Aggregate Planning, Linear Programming and Excel Solver
Input
Demand forecast for each quarter, number of workers available in the beginning,
hiring and firing costs per worker, straighttime and over time costs per hour,
inventory holding and backorder costs per unit, productivity (units per hour),
working hours (hours per day and days per quarter) and beginning inventory.
Output (Decision Variables)
Decide for each quarter: number of workers hired/fired, overtime hours, number
of units in the ending inventory and backorder.
Objective
Minimize the total cost of straight-time, overtime, hiring, firing, inventory holding
and backorder
Relationships
1. Inventory at the beginning of the current quarter (except the first quarter) =
positive inventory of the previous quarter – units backordered in the previous
quarter
2. Number of workers available in the current quarter = number of workers
employed in the previous quarter + number of workers hired in the current
quarter – number of workers fired in the current quarter
3. Regular production hours in a quarter = (Number of workers available in the
current quarter)  (Number of working hours per day)  (Number of working
days per quarter)
4. Actual number of units produced in a quarter = (Regular production hours +
overtime production hours)  (productivity in units per hour)
5. Inventory at the end of the current quarter = Actual number of units produced
in the current quarter + Inventory at the beginning of the current quarter –
Demand forecast of the current quarter
6. Inventory at the end of the current quarter = positive inventory of the current
quarter – units backordered in the current quarter
7. Costs:
 Straight-time cost=(regular production hours)(straight-time cost per hour)
 Overtime cost=(overtime production hours)(overtime cost per hour)
 Inventory holding cost=(ending inventory)(holding cost per unit)
 Backorder cost=(units backordered)(backorder cost per unit)
 Hiring cost=(number of workers hired)(hiring cost per worker)
 Firing cost=(number of workers fired)(firing cost per worker)
The Problem
Minimize total cost such that ending inventory computed in (5) equals ending
inventory computed in (6).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
A
Class Example
B
C
D
E
F
G
Plan v: Linear optimization
Number of Workers
Recruitment costs
Hiring cost
Firing cost
Wages
Straight time cost
Overtime cost
Inventory costs
Holding cost
Backorder cost
Productivity
Working hours
30
100 for each temporary worker
200 for each worker laid off
5 per hour
8 per hour
5
10
0.5
8
60
Forecast
Fall
Winter
Spring
Summer
Fall
Winter
Spring
Summer
10000
8000
7000
12000
Total cost
Beginning
Inventory
0
0
0
50000
368041.67
Workers
Hired
Workers
Fired
Total Overtime
Workers
Hours
500 9.58333333
0 39.58333
0
0
6.25 33.33333
0
0 4.166667 29.16667
0
0
0 29.16667
Regular
Actual
Production Production
Hours
Units
19000
9500
16000
8000
14000
7000
14000
7000
Bakorder
Cost
Fall
Winter
Spring
Summer
per unit-quarter
per unit
units per worker hour
hours per day
days per season
Overtime
Cost
0
0
0
0
Ending Positive Backorder Ending
Inventory Inventory
Inventory
Computed
Chosen
0
0
0
0
0
0
0
0
0
0
0
0
-5000
0
5000
-5000
Hiring
Cost
Firing
Cost
0 958.333333
0
0
0
1250
0
0 833.3333
0
0
0
Inventory StraightHolding
time
Cost
Cost
0
95000
0
80000
0
70000
0
70000
Setting up the Spreadsheet
A spreadsheet is set up to compute the total cost from the given values of inputs
and decision variables.
