You’ve recently joined a paper company and are now responsible for planning its
ID: 342878 • Letter: Y
Question
You’ve recently joined a paper company and are now responsible for planning its production and
workforce levels on January 1st, 2016, for the next 3 months. The company estimates the
following demand for its paper products (in cases) over the next three months:
Month
January
February
March
Demand Forecast
1000
800
1000
There are currently 10 workers working at the company and it is estimated that one worker can
produce 2 cases per day, where it can be assumed that each month has 20 working days. The
hiring cost is $1000 and the firing cost is $2000 per worker. Inventory cost is $10 per case per
month. The company will have 200 cases of paper in inventory at the beginning of January, and
would like to have at least 400 cases in inventory at the end of March. Assume that stockout or
backorders are not allowed.
a) Determine the optimal workforce plan (hiring and/or firing in each month) using Linear
Programming approach. You are only required to formulate the problem and you do not
need to solve this problem.
Next, consider the same problem but suppose now that you have an option of subcontracting part
of the workload. The contractor charges a fixed price of $1,000 for every month in which it
produces paper products for your company (for instance, if you decide to use the contractor in
Jan and March, the total fixed price is 2*1,000=$2,000). The contractor also charges $20 per
case of paper product which it produces.
b) Formulate this problem using Linear Programming approach; you are not required to
solve this problem.
Month
January
February
March
Demand Forecast
1000
800
1000
Explanation / Answer
(a)
Let Hi, Fi, and Ei be the hires, fires, and ending inventories in month-i; i=1,2,3. These are the decision variable.
Defined the following composite variables
Wi = Worker level in month-i = Wi-1 + Hi - Fi; i=1,2,3 given that W0 = 10
So, W1 = 10 + H1 - F1
W2 = W1 + H2 - F2 = 10 + H1 - F1 + H2 - F2
W3 = W2 + H3 + F3 = 10 + H1 - F1 + H2 - F2 + H3 - F3
Pi = production in any month = 40*Wi
Minimize Z = 1000(H1+H2+H3) + 2000(F1+F2+F3) + 10(E1+E2+E3)
Subject to,
Pi - (Ei - Ei-1) = Forecat of month-i
i.e.
40*(10+H1-F1+H2-F2+H3-F3) - E3 + E2 = 1000
40*(10+H1-F1+H2-F2) - E2 + E1 = 800
40*(10+H1-F1) - E1 = 200 = 1000
Ending inventory of Mach should be >= 400
So, E3 >= 400
Hi, Fi, Ei >= 0
Dressing up, we get,
Minimize Z = 1000H1+1000H2+1000H3+2000F1+2000F2+2000F3+10E1+10E2+10E3
Subject to,
40H1 - 40F1 + 40H2 - 40F2 + 40H3 - 40F3 - E3 + E2 = 600
40H1 - 40F1 + 40H2 - 40F2 - E2 + E1 = 400
40H1 - 40F1 - E1 = 400
E3 >= 400
Hi, Fi, Ei >= 0
---------------------------------------
(b)
In addition to the decision variables defined in part(a), assume that Si = subcontracting units in month-i and Yi be the set of binary integers such that Yi=1 when the subcontracting in the month-i is more than zero unit.
Minimize Z = 1000(H1+H2+H3) + 2000(F1+F2+F3) + 10(E1+E2+E3) + 2000(Y1+Y2+Y3) + 20(S1+S2+S3)
Subject to,
Pi + Si - (Ei - Ei-1) = Forecat of month-i
i.e.
40*(10+H1-F1+H2-F2+H3-F3) + S1 - E3 + E2 = 1000
40*(10+H1-F1+H2-F2) + S1 - E2 + E1 = 800
40*(10+H1-F1) + S1 - E1 = 200 = 1000
Linking Si and Yi
Si - 999Yi <= 0; i=1,2,3
Ending inventory of Mach should be >= 400
So, E3 >= 400
Hi, Fi, Ei, Si >= 0; Yi = {0,1}
Dressing up, we get,
Minimize Z = 1000H1+1000H2+1000H3+2000F1+2000F2+2000F3+10E1+10E2+10E3+2000Y1+2000Y2+2000Y3+20S1+20S2+20S3
Subject to,
40H1 - 40F1 + 40H2 - 40F2 + 40H3 - 40F3 +S3 - E3 + E2 = 600
40H1 - 40F1 + 40H2 - 40F2 +S2 - E2 + E1 = 400
40H1 - 40F1 + S1 - E1 = 400
S1 - 999Y1 <= 0
S2 - 999Y2 <= 0
S3 - 999Y3 <= 0
E3 >= 400
Hi, Fi, Ei, Si >= 0; Yi = {0,1}
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