Real-World Applications
3.9 Production Process Models
We now explain how to formulate an LP model of a simple production process.†The key step is to determine how the outputs from a later stage of the process are related to the outputs from an earlier stage.
Rylon Corporation manufactures Brute and Chanelle perfumes. The raw material needed to manufacture each type of perfume can be purchased for $3 per pound. Processing 1 lb of raw material requires 1 hour of laboratory time. Each pound of processed raw mate-rial yields 3 oz of Regular Brute Perfume and 4 oz of Regular Chanelle Perfume. Regular Brute can be sold for $7/oz and Regular Chanelle for $6/oz. Rylon also has the option of further processing Regular Brute and Regular Chanelle to produce Luxury Brute, sold at
$18/oz, and Luxury Chanelle, sold at $14/oz. Each ounce of Regular Brute processed fur-ther requires an additional 3 hours of laboratory time and $4 processing cost and yields 1 oz of Luxury Brute. Each ounce of Regular Chanelle processed further requires an ad-ditional 2 hours of laboratory time and $4 processing cost and yields 1 oz of Luxury Chanelle. Each year, Rylon has 6,000 hours of laboratory time available and can purchase up to 4,000 lb of raw material. Formulate an LP that can be used to determine how Ry-lon can maximize profits. Assume that the cost of the laboratory hours is a fixed cost.
Solution Rylon must determine how much raw material to purchase and how much of each type of perfume should be produced. We therefore define our decision variables to be
x1 number of ounces of Regular Brute sold annually x2 number of ounces of Luxury Brute sold annually x3 number of ounces of Regular Chanelle sold annually x4 number of ounces of Luxury Chanelle sold annually x5 number of pounds of raw material purchased annually Rylon wants to maximize
Contribution to profit revenues from perfume sales processing costs
costs of purchasing raw material
7x1 18x2 6x3 14x4 (4x2 4x4) 3x5
7x1 14x2 6x3 10x4 3x5
Thus, Rylon’s objective function may be written as
max z 7x1 14x2 6x3 10x4 3x5 (54) Rylon faces the following constraints:
Constraint 1 No more than 4,000 lb of raw material can be purchased annually.
Constraint 2 No more than 6,000 hours of laboratory time can be used each year.
Constraint 1 is expressed by
x5 4,000 (55)
Brute Production Process
E X A M P L E 1 3
†This section is based on Hartley (1971).
To express Constraint 2, note that
Total lab time used annually time used annually to process raw material
time used annually to process Luxury Brute
time used annually to process Luxury Chanelle
x5 3x2 2x4
Then Constraint 2 becomes
3x2 2x4 x5 6,000 (56)
After adding the sign restrictions xi 0 (i 1, 2, 3, 4, 5), many students claim that Ry-lon should solve the following LP:
max z 7x1 14x2 6x3 10x4 3x5
s.t. x5 4,000
3x2 2x4 x5 6,000 xi 0 (i 1, 2, 3, 4, 5)
This formulation is incorrect. Observe that the variables x1and x3do not appear in any of the constraints. This means that any point with x2 x4 x5 0 and x1and x3very large is in the feasible region. Points with x1and x3large can yield arbitrarily large prof-its. Thus, this LP is unbounded. Our mistake is that the current formulation does not in-dicate that the amount of raw material purchased determines the amount of Brute and Chanelle that is available for sale or further processing. More specifically, from Figure 10 (and the fact that 1 oz of processed Brute yields exactly 1 oz of Luxury Brute), it follows that
3x5
This relation is reflected in the constraint
x1 x2 3x5 or x1 x2 3x5 0 (57) Similarly, from Figure 10 it is clear that
Ounces of Regular Chanelle sold ounces of Luxury Chanelle sold 4x5
This relation yields the constraint
x3 x4 4x5 or x3 x4 4x5 0 (58) Constraints (57) and (58) relate several decision variables. Students often omit con-straints of this type. As this problem shows, leaving out even one constraint may very well
pounds of raw material purchased ounces of Brute produced
Ounces of Regular Brute Sold
ounces of Luxury Brute sold
x5 lb Raw material
3x5 oz
Brute x2 oz Reg. Brute processed into Lux. Brute x1 oz Reg. Brute sold
4x5 oz
Chanelle x4 oz Reg. Chanelle into Lux. Chanelle x3 oz Reg. Chanelle sold
F I G U R E 10 Production Process for Brute and Chanelle
lead to an unacceptable answer (such as an unbounded LP). If we combine (53)–(58) with the usual sign restrictions, we obtain the correct LP formulation.
