Readings
Readings
Chapter 4
Linear Programming Applications in Marketing, Finance,
and Operations Management
BA 452 Lesson A.9 Operations Management Applications
1
Overview
Overview
BA 452 Lesson A.9 Operations Management Applications
2
Overview
Production Scheduling Problems are Resource Allocation Problems when
outputs are fixed and when outputs and inputs occur at different periods in time.
The simplest problems consider only two time periods.
Production Scheduling Problems with Dynamic Inventory help managers find an
efficient low-cost production schedule for one or more products over several
time periods (weeks or months).
Workforce Assignment Problems are Resource Allocation Problems when labor
is a resource with a flexible allocation; some labor can be assigned to more
than one work center.
Make or Buy Problems are Linear Programming Profit Maximization problems
when outputs are fixed and when inputs can be either made or bought. Make
or Buy Problems help minimize cost.
Product Mix Problems are Resource Allocation Problems when outputs have
different physical characteristics. Product Mix Problems thus determine
production levels that meet demand requirements.
Blending Problems with Weight Constraints help production managers blend
resources to produce goods of a specific weight at minimum cost.
BA 452 Lesson A.9 Operations Management Applications
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Overview
Tool Summary





Do not make integer restrictions, and maybe the variables at an
optimum will be integers.
• First Example: Pi = (integer) number of producers in month i.
Use compound variables:
• First Example: Pi = number of producers in month i
Use dynamic or recursive constraints:
• First Example: Define the constraint that the number of
apprentices in a month must not exceed the number of recruits in
the previous month: A2 - R1 < 0; A3 - R2 < 0
Constrain one variable to be a proportional to another variable:
• First Example: Define the constraint that each trainer can train
two recruits: 2T1 - R1 > 0; 2T2 - R2 > 0
Use inventory variables:
• Second Example: P2 + I1–I2 = 150 (production-net inventory =
demand)
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling
Production Scheduling
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling
Overview
Production Scheduling Problems are Resource Allocation
Problems when outputs are fixed and when outputs and
inputs occur at different periods in time. Production
Scheduling Problems thus help managers find an efficient
low-cost production schedule for one or more products over
several periods in the future (weeks or months). The
manager determines the production levels that meet
demand requirements, given limitations on production
capacity, labor capacity, and storage space, while
minimizing total production and storage costs. The
simplest problems consider only two time periods.
BA 452 Lesson A.9 Operations Management Applications
6
Production Scheduling
Question: Chip Foose is the owner of Chip Foose
Custom Cars. Chip has just received orders for 1,000
standard wheels and 1,250 deluxe wheels next month
and for 800 standard and 1,500 deluxe the following
month. All orders must be filled.
The cost of making standard wheels is $10 and deluxe
wheels is $16. Overtime rates are 50% higher. There
are 1,000 hours of regular time and 500 hours of
overtime available each month. It takes .5 hour to make
a standard wheel and .6 hour to make a deluxe wheel.
The cost of storing a wheel from one month to the next
is $2.
Minimize total production and inventory costs for
standard and deluxe wheels.
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling

Define the production variables: Determine the regulartime and overtime production quantities in each month
for standard and deluxe wheels.
Wheel
Standard
Deluxe

Month 1
Reg. Time Overtime
SR1
SO1
DR1
DO1
Month 2
Reg. Time Overtime
SR2
SO2
DR2
DO2
Define the inventory variables: Determine the inventory
quantities for standard and deluxe wheels.
SI = number of standard wheels held in
inventory from month 1 to month 2
DI = number of deluxe wheels held in
inventory from month 1 to month 2
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling
Define the objective function: Minimize total production
and inventory costs for standard and deluxe wheels.
Min (production cost per wheel)
x (number of wheels produced)
+ (inventory cost per wheel)
x (number of wheels in inventory)
Min 10SR1 + 15SO1 + 10SR2 + 15SO2
+ 16DR1 + 24DO1 + 16DR2 + 24DO2
+ 2SI + 2DI
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling

Define the production month 1 constraint on (units
required) + (units stored).
Standard:
(1) SR1 + SO1 = 1,000 + SI or SR1 + SO1 - SI = 1,000
Deluxe:
(2) DR1 + DO1 = 1,250 + DI or DR1 + DO1 - DI = 1,250
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling

