Lab Section
♦ Excel Solver: Capacitated Plant Location
Model
♦ Lingo:
♦
♦
♦
Lingo Model Components
Transportation problem
Assignment problem
1. Capacitated Plant Location Model
=
number of potential plant locations/capacity
=
number of markets or demand points
=
annual demand from market j
=
potential capacity of plant i
=
annualized fixed cost of keeping plant i open
=
=
1 if plant i is open, 0
= otherwise
quantity shipped from plant i
to market j
cost of producing and shipping one unit from plant i to market j (cost includes
production, inventory, transportation, and tariffs)
subject to
Network Optimization Model
Capacitated Plant Location Model
Capacitated Plant Location Model
Capacitated Plant Location Model
Excel Solver Model
Excel Solver Results
2. The LINGO Computer Package
♦ LINGO is a user friendly computer package that can be used to
solve linear, integer, and nonlinear, stochastic programming
problems.
♦ LINGO assumes all variables are nonnegative
♦ View the LINGO Help file for syntax questions
♦ View the LINGO examples for studying different cases
♦ To solve the model, select the SOLVE command or click the red
bulls eye button.
♦ Some useful settings in OPTIONS
MODELING ON LINGO
♦ The first statement in a LINGO model is
MODEL:
END
♦ The objective function will be represented by MAX or MIN
commands to solve a maximization or minimization problem.
♦ Enter the constraints by typing directly on the next line.
♦ Ending command line in LINGO is “;” mark. The “!” mark is used for
comments.
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Example 1
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The Outputs
♦
The optimal z-value = 280
♦ VALUE gives the value of the
variable in the optimal LP
solution. Thus the optimal
solution calls for production of 2
desks, 0 tables, and 8 chairs.
♦ SLACK OR SURPLUS gives
(by constraint row) the value of
slack or excess in the optimal
solution.
♦ REDUCED COST gives the
coefficient in row 0 of the
optimal tableau (in a max
problem). The reduced cost of
each basic variables must be 0.
Reduced cost is the amount the objective function coefficient for
variable i would have to be increased for there to be an
alternative optimal solution.
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Shadow Prices
♦ Shadow prices are shown in the Dual Prices section of LINGO output.
♦ Shadow prices are the amount the optimal z-value improves if the rhs of a
constraint is increased by one unit (assuming no change in basis).
♦ ≥ constraints: nonpositive shadow prices.
♦ ≤ constraints: nonnegative shadow prices.
♦ = constraints: a positive, a negative, or 0 shadow price.
♦ For any inequality constraint, the product of the values of the constraint’s
slack/excess variable and the constraint’s shadow price must equal zero.
♦ This implies that any constraint whose slack or excess variable > 0 will have
a zero shadow price.
♦ Similarly, any constraint with a nonzero shadow price must be binding (have
slack or excess equaling zero).
♦ For constraints with nonzero slack or excess:
Type of Constraint
Allowable Increase for rhs
Allowable Decrease for rhs
≤
∞
= value of slack
≥
= value of excess
∞
2.1. LINGO Model Components
Variables & parameters
definition
Objective function
Constraints
Input Data
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2.2. Transportation Problem
♦ A transportation problem basically deals with the
problem, which aims to find the best way to fulfill
the demand of n demand points using the
capacities of m supply points.
♦ While trying to find the best way, generally a
variable cost of shipping the product from one
supply point to a demand point or a similar
constraint should be taken into consideration.
Example 2: ABC-Delivery Company
♦ ABC-Delivery has three warehouses that supply
the coffee needs of retailers (customers).
♦ The associated supply of each warehouse and
demand of each customer is given in the Table
1.
♦ The cost of sending 1000 boxes of coffee from a
warehouse to a customer depends on the
travelling distance
Ex. 2 - continued
♦ A transportation problem is specified by the
supply, the demand, and the shipping costs.
Relevant data can be summarized in a
transportation tableau.
From
To
C1
C2
C3
C4
Supply
(1000 boxes)
WH 1
$8
$6
$10
$9
35
WH 2
$9
$12
$13
$7
50
WH 3
$14
$9
$16
$5
40
Demand
(1000 boxes)
45
20
30
30
Example 2: Solution
♦ Decision Variables
♦
ABC must determine how much coffee is sent from each
warehouse to each customer so xij = Amount of coffee stocked
at warehouse i and sent to customer j
♦
x14 = Amount of coffee stocked at warehouse 1 and sent to customer 4
♦ Constraints
♦
♦
♦
A supply constraint ensures that the total quality stocked does
not exceed warehouse capacity. Each warehouse is a supply
point.
A demand constraint ensures that a location receives its
demand. Each customer/retailer is a demand point.
Since a negative amount of coffee can not be shipped all xij’s
must be non negative
LP Formulation
Min Z =
8x11+6x12+10x13+9x14+9x21+12x22+13x23+7x24+14x31+9x32+16x33+5x34
S.T.: x11+x12+x13+x14 <= 35
x21+x22+x23+x24 <= 50
x31+x32+x33+x34 <= 40
(Supply Constraints)
x11+x21+x31 >= 45
(Demand Constraints)
x12+x22+x32 >= 20
x13+x23+x33 >= 30
x14+x24+x34 >= 30
xij >= 0 (i= 1,2,3; j= 1,2,3,4)
♦ In general, a transportation problem is specified by
the following information:
♦
♦
♦
A set of m supply points from which a good is shipped.
Supply point i can supply at most si units.
A set of n demand points to which the good is shipped.
Demand point j must receive at least di units of the
shipped good.
Each unit produced at supply point i and shipped to
demand point j incurs a variable cost of cij.
♦ xij = number of units shipped from supply point i
to demand point j
The Lingo Model of The
Transportation Problem
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Transportation Problem
Characteristics
♦ If
then total supply equals to total demand, the problem is
said to be a balanced transportation problem.
♦ If total supply exceeds total demand, we can balance the problem by
adding dummy demand point. Since shipments to the dummy demand
point are not real, they are assigned a cost of zero.
♦ If a transportation problem has a total supply that is strictly less than total
demand the problem has no feasible solution.
♦ No doubt that in such a case one or more of the demand will be left
unmet.
♦ Generally in such situations a penalty cost is often associated with
unmet demand and as one can guess the total penalty cost is desired
to be minimum.
♦ The basic solution method for a balanced TP is the Northwest Corner
Method
2.3. Assignment Problems
♦ In general an assignment problem is balanced
transportation problem in which all supplies and
demands are equal to 1.
♦ The assignment problem’s matrix of costs is its cost
matrix.
♦ All the supplies and demands for this problem are
integers which implies that the optimal solution must
be integers.
♦ The Hungarian Method is usually used to solve
assignment problems.
Example 3: Machine Assignment Problem
♦ Machineco has four jobs to be completed.
♦ Each machine must be assigned to complete one job.
♦ The time required to setup each machine for completing
each job is shown.
Time (Hours)
Job1
Job2
Job3
Job4
Machine 1
14
5
8
7
Machine 2
2
12
6
5
Machine 3
7
8
3
9
Machine 4
2
4
6
10
♦ Machineco wants to minimize the total setup time needed to
complete the four jobs.
Solution
♦ Machineco must determine which machine should be assigned to
each job.
♦ i,j=1,2,3,4
♦ xij=1 (if machine i is assigned to meet the demands of job j)
♦ xij=0 (if machine i is not assigned to meet the demands of job j)
LINGO Model
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