1. A company has two warehouses A and B, and three retail outlets 1, 2 and 3. The warehouse capacities, retail outlet demands, and perunit shipping costs ($) are shown in the table below. Formulate a linear programming (LP) model of this transportation problem with the objective of minimizing total shipping cost.
Retail Outlets Warehouses 
1 
2 
3 
Total Supply 
A 
$5 
$8 
$3 
500 
B 
$7 
$4 
$6 
250 
Total Demand 
300 
400 
250 

2. A department head has four managers, and three projects (A, B, C) to be executed. The managers differ in efficiency, and the projects differ in their intrinsic complexity. The estimate of the times each manager would take to complete each project is given in the matrix below. Formulate a linear programming (LP) model for determining how the projects should be assigned, to a manager, so as to minimize the total project completion time. No manager should be assigned more than one project.
Manager
Project
1 
2 
3 
4 

A 
8 
14 
12 
11 
B 
6 
9 
10 
8 
C 
7 
10 
9 
11 
Please input answers into a Word Document, if possible.
Problem 1:
Let, x_{ij} = units shipped from warehouse ‘i’ to retail outlet j
Where, i = a, b for two warehouses
j = 1, 2, 3 for three retail outlets
Thus, there will be 2 x 3 = 6 decision variables in the LPP.
Objective Function:
Objective is to minimize the total shipping cost:
Let C_{ij} = per unitshipping cost time from warehouse i to outlet j
Min. Z = (C_{ij} x X_{ij})
Min. z = $5x_{a1} + $8x_{a2} + $3x_{a3} + $7x_{b1} + $4x_{b2} + $6x_{b3}
Subject To:
The constraint of problem are:
Warehouse cannot ship more than supply capacity
Demand of outlet has to be satisfied
Supply constraint of warehouses: Since the total supply capacity is less than total demand, all the capacity will be utilized.
Warehouse A: x_{a1} + x_{a2} + x_{a3} = 500
Warehouse B: x_{b1} + x_{b2} + x_{b3} = 250
Demand constraint of retail: Since demand is more than supply capacity, not all demand will be satisfied
Retail 1: x_{a1} + x_{b1} <= 300
Retail 2: x_{a2} + x_{b2} <= 400
Retail 1: x_{a3} + x_{b3} <= 250
Nonnegative Constraint: all Xij >= 0
Problem 2:
Let, x_{ij} = 1 if the manager ‘i’ is assigned to project j otherwise X_{ij} = 0
Where, i = 1, 2,…4 for four managers
j = a, b, c for three projects
Thus, there will be 4 x 3 = 12 decision variables in the LPP.
Objective Function:
Objective is to minimize the total project completion time:
Let C_{ij} = time required by manager i to complete project j
Min. Z = (C_{ij} x X_{ij})
Min. z = 8x_{1a} + 6x_{1b} + 7x_{1c} + 14x_{2a} + 9x_{2b} + 10x_{2c} + 12x_{3a} + 10x_{3b} + 9x_{3c} + 11x_{4a} + 8x_{4b} + 11x_{4c}
Subject To:
The constraint of problem is to assign one manager to one project.
Assignment of Managers: Since the number of manager is more than number of projects, one of the manager will remained unassigned.
Manager 1: x_{1a} + x_{1b} + x_{1c} <= 1
Manager 2: x_{2a} + x_{2b} + x_{2c} <= 1
Manager 3: x_{3a} + x_{3b} + x_{3c} <= 1
Manager 4: x_{4a} + x_{4b} + x_{4C} <= 1
Assignment of Projects
Project A: x_{1a} + x_{2a} + x_{3a} + x_{4a} = 1
Project B: x_{1b} + x_{2b} + x_{3b} + x_{4b} = 1
Project C: x_{1c} + x_{2c} + x_{3c} + x_{4c} = 1
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