Define the production month 2 constraint on (units
required) + (units stored).
Standard:
(3) SR2 + SO2 = 800 - SI or SR2 + SO2 + SI = 800
Deluxe:
(4) DR2 + DO2 = 1,500 - DI or DR2 + DO2 + DI = 1,500
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling




Define the constraint Reg. Hrs. Used Month 1 < Reg. Hrs. Avail. Month 1:
(5) .5SR1 + .6DR1 < 1000
Define the constraint OT Hrs. Used Month 1 < OT Hrs. Avail. Month 1:
(6) .5SO1 + .6DO1 < 500
Define the constraint Reg. Hrs. Used Month 2 < Reg. Hrs. Avail. Month 2:
(7) .5SR2 + .6DR2 < 1000
Define the constraint OT Hrs. Used Month 2 < OT Hrs. Avail. Month 2:
(8) .5SO2 + .6DO2 < 500
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling
Interpretation: Total cost $67,500, no storage, production
schedule:
Month 1
Month 2
Reg. Time Overtime Reg. Time Overtime
Standard
500
500
200
600
Deluxe
1250
0
1500
0
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling with Dynamic Inventory
Production Scheduling with
Dynamic Inventory
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling with Dynamic Inventory
Overview
Production Scheduling Problems with Dynamic Inventory
help managers find an efficient low-cost production
schedule for one or more products over several time
periods (weeks or months). A production scheduling
problem is resource-allocation problem for each of several
periods in the future. Complex problems consider more
than two time periods, so there are many periods of
inventory.
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling with Dynamic Inventory
Question: Wilson Sporting Goods produces baseballs.
Wilson must decide how many baseballs to produce each
month. It has decided to use a 6-month planning horizon.
 The forecasted demands for the next 6 months are
10,000; 15,000; 30,000; 35,000; 25,000; and 10,000.
 Wilson wants to meet these demands on time, knowing
that it currently has 5,000 baseballs in inventory and that
it can use a given month’s production to help meet the
demand for that month.
 During each month there is enough production capacity
to produce up to 30,000 baseballs, and there is enough
storage capacity to store up to 10,000 baseballs at the
end of the month, after demand has occurred.
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling with Dynamic Inventory




The forecasted production costs per baseball for
the next 6 months are $12.50, $12.55, $12.70, $12.80,
$12.85, and $12.95.
The holding cost per baseball held in inventory at the
end of the month is figured at 5% of the production cost
for that month: $0.625, $0.6275, $0.635, $0.64, $0.6425,
and $0.6475.
The selling price for baseballs is not considered relevant
to the production decision because Wilson will satisfy all
customer demand exactly when it occurs – at whatever
the selling price.
Therefore, Wilson wants to determine the production
schedule that minimizes the total production and holding
costs.
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling with Dynamic Inventory



The decision variables are the production quantities for
the 6 months, labeled P1 through P6. To keep quantities
small, all quantities are in hundreds of baseballs.
Constraints are easier to understand if we add variables
I1 through I6 to be the corresponding end-of-month
inventories (after meeting demand). For example, I3 is
the number of baseballs left over at the end of month 3.
The following constraints define inventories:
• P1 – I1 = 100-50 (production–inventory = net demand)
• P2 + I1–I2 = 150 (production-net inventory = demand)
• P3 + I2–I3 = 300 (production-net inventory = demand)
• P4 + I3–I4 = 350 (production-net inventory = demand)
• P5 + I4–I5 = 250 (production-net inventory = demand)
• P6 + I5–I6 = 100 (production-net inventory = demand)
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling with Dynamic Inventory


There are obvious constraints are on production and
inventory storage capacities: Pj  300 and Ij  100 for
each month j (j = 1, …, 6).
Finally, production and inventory storage are assumed
non-negative.
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling with Dynamic Inventory

