# Conceptual Review Questions

Document Sample

```					Chapter 13: Optimization Modeling

1. For each of the examples in this chapter, classify the constraints into the following categories. Are there
any other types of constraints that do not fall into these categories?
a. simple bounds
b. limitations
c. requirements
d. proportional relationships
e. balance constraints

Product mix example:
6G + 6P  3,000                                    limitations
3G + 5P  1750                                    limitations

Media Selection example:
500R + 2000T + 200N  40000                  limitations
0  R  15                                   simple bounds
T  12                                       simple bounds
6  N  30                                   simple bounds

Process Selection example:
D1 + P1 = 45000                              requirements
D2 + R2 + P2 = 76500                       requirements
D3 + R3 + P3 = 10000                       requirements
0.213D1 + 0.192D2 + 0.227D3  6552             limitations
0.192R2 + 0.227R3  32760                      limitations

Blending example:
X 1 + X 2 + X3 + X4 + X 5 + X6 + X7 + X8 = 1      requirements
0.169X1 + 0.12X2 + 0.085X3 + 0.154X4 + 0.085X5 +0 .12X6 + 0.18X7 + 0.119X8  0.13 relationships
proportional
0.26X1 + 0.041X2 + 0.038X3 + 0.063X4 + 0.038X5 + 0.017X6 + 0.179X7 + 0.04X8  0.15relationships
proportional
0.29X1 + 0.083X2 + 0.027X3 + 0.024X4 + 0.027X5 + 0.023X6 + 0.288X7 + 0.109X8 0.14
proportional relationships

Production Planning example:
PA + 0 = 150 + IA                                 balance constraints
PW + IA = 400 + IW                                 balance constraints
PS + IW = 50 + IS                                 balance constraints

Cash Management example:
200,000 – (A1 + B1 + C1 + 50,000)  100,000     requirements
1.005A1 + 12,000 – (A2 + B2)  100,000          requirements
 100,000
1.005A2 – (A3 + B3 + 23,000)                    requirements
1.005A3 + 1.0175B1 + 20,000 - (A4 + B4 )  100,000
requirements
1.005A4 + 1.0175B2 - (A5 + 41,000)  100,000 requirements
1.005A5 + 1.0175B3 + 13,000 - A6  100,000      requirements

Transportation Problem example:
X11 + X12 + X13 + X14  1200                    limitations
X21 + X22 + X23 + X24  800                     limitations
X11 + X21 = 150                                  requirements
X12 + X22 = 350                                  requirements
X13 + X23 = 500                                  requirements
X14 + X24 = 1000                                 requirements

2. Explain the following terms: optimization, objective function, optimal solution,
constraint, constraint function, feasible solution, infeasible solution.

Optimization - the process of selecting values for decision variables
that minimize or maximize some quantity of interest.

Objective function - The quantity we seek to minimize or maximize through optimization.

Optimal solution - the set of decision variable values that maximize or minimize the objective function.

Constraint - limitations or requirements that decision variables must satisfy.

Constraint function - Constraints are generally expressed mathematically as equations or inequalities.
When represented mathematically, a constraint function is a function of the decision variables.

Feasible solution - any solution that satisfies all constraints of a problem.

Infeasible solution - an optimization problem that has no feasible solutions.

3. Describe the differences among linear, integer, and nonlinear optimization problems.

Linear - has two basic properties. First, the objective function and all constraints are
linear functions of the decision variables. This means that each function is simply a sum of terms,
each of which is some constant multiplied by a decision variable and the second property
of a linear optimization problem is that all variables are continuous,
meaning that they may assume any real value (typically, nonnegative).

Integer - some or all of the variables are restricted to being whole numbers.

Nonlinear - the objective function and/or constraint functions are nonlinear functions
of the decision variables; that is, terms cannot be written as a constant times a variable.

4. List and describe the major categories in which constraints generally fall.

Simple Bounds. Simple bounds constrain the value of a single variable.

Limitations. Limitations usually involve the allocation of scarce resources.

Requirements. Requirements involve the specification of minimum levels of performance.

Proportional Relationships. Proportional relationships are often found in problems
involving mixtures or blends of materials or strategies.

Balance Constraints. Balance constraints essentially state that “input = output”
and ensure that the flow of material or money is accounted for at locations or between time periods.

5. List the important guidelines to follow for modeling optimization problems on spreadsheets.

Put the objective function coefficients, constraint coefficients, and right-hand-side values in a logical format in the spreads

Define a set of cells (either rows or columns) for the values of the decision variables.

Define separate cells for the objective function and each constraint function (the left-hand side of a constraint).

6. What Excel functions should you avoid when implementing linear optimization models on spreadsheets?

ABS
MIN
MAX
INT
ROUND
IF
COUNT

7. Explain the difference between general integer variables and binary variables.

Decision variables that we force to be integers are called general integer variables

Binary variables, are variables that are restricted to being either 0 or 1.

8. How can binary variables be used to model logical conditions? Provide several examples.

Binary variables enable us to model logical decisions in optimization models.
For example, binary variables can be used to model decisions
such as whether (x = 1) or not (x = 0) to place a facility at a certain location,
whether or not to run a production line, or whether or not to invest in a certain stock.
One common example is project selection or capital budgeting, in which a subset
of potential projects must be selected with limited resource constraints.
es. Are there
a logical format in the spreadsheet.

side of a constraint).

Chapter 13: Optimization Modeling

1. A manufacturer of office equipment produces two types of desks, standard and deluxe. Deluxe desks
have oak tops and more expensive hardware, and require additional time for finishing and polishing.
Standard desks require 80 square feet of pine wood, 12 hours of labor, while deluxe desks require 62
square feet of pine, 18 square feet of oak, and 18 hours of labor. For the next week, the company has
5000 square feet of pine, 700 square feet of oak, and 400 hours of labor available. Standard desks net a
profit of \$75 while deluxe desks net a profit of \$160. All desks can be sold to national chains such as
Staples or Office Depot. Develop a model to determine how many of each the company should make
next week to maximize profit contribution.

S = units of standard desk produced
D = units of deluxe desk produced

Max 75S + 160D
subject to: 80S + 62D <= 5000
18D <= 700
12S + 18D <= 400
S, D >= 0
Chapter 13: Optimization Modeling

2. Metropolitan Airport Services, Inc. is considering purchasing cars for transportation service between the
Municipal Airport and hotels in the downtown area. They are considering station wagons, minibuses,
and large buses. They are considering station wagons, minibuses, and large buses. Purchase prices
are \$45,000 for each station wagon, \$90,000 for each minibus, and \$200,000 for each large bus. The
Board of Directors has authorized a maximum budget of \$5,000,000 for purchases. Because of the
heavy air travel in the area, these vehicles would be utilized at maximum capacity regardless of the type
of vehicle purchased.
Expected net annual profit would be \$1,500 per station wagon, \$3,500 per minibus, and \$5,000 per large
bus. The company has hired 30 new drivers for these vehicles, each qualified to drive any one of the
three vehicle types. The maintenance department has the capacity to handle an additional 80 station
wagons. A minibus is equivalent to 1 2/3 station wagons for maintenance purposes, and a large bus is
equivalent to 2 station wagons for maintenance purposes. Develop a linear optimization model to
determine the optimal number of each type of vehicle to purchase in order to maximize profit.

S = number of station wagons purchased
M = number of mini buses purchased
L = number of large buses purchased

Max 1,500S + 3,500M + 5,000L
subject to: 45,000S + 90,000M + 200,000L <= 5,000,000
S + M + L <= 30
S + 5/3M + 2L <= 80
S, M, L >= 0
Chapter 13: Optimization Modeling

3. The International Chef, Inc. markets three blends of oriental tea: premium, Duke Grey, and breakfast.
The firm uses tea leaves from India, China, and new domestic California sources.

Tea Leaves
Quality          Indian      Chinese      California
Duke Grey         20%          30%          40%
Breakfast         20%          40%          40%

Net profit per pound for each blend is \$0.50 for premium, \$0.30 for Duke Grey, and \$0.20 for breakfast.
The firm's regular weekly supplies are 20,000 pounds of Indian tea leaves and 22,000 pounds of Chinese
tea leaves. Because it is new, and until it proves itself, no more than 16,000 pounds of California tea is
to be used in a week. The marketing research department reports that there is an almost unlimited
market for premium and fine blends. However, the maximum expected sale for breakfast blend is 2,000
pounds. Develop a linear optimization model to determine the optimal mix to maximize profit.

P = pounds of premium produced
D = pounds of Duke Grey produced
B = pounds of breakfast produced

Max .50P + .30D + .20B
subject to: .40P + .20D + .20B <= 20,000
.20P + .20D + .40B <= 22,000
.20P + .30D + .40B <= 16,000
B <= 2,000
P, D, B >= 0
Chapter 13: Optimization Modeling

4. You are in the business of producing and selling 100 pound bags of health food for farm animals.
You plan to advertise that each bag will provide a farm animal its minimum weekly requirements of
protein (200 grams), calcium (300 grams) and fiber (100 grams), and will contain no more
than 500 calories. You have found supplies at reasonable cost for three possible ingredients.

Cost        Protein      Calcium        Fiber         Calories
Corn            \$.03/lb       100 g/lb      2 g/lb        1 g/lb        50/lb
Fishbones       \$.005/lb        1 g/lb     50 g/lb        none           2/lb
Sawdust         \$.001/lb       none         none         200 g/lb        1/lb

You plan to sell the bag for \$1. Develop an optimization model for this problem.

