PRODUCTIONS/OPERATIONS MANAGEMENT

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Lecture 2 3
Linear Programming and Transportation Problem
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Linear Programming

• George Dantzig 1914 -2005
• Concerned with optimal allocation of limited
  resources such as
• Materials
• Budgets
• Labor
• Machine time
• among competitive activities
• under a set of constraints

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Linear Programming Example

• Maximize 60X1 50X2
• Subject to
• 4X1 10X2 lt 100
• 2X1 1X2 lt 22
• 3X1 3X2 lt 39
• X1, X2 gt 0

• What is a Linear Program?
• A LP is an optimization model that has
• continuous variables
• a single linear objective function, and
• (almost always) several constraints (linear
  equalities or inequalities)

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Linear Programming Model

• Decision variables
• unknowns, which is what model seeks to determine
• for example, amounts of either inputs or outputs
• Objective Function
• goal, determines value of best (optimum) solution
  among all feasible (satisfy constraints) values
  of the variables
• either maximization or minimization
• Constraints
• restrictions, which limit variables of the model
  
• limitations that restrict the available
  alternatives
• Parameters numerical values (for example, RHS of
  constraints)
• Feasible solution is one particular set of
  values of the decision variables that satisfies
  the constraints
• Feasible solution space the set of all feasible
  solutions
• Optimal solution is a feasible solution that
  maximizes or minimizes the objective function
• There could be multiple optimal solutions

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Another Example of LP Diet Problem

• Energy requirement 2000 kcal
• Protein requirement 55 g
• Calcium requirement 800 mg

Food Energy(kcal) Protein(g) Calcium(mg) Price per serving()
Oatmeal 110 4 2 3
Chicken 205 32 12 24
Eggs 160 13 54 13
Milk 160 8 285 9
Pie 420 4 22 24
Pork 260 14 80 13
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Example of LP Diet Problem

• oatmeal at most 4 servings/day
• chicken at most 3 servings/day
• eggs at most 2 servings/day
• milk at most 8 servings/day
• pie at most 2 servings/day
• pork at most 2 servings/day

Design an optimal diet plan which minimizes the
cost per day
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Step 1 define decision variables

• x1 of oatmeal servings
• x2 of chicken servings
• x3 of eggs servings
• x4 of milk servings
• x5 of pie servings
• x6 of pork servings

Step 2 formulate objective function

• In this case, minimize total cost
• minimize z 3x1 24x2 13x3 9x4 24x5
  13x6

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Step 3 Constraints

• Meet energy requirement
• 110x1 205x2 160x3 160x4 420x5 260x6
  ?2000
• Meet protein requirement
• 4x1 32x2 13x3 8x4 4x5 14x6 ? 55
• Meet calcium requirement
• 2x1 12x2 54x3 285x4 22x5 80x6 ? 800
• Restriction on number of servings
• 0?x1?4, 0?x2?3, 0?x3?2, 0?x4?8, 0?x5?2, 0?x6?2

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So, how does a LP look like?

• minimize 3x1 24x2 13x3 9x4 24x5 13x6
• subject to
• 110x1 205x2 160x3 160x4 420x5 260x6
  ?2000
• 4x1 32x2 13x3 8x4 4x5 14x6 ? 55
• 2x1 12x2 54x3 285x4 22x5 80x6 ? 800
• 0?x1?4
• 0?x2?3
• 0?x3?2
• 0?x4?8
• 0?x5?2
• 0?x6?2

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Guidelines for Model Formulation

• Understand the problem thoroughly.
• Describe the objective.
• Describe each constraint.
• Define the decision variables.
• Write the objective in terms of the decision
  variables.
• Write the constraints in terms of the decision
  variables
• Do not forget non-negativity constraints

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Transportation Problem

• Objective
• determination of a transportation plan of a
  single commodity
• from a number of sources
• to a number of destinations,
• such that total cost of transportation is
  minimized
• Sources may be plants, destinations may be
  warehouses
• Question
• how many units to transport
• from source i
• to destination j
• such that supply and demand constraints are met,
  and
• total transportation cost is minimized

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A Transportation Table
Table 8S.1
Warehouse
1
2
3
4
Factory
7
4
7
1
Factory 1 can supply 100 units per period
100
1
3
8
8
12
200
2
8
10
16
5
150
3
450
80
90
120
160
Demand
450
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LP Formulation of Transportation Problem

• minimize 4x117x127x13x1412x213x228x238x248
  x3110x32
• 16x335x34
• Subject to
• x11x12x13x14100
• x21x22x23x24200
• x31x32x33x34150
• x11x21x3180
• x12x22x3290
• x13x23x33120
• x14x24x34160
• xijgt0, i1,2,3 j1,2,3,4

Supply constraint for factories
Demand constraint of warehouses
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Assignment Problem

• Special case of transportation problem
• When of rows of columns in the
  transportation tableau
• All supply and demands 1
• Objective Assign n jobs/workers to n machines
  such that the total cost of assignment is
  minimized
• Plenty of practical applications
• Job shops
• Hospitals
• Airlines, etc.

