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1. Use the Simplex Method to solve the following LP:
max 𝑍 = 𝑥1 + 2𝑥2 + 4𝑥3
3𝑥1 + 𝑥2 + 5𝑥3 ≤ 10
𝑥1 + 4𝑥2 + 𝑥3 ≤ 8
+ 2𝑥3 ≤ 7
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
2. Solve the following LP using the Two-phase method:
max 𝑍 = 2𝑥1 + 3𝑥2
𝑥1 + 2𝑥2 ≤ 4
𝑥1 + 𝑥2 = 3
𝑥1 , 𝑥2 ≥ 0
3. Solve the following LP using the Two-phase method:
max 𝑍 = 3𝑥1 + 2𝑥2 + 4𝑥3
2𝑥1 + 𝑥2 + 3𝑥3 = 60
3𝑥1 + 3𝑥2 + 5𝑥3 ≥ 120
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
4. The Philbrick Company has two plants on opposite sides of the United States. Each
of these plants produces the same two products and then sells them to wholesalers
within its half of the country. The orders from wholesalers have already been received
for the next 2 months (February and March), where the number of units requested are
shown below. (The company is not obligated to completely fill these orders but will do
so if it can without decreasing its profits.)
Each plant has 20 production days available in February and 23 production days
available in March to produce and ship these products. Inventories are depleted at the
end of January, but each plant has enough inventory capacity to hold 1,000 units total
of the two products if an excess amount is produced in February for sale in March. In
either plant, the cost of holding inventory in this way is $3 per unit of product 1 and $4
per unit of product 2.
Each plant has the same two production processes, each of which can be used to
produce either of the two products. The production cost per unit produced of each
product is shown below for each process in each plant.
The production rate for each product (number of units produced per day devoted to that
product) also is given for each process in each plant below.
The net sales revenue (selling price minus normal shipping costs) the company receives
when a plant sells the products to its own customers (the wholesalers in its half of the
country) is $83 per unit of product 1 and $112 per unit of product 2. However, it also
is possible (and occasionally desirable) for a plant to make a shipment to the other half
of the country to help fill the sales of the other plant. When this happens, an extra
shipping cost of $9 per unit of product 1 and $7 per unit of product 2 is incurred.
Management now needs to determine how much of each product should be produced
by each production process in each plant during each month, as well as how much each
plant should sell of each product in each month and how much each plant should ship
of each product in each month to the other plant’s customers. The objective is to
determine which feasible plan would maximize the total profit (total net sales revenue
minus the sum of the production costs, inventory costs, and extra shipping costs).
a. Formulate a complete linear programming model in algebraic form that shows
the individual constraints and decision variables for this problem.
b. Formulate this same model on an Excel spreadsheet instead. Then use the Excel
Solver to solve the model.
c. Formulate this same model in python then use cplex solver to solve it.
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