C = lb corn in bag
F = lb fishbones in bag
S = lb sawdust in bag

Min .03C + .005F + .001S
subject to: 100C + F >= 200
2C + 50F >= 300
C + 200S >= 100
50C + 2F + 1S <= 500
C + F + S =100
C, F, S >= 0
Chapter 13: Optimization Modeling

5. Wu’s Chinese Cuisine is a small restaurant located in a shopping mall.
The restaurant is so popular because of its limited menu and fresh preparation
that even if some items sell out, customers will usually substitute something
else so that nearly all available food is sold each day. In planning a special
“lunch for two” promotion, the owner wishes to maximize the profit associated
with making three dishes: vegetable shrimp, broccoli beef,
and chicken green bean. Relevant data on pricing, amount of ingredients
used per serving, and cost of ingredients are given below.

Number of pounds/serving
Menu Item          Price/serving     Shrimp Broccoli Green beans            Beef
Mixed veggies        Chicken
Vegetable shrimp           \$8.89         0.3      0.2       0.1        0.4         0        0
Broccoli beef              \$7.59         0.4        0         0        0.3       0.3        0
Chicken green bean         \$7.29        0.05        0      0.45        0.1         0      0.4
Cost/lb                               \$5.50    \$1.20     \$1.20      \$1.20     \$5.00    \$4.00

The restaurant has available 55 hours of labor per day at a cost of \$12/hour,
and has a contract to purchase 30 lbs shrimp, 120 lbs broccoli, 120 lbs green beans,
120 lbs mixed veggies, 60 lbs beef, and 90 lbs chicken each day.
Develop a linear optimization model to maximize profit.

Max 8.89VS + 7.59BB + 7.29CG

Subject to:

VS = .3S+.2Broc+.1GB+.4MV
BB = .4S+.3MV+.3Beef
CG = .05S+.45GB+.1MV+.4C

L<=55(\$12)
S<=30(\$5.50)
Broc<=120(\$1.20)
GB<=120(\$1.20)
MV<=120(\$1.20)
Beef<=60(\$5.00)
Ch<=90(\$4.00)
Chapter 13: Optimization Modeling

6. A manufacturer of decorative candles has contracted with a national retailer to supply
a set of special holiday candles to its stores. These include large jars, small jars,
large pillars, small pillars, and a package of 4 votive candles. In negotiating the contract
for the display, the manufacturer and retailer agreed that 8 feet would be designated for the
display in each store, but that at least 2 feet be dedicated to large jars and large pillars,
and at least one foot to the votive candle packages. At least as many jars as pillars
must be provided. The manufacturer has obtained 200,000 pounds of wax,
250,000 feet of wick, and 100,000 ounces of holiday fragrance.
The amount of materials and display size required for each product is shown in the table below:

Large jar Small jar Large pillar Small pillar Votive pack
Wax                0.5      0.25       0.5         0.25       0.3125
Fragrance         0.24      0.12      0.24         0.12          0.15
Wick              0.43      0.22      0.58         0.33           0.8
Display feet      0.48      0.24      0.23         0.23          0.26
Profit/unit     \$0.25    \$0.20      \$0.24       \$0.21         \$0.16

How many of each product should be made to maximize the profit?

Max .25LJ+.2SJ+.24LP+.21SP+.16VP

subject to:
.48LJ+.24SJ+.23LP+.23SP+.26VP<=8ft
LJ>=2ft
LP>=2ft
VP>=1ft
(LJ+SJ)=(LP+SP)
.5LJ+.25SJ+.5LP+.25SP+.3125VP<=200,000
.24LJ+.12SJ+.24LP+.12SP+.15VP<=100,000
.43LJ+.22SJ+.58LP+.33SP+.8VP<=250,000
Chapter 13: Optimization Modeling

7. A city council is reviewing housing proposals for a new development area. There is some dispute among
various interest groups as to what the goals are. The zoning committee has recommended three types
of housing: single-family houses, deluxe condominiums, and apartments, and have also provided the
following data:

Family      Condos       Apartment
Land usage, acres/unit          0.25         0.4           0.125
Families housed/unit              1           4              6
Tax base generated/unit        \$50,000     \$100,000       \$25,000
Utility installation expense/unit    \$4,000      \$8,000         \$6,000

Twenty acres are available for zoning. Utility installation expense is to be held to no more than \$1 million.
A public opinion survey has been conducted, and the city council has reviewed this survey, important
issues are to provide housing to families, generate tax base, and minimize installation costs. Develop
linear optimization models to maximize families housed, and also for maximizing the tax base added

F = number of single family houses
C = number of deluxe condominiums
A = number of apartments

Max F + 4C + 6A
subject to: .25F + .4C + .125 A <= 20
4,000F + 8,000C + 6,000A <= 1,000,000
F, C, A >= 0

F = number of single family houses
C = number of deluxe condominiums
A = number of apartments

Max 50,000F + 100,000C + 25,000A
subject to: .25F + .4C + .125 A <= 20
4,000F + 8,000C + 6,000A <= 1,000,000
F, C, A >= 0
Chapter 13: Optimization Modeling

8. A department store chain is planning opening a new store. They need to decide how to allocate the
100,000 square feet of available floor space among seven departments. Data on expected performance
of each department per month, in terms of square feet (sf) are:

expected
Investment    Risk as a % minimum         maximum        profit
Department                  /sf           of \$ invested sf            sf             per sf
Electronics                     \$100           24%          6000         30000          \$12.00
Furniture                         50            12         10000         30000               6
Men's Clothing                    30             5          2000          5000               2
Clothing                         600            10          3000         40000              30
Jewelry                          900            14          1000          1000              20
Books                             50             2          1000          5000               1
Appliances                       400             3         12000         40000              13

The company has gathered \$20,000,000 to invest in floor stock. The risk element is a measure of risk
associated with investment in floor stock. The idea is that electronics loses \$24/\$100 invested per
month, based upon past records at other places for outdated inventory, pilferage, breakage, etc.
Electronics is the highest risk item. Expected profit is after covering risk. Develop a linear optimization
model to maximize profit.

E = sf allocated to electronics
F = sf allocated to furniture
M = sf allocated to men's clothing
C = sf allocated to clothing
J = sf allocated to jewelry
B = sf allocated to books
A = sf allocated to appliance
R = dollars at risk

Max 12E + 6F + 2M + 30C + 20J + B + 13A
subject to: 100E + 50F + 30M + 600C + 900J + 50B + 400A <= 20,000,000
E + F + M + C + J + B + A <= 100,000
(.24*100)E + (.12*50)F + (.05*30)M + (.10*60)C + (.14*900)J
+ (.02*50)B + (.03*400)A = R
E >= 6,000
F >= 10,000
M >= 2,000
C >= 3,000
J >= 1,000
B >= 1,000
A >= 12,000
E <= 30,000
F <= 30,000
M <= 5,000
C <= 40,000
J <= 1000
B <= 5,000
A <= 40,000
E, F, M, C, J, B, A >= 0
Chapter 13: Optimization Modeling

9. An energy company blends different types of coal to customer specifications.
Their current inventory consists of five types of coal; the table below shows the
properties of each type: BTU rating; percent of sulfur, ash, and moisture;
cost/ton; and availability of each type:

BTU           Sulfur       Ash   Moisture        Cost/ton Tons available
Eastern High Ash                    9500           0.55         31             8        \$24         100,000
Northern Appalachian               13000           2.75        8.5             7        \$45           75000
Central Appalachian                12500           0.75         12             7        \$54           25000
Big Sandy                          11500               1        15             8        \$47           25000
Southern River Basin               10400             0.6         7            14        \$40           12000

a. A utility has placed an order for a barge load of 1550 tons with a BTU requirement
of at least 11,900, and maximum of 0.715 percent sulfur, 13.5 percent ash,
and 8 percent moisture.
Develop a linear optimization model to find the least cost way of meeting this order.

b. Another utility wants 5 barges with a BTU rating of at least 12000,
and maximum of 1.6 percent sulfur, 15 percent ash, and 11 percent moisture.
Develop a linear optimization model to find the least cost way of
meeting this order (independent of part a).

