The management of the publishing company Book & Co\' wants to design its product
ID: 360673 • Letter: T
Question
The management of the publishing company Book & Co' wants to design its production schedule for the next quarter, where three books A, B and C can be published. Book A can only be published as a paperback, books B and C can also be published in a more fancy form with a hard cover. We label book A by the number 1, book B as paperback by 2 and as hard cover by 4, book C as paperback by 3 and as hard cover by 5. The profits (in S units) per book are listed in the second column of the table. Book Profit(in S) Paper Processing time Labor time P.cover H.cover 10 15 10 30 25 12 15 15 18 50 50 100 150 200 80 80 120 160 160 The objective is the maximization of the total profit. There are the fol- lowing constraints. The inventory of paper is limited to 15000 sheets of paper. In the table, the number of sheets needed for the different types of books is listed in the third column. Book & Co has only one machine, and all books are processed on this machine. The machine processing times for the various types of books are given in the fourth column of the table. During the next quarter, the machine capacity (the total time that the machine is available), is 80000 time units. The number of employees for finishing the books is limited. During the next quarter, there are 120,000 time units available for finishing the books. The finishing time per book is given in the fifth column of the table. Finally, each type of books needs a certain number of covers, while the total amount of covers for paperbacks and hard covers is limited to 1800 and 400 covers, respectively. The amounts of covers required per book are given in the last two columns of the table. Since the sale of hard cover books is expected to go more slowly than the sale of paperbacks, Book & Co produces less hard cover books than paperbacks in the same time period. The target rule is that the number of paperbacks for each different book must be at least 5 times the number of hard cover books. a. Write this problem as an LP-model, and calculate the maximum amount of profit and the optimum number of books to be produced.Explanation / Answer
Let X1, X2, X3, X4, X5 be the number of books types 1,2,3,4,5 respectively to be produced.
Therefore Objective is to have maximum profit is Z=10X1+15X2+10X3+30X4+25X5
Subject to the given constraints as follows:
Solution is as follows:
Detailed calculations are as follows:
X1 X2 X3 X4 X5 RHS Maximize 10 15 10 30 25 Max 10X1 + 15X2 + 10X3 + 30X4 + 25X5 Paper 9 12 15 15 18 <= 15,000 9X1 + 12X2 + 15X3 + 15X4 + 18X5 <= 15000 Machine time 50 50 100 150 200 <= 80,000 50X1 + 50X2 + 100X3 + 150X4 + 200X5 <= 80000 Labor time 80 80 120 160 160 <= 1,20,000 80X1 + 80X2 + 120X3 + 160X4 + 160X5 <= 120000 Pcover 1 2 1 0 0 <= 1,800 X1 + 2X2 + X3 <= 1800 Hcover 0 0 0 2 1 <= 400 2X4 + X5 <= 400Related Questions
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