# Question

A large paper manufacturing company, the Quality Paper Corporation, has 10 paper mills from which it needs to supply 1,000 customers. It uses three alternative types of machines and four types of raw materials to make five different types of paper. Therefore, the company needs to develop a detailed production distribution plan on a monthly basis, with an objective of minimizing the total cost of producing and distributing the paper during the month. Specifically, it is necessary to determine jointly the amount of each type of paper to be made at each paper mill on each type of machine and the amount of each type of paper to be shipped from each paper mill to each customer.

The relevant data can be expressed symbolically as follows:

Djk = number of units of paper type k demanded by customer j,

rklm = number of units of raw material m needed to produce 1 unit of paper type k on machine type l,

Rim = number of units of raw material m available at paper mill i,

ckl = number of capacity units of machine type l that will produce 1 unit of paper type k,

Cil = number of capacity units of machine type l available at paper mill i,

Pikl = production cost for each unit of paper type k produced on machine type l at paper mill i,

Tijk = transportation cost for each unit of paper type k shipped from paper mill i to customer j.

(a) Using these symbols, formulate a linear programming model for this problem by hand.

(b) How many functional constraints and decision variables does this model have?

(c) Use MPL to formulate this problem.

(d) Use LINGO to formulate this problem.

The relevant data can be expressed symbolically as follows:

Djk = number of units of paper type k demanded by customer j,

rklm = number of units of raw material m needed to produce 1 unit of paper type k on machine type l,

Rim = number of units of raw material m available at paper mill i,

ckl = number of capacity units of machine type l that will produce 1 unit of paper type k,

Cil = number of capacity units of machine type l available at paper mill i,

Pikl = production cost for each unit of paper type k produced on machine type l at paper mill i,

Tijk = transportation cost for each unit of paper type k shipped from paper mill i to customer j.

(a) Using these symbols, formulate a linear programming model for this problem by hand.

(b) How many functional constraints and decision variables does this model have?

(c) Use MPL to formulate this problem.

(d) Use LINGO to formulate this problem.

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