a.     Minimum Cost 1550Tons = EHA(24)+NA(45)+CA(54)+BS(47)+SR(40)

subject to:
(9,500)EHA+(13,000)NA+(12,500)CA+(11,500)BS+(10,400)SR >=11,900
(.55)EHA+(2.75)NA+(.75)CA+(1.0)BS+(.6)SR <=.715
(31)EHA+(8.5)NA+(12)CA+(15)BS+(7)SR <=13.5
(8)EHA+(7)NA+(7)CA+(8)BS+(14)SR <=8

b.     Minimum Cost 5,750 Tons = EHA(24)+NA(45)+CA(54)+BS(47)+SR(40)

subject to:
(9,500)EHA+(13,000)NA+(12,500)CA+(11,500)BS+(10,400)SR >=12,000
(.55)EHA+(2.75)NA+(.75)CA+(1.0)BS+(.6)SR <= 1.6
(31)EHA+(8.5)NA+(12)CA+(15)BS+(7)SR <=15
(8)EHA+(7)NA+(7)CA+(8)BS+(14)SR <= 11
Chapter 13: Optimization Modeling

10.   A South American honey farm makes five types of honey: cream, filtered, pasteurized,
mélange (a mixture of several types), and strained, which are sold in 1 or 0.5 kilogram
glass containers, one and 0.75 kg plastic containers, or in bulk. Key data are shown below

Selling prices (Chilean pesos)

.75 kg plastic              1 kg plastic   .5 kg glass      1 kg glass   bulk/kg
cream (a)                        744                       880           760             990       616
filtered (b)                     635                       744           678             840       521
pasteurized (c)                  696                       821           711             930       575
mélange (d)                      669                       787           683             890       551
strained (e)                     683                       804           697             910       563

Minimum demand

.75 kg plastic              1 kg plastic   .5 kg glass      1 kg glass
cream (a)                   300                            250           350             200
filtered (b)                250                            240           300             180
pasteurized (c)             230                            230           350             300
mélange (d)                 350                            300           250             350
strained (e)                360                            350           250             380

Maximum demand
.75 kg plastic                 1 kg plastic   .5 kg glass      1 kg glass
cream (a)                550                               350           470             310
filtered (b)             400                               380           440             300
pasteurized (c)          360                               390           490             400
mélange (d)              530                               410           390             430
strained (e)             480                               420           380             500

Package costs (Chilean pesos)
.75 kg plastic            1 kg plastic      .5 kg glass      1 kg glass
91                    112              276             351

Harvesting and production costs for each product per kilogram in pesos are
cream (a)                      322
filtered (b)                   255
pasteurized (c)                305
mélange (d)                    272
strained (e)                   287

Develop a linear optimization model to maximize profit.
Maximize Profit = Sales - (Packaging + Harvesting)

Cream Sales = a(744)+a(880)+a(760)+a(990)+a(616)
Filtered Sales = b(635)+b(744)+b(678)+b(840)+b(521)
Pasteurized Sales = c(696)+c(821)+c(711)+c(930)+c(575)
Melange Sales = d(669)+d(787)+d(683)+d(890)+d(551)
Strained Sales = e(683)+e(804)+e(697)+e(910)+e(563)

Package Costs

Cream = a(91)+a(112)+a(276)+a(351)+a(0)
Filtered = b(91)+b(112)+b(276)+b(351)+b(0)
Pasteurized = c(91)+c(112)+c(276)+c(351)+c(0)
Melange = d(91)+d(112)+d(276)+d(351)+d(0)
Strained = e(91)+e(112)+e(276)+e(351)+e(0)

Harvest and Production Costs = a(322) + b(255) + c(305) + d(272) + e(287)

Subject to:
Minimum Demand                       Maximum Demand
.75 kg plastic                       .75 kg plastic
300          cream (a)               550
250         filtered (b)             400
230       pasteurized (c)            360
350        mélange (d)               530
360        strained (e)              480

Minimum Demand                       Maximum Demand
1 kg plastic                          1 kg plastic
250          cream (a)                350
240         filtered (b)              380
230       pasteurized (c)             390
300        mélange (d)                410
350        strained (e)               420

Minimum Demand                       Maximum Demand
.5 kg glass                           .5 kg glass
350         cream (a)                 470
300        filtered (b)               440
350      pasteurized (c)              490
250       mélange (d)                 390
250       strained (e)                380

Minimum Demand                       Maximum Demand
1 kg glass                            1 kg glass
200        cream (a)                  310
180       filtered (b)                300
300   pasteurized (c)   400
350    mélange (d)      430
380    strained (e)     500
Chapter 13: Optimization Modeling

Jeremy Haller of Haller’s Pub & Brewery, has compiled data describing the amount of different
ingredients and labor resources needed to brew six different types of beers that the brewery makes.
He also gathered financial information and estimated demand over a 26-week forecast horizon.
These data are shown in the table below. The profits for each batch of each type of beer are:

Light Ale:               \$3,925.78
Golden Ale:              \$4,062.75
Freedom Wheat:           \$3,732.34
Berry Wheat:             \$3,704.49
Dark Ale:                \$3,905.79
Hearty Stout:            \$3,490.22

These values incorporate fixed overhead costs of \$7500 per batch. Use the data to validate the
profit figures and develop a linear optimization model to maximize profit.

Amounts for one batch (14 Barrels - 30 kegs - 4350 pints) of beer
Berry                                               Cost per
Light Ale    Golden Ale       Freedom Wheat Wheat             Dark Ale    Hearty Stout Availability Unit
A              G                FW             BW            DA           HS
Percent Demand                 27.00            22.00            19.00         10.00        11.00          11.00
American 2- Row
Grain (lb.)                  525.00           400.00           375.00        350.00       450.00       375.00       30,000       \$0.35
American 6- Row
Grain (lb.)                                   125.00           125.00        150.00       250.00       225.00        8,000       \$0.40
American Crystal
Grain (lb.)                  175.00                                                                    175.00        5,000       \$0.42
German Vienna
Grain (lb.)                  125.00                            200.00        175.00        50.00                     5,000       \$0.45
Flaked Barley (lb.)           75.00                            150.00        150.00                     75.00        5,000       \$0.47
Light Dry Malt
Extract (lb.)                 35.00                             45.00         50.00        25.00                     2,000       \$0.37
Hallertauer Hops
(lb.)                          4.00             3.00              2.00         2.00          8.00                      500       \$0.32
Kent Golding Hops
(lb.)                                           1.00                                         4.00        4.00          500       \$0.29
Tettnanger Hops
(lb.)                          4.00                               2.00                       4.00        2.00          500       \$0.31
Brewing Labor
(hrs.)                        70.00            72.00            81.00         83.00        75.00        96.00        4,032      \$18.00
Average # of Pints
per batch                      4350             4350             4350          4350         4350         4350

Beer Price (per pint)         \$3.00            \$3.00            \$3.00         \$3.00        \$3.00        \$3.00
Average Demand
(pints per week)              2,153            1,755            1,515           798          877          877
Average Demand
(batches per week)            0.495            0.403            0.348         0.183        0.202        0.202
Average Demand
(batches per 26
weeks)                       12.870           10.487            9.057         4.767        5.243        5.243

Yearly Gross            A          G                   FW            BW        DA         HS
\$3,925.78        \$4,062.75         \$3,732.34 \$3,705.11 \$3,905.79    \$3,490.22 This equals the data provided.

Maximize .27A+.22G+.19FW+.1BW+.11DA+.11HS

Constraints

(525*A)+(400*G)+(375*FW)+(350*BW)+(450*DA)+(375*HS)<=30,000
(125*G)+(125*FW)+(150*BW)+(250*DA)+(225*HS)<=8000
(175*A)+(175*HS)<=5000
(125*A)+(200*FW)+(175*BW)+(50*DA)<=5000
(75*A)+(150*FW)+(150*BW)+(75*HS)<=5000
(35*A)+(45*FW)+(50*BW)+(25*DA)<=2000
(4*A)+(3*G)+(2*FW)+(2*BW)+(8*DA)<=500
(1*G)+(4*DA)+(4*HS)<=500
(4*A)+(2*FW)+(4*DA)+(2*HS)<=500
(70*A)+(72*G)+(81*FW)+(83*BW)+(75*DA)+(96*HS)<=4032
Chapter 13: Optimization Modeling

12. Larry Doyle was recently named by Governor McGraw as campaign director for his upcoming reelection
campaign. Governor McGraw thinks that if he can get his message to 2 million people in the state, he
has a good chance to win a large chunk of votes at the state Libertarian Party Convention. Larry has

Voter Exposure              Cost per                   Maximum
Medium                     per \$1,000 spent            Insertion                   Units
Prime time TV                10,000                     \$5,000                       60
Non-prime time TV             4,000                     \$4,000                       60
Newspaper                     1,500                     \$2,000                      120
Billboards                     750                      \$1,000                      150

Governor McGraw has a campaign fund of \$6 million available, which according to state election law,
cannot be exceeded. As a traditionalist, Governor McGraw has specified that he wants to place at least
1 ad in each of the 50 largest newspapers in the state (just before it is time for them to make their
editorial recommendations to voters). Formulate a linear model to maximize voter exposure.

PTV = number of prime time TV ads
NTV = number of non-prime time TV ads
N = number of newspaper ads
B = number of billboard ads

Max 10,000PTV + 4,000NTV + 3,500R + 1,500N + 750B
subject to: 5,000PTV + 4,000NTV + 3,000R + 2,000N + 1,000B <= 6,000,000
PTV <= 60
NTV <= 60
R <= 100
N <= 120
B <= 150
N >= 50
PTV, NTV, R, N, B >= 0
Chapter 13: Optimization Modeling

13.   An international graduate student will receive a \$28,000 foundation scholarship and a
partial tuition scholarship. He must pay \$1500 in tuition for each of the autumn, winter,
and spring quarters, and \$500 in the summer, which are due on the first day of September,
December, March, and May, respectively. Living expenses are estimated to be \$1500 per month,
payable on the first day of the month. The foundation will pay him \$18000 on August 1,
and the remainder on May 1. To earn as much interest as possible,
the student wishes to invest the money. Three types of investments are available at his bank:
a 3-month CD, earning 0.75 percent (net 3-month rate); a 6-month CD, earning 1.9 percent;
and a 12-month CD, earning 4.2 percent. Develop a linear optimization model
to determine how can he best invest the money and meet his financial obligations?