max z 7x1 14x2 6x3 10x4 3x5
s.t. x5 4,000
s.t. 3x2 6x3 2x4 x5 6,000 x1 x2 6x3 2x4 3x5 0
s.t. x3 x4 4x5 0
xi 0 (i 1, 2, 3, 4, 5)
The optimal solution is z 172,666.667, x1 11,333.333 oz, x2 666.667 oz, x3 16,000 oz, x4 0, and x5 4,000 lb. Thus, Rylon should purchase all 4,000 lb of avail-able raw material and produce 11,333.333 oz of Regular Brute, 666.667 oz of Luxury Brute, and 16,000 oz of Regular Chanelle. This production plan will contribute
$172,666.667 to Rylon’s profits. In this problem, a fractional number of ounces seems rea-sonable, so the Divisibility Assumption holds.
We close our discussion of the Rylon problem by discussing an error that is made by many students. They reason that
1 lb raw material 3 oz Brute 4 oz Chanelle
Because x1 x2 total ounces of Brute produced, and x3 x4 total ounces of Chanelle produced, students conclude that
x5 3(x1 x2) 4(x3 x4) (59) This equation might make sense as a statement for a computer program; in a sense, the variable x5is replaced by the right side of (59). As an LP constraint, however, (59) makes no sense. To see this, note that the left side has the units “pounds of raw material,” and the term 3x1on the right side has the units
(ounces of Brute)
Because some of the terms do not have the same units, (59) cannot be correct. If there are doubts about a constraint, then make sure that all terms in the constraint have the same units. This will avoid many formulation errors. (Of course, even if the units on both sides of a constraint are the same, the constraint may still be wrong.)
P R O B L E M S
Group A
Ounces of Brute
1 Sunco Oil has three different processes that can be used to manufacture various types of gasoline. Each process involves blending oils in the company’s catalytic cracker.
Running process 1 for an hour costs $5 and requires 2 barrels of crude oil 1 and 3 barrels of crude oil 2. The output from running process 1 for an hour is 2 barrels of gas 1 and 1 barrel of gas 2. Running process 2 for an hour costs $4 and requires 1 barrel of crude 1 and 3 barrels of crude 2. The output from running process 2 for an hour is 3 barrels of gas 2. Running process 3 for an hour costs $1 and requires 2 barrels of crude 2 and 3 barrels of gas 2. The
output from running process 3 for an hour is 2 barrels of gas 3. Each week, 200 barrels of crude 1, at $2/barrel, and 300 barrels of crude 2, at $3/barrel, may be purchased. All gas produced can be sold at the following per-barrel prices:
gas 1, $9; gas 2, $10; gas 3, $24. Formulate an LP whose solution will maximize revenues less costs. Assume that only 100 hours of time on the catalytic cracker are available each week.
2 Furnco manufactures tables and chairs. A table requires 40 board ft of wood, and a chair requires 30 board ft of
wood. Wood may be purchased at a cost of $1 per board ft, and 40,000 board ft of wood are available for purchase. It takes 2 hours of skilled labor to manufacture an unfinished table or an unfinished chair. Three more hours of skilled labor will turn an unfinished table into a finished table, and 2 more hours of skilled labor will turn an unfinished chair into a finished chair. A total of 6,000 hours of skilled labor are available (and have already been paid for). All furniture produced can be sold at the following unit prices: unfinished table, $70; finished table, $140; unfinished chair, $60;
finished chair, $110. Formulate an LP that will maximize the contribution to profit from manufacturing tables and chairs.