There
BA 452 Lesson A.9 Operations Management Applications
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Production Scheduling with Dynamic Inventory
BA 452 Lesson A.9 Operations Management Applications
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Workforce Assignment
Workforce Assignment
BA 452 Lesson A.9 Operations Management Applications
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Workforce Assignment
Overview
Workforce Assignment Problems are Resource Allocation
Problems when labor is one of the resources, and labor
allocation has some flexibility; at least some labor can be
assigned to more than one department or work center.
Workforce Assignment Problems thus help when
employees have been cross-trained on two or more jobs or,
for instance, when sales personnel can be transferred
between stores.
BA 452 Lesson A.9 Operations Management Applications
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Workforce Assignment
Question: National Wing Company (NWC) is gearing up for the new B-48
contract. NWC has agreed to supply 20 wings in April, 24 in May, and 30 in
June. Wings can be freely stored from one month to the next.
Currently, NWC has 100 fully-qualified workers. A fully qualified worker can
either be placed in production or can train new recruits. A new recruit can be
trained to be an apprentice in one month. After another month, the apprentice
becomes a qualified worker. Each trainer can train two recruits. At the end of
June, NWC wishes to have at least 140 fully-qualified workers. (Note: NWC
must use firm-specific training. There is no outside market for fully-qualified
workers.) The production rate and salary per employee type is listed below.
Type of
Employee
Production
Trainer
Apprentice
Recruit
Production Rate
(Wings/Month)
.6
.3
.4
.05
Wage
Per Month
$3,000
$3,300
$2,600
$2,200
How should NWC optimize?
BA 452 Lesson A.9 Operations Management Applications
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Workforce Assignment
Answer:
 Define the Decision Variables
Pi = number of producers in month i (where i = 1, 2, 3 for April, May, June)
Ti = number of trainers in month i (where i = 1, 2 for April, May)
Ai = number of apprentices in month i (where i = 2, 3 for May, June)
Ri = number of recruits in month i (where i = 1, 2 for April, May)
 Define the objective function
Minimize total wage cost for producers, trainers,
apprentices, and recruits for April, May, and June:
Min 3000P1 + 3300T1 + 2200R1
+ 3000P2 + 3300T2 + 2600A2+2200R2
+ 3000P3 + 3300T3 + 2600A3+2200R3
BA 452 Lesson A.9 Operations Management Applications
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Workforce Assignment
Define the constraint that total production in Month 1 (April)
must equal or exceed contract for Month 1:
(1) .6P1 + .3T1 +.05R1 > 20
 Define the constraint that total production in Months 1-2 (April, May)
must equal or exceed total contracts for Months 1-2:
(2) .6P1 + .3T1 + .05R1 + .6P2 + .3T2 + .4A2 + .05R2 > 44
 Define the constraint that total production in Months 1-3 (April, May,
June) must equal or exceed total contracts for Months 1-3:
(3) .6P1+.3T1+.05R1+.6P2+.3T2+.4A2+.05R2+.6P3+.4A3 > 74
 Define the constraint that the number of producers and trainers in a
month (fully qualified workers) must not exceed the initial supply of
100, plus any apprentices employed in a previous month:
(4) P1 + T1 < 100
(5) P2 + T2 < 100
(6) P3 + T3 < 100 + A2

BA 452 Lesson A.9 Operations Management Applications
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Workforce Assignment
Define the constraint that the number of apprentices in a month
must not exceed the number of recruits in the previous month that
have not already become apprentices:
(7) A2 < R1; (8) A3 < (R1 - A2) + R2
 Note: Constraint (8) allows a recruit from Month 1 to be laid off
in Month 2, then rehired as an apprentice in Month 3.
 Define the constraint that each trainer can train two recruits:
(9) 2T1 - R1 > 0; (10) 2T2 - R2 > 0
 Define the constraint that at the end of June, there are to be at least
140 fully qualified workers:
(11) 100 + A2 + A3 > 140

