Question

Bloomington Publishers is considering publishing five different textbooks. The maximum number of copies of each textbook that can be sold, the variable cost of producing each textbook, the sales price of each textbook, and the fixed cost of a production run for each textbook are given in the file Prob3. For example, producing and selling 2000 copies of book 1 yields a revenue of $80(2000) = $160,000 but costs $80,000 + $44(2000) = $168,000. This company can produce at most 20,000 copies in total. Furthermore, it can publish no more than three different types of textbooks. Also, it knows that it cannot publish book 1 if it chooses to publish book 2. Finally, if this company publishes book 4 it must also publish book 5. Bloomington Publishers wants to find a production plan that maximizes total profit. Formulate and solve an integer programming model in Prob3 to help this publisher identify the best production plan.

Problem 3
Monetary data on types of books
	Book 1	Book 2	Book 3	Book 4	Book 5
Fixed cost	$80,000	$60,000	$100,000	$120,000	$160,000
Variable cost	$44	$36	$40	$30	$50
Selling price	$80	$64	$80	$76	$100
Maximum demand	6000	8000	8000	6000	10000
Production plan
	Book 1	Book 2	Book 3	Book 4	Book 5
						Total	Maximum Total Production (in copies)
Produced (in 1000s)							20000
Effective Demand (Logical upper bounds)
(a) No more than three different books can be published.
	Number published	Max number
(b) If Book 4 is published, then Book 5 must be published.
	Book 4	Book 5
(c) If Book 2 is published, then Book 1 cannot be published.
	Book 2	Book 1	Sum	Max sum
Summary of costs, revenue (all in $)
Fixed cost
Variable cost
Revenue
Profit
PLEASE show all formulas and solutions including solver, thank you!

Answer 1

Let indicate if the book i is going to be published, and 0 if not

Let indicate the number of copies of book i which will be produced

The above are the decision variables

Now the profit, (fixed cost only if we produce the book)

• Book 1: Profit from producing quantities is
  • ​
• Book 2: Profit from producing quantities is
  • ​
• Book 3: Profit from producing quantities is
  • 
• Book 4: Profit from producing quantities is
  • 
• Book 5: Profit from producing quantities is
  • 

The total profit is

The objective is to maximize the total profit hence the above is the objective function

Now the constraints

company can produce at most 20,000 copies in total.

• ​

Quantity produced cannot exceed the maximum demand for each book

it can publish no more than three different types of textbooks

• ​

it cannot publish book 1 if it chooses to publish book 2

• 

if this company publishes book 4 it must also publish book 5

• 

Now the LP model in full

Maximize

s.t

• ​

• ​
• 
• 
• 

The spreadsheet updated with formula

Get the following

Set up the solver using data--->solver

get the following

The maximizing the profit, the 3 books to be published are

book 1: 4000

book 4: 6000

book 5: 10,000