3 Suppose that in Example 11, 1 lb of raw material could be used to produce either 3 oz of Brute or 4 oz of Chanelle.
How would this change the formulation?
4 Chemco produces three products: 1, 2, and 3. Each pound of raw material costs $25. It undergoes processing and yields 3 oz of product 1 and 1 oz of product 2. It costs
$1 and takes 2 hours of labor to process each pound of raw material. Each ounce of product 1 can be used in one of three ways.
It can be sold for $10/oz.
It can be processed into 1 oz of product 2. This re-quires 2 hours of labor and costs $1.
It can be processed into 1 oz of product 3. This re-quires 3 hours of labor and costs $2.
Each ounce of product 2 can be used in one of two ways.
It can be sold for $20/oz.
It can be processed into 1 oz of product 3. This re-quires 1 hour of labor and costs $6.
Product 3 is sold for $30/oz. The maximum number of ounces of each product that can be sold is given in Table 23.
A maximum of 25,000 hours of labor are available.
Determine how Chemco can maximize profit.
6 Daisy Drugs manufactures two drugs: 1 and 2. The drugs are produced by blending two chemicals: 1 and 2. By weight, drug 1 must contain at least 65% chemical 1, and drug 2 must contain at least 55% chemical 1. Drug 1 sells for $6/oz, and drug 2 sells for $4/oz. Chemicals 1 and 2 can be produced by one of two production processes. Running process 1 for an hour requires 3 oz of raw material and 2 hours skilled labor and yields 3 oz of each chemical.
Running process 2 for an hour requires 2 oz of raw material and 3 hours of skilled labor and yields 3 oz of chemical 1 and 1 oz of chemical 2. A total of 120 hours of skilled labor and 100 oz of raw material are available. Formulate an LP that can be used to maximize Daisy’s sales revenues.
7† Lizzie’s Dairy produces cream cheese and cottage cheese. Milk and cream are blended to produce these two products. Both high-fat and low-fat milk can be used to produce cream cheese and cottage cheese. High-fat milk is 60% fat; low-fat milk is 30% fat. The milk used to produce cream cheese must average at least 50% fat and that for cottage cheese, at least 35% fat. At least 40% (by weight) of the inputs to cream cheese and at least 20% (by weight) of the inputs to cottage cheese must be cream. Both cottage cheese and cream cheese are produced by putting milk and cream through the cheese machine. It costs 40¢ to process 1 lb of inputs into a pound of cream cheese. It costs 40¢ to produce 1 lb of cottage cheese, but every pound of input for cottage cheese yields 0.9 lb of cottage cheese and 0.1 lb of waste. Cream can be produced by evaporating high-fat and low-fat milk. It costs 40¢ to evaporate 1 lb of high-fat milk.
Each pound of high-fat milk that is evaporated yields 0.6 lb of cream. It costs 40¢ to evaporate 1 lb of low-fat milk.
Each pound of low-fat milk that is evaporated yields 0.3 lb of cream. Each day, up to 3,000 lb of input may be sent through the cheese machine. Each day, at least 1,000 lb of cottage cheese and 1,000 lb of cream cheese must be produced. Up to 1,500 lb of cream cheese and 2,000 lb of cottage cheese can be sold each day. Cottage cheese is sold for $1.20/lb and cream cheese for $1.50/lb. High-fat milk is purchased for 80¢/lb and low-fat milk for 40¢/lb. The evaporator can process at most 2,000 lb of milk daily.
Formulate an LP that can be used to maximize Lizzie’s daily profit.
8 A company produces six products in the following fashion. Each unit of raw material purchased yields four units of product 1, two units of product 2, and one unit of product 3. Up to 1,200 units of product 1 can be sold, and up to 300 units of product 2 can be sold. Each unit of product 1 can be sold or processed further. Each unit of product 1 that is processed yields a unit of product 4.
Demand for products 3 and 4 is unlimited. Each unit of product 2 can be sold or processed further. Each unit of product 2 that is processed further yields 0.8 unit of product 5 and 0.3 unit of product 6. Up to 1,000 units of product 5 can be sold, and up to 800 units of product 6 can be sold.