BA 452 Lesson A.9 Operations Management Applications
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Workforce Assignment
Interpretation: Total wage cost =
$1,098,000, using the following
workforce assignment:
April
Producers
Trainers
Apprentices
Recruits
100
0
0
0
May
June
80
20
0
40
100
0
40
0
July
BA 452 Lesson A.9 Operations Management Applications
140
0
0
0
28
Make or Buy
Make or Buy
BA 452 Lesson A.9 Operations Management Applications
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Make or Buy
Overview
Make or Buy Problems are Linear Programming Profit
Maximization problems when outputs are fixed and when
inputs can be either made or bought. Make or Buy
Problems thus help production managers to minimize cost
by comparing, for some inputs, the lower cost of
manufacturing those inputs (rather then buying them) to the
opportunity cost of the scarce resources used in
manufacture.
BA 452 Lesson A.9 Operations Management Applications
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Make or Buy
Question: The Janders Company markets business and
engineering products. Janders is currently preparing to
introduce two new calculators: one for the business market
called the Financial Manager, and one for the engineering
market called the Technician. Each calculator has three
components: a base, an electronic cartridge, and a
faceplate or top. The same base is used for both
calculators, but the cartridges and tops are different. All
components can be manufactured by the company or
purchased from outside suppliers.
BA 452 Lesson A.9 Operations Management Applications
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Make or Buy
Company forecasters indicate that 3000 Financial Manager
calculators and 2000 Technician calculators will be
demanded. However, manufacturing capacity is limited.
The company has 200 hours of regular time manufacturing
time and 50 hours of overtime that can be scheduled for
the calculators. Overtime involves a premium at the
additional cost of $9 per hour.
BA 452 Lesson A.9 Operations Management Applications
32
Make or Buy
Determine how many units of each component to
manufacture and how many to buy given the cost,
purchase price and manufacturing time requirements:
Component
Manufacture cost
(regular time) per
unit
Purchase cost per
unit
Manufacturing Time
Base
$0.50
$0.60
1.0 minutes
Financial
cartridge
$3.75
$4.00
3.0 minutes
Technician
cartridge
$3.30
$3.90
2.5 minutes
Financial top
$0.60
$0.65
1.0 minutes
Technician top
$0.75
$0.78
1.5 minutes
BA 452 Lesson A.9 Operations Management Applications
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Make or Buy
Answer:
Define decision variables
BM = number of bases manufactured
BP = number of bases purchased
FCM = number of Financial cartridges manufactured
FCP = number of Financial cartridges purchased
TCM = number of Technician cartridges manufactured
TCP = number of Technician cartridges purchased
FTM = number of Financial tops manufactured
FTP = number of Financial tops purchased
TTM = number of Technician tops manufactured
TTP = number of Technician tops purchased
OT = number of overtime hours
BA 452 Lesson A.9 Operations Management Applications
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Make or Buy
Define the objective:
Min 0.5BM + 0.6BP + 3.75FCM + 4FCP+ 3.3TCM +
3.9TCP + 0.6FTM + 0.65FTP + 0.75TTM + 0.78TTP + 9OT
Component
Manufacture cost
(regular time) per
unit
Purchase cost per
unit
Manufacturing Time
Base
$0.50
$0.60
1.0 minutes
Financial
cartridge
$3.75
$4.00
3.0 minutes
Technician
cartridge
$3.30
$3.90
2.5 minutes
Financial top
$0.60
$0.65
1.0 minutes
Technician top
$0.75
$0.78
1.5 minutes
BA 452 Lesson A.9 Operations Management Applications
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Make or Buy
Demand Constraints:
BM + BP = 5000 Bases
FCM + FCP = 3000 Financial cartridges
TCM + TCP = 2000 Technician cartridges
FTM + FTP = 3000 Financial tops
TTM + TTP = 2000 Technician tops
Overtime Resource Constraint:
OT < 50
Manufacturing Time Constraint (right-side in minutes):
BM + 3FCM + 2.5TCM + FTM + 1.5TTM < 12000 + 60OT
BA 452 Lesson A.9 Operations Management Applications
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Make or Buy
The Management Scientist can solve this 11-variable, 7constraint linear program:
BM = 5000
BP = 0
FCM = 666.667
FCP = 2333.333
TCM = 2000.000
TCP = 0.000
FTM = 0.000
FTP = 3000.000
TTM = 0.000
TTP = 2000.000
OT = 0.000
BA 452 Lesson A.9 Operations Management Applications
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Product Mix
Product Mix
BA 452 Lesson A.9 Operations Management Applications
38
Product Mix
Overview
Product Mix Problems are Resource Allocation Problems
when outputs have different physical characteristics.
Product Mix Problems thus help managers determine the
production levels that meet demand requirements, given
limitations on production capacity and labor capacity, to
maximize profit or minimize cost.
BA 452 Lesson A.9 Operations Management Applications
39
Product Mix
Question: Floataway Tours has $420,000 that can be used to buy new
rental boats for hire during the summer. The boats can be bought from
two different manufacturers.
Floataway Tours would like to buy at least 50 boats, and would like to
buy the same number from Sleekboat as from Racer to maintain
goodwill. At the same time, Floataway Tours wishes to have a total
seating capacity of at least 200.
Boat
Builder
Speedhawk Sleekboat
Silverbird
Sleekboat
Catman
Racer
Classy
Racer
Maximum
Cost Seating
$6000
3
$7000
5
$5000
2
$9000
6
Expected
Daily Profit
$ 70
$ 80
$ 50
$110
How should Floataway Tours optimize?
BA 452 Lesson A.9 Operations Management Applications
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Product Mix