Input:
Known input values are entered in each of the cells below:
 B5:
Initial number of workers
 B8:
Hiring cost for each temporary workers
 B9:
Firing cost for each worker laid off
 B11:
Straight-time cost per hour
 B12:
Overtime cost per hour
 B14:
Holding cost per unit-quarter
 B15:
Backorder cost per unit
 B16:
Number of units produced per worker-hour
 B17:
Number of hours worked per day
 B18:
Number of days worked per quarter
 B23 to B26:
Forecast of demand in fall, winter, spring and summer
 C23:
Initial inventory, in the beginning of fall
Output (Decision Variables):
Excel solver will output the values of the decision variables in the following cells:
 D23 to D26:
numbers of workers hired in fall, winter, spring and summer
 E23 to E26:
numbers of workers fired in fall, winter, spring and summer
 G23 to G26:
overtime hours in fall, winter, spring and summer
 E31 to E34:
units in (positive) inventory in fall, winter, spring and summer
 F31 to F34:
units backordered in fall, winter, spring and summer
Relationships
Formulae are entered for relationships explained below:
1. Inventory at the beginning of the current quarter (except the first quarter) =
positive inventory of the previous quarter – units backordered in the previous
quarter
C24 = E31–F31 (copied to C25 and C26)
2. Number of workers available in the current quarter = number of workers
employed in the previous quarter + number of workers hired in the current
quarter – number of workers fired in the current quarter
F23 = $B$5+D23–E23
F24 = F23+D24–E24 (copied to C25 and C26)
3. Regular production hours in a quarter = (Number of workers available in the
current quarter)  (Number of working hours per day)  (Number of working
days per quarter)
B31 = F23*$B$17*$B$18 (copied to B32, B33 and B34)
4. Actual number of units produced in a quarter = (Regular production hours +
overtime production hours)  (productivity in units per hour)
C31 = (B31+G23)*$B$16 (copied to C32, C33 and C34)
5. Inventory at the end of the current quarter = Actual number of units produced
in the current quarter + Inventory at the beginning of the current quarter –
Demand forecast of the current quarter
D31 = C31+C23–B23 (copied to D32, D33 and D34)
6. Inventory at the end of the current quarter = positive inventory of the current
quarter – units backordered in the current quarter
G31 = E31–F31 (copied to G32, G33 and G34)
7. Costs:
 Straight-time cost=(regular production hours)(straight-time cost per hour)
G39 = B31*$B$11 (copied to G40, G41 and G42)
 Overtime cost=(overtime production hours)(overtime cost per hour)
C39 = G23*$B$12 (copied to C40, C41 and C42)
 Inventory holding cost=(ending inventory)(holding cost per unit)
F39 = E31*$B$14 (copied to F40, F41 and F42)
 Backorder cost=(units backordered)(backorder cost per unit)
B39 = F31*$B$15 (copied to B40, B41 and B42)
 Hiring cost=(number of workers hired)(hiring cost per worker)
D39 = D23*$B$8 (copied to D40, D41 and D42)
 Firing cost=(number of workers fired)(firing cost per worker)
E39 = E23*$B$9 (copied to E40, E41 and E42)
 Total cost
B44 = SUM(B39:E42)
The Excel Solver
1. Add-in Solver, if needed: Click on the “Tools” menu. If the “Solver” is not one
of the items of the “Tools” menu, choose the “Add-Ins.” Click on the box
adjacent to “Solver Add-In” (a check mark will appear) and click “OK.”
2. Define objective: From the “Tools” menu, choose “Solver.” The “Solver
Parameters” Window pops up. Click on the arrow adjacent to the box “Set
Target Cell.” Click on the Cell B44, that stores the total cost and click again
on the arrow. Click on the circle adjacent to “Min.”
3. Decision variables: The cell addresses for the decision variables are entered
in the box below “By Changing Cells:.” The decision variables are
$D$23:$E$26, $G$23:$G$26, $E$31:$F$34
4. Constraints: The constraints are entered in the box below “Subject to the
Constraints.” Click on “Add” and the “Add Constraint” window pops up.
Provide the information on constraint as shown below and click on “OK”:
5. Linearity and Non-negativity: From the “Solver Parameters” window click on
“Options.” The “Solver Options” window pops up. Click on a box adjacent to
“Assume Linear Model” and another adjacent to “Assume Non-Negativity.”
Click on “OK.”
6. Solve: Finally, from the “Solver Parameters” window click on “Solve.” A
“Solver Results” window pops up. Click on “OK.”
7. The Solution: See the changes to your spreadsheet. The cells for the decision
variables are expected to contain the optimal solution and its cost in the cell
for the total cost.
8. A limitation: Workers hired and fired are integer variables. But, the solution
obtained may not be integers.