Data:
3 Month CD       6 Month CD     12 Month CD
Return          0.75%           1.90%           4.20%

Net
Month      Expenditures Cash available
Aug             1           1,500        16,500
Sep             2           3,000        13,500
Oct             3           1,500        12,000
Nov             4           1,500        10,500
Dec             5           3,000         7,500      Starting Cash           18,000
Jan             6           1,500         6,000           mid cash           10,000
Feb             7           1,500         4,500        Min allowed             0
Mar             8           3,000         1,500
Apr             9           1,500           0
May            10           2,000         8,000
Jun            11           1,500         6,500
Jul            12           1,500         5,000

Model:                                                                        Cash       Int Earned
Month      3 Month CD       6 Month CD     12 Month CD        Balance        3 Month CD
1          1,000.00         1,500.00       5,000.00         9,000.00
2          1,000.00                                         5,000.00
3                                                           3,500.00            1007.5
4                                                           3,007.50            1007.5
5          1,000.00                                           15.00                  0
6           500.00         1,000.00                         7,015.00                 0
7                                                           7,043.50            1007.5
8          1,000.00                                         4,051.00            503.75
9                                                           3,054.75                 0
10                                                           1,054.75            1007.5
11                                                            562.25                  0
12                                                             81.25                  0
Ending          5,291.25

Model assumes complete withdrawal after investment (Total interest earned means investment interest)
Total
6 Month CD    12 Month CD Int Earned

1007.5
1007.5
0
1528.5                   1528.5
0                   1007.5
0                   503.75
0                        0
0                   1007.5
1019                     1019
0          5210       5210

stment interest)
Chapter 13: Optimization Modeling

14. You have been given \$1 million by a deceased relative. You plan to invest this money (in varying
amounts) in up to four long-term investment instruments (stocks, bonds, savings, and real estate). You
evaluate investments in stocks and bonds at the beginning of each of the next six years. Each dollar
invested in stocks at the beginning of the year is expected to return an average of \$1.15 in time for
immediate reinvestment one year later. Each dollar invested in bonds at the beginning of each year is
expected to return \$1.30 three years later (a profit of 30% in three years). Credit Union Savings returns
\$1.05 one year later, and each dollar invested in real estate is expected to return \$1.30 four years later.
A maximum of \$200,000 can be invested in real estate in any one year.

You want to diversify your investments to minimize risk. The total invested in stocks in a given year
should not exceed 30 percent of the total investment in the other alternatives for that year. At least 25
percent of the total investment should be in credit union savings certificates. You also want to have
\$150,000 available in cash (which can be immediately reinvested) at the end of the third year as leverage
in negotiating with your relatives. Develop a model to maximize the cash on hand at the end of the sixth
year.

Si = \$ invested in stocks at beginning of year i, i = 1-6
Bi = \$ invested in bonds at beginning of year i, i = 1-4
Ci = \$ invested in credit union savings at beginning of year i, i = 1-6
Ri = \$ invested in real estate at beginning of year i, i = 1-3

Max 1.15S6 + 1.30B4 + 1.05C6 + 1.30R3
subject to:   S1 + B1 + C1 + R1 = 1,000,000
S2 + B2 + C2 + R2 = 1.15S1 + 1.05C1
S3 + B3 + C3 + R3 = 1.15S2 + 1.05C2
S4 + B4 + C4= 1.15S3 +1.30B1+ 1.05C3
S5 + C5 = 1.15S4 +1.30B2+ 1.05C4 + 1.30R1
S6+ C6 = 1.15S5 +1.30B3+ 1.05C5 + 1.30R2

R1 <= 200,000
R2 <= 200,000
R3 <= 200,000

S1 <= .3 (B1 + C1 + R1)
S2 <= .3 (B2 + C2 + R2)
S3 <= .3 (B3 + C3 + R3)
S4 <= .3 (B4 + C4)
S5 <= .3 (C5 )
S6 <= .3 (C6 )

C1 >= .25(S1 + B1 + C1 + R1)
C2 >= .25(S2 + B2 + C2 + R2)
C3 >= .25(S3 + B3 + C3 + R3)
C4 >= .25(S4 + B4 + C4)
C5 >= .25(S5 + C5)
C6 >= .25(S6 + C6)

1.15S3 +1.30B1+ 1.05C3 >= 150,000

all Si, Bi, Ci, Ri >= 0
Chapter 13: Optimization Modeling

15. Liquid Gold, Inc. transports radioactive waste from nuclear power points to disposal sites around and
about the country. Each plant has an amount of material that must be moved each period. Each site
has a limited capacity per period. The cost of transporting between sites is given below (some
combinations of plants and storage sites are not to be used, and no figure is given).

cost to site:
Plant        Material       S51            S62           S73         S87           Site       Capacity
P1          20,876          105           86             x          23            S51        285,922
P2          50,870           86           58            41           x            S62        308,578
P3          38,652           93           46            65          38            S73        111,995
P4          28,951          116           27            94           x            S87        208,555
P5          87,423           88           56            82          89
P6          76,190          111           36            72           x
P7          58,237          169           65            48           x

Develop a transportation model for this problem.

xij = amount shipped from plant Pi to site j

Site           j
S51            1
S62            2
S73            3
S87            4

Min 105x11 + 86x12 + 23x14+ 86x21 + 58x22 + 41x23 + 93x31 + 46x32 + 65x33 + 38x34
+ 116x41 + 27x42 + 94x43 + 88x51 + 56x52 + 82x53 + 89x54 + 111x61
+ 36x62 + 72x63 + 169x71 + 65x72 + 48x73
subject to: x11 + x12 + x14 = 20,876
x21 + x22 + x23 = 50,870
x31 + x32 + x33 + x34 = 38,652
x41 + x42 + x43 = 28,951
x51 + x52 + x53 + x54 = 87,453
x61 + x62 + x63 = 76,190
x71 + x72 + x73 = 58,237
x11 + x21 + x31 + x41 + x51 + x61 + x71 <= 285,922
x12 + x22 + x32 + x42 + x52 + x62 + x72 <= 308,578
1
x23 + x33 + x43 + x53 + x63 + x73 <= 111,95511,995
x14 + x34 + x54 <= 208,555
all xij >= 0
Chapter 13: Optimization Modeling

16 A company in Victoria, TX produces bottles of aspirin products. Sales prices are given below:

PRODUCT                 SALES PRICE
Super Seltzer              \$3.00
Capsules                   \$3.50
Cheap Seltzer              \$2.00
Tablets                    \$2.50

The company ships these products to two distributors, located at Hearne, TX and Cuero, TX. There is
unlimited demand at each distributor. Shipping costs per bottle and contracted minimum quantities for
each distributor are:

Shipping cost/bottle                          Hearne       Cuero
\$0.21        \$0 .22

Minimum demand             Super Seltzer        700        1000
Capsules         800        1500
Cheap Seltzer       1000         800
Tablets        1800        5000

Raw material costs and maximum available quantities are given below:

Cost/ounce Maximum ounces
Acetylsalicylic acid     \$0.60      50,000
Sodium        \$0.30      25,000

The production costs per bottle and raw materials required per bottle are:

Production     Ounces      Ounces
cost/bottle   Acet. Acid   Sodium
Super Seltzer       \$0.25           2          3
Capsules         0.35           4          0
Cheap Seltzer        0.15           2          2
Tablets          0.1           3          0

Total number of bottles of capsules produced
is to be no more than 20% of the total number of bottles produced.

SSH = bottles of super seltzer shipped to Hearne
CH = bottles of capsules shipped to Hearne
CSH = bottles of cheap seltzer shipped to Hearne
TH = bottles of tablets shipped to Hearne
SSC = bottles of super seltzer shipped to Cuero
CC = bottles of capsules shipped to Cuero
CSC = bottles of cheap seltzer shipped to Cuero
TC = bottles of tablets shipped to Cuero

Max .44SSH + .54CH - .16 CS + .39TH + .43SSC + .53CC - .17CSC + .38TC
subject to: 2(SSH + SSC) + 4(CH + CC) + 2(CSH + CSC) + 3(TH + TC) <= 50,000
3(SSH + SSC) + 2(CSH + CSC) <= 25,000
CH + CC <= .2 (SSH + SSC + CH + CC + CSH + CSC + TH + TC)
SSH >= 700
CH >=800
CSH >= 1,000
TH >= 1,800
SSC >= 1,000
CC >= 1,500
CSC >= 800
TC >= 5,000
SSH, SSC, CH, CC, CSH, CSC, TH, TC >= 0

Note: the objective function coefficient are computed by subtracting raw material,
production, and shipping costs from the sale price.
Chapter 13: Optimization Modeling

17.   Mirza Manufacturing makes four electronic products, each of which is comprised of
3 main materials: magnet, wiring, and casing. The products are shipped to three distribution centers
in North America, Europe, and Asia. Marketing has specified that no location should receive
more than 3 times the demand. The material cost/unit are: magnet – \$0.59, wire – \$0.29,
and casing – \$0.31. The table below shows the number of units of each material
required in each unit of end product and the production cost per unit.

