more-lp-formulations - More LP Formulations Chapters 5 and...

More LP Formulations
Chapters 5 and 7
McCarl and Spreen

Types of Problem
 Joint Product

Assembly

Disassembly

Assembly-Dissassembly

Joint Product
 Wool and lamb

Trees: Wood, sawdust

Grain and straw

Joint Products Problem Formulation
Basic notation and the decision variable
Let us denote the set of
: the produced products
: the production process possibities
: the purchased inputs
: the available resources
Chapter 5: Joint Product

Joint Product
Let us define three fundamental decision variables as
: the set of produced product,
Sales,product
: the set of production possibilities,
Production,process
: the set of purchased inputs,
BuyInput,input

The objective function:
We want to maximize total profits across all of the possible
productions. To do so, we need to know sale prices of the
products, input purchase cost, inputs needed for different
production processes, and any other production costs
associated with production are needed.

product pr
prod
o
u
duct
ct
SalePrice Sales
∑
The Objective Function is to Maximize
input i
input
nput
InputCost BuyInput -
∑
process proces
process
s
OtherCost Production -
∑

1) demand and supply balance for which quantity sold of
each product is less than or equal to the quantity yielded
by production.
Constraints
product produ proce ,
process
ct ss process
Pr - q oducti Sale o s 0 n ≤
∑
2)demand and supply balance for which quantity purchased of
each fixed price input is greater than or equal to the quantity
utilized by the production activities.
process process input inp ,
proce
ut
ss
BuyI r - 0 Produc n t n put io ≤
∑
3) resource availability constraint insuring that the quantity
used of each fixed quantity input does not exceed the
resource endowments
resource, resou
proces
process pro
s
rce cess
Production s b ≤
∑
And non-negativity

Algebra
product pr
prod
o
u
duct
ct
SalePrice Sales
∑
input i
input
nput
InputCost BuyInput -
∑
process proces
process
s
OtherCost Production -
∑
product produ proce ,
process
ct ss process
Pr - q oducti Sale o s 0 n ≤
∑
process process input inp ,
proce
ut
ss
BuyI r - 0 Produc n t n put io ≤
∑
resource, resou
proces
process pro
s
rce cess
Production s b ≤
∑
Max
Subject to

Two products produced: Wheat and wheat straw
Three inputs: Land, fertilizer, seed
Seven production processes possible, producing different amounts
Of grain and straw and using different amounts of inputs
Example: wheat production

Information for each process
Outputs and Inputs Per Acre Process
1 2 3 4 5 6 7
Wheat yield in bu. 30 50 65 75 80 80 75
Wheat straw yield/bales 10 17 22 26 29 31 32
Fertilizer use in Kg. 0 5 10 15 20 25 30
Seed in pounds 10 10 10 10 10 10 10
Land 1 1 1 1 1 1 1

Other information
Wheat price = $4/bushel,
wheat straw price =$0.5/bale
Fertilizer = $2 per kg,
Seed = $0.2/lb.