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# Operations research

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Consider the following problem:
A supermarket store manager needs to determine how much to stock of two brands of soda: PC and CC, for Super Bowl Sunday. The store needs to tell the suppliers how many "units" are available and the suppliers then stock the store with their product. Units consist of different flavors and sizes depending on what the suppliers think will sell best. Fractions of units are OK. The store's makes a profit margin of 15 cents for each unit of PC and 10 cents for each unit of CC. In the soda aisle, there is a maximum of 36 linear feet of shelf space: each unit of PC takes up 9 linear feet of shelf space and each unit of CC takes up 4 linear feet of shelf space. In the end displays, there is a 20 linear feet of display space available: each unit of PC takes up 4 linear feet of display space and each unit of CC takes up 4 linear feet of display space. In other words, 1 unit of PC will require 9 feet in the soda aisle plus 4 feet in the end display area.

Problem 1
Set up the problem. What is the objective function? What are the decision variables? What are the constraints?
Problem 2
Solve the problem graphically. Identify all corner point (basic feasible) solutions.

Problem 3
Solve the problem again, this time using the Simplex Method. What is the optimal solution(s)?
In addition to the display area constraints, the manager must meet certain sales quotas from his district manager. The store manager gets 1 point for each unit of PC sold and 4 points for each unit of CC - he must meet a quota of 14 points.

What the district manager doesn't know is that the store manager has a side deal going with the supplier of PC. If the store sells at least half as much PC as CC, the store manager gets tickets to next year's Super Bowl.

Given these two additional constraints, how many units of each brand should the store manager stock?

Problem 4
Solve the problem with these additional two constraints, using either the LINDO or LINGO software. Using the LINDO or LINGO output (show the output) only and without solving the problem again, answer the following two questions.

https://brainmass.com/math/linear-programming/linear-programming-operations-research-358903

#### Solution Preview

Please refer to attached file for the complete and formatted solution.

Consider the following problem:
A supermarket store manager needs to determine how much to stock of two brands of soda: PC and CC, for Super Bowl Sunday. The store needs to tell the suppliers how many "units" are available and the suppliers then stock the store with their product. Units consist of different flavors and sizes depending on what the suppliers think will sell best. Fractions of units are OK. The store's makes a profit margin of 15 cents for each unit of PC and 10 cents for each unit of CC. In the soda aisle, there is a maximum of 36 linear feet of shelf space: each unit of PC takes up 9 linear feet of shelf space and each unit of CC takes up 4 linear feet of shelf space. In the end displays, there is a 20 linear feet of display space available: each unit of PC takes up 4 linear feet of display space and each unit of CC takes up 4 linear feet of display space. In other words, 1 unit of PC will require 9 feet in the soda aisle plus 4 feet in the end display area.

Problem 1
Set up the problem. What is the objective function? What are the decision variables? What are the constraints?
⇒ Problem statement: The supermarket needs to decide how much PC and CC soda to stock in order to maximize total profit by selling the two brands. Profit to be maximized considering soda aisle and end display area availability.
⇒ Decision variables: Let P be the units of PC soda to be stocked and C be the units of CC soda to be stocked
⇒ Objective: Maximize profit (M) where M = Profit margin per unit of PC soda*Units of PC soda stocked + Profit margin per unit of CC soda*Units of CC soda stocked
M = 0.15*P + 0.10*C
⇒ Constraints: First constraints is on the soda aisle space where total units of PC and CC soda units stocked should not consume more than 36 linear feet of shelf space
Total soda aisle space consumed is equal to space per unit of PC* number of PC units stocked plus space per unit of CC* number of CC units stocked. Therefore, constraints is 9*P + 4*C <= 36 (1)
Second constraints is on end display space and the constraint can be written as
4*P + 4*C<=20 (2)
As fractional units are acceptable no need to put integer constraints.
Non-negativity constraints that P and C >= 0 (mathematically not required in profit maximization)

Problem 2
Solve the problem graphically. Identify all corner point (basic feasible) solutions.
Following are the steps for solving this problem graphically.
⇒ Step 1: Plot the constraints on two dimensional graph. P (number of PC units) is on x-axis and C (number of CC units on y-axis. The constraints are plotted by finding intersection points at which constraint lines cross x-axis and y-axis.
Constraint (1) is set to equality and then each decision variable is set to 0 sequentially.
Constraint equation becomes 9*P + 4*C = 36
o Set P = 0 to get C = 9
o Set C= 0 to get P = 4
o We have two points (0,9) and (4,0). Connect these two points ...

#### Solution Summary

World document shows objective function, decision variables, the constraints, graphs and optimal solutions using lindo or lingo software.

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