Quantitative Models for Decision Making

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Quantitative Models for Decision Making

• ECE573 Data Structures and Algorithms
• Electrical and Computer Engineering Dept.
• Rutgers University
• http//www.cs.rutgers.edu/vchinni/dsa/

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PROBLEM SET

• (1) S.T. Powers Perfume Problem
• (2) Fabulous Nut Company
• (3) Pricing Out the Super Mix
• (4) GlobChem Production/Transportation Problem
• (5) Foresight Co. Production Planning Problem
• (6) Foreign Currency Trading
• (7) Cambridge Software Corporation

Rules Must attempt min of 4 problems. One
problem from the group of (problem 4, 5, 7) must
be attempted.
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(1) S.T. Powers Perfume Problem

• S.T. sells two products cologne and perfume
• Cologne sells for 3 per ounce each ounce
  requires
• 2 grams of fragrance
• 6 grams of intensifier
• Perfume sells for 8 per ounce. The recipe for
  one ounce of perfume is
• 4 grams of fragrance
• 2 grams of intensifier
• 1 gram of stabilizer
• S.T. has limited supplies. In particular, she
  has
• 1600 grams of fragrance
• 1800 grams of intensifier
• 350 grams of stabilizer
• She would like to use these ingredients
  immediately, before they spoil. What should she
  do to maximize the revenue earned from her
  supplies?

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Solution- Using Excel Linear Programming

• Given the decision variables
• C number of ounces of cologne produced
• P number of ounces of perfume produced
• Maximize the objective function value
• OFG 3C 8P
• Subject to the constraints
• 2c4P lt 1,600 (fragrance)
• 6C2P lt 1,800 (intensifier)
• 1P lt 350 (stabilizer)
• 1C gt 0 (non negativity)
• 1P gt 0 (non negativity)
• TOTAL PROFIT 3100 (C100 P350)

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Solution- Graphical Solution

• A two-variable LP (linear programming) can be
  represented on a plane
• The feasibility region is the common intersection
  of all constraints
• An optimal solution can always be found at a
  vertex

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Basics of Sensitivity Analysis

• Relaxing the RHS of a binding constraint improves
  the optimal OFV (Objective function value)
• A large enough change in an objective function
  coefficient can cause the optimal solution to
  change
• Uses of sensitivity analysis
• Testing the robustness of a models solution
• Answering what if questions concerning
  resources and products
• Understanding the marginal values of resources
  and products
• Sensitivity analysis (sensitivity to changes in
  underlying variables) is very important to
  understand the stability of optimal solutions

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(2) Fabulous Nut Company
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LP Formulation

• Decision variables
• R pounds of Regular Mix produced
• D pounds of Deluxe Mix produced
• H pounds of Holiday Mix produced
• Objective function
• Max 1.65 R 2.00 D 2.25 H
• Supply constraints
• 0.15 R 0.20 D 0.25 H lt 6,000 (Almonds)
• 0.25 R 0.20 D 0.15 H lt 7,500 (Brazil Nuts)
• 0.25 R 0.20 D 0.15 H lt 7,500 (Filberts)
• 0.10 R 0.20 D 0.25 H lt 6,000 (Pecans)
• 0.25 R 0.20 D 0.20 H lt 7,500 (Walnuts)
• Demand Constraints
• R gt 10,000 (Regular demand)
• D gt 3,000 (Deluxe demand)
• H gt 5,000 (Holiday demand)

TOTAL Contribution 61,375
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(3) Pricing Out the Super Mix

• Each pound of Super Mix requires
• 0.5 pounds of almonds
• 0.1 pounds each of filberts, Brazil nuts, pecans
• 0.2 pounds of walnuts
• The Fabulous Nut Company plans to price Super Mix
  to earn a contribution of 2.50 per pound
• How will the addition of Super Mix to the product
  line affect the quantities that should be
  produced?

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LP Formulation

• Decision variables
• R pounds of Regular Mix produced
• D pounds of Deluxe Mix produced
• H pounds of Holiday Mix produced
• S pounds of Super Mix produced
• Objective function
• Max 1.65 R 2.00 D 2.25 H 2.5 S
• Supply constraints
• 0.15 R 0.20 D 0.25 H 0.50 S lt 6,000
  (Almonds)
• 0.25 R 0.20 D 0.15 H 0.10 S lt 7,500
  (Brazil Nuts)
• 0.25 R 0.20 D 0.15 H 0.10 S lt 7,500
  (Filberts)
• 0.10 R 0.20 D 0.25 H 0.10 S lt 6,000
  (Pecans)
• 0.25 R 0.20 D 0.20 H 0.20 S lt 7,500
  (Walnuts)
• Demand Constraints
• R gt 10,000 (Regular demand)
• D gt 3,000 (Deluxe demand)
• H gt 5,000 (Holiday demand)
• S gt 0 (Super demand)

TOTAL Contribution 61,375 (Unchanged!)
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• Something hidden go and find it
• Go and look behind the Ranges
• Something lost behind the Ranges
• Lost and waiting for you. Go!
• Rudyard Kipling

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(4) GlobChem Production/Transportation Problem

• A US based multinational produces a specialty
  chemical in four dedicated plants, located in
  Newark Los Angeles, Rotterdam, and Kaula Lumpur
• The rpoduct is marketed worldwide, and the
  company has just instituted a regional sales
  organziation with sales offices / distribution
  centers based in Newark Rotterdam Sao Paulo
  and Tokyo.

