimore
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0.5
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0.3
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|
|
|
|
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Denver
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1.0
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0.5
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|
|
|
|
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Seattle
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0.2
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0.2
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|
|
|
|
|
|
|
|
|
|
|
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Customers
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Cl
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1.0
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2.0
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-
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1.0
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-
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-
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C2
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-
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-
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1.5
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0.5
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1.5
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-
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C3
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1.5
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-
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0.5
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0.5
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2.0
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0.2
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|
C4
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2.0
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-
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1.5
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1.0
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-
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1.5
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C5
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-
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-
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-
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0.5
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0.5
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0.5
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C6
|
1.0
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-
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1.0
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-
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1.5
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1.5
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(A dash indicates the impossibility of certain suppliers for certain depots or customers)
Each customer has a monthly requirement given below which must be met
C1 50,000 tons
C2 10,000 tons
C3 40,000 tons
C4 35,000 tons
C5 60,000 tons
C6 20,000 tons
Your boss would like you to construct a model that will find an optimal distribution pattern that minimizes overall cost.
First ignore the customer preferences. Suppose that you are free to use all "feasible" channels to ship the products to the customers. Answer the following questions:
a) Write down the mathematical model as an LP
b) Solve the problem using MS Excel Solver (submit only hard copies of the model and the solution sheet)
c) Solve the problem with AMPL (submit hardcopies of the model, data and output files). You should use compound sets in your AMPL model.
d) Would it be possible to meet all customer preferences regarding suppliers and if so what would be the extra cost of doing this? If it is not possible to satisfy all preferences what does the company need to do to make it possible?
e) Suppose the customer preferences are not hard constraints. That is, a customer may prefer being supplied from certain origin(s) even though she will still accept shipments from other origins. Giving at most importance to customer satisfaction, your company employs a policy that attempts to supply customers based on their preferences first, before utilizing any other (un-preferred) origins. Find the optimal distribution plan (algorithm) that minimizes the operational cost while maximizing the customer preferences. Add and/or assume parameters and constraints that you think necessary (Be reasonable and creative!). What is the optimal cost and how is it different than the one found in the original problem?
f) Now treat preferences as requirements. That is, a customer will never accept a shipment from an origin that is not included in her preferences list. Assuming that the capacity at Baltimore Depot has been increased to 60,000 units find the optimal distribution plan that minimizes the operational cost. Compare this result to the case where there is no customer preferences (as in part a but now with new Baltimore capacity).
You are expected to conduct your analysis using the features of AMPL whenever they are useful except for parts a and b.
2. Write down the mathematical model for Problem-
City 1 produces 500 tons of waste per day, and city 2 produces 400 tons of waste per day. Waste must be incinerated at incinerator 1 or 2, and each incinerator can process up to 500 tons of waste per day. The cost to incinerate waste is $40/ton at incinerator 1 and $30/ton at 2. Incineration reduces each ton of waste to 0.2 tons of debris, which must be dumped at one of two landfills. Each landfill can receive at most 200 tons of debris per day. It costs $3 per mile to transport a ton of material. Distances between locations are shown in the table. Formulate an LP that can be used to minimize the total cost of disposing of the waste of both cities.
3. Use display and indexing expressions to determine the membership of the following sets:
-Origin-destination links that have transportation cost less than $10 per ton.
-Destinations that can be served by GARY.
-Origins that can serve FRE.
-Links that are used for transportation in the optimal solution.
-Links that are used for transportation front CLEV in the optimal solution.
-Destinations to which the total cost of shipping, from all origins, exceeds $20,000.