Question: Create the Aggregate Operations Plan

You are the Chief Operating Officer (COO) in the aggressively growing WarpSpeed Inc company. Your corporation has for some time now developed the optimal operations plan from a linear programming formulation. The formulation is based on the following equations and algebraic relations.

There is an objective function which seeks to minimize the cost of hiring workers, firing workers, holding inventory, and producing your product. The cost of firing a worker is $500. The cost of hiring and training one worker is $100. The cost of carrying one unit in inventory for one month is $0.50. The cost of producing one unit of your product is $2.00.

You can meet demand in month t from production P _t in month t or from the inventory I _t-1 you start month t with (same as what you end month t-1 with). So, if demand in month t is measured by D _t , then the following identity will be correct.

I _t-1 + P _t – D _t = I _t

For month 1, 2, t = 1, 2 and, as t expands, the equation becomes

I ₀ + P ₁ – D ₁ = I ₁

I ₁ + P ₂ – D ₂ = I ₂

You have W _t workers in your employment in month t. In your production process each worker can produce 1000 units in each month. So for month t you produce the quantity P _t , and it is determined by

1000W _t = P _t

Starting with a work force of W _t-1 at the end of month t-1 (same as beginning of month t), adding by hiring H _t , reducing the work force by firing F _t will leave you with a number of workers in your work force of W _t at the end of month t (same as beginning of month t+1) according to the equation

W _t-1 + H _t – F _t = W _t

For month 1, 2, t = 1, 2 and, as t expands, this equation expands the way the inventory equation does. Your beginning work force W ₀ = 100 (here t = 1, so t-1 = 0).

We could next include further constraints which describe your determination of overtime production and outsourcing production. But, we shall no do so here and now. The description of your operations through these equations will be considered complete for now.

As a COO, your problem is to decide today at the beginning of month t = 1 how many workers you will hire, and fire in each of the four months, and how many units of your product you will produce in each of the 4 months (t = 1, 2, 3, 4) given demand forecasts for these months.

The demand forecast is that you will sell 80,000 units in month 1, 50,000 units in month 2, 120,000 in month 3, 150,000 units in month 4.

1. Formulate the linear program which will give you these decisions. Show all equations.
   Hint :
   A .: Define your decision variables H ₁ = number of workers to hire in month 1, H ₂ = number of workers to hire in month 2 and so on for months 3, and 4. Equivalent definitions apply to F _t = number of workers to fire in month t = 1, 2, 3, 4. Also use the inventory quantities I _t , the production quantities P _t and the number of workers on your force W _t as defined above as the remaining decision variables.
   B .: Formulate the objective function that measures total cost for each of the next four months.
   C .: Use the four equations defined above to form your constraints on the operations’ changes. Each of the four equations becomes four equations as t = 1, 2, 3, 4.
2. Solve the linear program with SOLVER. Having solved your linear program, you will now inform your managers and directors which actions they will be executing. That is, you will be informing them to:
   Execute P ₁ in month 1, and to set up for, and get ready to, produce P ₂ , P ₃ , P ₄ units
   Hire H ₁ in month 1, and to set up for, and get ready to, hire H ₂ , H ₃ , H ₄ workers
   Fire F ₁ in month 1, and to set up for, and get ready to, fire F ₂ , F ₃ , F ₄ workers
   Hold in inventory I ₁ in month 1, I ₂ in month 2, I ₃ in month 3, and I ₄ in month 4
   Hold W ₁ workers on your force in month 1, W ₂ , W ₃ , and W ₄ for the other months.
   What are the optimal values for these variables?
   
   These together constitute the optimal operations plan!
3. Suppose you learn that your beginning staff is not 100 (W ₀ is not 100), but that it is 80 (W ₀ = 80) due to unplanned retirements). Compute your optimal hiring and firing plan for the 4 months after you have learned of this change?

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