A Two Stage Production Planning Problem

In a production planning problem, the decision maker must decide how to purchase material, labor, and other resources in order to produce end products to maximize profit.

In this case study a company (GTC) produces wrenches and pliers, subject to the availability of steel, machine capabilities (molding and assembly), labor, and market demand. GTC would like to determine how much steel to purchase. Complicating the problem is that the available assembly capacity and the product contribution to earnings are unknown presently, but will be known at the beginning of the next period.

So, in this period, GTC must:

  • determine how much steel to purchase.

At the beginning of the next period, after GTC finds out how much assembly capacity is available and the revenue per unit of wrenches and pliers, GTC will determine

  • How many wrenches and pliers to produce.

The uncertainty is expressed as one of four possible scenarios, each with equal probability.

We begin by importing the PuLP package.


import pulp

Next, we will read in the data. Here, we read in the data as vectors. In actual use, this may be read from databases. First, the data elements that do not change with scenarios. These each have two values, one corresponding to wrenches, the other pliers.

products = ['wrenches', 'pliers']
price = [130, 100]
steel = [1.5, 1]
molding = [1, 1]
assembly = [0.3, 0.5]
capsteel = 27
capmolding = 21
LB = [0,0]
capacity_ub = [15, 16]
steelprice = 58

The next set of parameters are those that correspond to the four scenarios.

scenarios = [0, 1, 2, 3]
pscenario = [0.25, 0.25, 0.25, 0.25]
wrenchearnings = [160, 160, 90, 90]
plierearnings = [100, 100, 100, 100]
capassembly = [8, 10, 8, 10]

Next, we will create lists that represent the combination of products and scenarios. These will later be used to create dictionaries for the parameters.

production = [(j,i) for j in scenarios for i in products]
pricescenario = [[wrenchearnings[j], plierearnings[j]] for j in scenarios]
priceitems = [item for sublist in pricescenario for item in sublist]

Next, we use dict(zip(…)) to convert these lists to dictionaries. This is done so that we can refer to parameters by meaningful names.

price_dict = dict(zip(production, priceitems))
capacity_dict = dict(zip(products, capacity_ub*4))
steel_dict = dict(zip(products, steel))
molding_dict = dict(zip(products, molding))
assembly_dict = dict(zip(products, assembly))

To define our decision variables, we use the function pulp.LpVariable.dicts(), which creates dictionaries with associated indexing values.

production_vars = pulp.LpVariable.dicts("production", (scenarios, products), \
                                        lowBound=0, cat='Continuous')
steelpurchase = pulp.LpVariable("steelpurchase", lowBound=0, cat='Continuous')

We create the LpProblem and then make the objective function. Note that this is a maximization problem, as the goal is to maximize net revenue.

gemstoneprob = pulp.LpProblem("The Gemstone Tool Problem",pulp.LpMaximize)

The objective function is specified using the pulp.lpSum() function. Note that it is added to the problem using +=.

gemstoneprob += pulp.lpSum([pscenario[j] * (price_dict[(j,i)]*\
                            production_vars[j][i]) \
                            for (j,i) in  production] - \
                            steelpurchase * steelprice), "Total cost"

We then add in constraints. Constraints here in sets based on scenarios and products and are specified using the for i in list: notation. Within each constraint, summations are expressed using list comprehensions. Note that constraints are differentiated from the objective function as each constraint ends in a logical comparison (usually <= or >=, but can be ==) while Finally, here, the file gives each constraint a name which includes the specific scenario or product the constraint applies to.


for j in scenarios:
    gemstoneprob += pulp.lpSum([steel_dict[i] * production_vars[j][i] \
                                for i in products]) - \
                                steelpurchase <= 0, ("Steel capacity" + str(j))
    gemstoneprob += pulp.lpSum([molding_dict[i] * production_vars[j][i] \
                                for i in products]) <= capmolding, \
                                ("molding capacity" +str(j))
    gemstoneprob += pulp.lpSum([assembly_dict[i] * production_vars[j][i] \
                                for i in products]) <= capassembly[j], \
                                ("assembly capacity" +str(j))
    for i in products:
        gemstoneprob += production_vars[j][i] <= capacity_dict[i], \

The full file can be found here Two_stage_Stochastic_GemstoneTools.py

"""
Author: Louis Luangkesorn <lugerpitt@gmail.com> 2019
https://github.com/lluang

