Question

Q4. ABC Ltd. manufactures luxury golf bags in their two plants located in Augusta and Tupper Lake Warchouse facilities are located in Albany and Portsmouth. Distributors are located in Boston, New York and Philadelphia. The Augusta plant has a capacity of 300 units per month, and the Tupper Lake plant has a capacity of 100 units per month. Boston has a demand of 150 units per month, New York has a demand of 100 units per month, and Philadelphia has a demand of 150 units per month. The unit transportation costs (in dollars) for shipments from the two plants to the two warehouses are presented in Table 3 and those from the two warchouses to the three distributors are presented in Table 4 Table 3. Plant to Warehouse Information Warehouse Albany Portsmouth Piant Augusta Tupper Lake 3 4 Table 4. Warehouse to Distributor Information Distributor Boston New York Philadelphia Warehouse Albany Portsmouth 10 12 Marks a) Draw the network representation of the above Transhipment problem b) Formulate a linear programming model to determine the optimal quantity of bags to be transported from plants to warehouses, and from warehouses to distributors so that total transportation cost is minimized as well as total demand at distributors is satisfied. 5 Marks) e) An internationa! golftourzament is scheduled in New York which will increase the demanl of luxury golf bags in New York by 50 units next month. Formulate a feasible linear programming model by incorporating this increment in demand. 3 Marks) Q5. Quality Air Conditioning manufactures three types of home air conditioners: an economy model, a el The nrefits per unit are $63, $95, and $135, respectively. The

Answer 1

a) Network represetnation is as follows:

Plant Warehouse Distributor Boston-150 New York |-100 Philadelphia -150 300 Augusta Albany 100 Tupper Lake Portsmouth

b) LP model

Decision variable:

Xij = Quantity to be shipped from plant i to Warehousej

Xjk = Quantity to be shipped from Warehouse j to Distributor k

where  $i\in \left \{ 1,2 \right \}, j\in\left \{ 3,4 \right \}, k\in\left \{ 5,6,7 \right \}$

Objective:

$Min: \sum_{i=1}^{2}\sum_{j=3}^{4}X_{ij}*C_{ij} + \sum_{j=3}^{4}\sum_{k=5}^{7}X_{jk}*C_{jk}$

s.t.

X₁₃ + X₁₄ = 300

X₂₃ + X₂₄ = 100

X₁₃ + X₂₃ = X₃₅ + X₃₆ + X₃₇

X₁₄ + X₂₄ = X₄₅ + X₄₆ + X₄₇

X₃₅ + X₄₅ = 150

X₃₆ + X₄₆ = 100

X₃₇ + X₄₇ = 150

Xij >= 0

Solution using Excel Solver follows:

B24 Q fx -SUMPRODUCT(B3:C4,B14:C15)+SUMPRODUCT(B8:D9,B20:D21) 1 Table 3: Plant to Warehouse 2 Plant Warehouse 3 Augusta 4 Tupper Lake Alban S Solver Parameters Portsmouth Capaci 300 100 Set Objective SBS24 Max Min Value Of: 0 6 Table 4: Warehouse to Distributor 7 Warehouse Distributor Boston New YorkPhiladelphia 8 Augusta 9 Tupper Lake 10 Demand By Changing Variable Cells: SBS14:SCS15,SBS20:SDS21 Subject to the Constraints: 10 D$14:SDS15SDS3:SD$4 SBS16:SCS16= SES20:SES21 BS22:SDS22 SBS10 SDS10 Add 150 100 150 Change 12 Plant to Warehouse distribution plan 13 Plant Warehouse 14 Augusta 15 Tupper Lake 16 Column Total 17 18 Warehouse to Distributor distribution plan 19 Warehouse DistributorBston New York Philadelphia Row Total 20 Augusta 21 Tupper Lake 22 Demand 23 24 25 Alban 50 100 150 Portsmouth Row Total Delete 250 300 Reset All 100 250 Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP Options Solving Method Select the IPOPT Nonlinear engine for Solver Problems that are smooth nonlinear.Select the LP Simplex engine for linear Solver Problems. 150 150 150 150 100 100 250 150 Solve Close Total Cost4300

Formula:

B16 =SUM(B14:B15) copy to B16:C16, B22:D22

D14 =SUM(B14:C14) copy to D14:D15

E20 =SUM(B20:D20) copy to E20:E21

B24 =SUMPRODUCT(B3:C4,B14:C15)+SUMPRODUCT(B8:D9,B20:D21)

c) The revised model is as follows:

Objective:

$Min: \sum_{i=1}^{2}\sum_{j=3}^{4}X_{ij}*C_{ij} + \sum_{j=3}^{4}\sum_{k=5}^{7}X_{jk}*C_{jk}$

s.t.

X₁₃ + X₁₄ = 300

X₂₃ + X₂₄ = 100

X₁₃ + X₂₃ = X₃₅ + X₃₆ + X₃₇

X₁₄ + X₂₄ = X₄₅ + X₄₆ + X₄₇

X₃₅ + X₄₅ <= 150

X₃₆ + X₄₆ <= 150

X₃₇ + X₄₇ <= 150

Xij >= 0

a fx SUMPRODUCT(B3:C4,B14:C15)+SUMPRODUCT(B8:D9,B20:D21) B24 1 Table 3: Plant to Warehouse 2 Plant Warehouse 3 Augusta 4 Tupper Lake Alban Portsmouth Ca S Solver Parameters 300 100 Set Objective: SBS24 O Max (e) Min 0 Value Of: 0 6 Table 4: Warehouse to Distributor 7 Warehouse Distributor Boston New YorkPhiladelphia 8 Augusta 9 Tupper Lake 10 Demand By Changing Variable Cells: SBS14 SCS15,SBS20:SDS21 Subject to the Constraints: SDS14:SDS15 SDS3:SDS4 SBS16:SC$16 SES20:SES21 6 150 10 150 Add 150 SBS22:SD$22 <= SBS10:SDS10 Change 12 Plant to Warehouse distribution plan 13 PlantWarehouse 14 Augusta 15 Tupper Lake 16 Column Total 17 18 Warehouse to Distributor distribution plarn 19 Warehouse\ DistributorBoston New York Philadelphia Row Total 20 Augusta 21 Tupper Lake 22 Demand 23 Alban Portsmouth Row Total Delete 300 100 300 Reset All 100 100 300 Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP Options 100 100 Solving Method Select the IPOPT Nonlinear engine for Solver Problems that are smooth nonlinear.Select the LP Simplex engine for linear Solver Problems 150 150 150 150 100 Close Total Cost 4150 1 25 26