Product               Production Cost/unit Magnets             Wire          Casing
A                                     \$0.25                4             2            2
B                                     \$0.35                3             1            3
C                                     \$0.15                2             2            1
D                                     \$0.10                8             3            2

Min Demand
Product               NA                        EU             Asia
A                                        850            900            100
B                                        700            200            500
C                                       1100            800            600
D                                       1500           3500           2000

Max Demand
Product               NA                        EU             Asia
A                                       2550           2700            300
B                                       2100            600           1500
C                                       3300           2400           1800
D                                       4500          10500           6000

Shipping cost/unit
Product               NA                        EU             Asia
A                                         0.2           0.25          0.35
B                                         0.2            0.2           0.3
C                                         0.2            0.2           0.3
D                                         0.2            0.2          0.25

Unit Sales Revenue
Product               NA                        EU             Asia
A                                           4            4.5          4.55
B                                         3.7            3.9          3.95
C                                         2.7            2.9           2.4
D                                         6.8            6.5           6.9

Available Raw Material
Magnet                                120000
Wire                                   50000
Casing                                 40000

Develop an appropriate linear optimization model to maximize net profit.
Product A Needs   4M+2W+2C
Product B Needs   3M+1W+3C
Product C needs   2M+2W+1C
Product D needs   8M+3W+2C

A(4M)+B(3M)+C(2M)+D(8M)<=120000
A(2W)+B(1W)+C(2W)+D(3W)<=5000
A(2C)+B(3C)+(1C)+(2C)<=40000
A NA <= 3*2550
A EU <=3*2700
A ASIA <= 3*300
B NA <= 3*2100
B EU <=3*600
B ASIA <= 3*1500
C NA <= 3*3300
C EU <=3*2400
C ASIA <= 3*1800
D NA <= 3*4500
D EU <=3*10500
D ASIA <= 3*6000
A NA > 850
A EU > 900
A ASIA>100
B NA >700
B EU >200
B ASIA > 500
C NA> 1100
C EU>800
C ASIA >600
D NA > 1500
D EU > 3500
D ASIA > 2000
Minimize A(PC)+B(PC)+C(PC)+D(PC)
Minimize A(SC)+B(SC)+C(SC)+D(SC)
Maximize A(SR)+B(SR)+C(SR)+D(SR)
Chapter 13: Optimization Modeling

18.   Thermal transfer printing transfers ink from a ribbon onto paper through a combination of heat and pressure.
Different types of printers use different sizes of ribbons.
A ribbon manufacturer has forecasted demands for seven different ribbon sizes:

Size           Forecast (rolls)
60 mm                1620
83 mm                 512
102 mm                840
110 mm               2640
120 mm                502
130 mm                755
165 mm                680

The rolls from which ribbons are cut are 880 mm in length.
Scrap is valued at \$0.05 per mm.
Generate 10 different cutting patterns so that each size can be cut from at least one pattern.
Use your data to construct and solve an optimization model for finding the
number of patterns to cut to meet demand and minimize trim loss.

Number of Rolls
Pattern                 60          83         102           110          120          130       165       Scrap
A              1           1            1            2            1            1         1           0
B              2           0            0            0            0            2         3           5
C              2           1            1            5            0            0         0          25
D              3           0            0            5            0            1         0          20
E             13           1            0            0            0            0         0          17
F              5           3            2            0            1            0         0           7
G              6           0            0            0            0            0         3          25
H              8           0            0            0            1            2         0          20
I             8           1            0            1            0            0         1          42
J              9           1            0            0            2            0         0          17

Number of Feet of Product
Pattern                 60          83         102          110           120          130       165       Scrap
A             60          83         102          220           120          130       165           0
B            120           0            0           0             0          260       495           5
C            120          83         102          550             0            0         0          25
D            180           0            0         550             0          130         0          20
E            780          83            0           0             0            0         0          17
F            300         249         204            0           120            0         0           7
G            360           0            0           0             0            0       495          25
H            480           0            0           0           120          260         0          20
I           480          83            0         110             0            0       165          42
J            540          83            0           0           240            0         0          17

Minimize      0A+5B+25C+20D+17E+7F+25G+20H+42I+17J

Constraints
60 mm    1A+2B+2C+3D+13E+5F+6G+8H+8I+9J >= 1620
83 mm    1A+0B+1C+0D+1E+3F+0G+0H+1I+1J >= 512
102 mm   1A+0B+1C+0D+0E+2F+0G+0H+0I+0J >= 840
110 mm   2A+0B+5C+5D+0E+0F+0G+0H+1I+0J >= 2640
120 mm   1A+0B+0C+0D+0E+1F+0G+1H+0I+2J >= 502
130 mm   1A+2B+0C+1D+0E+0F+0G+2H+0I+09 >= 755
165 mm   1A+3B+0C+0D+0E+0F+3G+0H+1I+0J >= 680
Chapter 13: Optimization Modeling

19. The personnel director of a company that recently absorbed another firm, and is now downsizing, must
relocate four information systems analysts from recently closed locations. Unfortunately, there are only
three positions available for five people. Salaries are fairly uniform among this group (those with higher
pay were already given the opportunity to begin anew). Moving expenses will be used as the means of
determining who will be sent where. Estimated moving expenses are:

Analyst                  moving cost to:     Gary       Salt Lake City    Fresno
Arlene                                      \$8,000         \$7,000         \$5,000
Bobby                                       \$5,000         \$8,000        \$12,000
Charlene                                    \$9,000        \$15,000        \$16,000
Douglas                                     \$4,000         \$8,000        \$13,000
Emory                                       \$7,000         \$3,000         \$4,000

Model this as an integer optimization model to minimize cost and determine which analysts to relocate
to the three locations.

Xij = 1: relocate analyst i to city j
0: do not relocate analyst i to city j

Analyst          i
Arlene           1
Bobby            2
Charlene          3
Douglas           4
Emory            5

City           j
Gary           1
Salt Lake City       2
Fresno           3

Min 8,000X11 + 7,000X12 + 5,000X13 + 5,000X21 + 8,000x22 + 12,000X23
+ 9,000X31 + 15,000X32 + 16,000X33 + 4,000X41 + 8,000X42
+ 13,000X43 + 7,000X51 + 3,000X52 + 4,000X53
subject to: X11 + X12 + X13 <= 1
X21 + X22 + X23 <= 1
X31 + X32 + X33 <= 1
X41 + X42 + X43 <= 1
X51 + X52 + X53 <= 1
X11 + X21 + X31 + X 41 + X51 = 1
X12 + X22 + X32 + X 42 + X52 = 1
X13 + X23 + X33 + X 43 + X53 = 1
all Xij = 1 or 0

Arlene to Fresno
Douglas to Gary
Emory to Salt Lake City
Chapter 13: Optimization Modeling

20.   A medical device company is allocating next year’s budget among its divisions.
As a result, the R&D Division needs to determine which research and
development projects to fund. Each project requires various software
and hardware and consulting expenses, along with internal human resources.

Project                        Estimate NPV engineers          costs
1                         \$600,000               9       \$196,000
2                         \$680,000              12       \$400,000
3                         \$550,000               7         \$70,000
4                         \$400,000               4       \$180,000
5                         \$350,000               8       \$225,000
6                         \$725,000              10       \$275,000
7                         \$340,000               8       \$130,000

A budget allocation of \$1,250,000 has been approved, and 32 engineers
are available to work on the projects. The R&D group has determined that
at most one of projects 1 and 2 should be pursued, and that if project 4 is chosen,
then project 2 must also be chosen. Develop a model to select the best projects within the budget.

NPV          Eng                Costs        Decision
2,3,4,5                          \$1,980,000               31       \$875,000          1 Yes
2,3,4,7                          \$1,970,000               31       \$780,000          1 Yes
1,5,6                            \$1,675,000               27       \$696,000          1 Yes
1,6,7                            \$1,665,000               27       \$601,000          1 Yes
2,3,4                            \$1,630,000               23       \$650,000          1 Yes
1,6                              \$1,325,000               19       \$471,000          1 Yes
2,4                              \$1,080,000               16       \$580,000          1 Yes
1,5                                \$950,000               17       \$421,000          1 Yes
1,7                                \$940,000               17       \$326,000          1 Yes
1                                \$3,645,000               58     \$1,476,000          0 No
2,3,4,5,6,7                      \$3,045,000               49     \$1,280,000          0 No
2,3,4,5,6                        \$2,705,000               41     \$1,150,000          0 No
2,3,4,6,7                        \$2,695,000               41     \$1,055,000          0 No
1,3,5,6,7                        \$2,565,000               42       \$896,000          0 No
2,4,5,6,7                        \$2,495,000               42     \$1,210,000          0 No
2,4,5,6                          \$2,155,000               34     \$1,080,000          0 No
1,5,6,7                          \$2,015,000               35       \$826,000          0 No

Project 1    Project 2    Project 3    Project 4   Project 5
NPV                             \$ 600,000 \$ 680,000 \$ 550,000 \$ 400,000 \$ 350,000
Additional Costs                     196000      400000         70000     180000      225000
Engineers                                 9           12            7           4           8

Model                                                                     1

Project selection decisions                  0             1              1            1             1
Costs               0      400000       70000    180000    225000
Engineers           0           12           7         4         8
NPV         \$   -       \$ 680,000 \$   550,000 \$ 400,000 \$ 350,000
Project 6        Project 7   Resources
\$      725,000 \$ 340,000
275000     130000      1250000
10           8           32

0           0     Total
0           0        875000
0           0             31
\$   -       \$   -       \$ 1,980,000
Chapter 13: Optimization Modeling

21.   A call center has five categories of employees: Level 1 managers who spend 20 percent
of their time in the call center and the remainder doing project work; Level 2 managers who
spend 50 percent of their time in the call center and the rest on project work; Technical Specialist 1,
who spend 50 percent of their time in the call center; 30 percent on project work, and the remainder
on other tasks; Technical Specialist 2, who spend 80 percent in the call center, and 10 percent
on project work; and outsourced contractors, who spend 100 percent of their time on call center work.