Title: Gemstone Optimization problem

Problem taken from Data, Models, and Decisions by Bertsimas and Freund, 4th Edition
DMD 7.2

## 2 stage problem

- **Scenarios:** $s \in S = (1, 2, 3, 4)$
- **Probability scenario occuring:** $p^s$
- **Cost of steel:** $c$
- **Total steel:** $cap_{steel}$
- **Total molding and assembly hours:** $cap_{molding}, cap_{assembly}^s$
- **Wrench and plier earnings by scenario:** $w^s, p^s$
- **Max demand wrenches and pliers:** $UB_w, UB_p$
- **Decision variables**
  - $(W_{t+1}^s, P_{t+1}^s)$
- **Objective**   $Max \sum_s (p^s * (w^s W_{t+1}^s + p^s P_{t+1}^s) - c$
- **Steel Constraint:** $1.5W_{t+1}^1 + P_{t+1}^1 - C \le 0$
- **Molding Constraint:** $W_{t+1}^1 + P_{t+1}^1 \le cap_{molding}$
- **Assembly Constraint:** $0.3 W_{t+1}^1 + 0.5 P_{t+1}^1  \le cap_{molding}^s$
- **Demand Limit W:** $W \le UB_w$
- **Demand Limit P:** $P \le UB_p$
- **Nonnegativity:** $W, P \ge 0$
"""

import pulp

# parameters
products = ['wrenches', 'pliers']
price = [130, 100]
steel = [1.5, 1]
molding = [1, 1]
assembly = [0.3, 0.5]
capsteel = 27
capmolding = 21
LB = [0,0]
capacity_ub = [15, 16]
steelprice = 58
scenarios = [0, 1, 2, 3]
pscenario = [0.25, 0.25, 0.25, 0.25]
wrenchearnings = [160, 160, 90, 90]
plierearnings = [100, 100, 100, 100]
capassembly = [8, 10, 8, 10]

production = [(j,i) for j in scenarios for i in products]
pricescenario = [[wrenchearnings[j], plierearnings[j]] for j in scenarios]
priceitems = [item for sublist in pricescenario for item in sublist]

# create dictionaries for the parameters
price_dict = dict(zip(production, priceitems))
capacity_dict = dict(zip(products, capacity_ub*4))
steel_dict = dict(zip(products, steel))
molding_dict = dict(zip(products, molding))
assembly_dict = dict(zip(products, assembly))

# Create variables and parameters as dictionaries
production_vars = pulp.LpVariable.dicts("production", (scenarios, products), \
                                        lowBound=0, cat='Continuous')
steelpurchase = pulp.LpVariable("steelpurchase", lowBound=0, cat='Continuous')

# Create the 'gemstoneprob' variable to specify
gemstoneprob = pulp.LpProblem("The Gemstone Tool Problem",pulp.LpMaximize)

# The objective function is added to 'gemstoneprob' first
gemstoneprob += pulp.lpSum([pscenario[j] * (price_dict[(j,i)]*\
                            production_vars[j][i]) \
                            for (j,i) in  production] - \
                            steelpurchase * steelprice), "Total cost"

for j in scenarios:
    gemstoneprob += pulp.lpSum([steel_dict[i] * production_vars[j][i] \
                                for i in products]) - \
                                steelpurchase <= 0, ("Steel capacity" + str(j))
    gemstoneprob += pulp.lpSum([molding_dict[i] * production_vars[j][i] \
                                for i in products]) <= capmolding, \
                                ("molding capacity" +str(j))
    gemstoneprob += pulp.lpSum([assembly_dict[i] * production_vars[j][i] \
                                for i in products]) <= capassembly[j], \
                                ("assembly capacity" +str(j))
    for i in products:
        gemstoneprob += production_vars[j][i] <= capacity_dict[i], \
                                            ("capacity " + str(i) + str(j))

# Print problem
print(gemstoneprob)

# The problem data is written to an .lp file
gemstoneprob.writeLP("gemstoneprob.lp")
# The problem is solved using PuLP's choice of Solver
gemstoneprob.solve()
# The status of the solution is printed to the screen
print("Status:", pulp.LpStatus[gemstoneprob.status])

# OUTPUT

# Each of the variables is printed with it's resolved optimum value
for v in gemstoneprob.variables():
    print(v.name, "=", v.varValue)
production = [v.varValue for v in gemstoneprob.variables()]

# The optimised objective function value is printed to the console
print("Total price = ", pulp.value(gemstoneprob.objective))