Up to 3,000 units of raw material can be purchased at $6 per unit. Leftover units of products 5 and 6 must be destroyed.
It costs $4 to destroy each leftover unit of product 5 and $3
†Based on Sullivan and Secrest (1985).
T A B L E 23
Product Oz
1 5,000
2 5,000
3 3,000
Group B
5 A company produces A, B, and C and can sell these products in unlimited quantities at the following unit prices:
A, $10; B, $56; C, $100. Producing a unit of A requires 1 hour of labor; a unit of B, 2 hours of labor plus 2 units of A; and a unit of C, 3 hours of labor plus 1 unit of B. Any A that is used to produce B cannot be sold. Similarly, any B that is used to produce C cannot be sold. A total of 40 hours of labor are available. Formulate an LP to maximize the company’s revenues.
to destroy each leftover unit of product 6. Ignoring raw material purchase costs, the per-unit sales price and production costs for each product are shown in Table 24.
Formulate an LP whose solution will yield a profit-maximizing production schedule.
9 Each week Chemco can purchase unlimited quantities of raw material at $6/lb. Each pound of purchased raw material can be used to produce either input 1 or input 2.
Each pound of raw material can yield 2 oz of input 1, requiring 2 hours of processing time and incurring $2 in processing costs. Each pound of raw material can yield 3 oz of input 2, requiring 2 hours of processing time and incurring
$4 in processing costs.
Two production processes are available. It takes 2 hours to run process 1, requiring 2 oz of input 1 and 1 oz of in-put 2. It costs $1 to run process 1. Each time process 1 is run 1 oz of product A and 1 oz of liquid waste are produced.
Each time process 2 is run requires 3 hours of processing time, 2 oz of input 2 and 1 oz of input 1. Process 2 yields 1 oz of product B and .8 oz of liquid waste. Process 2 in-curs $8 in costs.
Chemco can dispose of liquid waste in the Port Charles River or use the waste to produce product C or product D.
Government regulations limit the amount of waste Chemco is allowed to dump into the river to 1,000 oz/week. One ounce of product C costs $4 to produce and sells for $11.
One hour of processing time, 2 oz of input 1, and .8 oz of liquid waste are needed to produce an ounce of product C.
One unit of product D costs $5 to produce and sells for $7.
One hour of processing time, 2 oz of input 2, and 1.2 oz of liquid waste are needed to produce an ounce of product D.
At most 5,000 oz of product A and 5,000 oz of product B can be sold each week, but weekly demand for products C and D is unlimited. Product A sells for $18/oz and prod-uct B sells for $24/oz. Each week 6,000 hours of process-ing time is available. Formulate an LP whose solution will tell Chemco how to maximize weekly profit.
10 LIMECO owns a lime factory and sells six grades of lime (grades 1 through 6). The sales price per pound is given in Table 25. Lime is produced by kilns. If a kiln is run for an 8-hour shift, the amounts (in pounds) of each grade
of lime given in Table 26 are produced. It costs $150 to run a kiln for an 8-hour shift. Each day the factory believes it can sell up to the amounts (in pounds) of lime given in Table 27.
Lime that is produced by the kiln may be reprocessed by using any one of the five processes described in Table 28.
For example, at a cost of $1/lb, a pound of grade 4 lime may be transformed into .5 lb of grade 5 lime and .5 lb of grade 6 lime.
Any extra lime leftover at the end of each day must be disposed of, with the disposal costs (per pound) given in Table 29.
Formulate an LP whose solution will tell LIMECO how to maximize their daily profit.
11 Chemco produces three products: A, B, and C. They can sell up to 30 pounds of each product at the following prices (per pound): product A, $10; product B, $12; product C, $20. Chemco purchases raw material at $5/lb. Each pound of raw material can be used to produce either 1 lb of A or 1 lb of B. For a cost of $3/lb processed, product A can be converted to .6 lb of product B and .4 lb of product C. For a cost of $2/lb processed, product B can be converted to .8 lb of product C. Formulate an LP whose solution will tell Chemco how to maximize their profit.