Based on analysis of data at a call center, the call center manager has determined that she needs
at least 5 full time equivalent people on project work and 30 on call center work.
Level 1 managers earn \$100,000 per year; Level 2 managers, \$90,000;
Technical Specialists earn \$65,000 for both types; and contractors are paid \$75,000.
What is the best staffing mix for this operation?

need 5 for project
need 30 for Call Center

Call             Project         Other            Salary
L1M                        0.2             0.8                              100000
L2M                        0.5             0.5                               90000
TS1                        0.5             0.3              0.2              65000
TS2                        0.8             0.1              0.1              65000
Cont                         1                                               75000

L1M              L2M             TS1              TS2                 Cont
#needed                      1            1                   1                  34               2
Call                       0.2          0.5                 0.5                27.2               2           30.4
Project                    0.8          0.5                 0.3                 3.4               0              5
Salaries               100000        90000               65000             2210000           150000       2615000
Chapter 13: Optimization Modeling

22.   A software support division of a major corporation has 8 projects that can be performed.
Each project requires different amounts of development time and testing time.
In the coming planning period, 1190 hours of development time and 1000 hours of testing time
are available, based on the skill mix of the staff. The internal transfer price
(revenue to the support division) and the times required for each project are shown below.
Which projects should be selected to maximize revenue?

Project                              Development Time        Testing Time     Transfer Price
1                     80                67           \$23,520
2                    248               208           \$72,912
3                     41                34           \$12,054
4                    110                92           \$32,340
5                    240               202           \$70,560
6                    195               164           \$57,232
7                    269               226           \$79,184
8                    110                92           \$32,340

Project 1           Project 2     Project 3     Project 4
Transfer Price                        \$          23,520 \$      72,912 \$      12,054 \$ 32,340
Development Time                                      80           248            41       110
Testing Time                                          67           208            34         92

Model

Project selection decisions                              0               1                  0           1
Development Time                                         0             248                  0         110
Testing Time                                             0             208                  0          92
Transfer Price                       \$               -       \$     72,912 \$             -       \$ 32,340
Project 5 Project 6 Project 7 Project 8 Resources
\$ 70,560 \$ 57,232 \$ 79,184 \$ 32,340
240      195       269       110           1190
202      164       226         92          1000

1        1        1        1        Total
240      195      269      110             1172
202      164      226       92               984
\$ 70,560 \$ 57,232 \$ 79,184 \$ 32,340 \$         344,568
Chapter 13: Optimization Modeling

23.   Toward the end of the fiscal year, a company division has \$75,000 remaining in its budget,
with a “use it or lose it” policy. Four potential projects are available for consideration.
Each project requires a fixed amount of cash outlay, project management labor time,
client labor time, and analyst time. Relevant data are shown in the table below.
Sixty five hours of project management time, 30 hours of client labor time,
and 85 hours of analyst time are available.

Determine which projects should be selected to maximize the total return.

Project 1       Project 2        Project 3      Project 4
Return                                   \$80,000            70,000         50,000         40,000
Cash outlay                              \$25,000            40,000         12,000         27,500
Project management labor                       25               20             15             12
Client labor                                   15               12             10              8
Analyst labor                                  60               50             35             30

Data                                                                                          Available
Project 1       Project 2       Project 3     Project 4     Resources
Expected Return (NPV)              \$      80,000 \$        70,000 \$      50,000 \$      40,000
Cash requirements                  \$      25,000 \$        40,000 \$      12,000 \$      27,500 \$     75,000
Project management labor                       25              20            15            12           65
Client labor                                   15              12            10             8           30
Analyst labor                                  60              50            35            30           85

Model

Project selection decisions                       0               1                1               0       Total
Cash Used                          \$          -       \$     40,000 \$         12,000 \$          -       \$    52,000
Project management labor                          0              20               15               0             35
Client labor                                      0              12               10               0             22
Analyst labor                                     0              50               35               0             85

Return                             \$          -       \$     70,000    \$      50,000   \$        -       \$   120,000
Chapter 13: Optimization Modeling

24.   Many high technology products such as crystals and alloys can be manufactured more efficiently
in the weightless environment of earth orbit. You are planning production operations for a space flight.
Five products are being considered. There are unit profits, volumes, weights, labor hours per unit,
and maximum demands. Model this problem as an integer optimization problem.

Products                             Alloy1       Alloy2     Crystal 1    Crystal2    Interferon
Profit (\$/unit)                        10          1.7         3.5          1.6           2.6
Volume (CF/unit)                        9            3          10           7            13
Weight (lb/unit)                       59           18          26          26            10
Labor (/unit)                         2.2          0.5         0.7          0.2           1.1
Demand ()                             22           69          90          40            85

Volume available: 600 CF
Weight allowable: 2100 lb
Labor available: 40 hr

A1 = units of Alloy1
A2 = units of Alloy2
C1 = units of Crystal1
C2 = units of Crystal2
I = units of Interferon

Max 10A1 + 1.7A2 + 3.5C1 + 1.6C2 + 2.6I
subject to: 9A1 + 3A2 + 10C1 + 7C2 + 13I <= 600
59A1 + 18A2 + 26C1 + 26C2 + 10I <= 2100
2.2A1 + 0.5A2 + 0.7C1 + 0.2C2 + 1.1I <= 40
A1 <= 22
A2 <= 69
C1 <= 90
C2 <= 40
I <= 85
A1, A2, C1, C2, I >= 0 and integer
Chapter 13: Optimization Modeling

25.   Soapbox is a local band that plays classic and contemporary rock. The band members charge \$600
for a three-hour gig. They would like to play at least 30 gigs per year, but need to determine the best way
to promote themselves. The most they are willing to spend on promotion is \$5000.
The possible promotion options are:

·    Playing free gigs
·    Making a demo CD
·    Hiring an agent
·    Handing out fliers
·    Creating a web site

Each free gig costs them about \$150 for travel and equipment, but generates about 4 paying gigs.
A high-quality studio demo CD should help the band book about 20 gigs, but will cost about \$750.
A demo CD made on home recording equipment will only cost \$250, but may only result in
6 bookings. A good agent will get the band about 15 gigs, but agents typically charge
20 percent of all revenue, whether booked by the agent or not. The band can create a web site
for \$150 and would expect to generate 6 gigs from this exposure. They also estimate that
they may book one gig for every 100 fliers they hand out, which would cost 8 cents each.
Develop an optimization model to find the best promotion strategy to maximize their profit.

Gigs Gen
Promotion Budget                5000          30
"Free" Costs                     150           4
Pro CD                           750          20
Home CD                          250           6
Agent                           20%           15
Web                              150           6
Fliers                             8           1

Free Gigs Pro CD    Home CD Agent                     Web         Fliers
Number              1         1        1                      0           1             1
Costs             150       750      250                      0         150             8           1308
Gig Gen             4        20        6                      0           6             1             37
Revenue              22200
Chapter 13: Optimization Modeling

26.   Dannenfelser Design works with clients in three major project categories:
architecture, interior design, and combined. Each type of project requires
an estimated number of hours for different categories of
employees as shown in the table below.

Architecture        Interior Design Combined Hourly Rate
Principal                             15                  5       18     \$115
Sr. Designer                          25                 35       40      \$85
Draftsman                             40                 30       60      \$60

In the coming planning period, 184 hours of Principal time, 414 hours of
Sr. Designer time, 588 hours of draftsman time, and 72 hours of administrator
time are available. Profit per project average \$1290 for architecture,
\$1110 for interior design, and \$1878 for combined projects.
The firm would like to work on at least one of each type of project for
exposure among clients. Assuming that the firm has more demand than
they can possibly handle, find the best mix of projects to maximize profit.

Architecture        Interior Design Combined Hourly Rate Time Available
Principal                           15                    5       18     \$115         184
Sr. Designer                        25                   35       40      \$85         414
Draftsman                           40                   30       60      \$60         588
Administrator                        5                    5        8      \$40          72
Profit                            1290                1110      1878

Architecture        Interior Design Combined Totals
Number of Projects                 10                     2        1        13
Principal                         150                    10       18       178
Sr. Designer                      250                    70       40       360
Draftsman                         400                    60       60       520
Profit                          12900                 2220      1878     16998
Chapter 13: Optimization Modeling

27. A young entrepreneur has invented a new an air--adjustable basketball shoe with pump,
similar to those advertised widely by more expensive brand names. He contacted a supplier
of Victor basketball shoes, a little-known brand with low advertising. This supplier would
provide shoes at the nominal price of \$6 per pair of shoes. He needs to know the best price
at which to sell these shoes. As a business student with strong economics training,
he remembered that the volume sold is affected by the product’s price—the higher the price,
the lower the volume. He asked his friends and acquaintances what they would pay
for a premium pair of basketball shoes that were a “little off-brand.”
Based on this data, he developed the formula

Volume = 1000 - 20 Price

There are some minor expenses involved, including a \$50 fee for selling shoes in the neighborhood (a
fixed cost), as well as his purchase price of \$6 per shoe. Develop an appropriate objective function to
find the optimal price level.

