LP Modeling

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LP Modeling

• Examples discussed in class

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Example Basketball Team Selection

• Coach Jack is trying to choose the starting
  line-up for the basketball team. The team
  consists of seven players who have been rated (on
  a scale of 1poor to 3excellent) according to
  their ball-handling, shooting, rebounding, and
  defensive abilities. The positions that each
  player is allowed to play and the players
  abilities are listed in the following table
• The five-player starting line-up must satisfy the
  following restrictions
• At least 4 members must be able to play guard, at
  least 2 members must be able to play forward, and
  at least 1 member must be able to play centre.
• The average ball-handling, shooting, and
  rebounding level of the starting line-up must be
  at least 2.
• If player 3 starts, then player 6 cannot start.
• If player 1 starts, then player 4 and 5 must both
  start.
• Either player 2 or player 3, or both, must start.
• Given these constraints, Coach Jack wants to
  maximize the total defensive ability of the
  starting team. Formulate an IP that will help him
  choose his starting team.

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Categorization of OR
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The General Structure of a Mathematical Model
Inputs
Output
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A Simplified Model of a Manufacturing Situation
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Example 1 LP Formulation

• Papa Louis manufacturers wooden tables and chairs
  for small kids. Each "table" built Sells for 27
  and uses 10 worth of raw materials, increases
  Papa Louiss variable labor/overhead costs by
  14. Requires 2 hours of finishing labor and 1
  hour of carpentry labor. Each "chair" built
  Sells for 21 and uses 9 worth of raw materials,
  increase Papa Louiss variable labor/overhead
  costs by 10. Requires 1 hours of finishing labor
  AND 1 hour of carpentry labor. Each week Papa
  Louis can obtain only 100 finishing hours and
  only 80 carpentry hours. Also demand for the
  chairs is unlimited. However, at most 40 tables
  are bought each week. Papa Louis wants to
  maximize weekly profit (revenues - expenses).

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Example 2 LP Formulation

• Production Planning A manufacturing firm has
  discontinued production of a certain unprofitable
  product line. This created considerable excess
  production capacity. Management is considering
  devoting this excess capacity to one or more of
  the three products call them products 1, 2, and
  3. The available capacity on the machines that
  might limit output is summarized in the following
  table
• The number of machine hours required for each
  unit of the respective products is given as
• The sales department indicates that the sales
  potential for products 1, and 2 exceeds the
  maximum production rate and that the sales
  potential for product 3 is 20 units per week.
  The unit profit would be 30.0, 20, and 15,
  respectively, on products 1, 2, and 3. Formulate
  the linear programming model for determining how
  much of each product to produce in order to
  maximize profits.

Total Availability of resources lt500 hr/week lt350
hr/week lt150 hr/week
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Example 3 Crude Petroleum

• Shell Petroleum runs a small refinery on the
  Texas coast. The refinery distills crude
  petroleum from two sources, Saudi Arabia and
  Venezuela, into three main products gasoline,
  jet fuel, and lubricants. The two crudes differ
  in chemical composition and thus yield different
  product mixes. Each barrel of Saudi crude yields
  0.3 barrel of gasoline, 0.4 barrel of jet fuel,
  and 0.2 barrel of lubricants. On the other hand,
  each barrel of Venezuelan crude yields 0.4 barrel
  of gasoline but only 0.2 barrel of jet fuel and
  0.3 barrel of lubricants. The remaining 10 of
  each barrel is lost to refining.
• The crudes also differ in cost and availability,
  Shell can purchase up to 9000 barrels per day
  from Saudi Arabia at 20 per barrel. Up to 6000
  barrels per day of Venezuelan petroleum are also
  available at the lower cost of 15 per barrel
  because of the shorter transportation distance.
  Shell's contracts with independent distributors
  require it to produce 2000 barrels per day of
  gasoline, 1500 barrels per day of jet fuel, and
  500 barrels per day of lubricants. How can these
  requirements be fulfilled most efficiently?

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Example 4 LP Formulation

• Manpower Planning Certain types of facilities
  operate seven days each week and face the problem
  of allocating manpower during the week as
  staffing requirements change as a function of the
  day of the week. Perhaps the most fundamental
  staffing problem is the assignment of days off to
  full-time employees. In particular, it is
  regularly the case that each employee is entitled
  to two consecutive days off per week. If the
  number of employees required on each of the seven
  days of the week is given, then the problem is to
  find the minimum workforce size which allows
  these demands to be met and then to determine the
  days off for the people in this work-force. To be
  specific, let us study the problem faced by Grand
  River Transit Bus Company. The number of drivers
  required for each day of the week is as follows
• How many drivers should be scheduled to start a
  five-days stint on each day of the week?
  Formulate this problem as a linear program.
• If daily pay is 50 per person on weekdays, 75
  on Saturday, 90 on Sundays. Modify the LP
  formulation so that the objective is now to
  minimize the weekly payroll costs rather than to
  minimize the workforce size.

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Example 5 LP Formulation

• Target Shirt Company makes three varieties of
  shirts Collegiate, Traditional and European.
  These shirts are made from different combinations
  of cotton and polyester. The cost per yard of
  unblended cotton is 5 and for unblended
  polyester is 4. Target can receive up to 4,000
  yards of raw cotton and 3,000 yards of raw
  polyester fabric weekly. The table shows below
  pertinent data concerning the manufacture of the
  shirts. (Meet weekly contracts or more while not
  exceeding weekly demand).
•  
•  
• Formulate and solve a linear program that would
  give a manufacturing policy for Target Shirt
  Company.

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Example 5 LP Formulation

• Finco has the following investments available
• Investment A For each dollar invested at time 0,
  we receive 0.10 at time 1 and 1.30 at time 2.
  Time 0 now time 1 one year from now and so
  on.)
• Investment B For each dollar invested at time 1,
  we receive 1.60 at time 2.
• Investment C For each dollar invested at time 2,
  we receive 1.20 at time 3.
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• At any time, leftover cash may be invested in
  T-bills, which pay 10 per year. At time 0, we
  have 100. At most, 50 can be invested in each
  of investments A, B, and C. Formulate an LP that
  can be used to maximize Fincos cash on hand at
  time 3.