The distribution system for the Herman Company consists of four plants, three wa
ID: 351940 • Letter: T
Question
The distribution system for the Herman Company consists of four plants, three warehouses and four customers. Plant capacities and shipping costs per unit (in $) from each plant to each warehouse are as follows:
______________________Warehouse
Plant_________1(5)_________2(6)_______3(7)__________Capactiy
1 ____________11 __________13 ________11___________ 500
2 ____________14 __________11 ________12___________ 550
3 ____________11__________ 12________ 11___________ 550
4 ____________13__________ 12________ 11___________ 500
Customer demand and shipping costs per unit (in $) from each warehouse to each customer are as follows:
_________________________________Customer
Warehouse___________1(8) ____________2(9) ____________3(10) ____________4(11)
1(5) _________________8______________ 6______________ 10________________ 6
2(6)_________________ 5______________ 8_______________ 9________________ 9
3(7) _________________6______________ 8______________ 10________________ 7
Demand____________ 450 ____________500_____________ 450______________ 550
(a) Formulate the linear programming model–do not solve it.
(b) Suppose items can be shipped from Plant 3 (node 3) to customer 4 (node 11) at a cost of $10 each, would you need to change the linear programming model)? If so, indicate all the changes you will need to make to the model you do not need to reformulate the entire model.
Explanation / Answer
A.
Let Xij be the amount shipped form node i to node j
Decision variables
Plant( 1,2,3,4) to warehouse (5,6,7)
X15,X16, X17
X25, X26, X27
X35, X36, X37
X45, X46, X47
warehouse (5,6,7) to customer ( 8,9,10,11)
X58, X59, X510,X511
X68,X69, X610,X611
X78,X79,X710,X711
Objective
Minimize ( plant to warehouse transport cost+ warehouse to customer transport cost )
Minimize
11X15+13 X16+11X17+
14X25+11X26+12X27+
11X35+12X36+11X37+
13X45+12X46,+11X47+
8X58+6X59+10X510+6X511+
5X68+8X69+9X610+9X611+
6X78+8X79+10X710+7X711
Constraints
Plant capacity
Demand
Amount entering each warehouse is equal to amount exiting the warehouse to customers .
(X15+ X25+X35+X45=X58+X59+X510+X511,
X16+X26+X36+X46=X68+X69+X610+X611,
X17 +X27+X37+X47=X78+X79+X710+X711 )
Non negativity
B.
Objective
The objective function becomes
Minimize
11X15+13 X16+11X17+
14X25+11X26+12X27+
11X35+12X36+11X37+
13X45+12X46,+11X47+
8X58+6X59+10X510+6X511+
5X68+8X69+9X610+9X611+
6X78+8X79+10X710+7X711
+10X311
Capacity
Now the plant capacity should also include the amount being directly from plant 3 to customer 11
X35+X36+X37 +X311550
Demand
The demand constraint should also include the amount being directly from plant 3 to customer 11
X511+X611+X711 +X311=550
Non negativity
Additional constraints X311 is introduced
X15,X16,X17,X25,X26,X27,X35,X36,X37,X45,X46,X47,X58,X59,X510,X511,X68,X69,X610,X611,X78,X79,X710,X711,X311 0
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