12 Chemco produces 3 chemicals: B, C, and D. They begin by purchasing chemical A for a cost of $6/100 liters. For an
T A B L E 24
Sales Production Product Price ($) Cost ($)
1 7 4
2 6 4
3 4 2
4 3 1
5 20 5
6 35 5
T A B L E 25
Grade 1 2 3 4 5 6
Price($) 12 14 10 18 20 25
T A B L E 26
Grade 1 2 3 4 5 6
Amount produced 2 3 1 1.5 2 3
T A B L E 27
Grade 1 2 3 4 5 6
Maximum demand 20 30 40 35 25 50
T A B L E 28
Input (1 Lb) Output Cost ($ per Lb of Input) Grade 1 .3 lb Grade 3
.2 lb Grade 4 2
.3 lb Grade 5 .2 lb Grade 6
Grade 2 .1 lb Grade 6 1
Grade 3 .8 lb Grade 4 1
Grade 4 .5 lb Grade 5 1
.5 lb Grade 6
Grade 5 .9 lb Grade 6 2
T A B L E 29
Grade 1 2 3 4 5 6
Cost of Disposition ($) 3 2 3 2 4 2
additional cost of $3 and the use of 3 hours of skilled labor, 100 liters of A can be transformed into 40 liters of C and 60 liters of B. Chemical C can either be sold or processed further. It costs $1 and takes 1 hour of skilled labor to process 100 liters of C into 60 liters of D and 40 liters of B. For each chemical the sales price per 100 liters and the maximum amount (in 100s of liters) that can be sold are given in Table 30.
A maximum of 200 labor hours are available. Formulate an LP whose solution will tell Chemco how to maximize their profit.
13 Carrington Oil produces two types of gasoline, gas 1 and gas 2, from two types of crude oil, crude 1 and crude 2. Gas 1 is allowed to contain up to 4% impurities, and gas 2 is allowed to contain up to 3% impurities. Gas 1 sells for
$8 per barrel, whereas gas 2 sells for $12 per barrel. Up to 4,200 barrels of gas 1 and up to 4,300 barrels of gas 2 can be sold. The cost per barrel of each crude, availability, and the level of impurities in each crude are as shown in Table 31. Before blending the crude oil into gas, any amount of each crude can be “purified” for a cost of $0.50 per barrel.
Purification eliminates half the impurities in the crude oil.
Determine how to maximize profit.
14 You have been put in charge of the Melrose oil refinery.
The refinery produces gas and heating oil from crude oil.
Gas sells for $8 per barrel and must have an average “grade level” of at least 9. Heating oil sells for $6 a barrel and must
T A B L E 30
B C D
Price ($) 12 16 26
Maximum demand 30 60 40
have an average grade level of at least 7. At most, 2,000 barrels of gas and 600 barrels of heating oil can be sold.
Incoming crude can be processed by one of three methods.
The per barrel yield and per barrel cost of each processing method are shown in Table 32. For example, if we refine 1 barrel of incoming crude by method 1, it costs us $3.40 and yields .2 barrels of grade 6, .2 barrels of grade 8, and .6 barrels of grade 10.
Before being processed into gas and heating oil, processed grades 6 and 8 may be sent through the catalytic cracker to improve their quality. For $1.30 per barrel, a bar-rel of grade 6 may be “cracked” into a barbar-rel of grade 8. For
$2 per barrel, a barrel of grade 8 may be cracked into a bar-rel of grade 10. Any leftover processed or cracked oil that cannot be used for heating oil or gas must be disposed of at a cost of $0.20 per barrel. Determine how to maximize the refinery’s profit.
T A B L E 31
Cost per Impurity Availability
Oil Barrel ($) Level (%) (Barrels)
Crude 1 6 10% 5,000
Crude 2 8 2% 4,500
T A B L E 32
Method Grade 6 Grade 8 Grade 10 Cost ($)
1 .2 .2 .6 3.40
2 .3 .3 .4 3.00
3 .4 .4 .2 2.60