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Cost Table for Assignment Problem
Machine (j)
1 2 3 4
1 1 4 6 3
2 9 7 10 9
3 4 5 11 7
4 8 7 8 5
Worker (i)
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LP Formulation of Assignment Problem

• minimize x114x126x133x14 9x217x2210x239x24
  4x315x3211x337x34 8x417x428x435x44
• subject to
• x11x12x13x141
• x21x22x23x241
• x31x32x33x341
• x41x42x43x441
• x11x21x31x411
• x12x22x32x421
• x13x23x33x431
• x14x24x34x441
• xij 1, if worker i is assigned to machine j,
  i1,2,3,4 j1,2,3,4
• 0 otherwise

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Product Mix Problem

• Floataway Tours has 420,000 that can be used to
  purchase new rental boats for hire during the
  summer.
• The boats can be purchased from two
• different manufacturers.
• Floataway Tours would like to purchase at least
  50 boats.
• They would also like to purchase the same number
  from Sleekboat as from Racer to maintain
  goodwill.
• At the same time, Floataway Tours wishes to have
  a total seating capacity of at least 200.
• Formulate this problem as a linear program

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Product Mix Problem

• Maximum Expected Daily
• Boat Builder Cost Seating
  Profit
• Speedhawk Sleekboat 6000 3
  70
• Silverbird Sleekboat 7000
  5 80
• Catman Racer 5000
  2 50
• Classy Racer 9000
  6 110

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Product Mix Problem

• Define the decision variables
• x1 number of Speedhawks ordered
• x2 number of Silverbirds ordered
• x3 number of Catmans ordered
• x4 number of Classys ordered
• Define the objective function
• Maximize total expected daily profit
• Max (Expected daily profit per unit) x
  (Number of units)
• Max 70x1 80x2 50x3 110x4

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Product Mix Problem

• Define the constraints
• (1) Spend no more than 420,000
• 6000x1 7000x2 5000x3 9000x4 lt 420,000
• (2) Purchase at least 50 boats
• x1 x2 x3 x4 gt 50
• (3) Number of boats from Sleekboat equals
  number of boats from Racer
• x1 x2 x3 x4 or x1 x2 - x3 -
  x4 0
• (4) Capacity at least 200
• 3x1 5x2 2x3 6x4 gt 200
• Nonnegativity of variables
• xj gt 0, for j 1,2,3,4

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Product Mix Problem - Complete Formulation

• Max 70x1 80x2 50x3 110x4
• s.t.
• 6000x1 7000x2 5000x3 9000x4 lt 420,000
• x1 x2 x3 x4 gt 50
• x1 x2 - x3 - x4 0
• 3x1 5x2 2x3 6x4 gt 200
• x1, x2, x3, x4 gt 0

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Applications of LP

• Product mix planning
• Distribution networks
• Truck routing
• Staff scheduling
• Financial portfolios
• Capacity planning
• Media selection marketing

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Graphical Solution of LPs

• Consider a Maximization Problem
• Max 5x1 7x2
• s.t. x1
  lt 6
• 2x1
  3x2 lt 19
• x1
  x2 lt 8
• x1, x2 gt
  0

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Graphical Solution Example

• Constraint 1 Graphed

x2
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
x1 lt 6
(6, 0)
x1
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Graphical Solution Example

• Constraint 2 Graphed

x2
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
(0, 6 1/3)
2x1 3x2 lt 19
(9 1/2, 0)
x1
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Graphical Solution Example

• Constraint 3 Graphed

x2
(0, 8)
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
x1 x2 lt 8
(8, 0)
x1
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Graphical Solution Example

• Combined-Constraint Graph

x2
x1 x2 lt 8
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
x1 lt 6
2x1 3x2 lt 19
x1
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Graphical Solution Example

• Feasible Solution Region

x2
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
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Feasible Region
x1
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Graphical Solution Example

• Objective Function Line

x2
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
(0, 5)
Objective Function 5x1 7x2 35
(7, 0)
x1
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Graphical Solution Example

• Optimal Solution

x2
Objective Function 5x1 7x2 46
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
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Optimal Solution (x1 5, x2 3)
x1
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Graphical Linear Programming

1. Set up objective function and constraints in
   mathematical format
2. Plot the constraints
3. Identify the feasible solution space
4. Plot the objective function
5. Determine the optimum solution

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Possible Outcomes of a LP

• A LP is either
• Infeasible there exists no solution which
  satisfies all constraints and optimizes the
  objective function
• or, Unbounded increase/decrease objective
  function as much as you like without violating
  any constraint
• or, Has an Optimal Solution
• Optimal values of decision variables
• Optimal objective function value

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Infeasible LP An Example

• minimize 4x117x127x13x1412x213x228x238x248
  x3110x3216x335x34
• Subject to
• x11x12x13x14100
• x21x22x23x24200
• x31x32x33x34150
• x11x21x3180
• x12x22x3290
• x13x23x33120
• x14x24x34170
• xijgt0, i1,2,3 j1,2,3,4

Total demand exceeds total supply
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Unbounded LP An Example

• maximize 2x1 x2
• subject to
• -x1 x2 ? 1
• x1 - 2x2 ? 2
• x1 , x2 ? 0

x2 can be increased indefinitely without
violating any constraint gt Objective function
value can be increased indefinitely
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Multiple Optima An Example

• maximize x1 0.5 x2
• subject to
• 2x1 x2 ? 4
• x1 2x2 ? 3
• x1 , x2 ? 0

• x1 2, x20, objective function 2
• x1 5/3, x22/3, objective function 2

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Practice Example