or eXTRA CREDIT HW1
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1. Chemical Processing - Input/Output Processing - Blending
The Massoil Refinery processes two different kinds of crude oil, Venezuelan and Saudi to
produce two general classes of products, Light and Heavy. Either crude oil can be
processed by either of two modes of processing, Short or Regular. The processing cost
and the amounts of Heavy and Light produced depend upon the mode of processing used
and the type of crude oil used. Costs vary, both across crude oils and across processing
modes. The relevant characteristics are summarized in the table below. For example, the
short process converts each unit of Venezuelan crude to .45 units of Light product, .52
units of Heavy product and .03 units of waste.
SHORT PROCESS REGULAR PROCESS
Venezuelan Saudi Venezuelan Saudi
Light product fraction .45 .60 .49 .68
Heavy product fraction .52 .36 .50 .32
Unusable product fraction .03 .04 .01 .00
Saudi crude costs $22 per unit while Venezuelan crude costs $21 per unit. The
short process costs $2.10 per unit processed while the regular process costs $2.50 per
unit. During any week, Massoil can process at most 10,000 units of crude at the regular
mode and 13,000 units of crude at the short mode.
The respective market values of light and heavy products are $29 and $27 per
unit, while the unusable product is disposed of at the cost of $1 per unit. Formulate a LP
problem that optimally determines the amounts of crudes to buy and the processes to run.
What are the optimal purchasing and operating decisions?
Vene short,x=13,000
Vene Regular, z=10,000
Objective profit=99140
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The Kommet Machine Tool Company produces two products, muffler bearings and torque amplifiers.
One muffler bearing requires .20 hours of assembly labor, .45 hours in the stamping department and
9 square feet of sheet steel. Each torque amplifier requires .50 hours in assembly and .30 hours in
stamping and uses 8 square feet of sheet steel. Current weekly capacities in the two departments are
400 hours of assembly labor and 500 hours of stamping capacity. Sheet steel costs 35 cents per
square foot. Muffler bearings can be sold for $20.00 each. Torque amplifiers can be sold for $17.00
each. Unused capacity in either department cannot be laid off or otherwise fruitfully used. a.
Formulate and solve the LP useful in maximizing the weekly profit contribution. Determine the
optimal schedule, i.e. production levels of the two products, assembly labor used, time required in
the stamping department and sheet steel required. b. It has just been discovered that up to 150
hours of overtime assembly labor can be scheduled at a cost of $7 per hour. Modify the formulation
and solve. What if an overtime assembly hour costs $10.00?
) LP formulation is as follows:
Profit contribution for each muffler bearing = selling price - cost of steel sheet = 20 - 0.35*9 = 16.85
Profit contribution for each torque amplifier = selling price - cost of steel sheet = 17 - 0.35*8 = 14.2
Let M and T be the number of muffler bearings and torque amplifiers to be produced
Maximize 16.85M + 14.2T
s.t.
.2M + .5T <= 400
.45M + .3T <= 500
M, T >= 0
Production levels:
M = 787.88
T = 484.85
Assembly labor used = 787.88*.2+484.85*.5 = 400
time used in stamping dept = 787.88*.45+484.85*.3 = 500
Total profit contribution = $
20160.61
Solution using LINDO is following:
b) Let O be the overtime assembly hours to be scheduled
Revised LP model is following:
Maximize 16.85M + 14.2T - 7O
s.t.
.2M + .5T - O <= 400
.45M + .3T <= 500
O <= 150
M, T, O >= 0
If assembly hour overtime cost $ 10 per hour, then there will be no change in the optimal solution,
because $ 10 is greater than the dual(shadow) price of assembly hour constraint.
Solution of the revised model is following:
3. Blending A federal regulation required that the average of the miles per gallon computed over all
automobiles sold by an automobile company in a specific year be at least 23 miles per gallon. Let us
consider a hypothetical case for the Ford Motor Company. Assume that Ford sold only the four car
types: Crown Victoria, Taurus, Escort, and Fiesta. Various parameters of theses cars are listed below:
CAR MILES/GALLON MARGINAL PRODUCT COST SELLING PRICE Fiesta 33 7000 8500 Escort 26 10250
13250 Taurus 24 15000 19500 Crown Vic. 13 23000 28500 There is some flexibility in the production
facilities so capacities may apply to pairs of car types. These limitations are: YEARLY CAPACITY IN
UNITS CAR TYPES LIMITED 1,000,000 Fiesta 3,000,000 Taurus plus Escorts 2,000,000 Crown Vic. plus
Taurus There is a sales capacity limit of 4,500,000 on the total of all cars sold. How many of each car
type should Ford plan to sell?
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