Profit = (Price - 6) x (1000 - 20Price) - 50
2
= -20Price + 1120Price - 6050
Chapter 13: Optimization Modeling

28. The entrepreneur in the previous problem did very well selling Victor shoes. His shoe supplier told
him of a new product,Top Notch, that was entering the market. This shoe would be a product substitute
for Victors, so that the higher the price of either shoe, the greater the demand for the other.
He interviewed more potentialclients to determine price response and cross-elasticities.
This yielded the following relationships:

Volume of Victors = 1000 - 20 Pv + 1 Pa
Volume of Top Notch Shoes = 800 + 2 Pv - 18 Pa

where Pv = price of Victors, and Pa = price of Abibas. Develop a new profit function to include this
information.

Profit Victors = (Pv - 6) x (1000 - 20 Pv + 1 Pa) - 50
2
= -20Pv + 1120Pv + PvPa - 6Pa - 6050

Profit Abibas = (Pa - 6) x (800 + 2Pv - 18Pa) - 50
2
= -18Pa + 908Pa + 2PvPa - 12Pv - 4850

2      2
Total Profit = -20Pv - 18Pa + 1108Pv + 902Pa + 3PvPa - 10,900
Chapter 13: Optimization Modeling

29. The Hal Chase Investment Planning Agency is in business to help investors optimize their return from
investment, to include consideration of risk. Hal deals with three investment mediums: a stock fund, a
bond fund, and his own Sports and Casino Investment Plan (SCIP). The stock fund is a mutual fund
investing in openly traded stocks. The bond fund focuses on the bond market, which has a much stabler
but lower expected return. SCIP is a high-risk scheme, often resulting in heavy losses, but occasionally
coming through with spectacular gains. Average returns, their variances, and covariances, are given
below.

Stock         Bond           SCIP
Average return                 0.148          0.06          0.152
Variance                     0.014697      0.000155       0.160791
Covariance with Stock                      0.000468      -0.002222
Covariance with Bond                                     -0.000227

Negative covariance indicates that SCIP tends to move in the opposite direction as stocks or bonds.
Develop a portfolio optimization model for this situation.

Stock = percentage of Stock in portfolio
Bond = percentage of Bond in portfolio
SCIP = percentage of SCIP in portfolio

Min .014697Stock2 + .000155Bond2 + .160791SCIP2 + 2(.000468)StockBond
+ 2(-.002222)StockSCIP + 2(-.000227)BondSCIP
subject to: Stock + Bond + SCIP = 1
.148Stock + .06Bond + .152SCIP >= .10
Stock, Bond, SCIP >= 0

Note: the second constraint assumes a minimum yield of .10.
Chapter 13: Optimization Modeling

30. Develop a spreadsheet model for the media selection problem described in this chapter.

Minimum Units         0           12            6
Maximum Units         15            -           30

Number of ads       15            18          30
Total
Budget           7,500        36,500        6,000        50,000

Total
Exposure        30,000        63,875        81,000       174,875

Note: use decision variables (changing cells) for Max/Min Units.
Chapter 13: Optimization Modeling

31. Develop a spreadsheet model for the process selection problem described in this chapter.

Outsourcing
Demand        Mill cost      cost
Data:            Fabric       (yards)      (\$/yard)     (\$/yard)
1          45,000         0.65         0.85
2          76,500         0.61         0.75
3          10,000         0.50         0.65

Dobbie       Dobbie      Regular      Regular
loom        loom         loom          loom
capacity     capacity     capacity     capacity
Fabric   (yards/hour) (hours/yard) (yards/hour) (hours/yard)
1            4.7       0.213          0.0        99,999
2            5.2       0.192          5.2         0.192
3            4.4       0.227          4.4         0.227
Hours available   6,552                     32,760

Model:                        Fabric       Dobbie       Regular      Outsourcing
1          30,794         0            14,205
2            0          76,500            0
3            0          10,000            0
Total
Fabric 1: yards produced     30,794         0           14,205      45,000
Fabric 2: yards produced        0         76,500          0         76,500
Fabric 3: yards produced        0         10,000          0         10,000
Dobbie: hours used       6,552
Regular: hours used                   32,760

Fabric 1: cost    20,016          0          12,075      32,091
Fabric 2: cost      0          46,665          0         46,665
Fabric 3: cost      0           5,000          0          5,000
Total cost   83,756
Chapter 13: Optimization Modeling

32. Develop a spreadsheet model for the blending problem described in this chapter.

Data:
Ingredient         Protein     Fat          Fiber       Cost/lb.
Sunflower seeds         0.169     0.260         0.290        0.22
White millet       0.120     0.041         0.083        0.19
Kibble corn       0.085     0.038         0.027        0.10
Oats       0.154     0.063         0.024        0.10
Cracked corn         0.085     0.038         0.027        0.07
Wheat         0.120     0.017         0.023        0.05
Safflower       0.180     0.179         0.288        0.26
Canary grass seed        0.119     0.040         0.109        0.11
Requirement          0.13      0.15          0.14

Total weight     1

Model:                          Pounds
in Mixture     Protein      Fat         Fiber        Cost
Sunflower seeds         0.547        0.09       0.14         0.16         0.12
White millet       0.000        0.00       0.00         0.00         0.00
Kibble corn       0.000        0.00       0.00         0.00         0.00
Oats       0.000        0.00       0.00         0.00         0.00
Cracked corn         0.000        0.00       0.00         0.00         0.00
Wheat         0.453        0.05       0.01         0.01         0.02
Safflower       0.000        0.00       0.00         0.00         0.00
Canary grass seed        0.000        0.00       0.00         0.00         0.00
Total      1.000        0.15       0.15         0.17         0.14
Chapter 13: Optimization Modeling

33. Develop a spreadsheet model for the production planning problem described in this chapter.

Data:                        Forecasted     Labor            Labor
Sales       per Hour        per Unit
Autumn          150          5.50         11.00
Winter          400          7.00         14.00
Spring           50          6.25         12.50

Holding cost      1.20

Model:
Production    Inventory     Available     Cost
Autumn              550           400      150         6,530
Winter                 0             0     400           0
Spring                50             0      50          625
7,155
Chapter 13: Optimization Modeling

34. Develop a spreadsheet model for the cash management problem described in this chapter.

Data:
1 Month CD 3 Month CD 6 Month CD
Return          0.50%      1.75%      2.30%

Net
Month     Expenditures
1           50,000
2          -12,000
3           23,000
4          -20,000
5           41,000                Starting Cash     200,000
6          -13,000               Minimum Cash       100,000

Model:                                                               Cash
Month     1 Month CD 3 Month CD 6 Month CD          Balance
1         50,000.00     0.00      0.00            100,000.00
2         22,885.57  39,364.43    0.00            100,000.00
3            0.00       0.00      0.00            100,000.00
4          941.98    19,058.02    0.00            100,000.00
5            0.00       0.00      0.00            100,000.00
6         13,000.00     0.00      0.00            100,000.00
Ending           132,456.53

Note: in the model above, the end of month cash balance is re-invested in the next month.
Chapter 13: Optimization Modeling

35. Develop a spreadsheet model for the distribution center location problem described in this chapter.

Data:

Plant/D.C.       Cleveland      Baltimore       Chicago        Phoenix        Capacity
Marietta           12.60         14.35           11.52          17.58          1,200
Minneapolis         9.75         12.63            8.11          15.88           800
Fayetteville       10.41         11.54            9.87           8.32          1,500
Chico              13.88         16.95           12.51          11.64          1,500
Demand              300           500             700           1,800

Maximum new plants           1

Model:
Total
Plant/D.C.       Cleveland      Baltimore       Chicago        Phoenix        Shipped                     Capacity
Marietta            200           500              0             300           1,000                       1,200
Minneapolis         100             0            700              0             800           Build         800
Fayetteville         0              0              0            1,500          1,500           1           1,500
Chico                0              0              0              0              0             0             0
Demand met         300           500            700            1,800                          1

Total Cost =   34,101
Chapter 13: Optimization Modeling

36. Develop a spreadsheet model for the cutting stock problem described in this chapter.

Size of end item

Pattern     12”           15”         30”           Scrap
1           0             6           0            10
2           0             0           3            10
3           8             0           0            4
4           2             1           2            1
5           7             1           0            1

Demands this week are 870 12” rolls, 450 15” rolls, and 650 30’ rolls. The problem is to develop
a model that will determine how many 100” rolls to cut into each of the five patterns
in order to meet demand and scrap.

