# Manufacturing and Resource Allocation

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1 MA 2071 A 05 Bill Farr September 2, 2005

2 1 2

3 Outline Many industrial problems in manufacturing and resource allocation can be expressed as linear systems. This presentation provides an introduction to a simple class of these problems.

4 Outline Your company produces m products P 1, P 2,..., P m in n manufacturing facilities, F 1, F 2,..., F n. For simplicity, assume that the input to each facility consists of a single raw material. Given the demand or production targets for each of your products, determine the allocation of your raw material among the manufacturing facilities that will produce the desired output.

5 Production Vectors Outline Definition For each manufacturing facility F i, the vector v i is the vector in R m whose j th entry is the amount of product P j it can produce per unit of raw material.

6 Production Vectors Outline Definition For each manufacturing facility F i, the vector v i is the vector in R m whose j th entry is the amount of product P j it can produce per unit of raw material. Example Suppose that a particular facility produces 100 widgets, 300 spindles, and 533 brackets per unit of input. The production vector would be as follows

7 Target or Demand Vector Definition The Target or Demand vector, b, is the vector in R m whose j th entry is the production target or demand for the j th product.

8 Target or Demand Vector Definition The Target or Demand vector, b, is the vector in R m whose j th entry is the production target or demand for the j th product. Example Suppose that you wanted to produce 5000 widgets, 2000 spindles, and 7299 brackets. The demand vector would be

9 Resource Allocation Vector Definition The Resource Allocation Vector, x, is the vector in R n whose j th entry is the number of units of raw material to be allocated to the j th manufacturing facility.

10 Resource Allocation Vector Definition The Resource Allocation Vector, x, is the vector in R n whose j th entry is the number of units of raw material to be allocated to the j th manufacturing facility. Example Suppose that you have four manufacturing facilities. The vector below shows an example resource allocation vector. It says that facility F 1 requires 205 units of raw material, F 2 requires 45 units of raw material, and so on

11 Output Vector Outline Definition The Output Vector from facility F i is the vector whose j th entry is the amount of product P j produced from a given resource vector x. Since x i is the number of units of input to facility F i and v i is the production vector for F i, the output vector from facility F i is just x i v i

12 Linear System - Vector form To get the linear system, you just add up the outputs from each facility and set the sum equal to the demand vector. This gives the following. x 1 v 1 + x 2 v x n v n = b

13 Linear System - Matrix form If we define A to be the matrix whose columns are the production vectors v 1, v 2,..., v n then we have the linear system given below. Ax = b

14 Outline Example A small petroleum refining company has three refineries. Each refinery starts with crude oil and produces three products: motor oil, diesel fuel, and gasoline. A waste product, called parrafin, is also produced. Given the demand for each of the products, you want to determine the allocation of crude oil to each refinery.

15 Production Data Outline Example The numbers in the table are gallons of product or waste produced from each barrel of crude oil. (A barrel is a unit of measurement, equal to 42 gallons, used mainly for historical reasons.) Product Refinery 1 Refinery 2 Refinery 3 motor oil diesel fuel gasoline paraffin 3 5 2

16 Demand for Products Example Suppose that the current daily demand from your distributor for products is as follows gallons of motor oil 3100 gallons of diesel fuel 5100 gallons of gasoline.

17 Linear System Outline Example Using the entries from each column of the table for the three products and the demand numbers from the previous slide, you get the following linear system in augmented matrix form

18 RREF and Solution Example To solve for the crude oil inputs for each refinery, obtain the RREF for the linear system on the previous slide The entries in the last column are the crude oil allocations for the three refineries.

19 Problems for Further Investigation 1 Suppose that the demand for each product doubled simultaneously. How would your answer change? Explain your result mathematically. 2 Now suppose that another distributor has come forward and says that it would require 750 gallons of motor oil, 2000 gallons of diesel and 2000 gallons of gasoline per day. How would you set up production to satisfy this distributor only? That is, suppose this is the only distributor to which the company is selling. Is there only one way of doing this? 3 Now calculate the needs of each refinery, in barrels of crude oil per day, if both distributors are to be satisfied. How does this compare to your two previous answers? What mathematical conclusion can you draw?

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