Question

AGGREGATE PLANNING
Garden Hub makes a single product, a garden tool that sells for $40. Garden Hub starts with 1000 of these tools in inventory and is expected to end with at least 500 in stock. Garden Hub can temporarily backlog demand for a cost, but at the end of the time horizon, they require their backlog to be zero – This is an important constraint to remember- if we forget it, we will get strange results . Production costs are based on parts and labor with no machine capacity issues – They start with 80 employees can hire or fire workers for a cost. – Workers get regular pay whether they are producing or not. There are 20 days of production per month, each month. – We can have workers work overtime (no more than 10 hrs/mo per worker) for extra $ – We can also subcontract production out and pay a flat fee (in lieu of labor + materials). Garden Hub would like to generate a 6-month plan that maximizes profits (revenue net of costs) – For now we can just minimize costs, if we have no influence over demand. Here’s the demand - see the seasonality?

Note: subcontracting costs includes all materials and labor.

ONLY FORMULATE THE PROBLEM, NO NEED TO SOLVE.

Month January February Item Cost Demand Forecast $10/unit Materials 1600 $2/unit/month Inventory holding cost 3,000 Marginal cost of$5/unit/month March 3,200 a stockout Hiring and training costs Layoff cost Regular time cost Labor hours required Over time cost Cost of subcontracting $300/worker 3,800 April $500/worker $4/hour 4/unit May 2,200 2,200 June $6/hour $30/unit

Answer 1

SOLUTION

Firstly, our given data is:

Item Materials Inventory Cost Month January February Demand Forecast $10/unit 1600 $2/unit/month 3,000 holding cost Marginal cost of$5/unit/month March 3,200 a stockout Hiring and training costs Layoff cost Regular time cost Labor hours required Over time cost Cost of subcontracting $300/worker April 3,800 2,200 2,200 $500/worker $4/hour 4/unit May June $6/hour $30/unit

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The decision variables for month, t = 1 to 6 (January to June) will be:

Wt=Workforce size for month t

Ht=Number of employees hired at the beginning of month t

Lt=Number of employees laid off at the beginning of month t

Pt=Production in month t

It=Inventory at the end of month t

St=Number of units stocked out or backlogged at the end of month t

Ct=Number of units subcontracted for month t

Ot=Number of overtime hours worked in month t

Total cost = (Regular-time labor cost + Overtime labor cost + Cost of hiring and layoffs + Cost of holding inventory + Cost of stocking out + Cost of subcontracting + Material cost)

Regular-time labor cost per worker = $4/hour*8hrs/day*20days/month = $640/month

So, total regular-time labor cost = $640/month*Number of workers = $640*W_t

Overtime labor cost = $6*O_t

Cost of hiring and layoffs = $300*H_t + $500L_t [It costs $300 to train any new hire and $500 for every worker fired]

Cost of holding inventory = $2*I_t [Units at end of month is It and cost of holding one unit of inventory is $2]

Similarly, Cost of stocking out = $5*S_t

              Cost of subcontracting = $30*C_t

Materials cost = $10/unit * Production number of units = $10*P_t

So overall cost = 640W_t + 6O_t + 300H_t + 500L_t + 2I_t + 5S_t + 30C_t + 10P_t

Hence, objective function is:

Minimize Z = 640W_t + 6O_t + 300H_t + 500L_t + 2I_t + 5S_t + 30C_t + 10P_t

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The problem of minimization is subject to constraints:

1. Workforce constraints:

W_t = W_t-1 + H_t - L_t (Workforce at any month is the workforce in the previous month + Number of workers hired - NUmber of workers fired) (With W₀ = 80 as initially we have 80 workers)

2. Production capacity constraints:

Pt = 40W_t + O_t/4 [Wt are the number of workers and each can worker can make 2 units in a day and hence 40 units in a month. Labor hours for each unit is 4 hours and hence each day a worker can make 2 units and there are 20 production days in a month] [Every 4 hours will give 1 unit, hence number of overtime hours/4 will give number of units through overtime labor]

3. Inventory balance constraint:

I_t-1 + P_t + C_t - D_t - S_t-1 = I_t - S_t [Inventory at the beginning of the month + Number of units produced + Number of units through subcontract - Demand fulfilled for that month - Demand for previous month fulfilled due to stockout in previous month = Ending inventory for this month - Stockout in this month] [Also I₀ = 1000 as we start with 1000 units]

I₆ >= 500 [There need to be at least 500 units at the end of the horizon time period that is six months in this case]

4. Overtime limit constraints:

O_t <= 10W_t [Overtime constraints say that overtime hours cannot be more than 10 hours per worker]

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So full problem formulation is:

Minimize Z = 640W_t + 6O_t + 300H_t + 500L_t + 2I_t + 5S_t + 30C_t + 10P_t

subject to

W_t = W_t-1 + H_t - L_t

W₀ = 80

Pt = 40W_t + O_t/4

I_t-1 + P_t + C_t - D_t - S_t-1 = I_t - S_t

I₆ >= 500

O_t <= 10W_t

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Kindly comment in case of any questions. Thanks :)