12"                 870
15"                 450
30"                 650                                            Change the number cell value

Pattern     12”           15”         30”           Scrap          Number
1           0             342         0            570                    57
2           0             0           630          2100                  210
3           800           0           0            400                   100
4           80            57          20           10                     10
5           0             57          0            0
Demand              880         456          650            3080
Chapter 13: Optimization Modeling

37. Develop a spreadsheet model for the computer configuration described in this chapter.

Option                        Variable       Utility
slower processor              Xp1            0.2
faster processor              Xp2            0.7
fastest processor             Xp3            0.9
2 year warranty               Xw1            0.5
3 year warranty               Xw2            0.55
512MB memory                  Xm1            0.7
1 GB memory                   Xm2            0.8
30 GB hard drive              Xd1            0.1
60 GB hard drive              Xd2            0.3
80 GB hard drive              Xd3            1
CD only                       Xc1            0
CDRW                          Xc2            0.95
DVD/RW                        Xc3            0.45
8 cell battery                Xb             0.15
music and photo software      Xs             0.85

Base Unit                             1000

The following options are available, along with price changes:

Modular bay: CD-ROM only (subtract \$39): CD/RW (add \$79), DVD/RW (add \$179)
Battery: 8-cell Lithium ion (add \$59)
Enhanced music and photo software (add \$79)

1       2           3
Processor                     p                        -179     100         300
Warranty                      w                          0      129         269
Memory                        m                          0       50         500
HD                            d                         -29      39          79
Mod Bay                       c                         -39      79         179
Batt                          b                         59        0            0
Music                         s                         79        0            0

Data
Base          1000   Change Data (1, 2, 3)
Processor        2
Warranty         1
Memory           1
HD               3
Mod Bay          3
Batt             2
Mus              2

Model

Base          1000
Processor      100
Warranty         0
Memory           0
HD             79
Mod Bay       179
Batt            0
Music           0
Total Price   1358
Chapter 13: Optimization Modeling

38. Develop a spreadsheet model for the distribution center location described in this chapter.
SAME AS 35
Data:

Plant/D.C.       Cleveland   Baltimore     Chicago      Phoenix     Capacity
Marietta           12.60      14.35         11.52        17.58       1,200
Minneapolis         9.75      12.63          8.11        15.88        800
Fayetteville       10.41      11.54          9.87         8.32       1,500
Chico              13.88      16.95         12.51        11.64       1,500
Demand              300        500           700         1,800

Maximum new plants          1

Model:
Total
Plant/D.C.   Cleveland       Baltimore     Chicago      Phoenix     Shipped                   Capacity
Marietta        200            500            0           300        1,000                     1,200
Minneapolis     100              0          700            0          800          Build        800
Fayetteville     0               0            0          1,500       1,500          1          1,500
Chico            0               0            0            0           0            0            0
Demand met    300            500          700          1,800                      1

Total Cost =   34,101
Chapter 13: Optimization Modeling

39. Develop a spreadsheet model for the direct marketing problem described in this chapter.

Data:
Magazine         1             2            3             4      5
Customer        Cost          900          1000         1100          1500   1500
1            10            0             0            0             0      0
2            1,4           1             0            0             1      0
3             1            1             0            0             0      0
4            5,6           0             0            0             0      1
5             5            0             0            0             0      1
6            10            0             0            0             0      0
7            2,9           0             1            0             0      0
8            5,8           0             0            0             0      1
9          1,5,10          1             0            0             0      1
10         4,6,8,10         0             0            0             1      0
11             6            0             0            0             0      0
12             3            0             0            1             0      0
13             5            0             0            0             0      1
14            2,6           0             1            0             0      0
15             8            0             0            0             0      0
16             6            0             0            0             0      0
17            4,5           0             0            0             1      1
18             7            0             0            0             0      0
19            5,6           0             0            0             0      1
20            2,8           0             1            0             0      0
21            7,9           0             0            0             0      0
22             6            0             0            0             0      0
23          3,6,10          0             0            1             0      0
24          NONE            0             0            0             0      0
25            5,8           0             0            0             0      1
26           3,10           0             0            1             0      0
27            2,8           0             1            0             0      0
28            4,7           0             0            0             1      0
29             6            0             0            0             0      0
30         3,4,5,10         0             0            1             1      1
31             4            0             0            0             1      0
32             8            0             0            0             0      0
33          1,3,10          1             0            1             0      0
34            4,5           0             0            0             1      1
35           1,5,6          1             0            0             0      1
36            1,3           1             0            1             0      0
37           3,5,8          0             0            1             0      1
38             3            0             0            1             0      0
39            2,7           0             1            0             0      0
40            2,7           0             1            0             0      0
41             7            0             0            0             0      0
42           4,5,6          0             0            0             1      1
43          NONE            0             0            0             0      0
44           5,10           0             0            0             0      1
45            1,2           1             1            0             0      0
46          7            0           0      0   0   0
47       1,5,10          1           0      0   0   1
48          3            0           0      1   0   0
49        1,3,4          1           0      1   1   0
50       NONE            0           0      0   0   0
51         2,6           0           1      0   0   0
52       NONE            0           0      0   0   0
53     2,5,8,9,10        0           1      0   0   1

Budget Limit   2500

Model:
Magazine      x              y         Budget
1        1              9           900
2        0              0            0
3        0              0            0
4        0              0            0
5        1             16          1500
6        0              0            0
7        0              0            0
8        0              0            0
9        0              0            0
10        1             11          1100
Maximize        36          3500
6      7      8     9     10
1500   1000   1200   500   1100
0      0      0     0      1
0      0      0     0      0
0      0      0     0      0
1      0      0     0      0
0      0      0     0      0
0      0      0     0      1
0      0      0     1      0
0      0      1     0      0
0      0      0     0      1
1      0      1     0      1
1      0      0     0      0
0      0      0     0      0
0      0      0     0      0
1      0      0     0      0
0      0      1     0      0
1      0      0     0      0
0      0      0     0      0
0      1      0     0      0
1      0      0     0      0
0      0      1     0      0
0      1      0     1      0
1      0      0     0      0
1      0      0     0      1
0      0      0     0      0
0      0      1     0      0
0      0      0     0      1
0      0      1     0      0
0      1      0     0      0
1      0      0     0      0
0      0      0     0      1
0      0      0     0      0
0      0      1     0      0
0      0      0     0      1
0      0      0     0      0
1      0      0     0      0
0      0      0     0      0
0      0      1     0      0
0      0      0     0      0
0      1      0     0      0
0      1      0     0      0
0      1      0     0      0
1      0      0     0      0
0      0      0     0      0
0      0      0     0      1
0      0      0     0      0
0   1   0   0   0
0   0   0   0   1
0   0   0   0   0
0   0   0   0   0
0   0   0   0   0
1   0   0   0   0
0   0   0   0   0
0   0   1   1   1
Chapter 13: Optimization Modeling

CASE: DISTRIBUTION CENTER LOCATION FOR TRACWAY

Tracway produces its most popular model of lawn tractor in its St. Louis, Greenwood, and Camarillo plants
and ships these units to major distribution centers in Atlanta, Chicago, and San Diego. To support its
growing presence in the Pacific Rim, Tracway is considering adding two distribution centers.
Locations being considered are Taiwan; Singapore; Sydney, Australia; and Auckland, New Zealand.
Tracway anticipates locating distribution centers in two of these new locations. Data acquired from
accounting, marketing, and production are shown in Figure 13.10. Total demand in the Pacific Rim is
expected to be 5000 units. Develop an appropriate optimization model to identify the best location
for the new DCs and transportation allocations to meet demand.

Distribution Center
Plant         Atlanta       Chicago     San Diego Taiwan        Singapore Sydney     Auckland
St Louis               35          40             60        130        120       148          145
Greenwood              30          30             45        136        130       160          170
Camarillo              60          65             50        115        100       120          170
Demand               9000        3000          9500

Need two plants to handle 5000 increase assuming equal facilities of 2500

Model:

Plant/D.C.  Atlanta         Chicago     San Diego Taiwan        Singapore Sydney           Auckland
St Louis       9,000
Greenwood                     2,000       7,000
Camarillo                     1,000       2,500        2,500          2500
Demand met    9,000          3,000       9,500        2,500       2,500          0              0

Total Cost =     1,417,500

Plant/D.C.  Atlanta         Chicago     San Diego Taiwan        Singapore    Sydney        Auckland
St Louis       9,000          2,000
Greenwood                     1,000       7,000
Camarillo                                 2,500        2,500                    2500
Demand met    9,000          3,000       9,500        2,500         0          2,500            0

Total Cost =     1,452,500

Plant/D.C.    Atlanta       Chicago     San Diego Taiwan        Singapore    Sydney        Auckland
St Louis         9,000                                                                         2,500
Greenwood                     3,000       5,000
Camarillo                                 4,500        2,500
Demand met    9,000     3,000     9,500     2,500       0             0            2,500

Total Cost =     1,505,000

Plant/D.C.  Atlanta    Chicago   San Diego Taiwan    Singapore Sydney           Auckland
St Louis       9,000     2,000
Greenwood                1,000     7,000
Camarillo                          2,500                  2500       2500
Demand met    9,000     3,000     9,500       0       2,500         2,500            0

Total Cost =     1,415,000

Plant/D.C.  Atlanta    Chicago   San Diego Taiwan    Singapore Sydney           Auckland
St Louis       9,000                                                                2,500
Greenwood                3,000     5,000
Camarillo                          4,500                  2500
Demand met    9,000     3,000     9,500       0       2,500           0            2,500

Total Cost =     1,467,500

Plant/D.C.  Atlanta    Chicago   San Diego Taiwan    Singapore    Sydney        Auckland
St Louis       9,000                                                                2,500
Greenwood                3,000     5,000
Camarillo                          4,500                             2500
Demand met    9,000     3,000     9,500       0         0           2,500          2,500

Total Cost =     1,517,500
amarillo plants

Capacity
12000
8000
7500

Total
Shipped     Capacity
9,000          12000
9,000            8000
8,500            7500

Total
Shipped     Capacity
11,000          12000
8,000            8000
7,500            7500

Total
Shipped     Capacity
11,500          12000
8,000            8000
7,000            7500
Total
Shipped   Capacity
11,000        12000
8,000          8000
7,500          7500

Total
Shipped   Capacity
11,500        12000
8,000          8000
7,000          7500

Total
Shipped   Capacity
11,500        12000
8,000          8000
7,000          7500

```
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