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Prices and Costs Vary by Region and Plant
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GlobChems Problem Statement

• GlobChem wants to determine its annual production
  and distribution schedule in order to maximize
  profits while observing the limits on each
  plants capacity and the maximum sales possible
  in each region. In determining this production
  and distribution schedule, consider the locations
  and capacities of the plans to be fixed.

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Solution

• Network description
• Ai available capacity at plat i
• Ci unit cost at plant i
• Tij unit transportation cost from plant i to
  market j
• Dj demand in market j
• Pj unit price in market j
• (Index I refers to plants, where 1 is Newark, 2
  Los Angeles, 3 Rotterdam and 4 Kaula Lumpur. The
  index j refers to markets, where 1 is Newark, 2
  Sao Paulo, 3 Rotterdam, and 4 Tokyo).

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Solution

• Objective function Maximize the annual profit
  contribution
• Maximize
• profit contribution revenues production costs
  Transportation Costs
• (Fixed costs are not included as they are sunk
  costs and will not contribute in determining the
  optimal solution)
• Compact LP Notation
• Let Yij denote the unit profit contribution of
  the product produced at plant I and sold in
  market j.
• Yij Pj Ci Tij
• Max ?? Yij Xij (First ? running from I1 to N,
  second ? running from j1, M)
• Subject to
• (capacity) ? Xij lt Ai i1,N (? running
  from j1.M)
• (demand) ?xij lt Dj j 1, ,M (? running
  from I1, N)
• (nonneg) xij gt 0 I1,..,N j1,,M

Solution Gross Profit 11,616,000
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(5) Foresight Co. Production Planning Problem
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Solution

• We need to determine Pt production in month t
  (t1,..,6)
• Objective function Minimize total production
  cost holding cost over the 6-month horizon
• Define It ending inventory for month t,
  t1,2,,6

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Complete LP formulation

• Minimize
• 14.5 P1 15.25 P2 16 P3 16.7 P4 17.25 P5
  17 P6 (production cost)
• 0.6 I1 0.65 I2 0.65 I3 0.65 I4 0.7 I5
  0.65 I6
• Such that
• P1 lt10000 P2 lt 10,000 P3 lt 11000
• P4 lt 10,000 P5 lt 11000 P6 lt 10000
  (production capacity)
• I0 P1 9000 I1
• I1 P2 7500 I2
• I2 P3 12500 I3
• I3 P4 9000 I4
• I4 P5 8000 I5
• I5 P6 11500 I6
• I1 gt0 I2gt0 ..I6gt0 (no backlogging
  allowed)
• P1 gt 0 P2gt0P6gt 0 (production must be
  non-negative)
• Can you see why the optimal soluton will always
  set the last It variable to zero?

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(6) Foreign Currency Trading
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(6) Foreign Currency Trading- Contd
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FX Arbitrage LP Formulation Decision Variables

• Let I 1, 2, 3, 4, 5 represent the currencies of
  US, Britain, France, Germany and Japan
  respectively.
• Decision variables
• Xij amount of currency I to be converted in to
  currency j, measured in units of currency I for
  all I and j where I is not equal to j
• Ex X12 the amount of US Dollars converted into
  British pounds
• Fi the final net amount in currency I
  currency I flow in Currency I Flow out
• Ex f1 sum of currencies converted into doalls
  (inflow) sum of dollars converted into other
  currencies (outflows)

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Constraints

• Flow constraints denote where the currencies come
  from and go to
• F1 1.5648 X21 0.1856 X310.6361 X41 0.01011
  X51 (X12X13X14X15)
• Write similarly the other four constraints
• Non negativity
• Xij gt 0 for all I and J and I is not equal to j
• Fi gt 0 , I 1,.., 5

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Objective Function

• Maximize the final net worth in US Dollars Max
  F1
• Additional constraint
• If an arbitrage exists, then the linear program
  will be unbounded, and the optimizer will not
  return an optimal solution indicating how
  arbitrage profits could be obtained.
• This add the constraint f1 lt1 (bound on total
  arbitrage)

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Solution
1 US Dollar
1069.21
2 British Pound
5 Japanese Yen
1682.46
683.23
4 German D-mark
3 French Franc
5759.87
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(7) Cambridge Software Corporation
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Solution

• Not posted