LP Formulation: Process selection and product mix
Structure of Problem:
 Several products to be produced
 Several machines to produce them, but differ in efficiency
 Find:
1. Product mix: the number of each product to be produced
2. How to produce them (amount in which machine)?
Decision Variables:
Xij -- # of product i to be produced in machine j.
Example 1: Crop Planning
A farm owner wants to know how many acres of three different crops to plant on three
different plots in order to maximize profit.
The farmer's tract of land consists of 2000 acres. The farmer has subdivided the tract into
three plots and has contracted with three local farm families to operate the plots. The farm
owner has instructed each sharecropper to plant three crops: corn, peas, and soybeans.
The size of each plot has been determined by the capabilities of each local farmer. Plot
sizes, crop restrictions, and profit per acre are given in the following tables.
Plot
Acreage
1
500
Crop
Corn
Peas
Soybeans
2
800
Max Acreage
900
700
1000
3
700
Profit($)/Acre
600
450
300
Any of the three crops may be planted on any of the plots; however, the farm owner has
placed the following restrictions on the farming operation. At least 60% of each plot must
be under cultivation. To ensure that each sharecropper works according to his or her
potential and resources (which determined the acreage allocation), the owner wants the
same proportion of each plot to be under cultivation. The owner's objective is to
determine how much of each crop to plant on each plot in order to maximize profit.
Crop Planning Excel Files: Edition1
Edition2
Example 2: Cargo Balancing
A ship has three cargo holds: forward, center, and rear. The capacity limits are:
Hold
Weight Capacity
(tons)
3000
4000
2500
Forward
Center
Rear
Volume Capacity
(cubic feet)
155,000
185,000
145,000
Because of balance considerations, the weight in the forward hold must be within 5% of
the weight in the rear hold; the weight in the center hold must be at least 30% greater than
the weight in the forward hold, at least 30% greater that the weight in the rear hold, and at
most 90% of the sum of the weights in the forward and rear holds.
The following cargoes are offered:
Commodity
1
2
3
4
Amount
(tons)
5000
3000
1000
1500
Volume
(cubic feet) /ton
60
50
25
40
Profit
($)/ton
6
8
5
7
The ship owner may accept all or any part of each commodity. (Assume, for example, the
commodities are grain and/or liquids.) The ship owner wishes to find the number of tons
of each commodity which should be placed in each hold in order to maximize profits.
Cargo Balancing Excel Files: Edition1
Edition2