Define the Decision Variables
x1 = number of Speedhawks ordered
x2 = number of Silverbirds ordered
x3 = number of Catmans ordered
x4 = number of Classys ordered
Define the Objective Function
Maximize total expected daily profit:
Max (Expected daily profit per unit)
x (Number of units)
Max 70x1 + 80x2 + 50x3 + 110x4
BA 452 Lesson A.9 Operations Management Applications
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Product Mix




Define the constraint to spend no more than $420,000:
(1) 6000x1 + 7000x2 + 5000x3 + 9000x4 < 420,000
Define the constraint to buy at least 50 boats:
(2) x1 + x2 + x3 + x4 > 50
Define the constraint that the number of boats from Sleekboat
equals the number of boats from Racer:
(3) x1 + x2 = x3 + x4 or x1 + x2 - x3 - x4 = 0
Define the constraint that seating capacity be at least 200:
(4) 3x1 + 5x2 + 2x3 + 6x4 > 200
BA 452 Lesson A.9 Operations Management Applications
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Product Mix
Interpretation: Expected daily profit is
$5,040.00.
• Buy 28 Speedhawks from
Sleekboat; buy 28 Classy’s from
Racer.
• The minimum number of boats was
exceeded by 6 (surplus for
constraint #2).
• The minimum seating capacity was
exceeded by 52 (surplus for
constraint #4).
BA 452 Lesson A.9 Operations Management Applications
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Blending with Weight Constraints
Blending with Weight Constraints
BA 452 Lesson A.9 Operations Management Applications
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Blending with Weight Constraints
Overview
Blending Problems with Weight Constraints help production
managers blend resources to produce goods of a specific
weight at minimum cost.
BA 452 Lesson A.9 Operations Management Applications
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Blending with Weight Constraints
Question: The Maruchan Corporation receives four raw grains from
which it blends its Maruchan Ramen Noodle Soup. Maruchan
advertises that each 8-ounce packet meets the minimum daily
requirements for vitamin C, protein and iron. The following is the cost
of each raw grain, the vitamin C, protein, and iron units per pound of
each grain.
Grain
1
2
3
4
Vitamin C
Units/lb
9
16
8
10
Protein
Units/lb
12
10
10
8
Iron
Units/lb
0
14
15
7
Cost/lb
.75
.90
.80
.70
Maruchan is interested in producing the 8-ounce mixture at
minimum cost while meeting the minimum daily requirements of 6
units of vitamin C, 5 units of protein, and 5 units of iron.
BA 452 Lesson A.9 Operations Management Applications
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Blending with Weight Constraints
Answer:
 Define the decision variables.
xj = the pounds of grain j (j = 1,2,3,4) used in 8-ounce mixture
 Define the objective. Minimize the total cost for an 8-ounce mixture:
Min .75x1 + .90x2 + .80x3 + .70x4
BA 452 Lesson A.9 Operations Management Applications
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Blending with Weight Constraints




Constrain the total weight of the mix to 8-ounces (.5 pounds):
(1) x1 + x2 + x3 + x4 = .5
Constrain the total amount of Vitamin C in the mix to be at least 6 units:
(2) 9x1 + 16x2 + 8x3 + 10x4 > 6
Constrain the total amount of protein in the mix to be at least 5 units:
(3) 12x1 + 10x2 + 10x3 + 8x4 > 5
Constrain the total amount of iron in the mix to be at least 5 units:
(4) 14x2 + 15x3 + 7x4 > 5
BA 452 Lesson A.9 Operations Management Applications
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Blending with Weight Constraints
Interpretation: The mixture
costs Frederick’s 40.6 cents.
Optimal blend:
 0.099 lb. of grain 1
 0.213 lb. of grain 2
 0.088 lb. of grain 3
 0.099 lb. of grain 4
BA 452 Lesson A.9 Operations Management Applications
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BA 452
Quantitative Analysis
End of Lesson A.9
BA 452 Lesson A.9 Operations Management Applications
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