Question

QDM Question 4 : In the planning of the monthly production for the next four months, in each month a company must operate either a normal shift or an extended shift (but not both) if it produces. It may choose not to produce in a month. A normal shift costs $100,000 per month and can produce up to 5,000 units per month. An extended shift costs $140,000 per month and can produce up to 7,500 units per month.

The cost of holding inventory is estimated to be $2 per unit per month (based on the average inventory held during each month) and the initial inventory is 3,000 units (i.e., inventory at the beginning of Month 1). The inventory at the end of month 4 should be at least 2,000 units. The demand for the company's product in each of the next four months is estimated to be as shown below:

Month	1	2	3	4
Demand	6000	6500	7500	7000

Production constraints are such that if the company produces anything in a particular month it must produce at least 2,000 units. The company wants a production plan for the next four months to meet its demands.  Formulate an integer programming model to solve the problem at minimum cost.

Decision variables :

Objective function :

Constraints :

Additional constraint 1 : The company can operate an extended shift in Month 1 only if it operates a normal shift in each of Month 2, Month 3 and Month 4.

Additional constraint 2 : The company must produce in either Month 1 or Month 2 (or both) if it does not produce in Month 3.

Additional constraint 3 : For each of Month 2, 3 and 4, the company cannot operate an extended shift in a month if it operates one in the previous month.

Answer 1

Decision Variables:

• Production amount, let's represent it with Pi, in ith month.
• wether to operate normal production shift in ith month, ni, such that, ni=1 if normal shift operates.& ni=0 if not
• wether to operate extended production shift in ith month, ei, such that, ei=1 if extended shift operates & ei=0 if not
• wether to produce in ith month, xi, such that, xi=1 if production shift operates & xi=0 if not
• Closing Inventory for ith month, Ii

where i {1,2,3,4}

Objective function:

To minimise total cost of production and holding inventory over the four months

Production cost for ith month = 100000*ni + 140000*ei

Inventory Holding Cost for ith month = (Average inventory for ith month)*2
= ((Ii + Ii-1)/2)*2

Minimize Total Cost, Z= [100000*ni + 140000*ei + ((Ii + Ii-1)/2)*2]

Constraints:

1. Operation of only one shift each month
   ni + ei 1
2. Production limits
   Pi   5000*ni + 7500*ei
3. Inventory equation
   Ii= Ii-1 + Pi - Di ; where Di is the demand for the ith month
4. Closing inventory at end of each month should be greater than zero
   Ii 0
5. I0 = 3000
6. I4 2000
7. Minimum production of 2000 units in case production run starts in any month
   Pi   2000xi
   

Additional constraint 1:
Since for the option to produce in extended shift to be available for first month we'd have to fix the production shift as normal for months 2,3,4
n2, n3, n4 = 1

Additional Constraint 2:
This constraints can be reframed as the production ust run atleast once in first 3 months and not more than 3 times (as only one production run is allowed each month)
n1 + e1 + n2 + e2 + n3 + e3 1

Additional Constraint 3:
extended production shift cannot run for consecutive months
ei + ei-1 1