Problem: Explicitly state the following:

1. Decision Variables: Express the decision variable in mathematical notation, and state what each decision variable stands for in words.
2. Constraints: Write each constraint in mathematical notation, and state what it stands for in words.
3. Objective function: Write the objective in mathematical notation, and state what it stands for in words.
4. Following steps (1) – (3), write the entire model as a linear programming (LP) formulation.
5. Solve the LP using GAMS, and determine the optimal values of: decision variables, objective function, and constraints.

A steel manufacturer produces four sizes of I-beams: small, medium, large, and extra large. These beams can be produced on any one of the three machine types: A, B, C. The length in feet of the I-beams that can be produced on the machines per hour is summarized.

Assume that each machine can be used up to 50 hours/week, and that the hourly operating costs of these machines are $30, $50, and $80, respectively. Further suppose that 10000, 8000, 6000, and 6000 feet of the different size I-beams are respectively required weekly. Formulate this machine scheduling problem as a linear program to minimize operating costs.

Problem: Explicitly state the following:

1. Decision Variables: Express the decision variable in mathematical notation, and state what each decision variable stands for in words.
2. Constraints: Write each constraint in mathematical notation, and state what it stands for in words.
3. Objective function: Write the objective in mathematical notation, and state what it stands for in words.
4. Following steps (1) – (3), write the entire model as a linear programming (LP) formulation.
5. Solve the LP using GAMS, and determine the optimal values of: decision variables, objective function, and constraints.

A trucking company with $2,000,000 to spend on new equipment is contemplating purchasing three types of vehicles: A, B, and C. Vehicle A has a 10-ton payload and is expected to cover 55 mi/hr, and costs $40,000. Vehicle B has a 20-ton payload and is expected to cover 50 mi/hr, and costs $65000. Vehicle C is a modified form of B; it carries sleeping quarters for one driver, and this reduces its capacity to 18-tons and raises the cost to $75000.

Vehicle A requires a crew of one man, and if driven on three shifts per day, could be run for an average of 18 hours/day. Vehicles B and C require a crew of two men each, but whereas B can be driven 18 hours/day with three shifts, C can be run an average of 21 hours/day. The company has 150 drivers available each day and would find it difficult to obtain further crews. Maintenance facilities are such that the total number of vehicles must not exceed 30.

How many vehicles of each type should be purchased if the company wishes to maximize its capacity in ton-mi per day?

Problem: Explicitly state the following:

1. Decision Variables: Express the decision variable in mathematical notation, and state what each decision variable stands for in words.
2. Constraints: Write each constraint in mathematical notation, and state what it stands for in words.
3. Objective function: Write the objective in mathematical notation, and state what it stands for in words.
4. Following steps (1) – (3), write the entire model as a linear programming (LP) formulation.
5. Solve the LP using GAMS, and determine the optimal values of: decision variables, objective function, and constraints.

The Lehigh Railroad Company will have four locomotives at IE junction, one locomotive at Centerville junction, and two locomotives at Wayover junction. Student trains, each requiring one locomotive, are at A-station, Fine Place, Goodville, and Somewhere Street. The local map gives the following distances.

FP G

Formulate this problem as a transportation problem , clearly identifying the supply and demand nodes, and determine the optimal allocation of locomotives to stations so that the total distance is minimized.

Problem: Explicitly state the following:

1. Decision Variables: Express the decision variable in mathematical notation, and state what each decision variable stands for in words.
2. Constraints: Write each constraint in mathematical notation, and state what it stands for in words.
3. Objective function: Write the objective in mathematical notation, and state what it stands for in words.
4. Following steps (1) – (3), write the entire model as a linear programming (LP) formulation.
5. Solve the LP using GAMS, and determine the optimal values of: decision variables, objective function, and constraints.

Larry Edison is the director of the Computer Center at Buckley College. He now needs to schedule the staffing of the center. The center is open from 8:00 a.m. until midnight. Based on previous usage of the center, Larry has determined that the following number of computer consultants are required to monitor the center at various times of the day:

Problem: Two types of computer consultants can be hired: full-time and part-time. Full-time consultants work for 8 consecutive hours in any of the following shifts: morning (8:00 a.m. – 4:00 p.m.), afternoon (Noon – 8:00 p.m.), or evening (4:00 p.m. – Midnight), and are paid $14/hour. Part-time consultants can be hires to work in any of the four-hour shifts listed in the above table, and they are paid $12/hour. An additional requirement is that there must be at least 2 full-time consultants on duty for every part-time consultant on duty. Determine the number of full-time and part-time consultants that must work each shift to meet the above demand at a minimum cost.

Problem: Consider the following data for a linear programming problem where the objective is to maximize profit from allocating three resources to two nonnegative activities.

1. Formulate a linear program for this problem
2. Use the graphical method to solve the problem
3. Verify your answer using GAMS

Problem: Consider the following linear program:

Minimize

z  x 1  x 2

subject to:

x 1  x 2  6

x 1  x 2  0

x 1  x 2  3

 x 1, x 2   0 .

Draw the feasible region, clearly indicating the axes, constraints, extreme points, and contours of the objective function. Graphically determine all the optimal solutions to the LP.

Problem: Consider the following LP:

(a) Write the dual to this LP.

(b) Convert the LP given above into standard form.

(c) Write the dual to the standard form of the LP derived in part (b).

(d) Convert the dual from part (c) into standard form.

Problem: Consider the following data for a transportation problem:

(a) Determine a starting basic feasible solution by the Northwest Corner rule. Give the corresponding graphical solution representation.

(b) Starting with this solution, find the optimum using the transportation simplex method.

Problem: Consider the following data for a transportation problem:

(a) Starting with the Northwest Corner rule, solve this problem using the transportation algorithm.

(b) Verify the answer in part (a) using GAMS.

(c) Suppose that c ₂₄ is replaced by 4. Without resolving the problem from the beginning, find the new optimal solution.

(d) How large can c ₁₂ be made before the optimality of the solution in part (a) is violated?

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