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					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).


preadsheets?
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
                 Premium           40%          20%          40%
                 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
    obtained the following information about advertising media availability and cost:

                              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
   Radio                         3,500                     $3,000                      100
   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
                 R = number of radio 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

                            Additional information is provided below.

      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.



                                                    Internal         Additional
      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
      Administrator                          5                  5        8      $40

      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
      Administrator                      50                    10        8        68
      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.

   Data:                          Radio          TV        Magazine       Budget
                 Cost/ad           500          2,000        200          50,000
            Exposure value/ad     2,000         3,500       2,700
               Minimum Units         0           12            6
              Maximum Units         15            -           30

   Model:                         Radio          TV        Magazine
               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:

     Processor: slower (subtract $179), faster (add $100), fastest (add $300)
     Warranty: 2 year (add $129), 3 year (add $269),
     Memory: 512MB (add $50), 1 GB (add $500)
     Hard drive: 30GB (subtract $29), 60GB (add $39), 80GB (add $79)
     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